Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Wen, Rui
Doctor of Philosophy - PhD
University of British Columbia
Despite great success we have achieved in the past two decades on systematically understanding gapped topological phases, a firm foundation for understanding gapless topological phases is still lacking. In this dissertation I develop a general framework to describe, classify and construct gapped and gapless phases with or without symmetry. I then apply this framework to construct new gapless phases, and classify some classes of them completely. The first chapter is a non-technical summary of the subjects, ideas and results of the rest of the dissertation. In the first chapter I introduce the notion of gapless topological phases, and briefly discuss their general structures. We also discuss how topological holography can be useful for classifying and constructing gapless phases with generalized symmetries. In chapter two I develop a framework for understanding general quantum phases and phase transitions with generalized symmetries. After reviewing the established theory of gapped phases, we introduce two key notions: quantum order and symmetry-enriched quantum order. These concepts are the correct generalized of topological order and symmetry-enriched topological orders. I show that these new notions subsume conventional phase transitions, gapless spin liquids, and gapless symmetry-protected topological phases. For a class of symmetry-enriched quantum orders with finite many topological defects, we present a general classification using higher tensor category theory. The rest of the dissertation provides more detailed studies of specific gapless topological phases, including classification and lattice model construction of gapless symmetry-protected topological phases and the connection to the theory of topological holography.
ON GAPLESS TOPOLOGICAL PHASES AND THEIR HOLOGRAPHIC DUALSbyRui WenA DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2025©Rui Wen, 2025The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:On Gapless Topological Phases and Their Holographic Dualssubmitted by Rui Wen in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Andrew Potter, Assistant Professor, Physics, UBCSupervisorMoshe Rozali, Professor, Physics, UBCSupervisory Committee MemberFei Zhou, Professor, Physics, UBCUniversity ExaminerBen Williams, Associate Professor, Mathematics UBCUniversity ExaminerFiona Burnell , Professor, Physics, University of MinnesotaExternal ExaminerAdditional Supervisory Committee Members:Ziliang Ye, Assistant Professor, Physics, UBCSupervisory Committee MemberJulio Parra Martinez, Assistant Professor, Physics, UBCSupervisory Committee MemberiiAbstractDespite great success we have achieved in the past two decades on systematically un-derstanding gapped topological phases, a firm foundation for understanding gaplesstopological phases is still lacking. In this dissertation I develop a general framework todescribe, classify and construct gapped and gapless phases with or without symmetry.I then apply this framework to construct new gapless phases, and classify some classesof them completely.The first chapter is a non-technical summary of the subjects, ideas and resultsof the rest of the dissertation. In the first chapter I introduce the notion of gaplesstopological phases, and briefly discuss their general structures. We also discuss howtopological holography can be useful for classifying and constructing gapless phaseswith generalized symmetries.In chapter two I develop a framework for understanding general quantum phasesand phase transitions with generalized symmetries. After reviewing the establishedtheory of gapped phases, we introduce two key notions: quantum order and symmetry-enriched quantum order. These concepts are the correct generalized of topological orderand symmetry-enriched topological orders. I show that these new notions subsumeconventional phase transitions, gapless spin liquids, and gapless symmetry-protectedtopological phases. For a class of symmetry-enriched quantum orders with finite manytopological defects, we present a general classification using higher tensor categorytheory.The rest of the dissertation provides more detailed studies of specific gapless topolog-ical phases, including classification and lattice model construction of gapless symmetry-protected topological phases and the connection to the theory of topological holography.iiiLay SummaryMatter organizing into different phases is a phenomenon we see everyday. Water canbe in three phases: fluid, solid and gas. When temperature is low enough quantumentanglement among different atoms and electrons of a material become so strong thata new kind of quantum phase emerges. These exotic quantum phases have been acentral subject of condensed matter physics.The goal of this thesis is to develop a framework for systematically understand-ing quantum phases, and apply this framework to construct new models of quantumphases. Quantum orders exhibit intriguing features such as topological defects andunconventional symmetries. Leveraging tools from higher category theory, we developa general mathematical theory for describing how the topological defects interact withthe symmetries of quantum phases.ivPrefaceâ¢ Chapters 1-3 are based on unpublished original works of the author Rui Wen.â¢ Chapter 4 has been published[Rui Wen and Andrew C. Potter Phys. Rev. B 107,245127]. Professor Andrew C. Potter and I contributed equally to the work.â¢ Chapter 5 has been published[Rui Wen and Andrew C. Potter Phys. Rev. B 111,115161]. Professor Andrew C. Potter and I contributed equally to the work.â¢ Chapter 6 is based on unpublished original work of mine.No Generative Artificial Intelligence tools were used in the research process, devel-opment, or writing of the thesis.vContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Gapless Topological Phases . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Topological Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Structure of Gapless Phases . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Gapped Phases and Topological Defects . . . . . . . . . . . . . . . . . 102.2 Symmetry-Enriched Topological Orders . . . . . . . . . . . . . . . . . . 142.3 Comments on the Notion of Symmetry . . . . . . . . . . . . . . . . . . 162.4 Topological Defects in Gapless Phases . . . . . . . . . . . . . . . . . . . 172.5 Quantum Order and Symmetry-enriched Quantum Order . . . . . . . . 233 Gapless Symmetry-Protected Topological Phases . . . . . . . . . . . 303.1 Classification of gSPT . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Gauging of 2+1D gSPT . . . . . . . . . . . . . . . . . . . . . . . . . . 39vi3.3 gSPT with generalized symmetries . . . . . . . . . . . . . . . . . . . . 424 Construction of Gapless SPT from Group-Cohomology . . . . . . . . 454.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Gapless SPTs: Definitions and Taxonomy . . . . . . . . . . . . . . . . 484.3 Low dimensional igSPTs . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Constructing igSPTs from group cohomology data . . . . . . . . . . . 644.5 A 2d Time-Reversal Symmetric igSPT . . . . . . . . . . . . . . . . . . 764.6 A 3d Ising igSPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.8 Additional 1d igSPT examples . . . . . . . . . . . . . . . . . . . . . . 904.9 Other 2d igSPTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.10 The SPT pumping action of N . . . . . . . . . . . . . . . . . . . . . . 1014.11 Gauging the extending symmetry . . . . . . . . . . . . . . . . . . . . . 1024.12 Proof of Intrinsicness of igSPT with group cohomology anomalies . . . 1104.13 Relating anomaly cancellation and SPT pumping pictures of igSPT edgestates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.14 Perspective from Gauge-Response Theory from Gapped SPTs . . . . . 1135 Classification of 1+1D gSPT via SymTFT . . . . . . . . . . . . . . . . 1175.1 Background: gapless SPTs, and SymTFT . . . . . . . . . . . . . . . . . 1175.2 SymTFT for GSPTS? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Structure of 1+1D bosonic gSPTs . . . . . . . . . . . . . . . . . . . . . 1295.4 Condensation of quantum double and gSPT . . . . . . . . . . . . . . . 1365.5 Special cases of gSPTs and their SymTFT description . . . . . . . . . . 1415.6 Consistency conditions on the data η, ϵ . . . . . . . . . . . . . . . . . . 1475.7 Connection to the LHS spectral sequence . . . . . . . . . . . . . . . . . 1485.8 The bulk-edge correspondence for 1+1D gSPTs . . . . . . . . . . . . . 1505.9 Equivalence between the post-condensation twist and gSPT emergentanomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516 2+1D Phases with Symmetries and SymTFT . . . . . . . . . . . . . . 1536.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.2 Bosonic fusion 2-categories and their SymTFT . . . . . . . . . . . . . . 157vii6.3 Topological holography for 2+1-D gapped phases . . . . . . . . . . . . 1586.4 2+1-D Gapped phases with generalized symmetries from SymTO . . . 1616.5 Topological holography for 2+1-D gapless phases . . . . . . . . . . . . 1746.6 Higher condensation theory for 3+1D topological orders . . . . . . . . . 1766.7 String condensation in Z1[2VecG] . . . . . . . . . . . . . . . . . . . . . 1786.8 Minimal condensable algebras: Cohomology method . . . . . . . . . . . 1826.9 Physics of 2+1-D G-gSPT: decorated domain wall and spectral sequence 1886.10 Examples of minimal condensable algebras and G-gSPTs . . . . . . . . 1916.11 2+1-D Gapless SPT with generalized symmetry . . . . . . . . . . . . . 1957 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.1 Lattice model of symmetry-enriched quantum orders . . . . . . . . . . 2007.2 Gauging of quantum order and symmetry-enriched quantum order . . . 2017.3 The theory of local operator algebra. . . . . . . . . . . . . . . . . . . . 2027.4 Irrational quantum orders . . . . . . . . . . . . . . . . . . . . . . . . . 2027.5 Continuous symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.6 Spacetime symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214A.1 Some useful higher linear algebra . . . . . . . . . . . . . . . . . . . . . 214A.2 Condensable E2-algebras in braided fusion 2-categories . . . . . . . . . 218A.3 Extension theory of (braided)fusion categories . . . . . . . . . . . . . . 220viiiList of Tables4.1 Examles of igSPTs from group-cohomology . . . . . . . . . . . . . . . . 754.2 Edge-SPT pumping symmetry from dimensional reduction . . . . . . . 1166.1 Classification of 2+1-D gSPT with exact symmetry G and gapped sym-metry N and whether the symmetry allows for igSPT. . . . . . . . . . 1566.2 Fusion rules of objects in C[S4, S3, 0, 0] . . . . . . . . . . . . . . . . . . 173ixList of Figures4.1 Schematic of an igSPT with symmetry group GUV . . . . . . . . . . . . 464.2 Anatomy of a 1d igSPT . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Lattice models with anomalous GIR-symmetry from group cohomology 654.4 The igSPT lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.5 Decorated domain wall picture . . . . . . . . . . . . . . . . . . . . . . . 814.6 Symmetry properties of 2d QSETs . . . . . . . . . . . . . . . . . . . . 834.7 Fully gapped 3d QSET . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.8 Gauging the extending symmetry. . . . . . . . . . . . . . . . . . . . . . 1034.9 Symmetry fractionalization . . . . . . . . . . . . . . . . . . . . . . . . . 1085.1 SymTFT description of Z2 symmetry. . . . . . . . . . . . . . . . . . . . 1225.2 SymTFT description of the cluster chain. . . . . . . . . . . . . . . . . . 1235.3 String order parameter of the Z4-igSPT in the thin slab construction. . 1255.4 Edge modes of the Z4-igSPT in the thin slab construction. . . . . . . . 1265.5 Stability of the Z4-igSPT from SymTFT . . . . . . . . . . . . . . . . . 1285.6 The algebra of symmetry action of a gSPT. . . . . . . . . . . . . . . . 1315.7 Edge modes of igSPTs of SPT-pump type in the thin slab construction. 1426.1 Dimension reduction of sandwich . . . . . . . . . . . . . . . . . . . . . 1626.2 Domain walls in the toric code topological order. . . . . . . . . . . . . . 1676.3 The club-sandwich construction . . . . . . . . . . . . . . . . . . . . . . 1756.4 The club-sandwich construction of a (Z[1]2 Ã Z[1]2 )â Z[0]2 -igSPT. . . . . . 1976.5 The club-sandwich construction of a 2VecZ4 â 2Rep(D4)-igSPT. . . . . 199xAcknowledgementsI thank my parents for their support. Only in recent years have I begun to appreciate theimmense sacrifices they made for meâthe depth of their love is something I can neverfully repay. I couldnât have achieved what I have today without their unconditionalsupport.I thank my supervisor Andrew Potter for his exceptional guidance and expertise.His support has meant a lot to me.To my life-long friend Yan Zhang: No words can express how grateful I am for yourpresence in my life. Thank you for supporting me, believing in me and loving me.I thank my dear friend Victor Li for his company and all the rides to the mountains.And finally, I thank myselfâfor not giving up, and for staying true to who I am.xiChapter OneIntroductionThis chapter is intended as a non-technical overview of the subjects, ideas, and results ofthis dissertation. The dissertation focuses on two topicsâgapless topological phases andtopological holography. Here, we briefly describe these topics, explain our motivationfor studying them, outline the tools and methods employed, and summarize the mainachievements presented in this dissertation.1.1 Gapless Topological PhasesThe term âgapless topological phasesâ involves three elementary concepts: phase, topo-logical and gapless. Let me explain their meanings. The study of phases is almost thedefinition of condensed matter physics. There are two basic types of phases, classicalphases and quantum phases. Classical phases are phases that can be described by clas-sical statistical mechanics, where the fundamental degrees of freedom are assumed toobey classical mechanics. Classical phases are described by Landauâs symmetry break-ing theory. For example, ice breaks spatial translation symmetry while water does not.A classical phase transition is usually driven by change of temperature or pressure, andall classical phases become identical when temperature is close to absolute zero unlesssymmetry constraint is enforced. Near zero temperature, thermal fluctuation vanishesand quantum fluctuation dominates, giving rise to the notion of quantum phases. Inthe rest of this dissertation phase will mean exclusively quantum phase. Quantumphases are phases defined at zero temperature. For example, superconductor is a kind1of quantum phase, whose unusual behavior persists all the way to zero temperature.Superfluid, quantum Hall states, spin liquids, are also different quantum phases. Theirunusual properties all persist approaching zero temperature. It is actually more ac-curate to say that it is the zero temperature properties that persist at finite smalltemperature. Near zero temperature, only very low energy states of the system mat-ter. Therefore different quantum phases correspond to different properties of their lowenergy quantum states. This property is quantum entanglement. To understand orclassify possible quantum phases is the same as to understand possible ways a largeamount of particles can entangle with each other.Landauâs theory of symmetry-breaking was so successful that people once believedthat quantum phases can be characterized by their symmetries as well. From ourprevious discussion it is clear that this is not the case. Quantum entanglement isa concept that has nothing to do with symmetry. Indeed soon it was realized thatquantum phases can not be distinguished by their symmetries. This was first discoveredin gapped phases 1. A gapped phase has a gap between the ground state and the firstexited state. An insulator or a superconductor is gapped, while a conductor or asuperfluid is gapless. Gapped phases are arguably the simplest quantum phases. Atlow temperatures, a gapped phase is almost featureless, since all the local correlationfunctions decay exponentially. For this reason, it was believed that there were nointeresting gapped phases. However the bulk gap does not rule out possible bulk non-local properties and boundary local properties. The notion of topological order was thenintroduced to describe gapped phases with nontrivial nonlocal or boundary features [18,132, 133]. Topological orders are called topological for good reasons. Topology is abranch of mathematics that studies properties of geometric shapes that do not changeunder continuous deformation. It was realized that a gapped phase is characterized bycertain macroscopic properties that are robust to any perturbation. For instance, theHall conductivity of a quantum Hall state is a robust property that does not changewhen we add impurities. A topological order is often characterized by a finite setof invariants that do not change within the same phase. These invariants to gappedphases are just like topological invariants to geometric shapes. A typical example is theChern number of a band insulator, which is a topological invariant of energy bands.This is also more than just an analogy. If we place a gapped phase on a manifold, the1See [136] for a review of this period of history.2ground state degeneracy will only depend on the topology of the manifold. In this way atopological order is literally a topological invariant of geometric shapes. Another reasonfor the name is that a topological order often has bulk topological defects, i.e., defectswhose properties do not change when we move them around or deform their shapes.A typical example is the fractional charge of a flux threaded through a fractional Hallstate. In fact, a topological order is almost determined completely by its topologicaldefects. In 2 dimensional systems, point-like topological defects are also called anyons.It is widely accepted that a 2 dimensions topological order is determined by its bulkanyons and its boundary conductance. A final reason for the name is that it is believeda topological order is described by a topological quantum field theory(TQFT) in thecontinuous limit.We have now explained the meaning of phases and gapped topological phases(i.e.,topological orders). A gapless topological phase is, obviously, a quantum phase that isgapless and topological. The meaning of a gapless quantum phase is clear, but whatis a topological gapless phase? Be aware that gapped phases and topological ordersare really synonymous: there are no gapped phases that are ânot topologicalâ 2. Theterm topological order is widely used because of the reasons we explained above. Inthe same way gapless topological phase really just means gapless quantum phases. Thevery notion of phase is a topological notion. When we say two systems are in thesame phase, we mean they look similar from a large scale. This means we have somemacroscopic measures that are identical for two microscopically different systems. Thisis already a notion of topology. Going from microscopic systems to macroscopic phasesis just like going from geometric shapes to topological shapes.As stated at the beginning, understanding all possible phases is the ultimate goalof condensed matter physics. The theory of gapped phases is largely complete by now,the remaining unsolved mystery is gapless phases. This is why we choose to study themin this dissertation.A complete theory of all gapless quantum phases is a rather ambitious goal, andis expected by many to be very difficult. What is often forgotten is that we do noteven have a good definition of gapless phases. For gapped phases, there is a widely2The phases considered here and in the rest of dissertation are the so-called âliquid-like â phases [136,58]. Non-liquid-like phases can indeed be non-topological, however I believe non-liquid-like phases areless fundamental and more artificial.3accepted working definition(see Definition. 2.1.1). Without a clear definition, we cannot get anywhere far. Therefore our first goal in this dissertation is to provide aframework with which we can start to develop the theory of gapless phases. Thisgoal is partially achieved in Chapter. 2. We (re)introduce the notion of quantum order,which generalizes topological order to include gapless phases. The general notion ofquantum order is due to Xiao-gang Wen [135, 134]. The key insight of Xiao-gang Wenis that quantum phase is a notion about quantum entanglement, not symmetry. Anyattempt to understand quantum phases using symmetry is doomed to failure. Ourcontribution here is that we make this concept more rigorous. The definition givenin Sec. 2.5 makes it clear that quantum order, just like topological order, is a newkind of phase beyond Landauâs theory of symmetry breaking. This however does notmean that symmetry does not interact with quantum orders or topological orders.For gapped phases, it is known that symmetry leads a new notion of phase, calledsymmetry-enriched topological orders(SET). Although topological orders themselvesare not defined with any notion of symmetry, we can equip them with symmetries.The same symmetry can act on the same topological order in different ways, leadingto different SETs with the same underlying topological order. In a similar fashion,we introduce the notion of symmetry-enriched quantum orders. Different symmetry-enriched quantum orders are characterized by different ways symmetries act on quantumorders.Recall that a topological order is largely characterized by its topological defects.It is natural to expect this is still partially true for gapless phases. This means itis possible for a gapless phase to have topological defects, and the structure of thesetopological defects should partially characterize the phase. In Section 2.4, we examineseveral examples to show that this is indeed true. We observe that the structure ofa general quantum order consists of a local part and a global part. The local partconsists of information of local operators and their correlation functions, which we calllocal operator algebra, while the global part consists of the information of topologicaldefects and their fusion and braiding rules, which we call topological skeleton. Wepropose that the theory of quantum order should consist of separate theories of localoperator algebra and topological skeleton, as well as rules for the two structures to becompatible with each other. It is expected that the theory of topological skeleton willbe much simper than that of local operator algebras. We explain in Sec 2.4 why fusion4higher category is the natural mathematical language for describing topological defectsin any phase, gapped or gapless. With the tools of higher fusion category theory, wepropose a theory of topological skeleton via abstract nonsense. Unlike the theory ofquantum orders themselves, the theory of symmetry-enriched quantum orders turnsout to be quite simple. We describe this theory in Sec. 2.5, again using the tools fromhigher fusion category. The key insight is that the action of symmetry on a systemcorresponds mathematically to a functor between fusion higher categories. Our theorysubsumes the usual theory of SETs and is supported by many known gapless examples,the most well-known ones being the so-called âgapless symmetry-protected topologicalphases(gSPT)â, which is also the main focus of the rest chapters of this dissertation.One important guidance for us in developing this framework of gapless phases isthe well-established theory of 1+1D 3 rational conformal field theories(RCFT). Con-formal field theories are believed to describe continuous phase transitions and manyother gapless phases. 1+1D RCFTs are among all CFTs the simplest and most well-understood ones. A precise mathematical theory has been established to describe them,and is believed by many to be a complete description. Therefore the theory of generalquantum orders must contain the theory of 1+1D RCFTs. This is indeed the case forour theory. In fact our notions of topological skeleton and local operator algebra arelargely inspired by the structure of 1+1D RCFTs. Indeed, a 1+1D RCFT is describedby a vertex operator algebra and a category of topological defects, such that they arecompatible with each other. We believe this is an instance of a more general structureof quantum orders. Some higher dimensional examples are also examined in Sec 2.4 tosupport this.We should note here that the idea of separating the local and global structures ofgapless phases, as well as the notion of topological skeleton, are due to Kong and hiscollaborators [64, 66, 68]. However the framework developed in Chapter 2 is much morecomplete and physically grounded than their works.We make no claim that a complete theory of quantum orders is achieved in thisdissertation. What is answered in this dissertation can be viewed as the âeasy partâof the problem of gapless phases: the theory of topological skeleton and symmetry-enrichment 4. The difficult part is the theory of local operator algebra and its com-3In this dissertation d+ 1D means d spatial dimension and one time dimension.4Even for these two problems we really only consider the so-called ârational casesâ, where the types5patibility relation with topological skeleton, which we look forward to develop in thefuture. Nonetheless we believe the framework proposed in Chapter 2 is a valuable firststep towards an ultimate theory. We discuss some of the future directions and chal-lenges for this framework in Sec 7. An immediate application of our framework is theprediction of new symmetry-enriched quantum orders.When we restrict to a subclass of phases, we often have a more refined and detailedtheory. Chapter 3 is devoted to develop a more refined theory for a class of symmetry-enriched quantum orders called symmetry-protected gapless topological phases(gSPT).For gSPTs there are many more elementary tools at our disposal, such as group theoryand group-cohomology. We follow a more traditional path in Sec 3 and develop a theoryof gSPTs from scratch. We then show that this theory is really just a special case ofthe theory of symmetry-enriched quantum orders developed in Chapter 2. Nevertheless,the more elementary approach of Chapter 3 allows us to describe things more explicitly,such as gauging and lattice model construction of gSPTs. Chapter 4 is the authorâswork on lattice model construction of gSPTs.1.2 Topological HolographyTopological holography, also known as Symmetry Topological Field Theory(SymTFT),has become quite popular in recent years [38, 95, 5]. Its central idea can be summarizedin a single sentence: a d+1-D system with symmetry S is the same as a boundary of a(d + 1) + 1-D topological order Z(S). Here the symmetry S can be quite general, butis assumed to be finite in order to apply the fusion higher category tools. The symbolZ stands for the center construction in fusion higher category theory. The topologicalorder Z(S) is called the symmetry topological field theory(SymTFT) of the symmetryS. The most familiar case is when the symmetry is a finite group G that acts as aninternal symmetry, then the SymTFT is simply the G-gauge theory in one dimensionhigher. In this case the idea of topological holography is simply that any boundary of aG-gauge theory naturally has a (possibly spontaneously broken)G-symmetry. Beyondthis trivial example, SymTFT is useful for three reasons.of topological defects and symmetry generators are finite. This is because the fusion higher categorymethod only applies to such cases. Some of our results can still be generalized to non-rational cases.This is discussed in Sec. 76â¢ For a generalized symmetry, it is often difficult to construct phases with that sym-metry systematically and directly. The SymTFT turns this problem into construct-ing boundary conditions for a higher dimensional topological order, a problem thatis more under control.â¢ There are many duality relations among symmetries and phases with symmetries.The most famous examples are the KramersâWannier duality and the Jordan-Wigner transformation. Dualities have an elegant interpretation in SymTFT as aâchange of basisâ. It is a mathematical fact that if two symmetries S, Sâ² are dualto each other then they have the same center: Z(S) â Z(Sâ²) 5. This makes theSymTFT an invariant under duality transformations. This leads to a new methodof constructing dualities and analyzing the effect of duality transformation onphases.â¢ SymTFT has been used by many, including the author, to study gapless phases,especially gSPTs. From the principle of SymTFT, a gapless phase with symmetryS is the same as a gapless boundary of the topological order Z(S). Although thisis not a solved problem, a partial solution does exists. More precisely, similar toquantum orders, the structure of a gapless boundary of a topological order alsoconsists of a topological skeleton part and a local operator algebra part. Thetopological skeleton has a simple and complete classification in terms of the so-called âcondensable algebrasâ. It can then be shown that the topological skeletonof a gapless boundary corresponds to the topological skeleton and the symmetryaction on the topological skeleton of a gSPT.Chapter 5 and 6 are the authorâs works on the relation between SymTFT andgSPT [131, 129]. Note that at the time of writing those papers, the concept of quantumorder and symmetry-enriched quantum order had not been introduced. What the works[131, 129] found was that there was a deep and precise connection between a yet-to-be-discovered notion of gapless phases with symmetry, and certain yet-to-be-discoveredstructure of gapless boundaries of the SymTFT. This mathematical relation is a rigorousone, despite its ambiguous physical meaning at that time. With the framework ofquantum order and symmetry-enriched quantum order developed in Chapter 2, I hope5Here by duality we mean weak duality. Two symmetries are weakly dual to each other if there isa one-one correspondence between their categories of gapped phases.7the physical meaning of those works is now clear. Due to limited energy and time, I amnot able to rewrite my works to be consistent with the framework of Chapter 2, thusChapter 5 and 6 should be read with caution. The physical understanding there mayhave been outdated, but the technical results are still sound.8Chapter TwoStructure of Gapless PhasesDiscovery is a childâs privilege. I mean the small child, the child who is notafraid to be wrong, to look silly, to not be serious, and to act differently fromeveryone else.â Alexander GrothendieckThis chapter introduces the notion of gapless topological phases and outlines asystematic framework for understanding them. A central goal of condensed matterphysics is to classify all quantum phases. A major achievement on this problem in thepast decade is that we have solved it in the gapped case. In the theory of gapped phases,the notion of topological order plays a crucial role. Originally it was realized in the studyof spin liquid that certain nontrivial gapped phases do not have any symmetry thereforefall outside Landauâs paradigm of symmetry breaking. The notion of topological orderwas then introduced to describe such gapped phases. One important consequence of thisnotion is that it allows us to divide the problem of classifying all gapped phases into twosubproblems: the classification of topological orders, and the classification of symmetry-enriched topological orders. Both the notion of topological order(TO) and symmetry-enriched topological order(SET) can be defined rigorously and microscopically. Thiswas the first time in history that the notion of âphaseâ was given a precise definition.After three decades of efforts since their discovery, a magnificent mathematical theoryhas been built to describe gapped topological phases [58, 47]. This theory is brieflyreviewed in section 2.1.We then illustrate through familiar physical models that topological phenomena,9especially topological defects, are ubiquitous in gapless phases as well. These examplesalso demonstrate that the framework we have developed in the past to describe gappedtopological phases, namely category theory, can largely be adapted to describe topologi-cal defects in gapless phases. As more cases are examined, two notions begin to emerge:first, that there exists a notion of gapless phase not protected by any symmetry, akinto topological order; and second, that the global topological aspects of such phases ad-mit a simple, categorical description much like that of topological orders. The conceptâquantum orderâ 1 is then introduced to describe gapless phases not protected by anysymmetry, and the concept âtopological skeletonâ is introduced to describe the topo-logical aspects of quantum orders. A quantum order also has local dynamics that is notfully decoupled from the topological skeleton, but we should be able to study the globalaspect and the local aspect separately before sewing them into one single theory. Theclassification of all quantum phases is thus divided into two subproblems: the classifi-cation of quantum orders, and the classification of symmetry-enriched quantum orders.Supported by many examples, we argue that the latter problem is solved by simple(categorical)representation theory, similar to the theory of SETs. This completes ourparadigm of classifying all quantum phases. The discussion in this chapter is neithercomplete nor fully rigorous, but we believe our framework is robust and the languageintroduced here is necessary for developing a more complete theory. Challenges andpromises of our framework are discussed in the final section of this chapter.2.1 Gapped Phases and Topological DefectsGapped phases are undoubtedly the simplest quantum phases. A system is calledgapped if the energy gap between the ground state and the first excited state doesnot vanish in the thermodynamic limit. Then in the low energy limit, the energy gapis essentially infinite, and only the ground state is âvisibleâ to us. This simplicityof gapped systems is why a systematic theory is possible. To understand universalproperties of gapped phases we must give a clear definition of âphaseâ. The followingdefinition is well-known.1The notion of quantum order was initially proposed by Xiao-gang Wen to describe gapless quantumspin liquids. My notion of quantum order is intuitively the same as Xiao-gangâs. However my definitionof quantum order is a microscopic one and is more rigorous.10Definition 2.1.1: Two gapped Hamiltonians H1, H2 are said to be in the same phaseif there is a continuous path of gapped Hamiltonian H(t), t â [0, 1], such that H(0) =H1, H(1) = H1. This defines an equivalence relation on the space of gapped Hamilto-nians, a topological order is then an equivalence class of this equivalence relation.Notice that in this definition no symmetry of any sense is mentioned. In this sensea topological order is a gapped phase with no symmetry. Indeed, a topological order isvery robust: any perturbation of the system does not change the topological order aslong as the gap stays open. We do not need to respect any prescribed symmetry actionin the perturbation. A topological order can not have any interesting local dynamics:in the IR all local correlation functions go to zero. Formally, if we consider a correlationfunctionG(x1, · · · , xn) := â¨Ï1(x1) · · Â·Ïn(xn)â© (2.1)then its IR scaling limit islimÎ»â+âÎ»âG(Î»x1, · · · , λxn) = 0 (2.2)for any choice of â â R. Therefore if there is anything nontrivial about a topologicalorder, it can only be non-local, or appear on the boundary of the system.Example 2.1.1: The integer quantum Hall effect(IQH) is a topological order witha trivial bulk, however its boundary has gapless chiral modes. This is an exampleof a gapped phase with no nontrivial bulk observables but has nontrivial boundaryobservables.Example 2.1.2: The fractional quantum Hall effect(FQH) is a topological order withboth nontrivial bulk observables and boundary observables. In the bulk, it has fractionalcharged quasi-particles, which can be reviewed as point-like defects(holes) in the system.These quasi-particles are non-local: they can not be created by any local operator. Onthe boundary it also has gapless chiral modes.Example 2.1.3: An ordinary(s-wave) superconductor is a topological order with non-trivial bulk observables and a trivial boundary. In the bulk the electrons and magneticfluxes are nontrivial topological excitations. Due to the copper-pair condensation theelectron now becomes a quasi-particle of order 2. The minimal deconfined magnetic11flux is 12Φ0 where Φ0 is the flux quanta. It is an order-2 topological defect. UnlikeIQH/FQH, its boundary is gapped and trivial.Remark. Technically a 2+1D superconductor is not simply a 2+1D topological order,but rather a 2+1D topological order coupled to a 3+1D gapless bulk(the real worldquantum electrodynamics.). However a 3+1D superconductor is a well-defined 3+1Dtopological order on its own.Systems with a completely trivial bulk, such as the IQH, are called invertible gappedphases. Such systems are quite rare and highly restricted. In dimensions relevant tocondensed matter physics, there are only two families of them:â¢ Fermionic 2+1D invertible phases are generated by the p + ip topological super-conductor.â¢ Bosonic 2+1D invertible phases are generated by the Kitaevâs E8-state.The IQH can be viewed as two copies of p + ip, stacked on top of each other. TheE8-state has not been observed in nature, even a lattice model for it is not availableyet.Since invertible gapped phases are relatively simple 2 and well-understood, we canfocus only on gapped phases with nontrivial bulk non-local observables. The non-localobservables are encoded in the topological defects of the system. By a topological defectwe mean a defect whose IR properties do not change as we deform its shape or locationin space. For example, if we move it around the system and return it to the initiallocation, we should be able to say it is the âsameâ defect as before.In 2+1D, the topological defects often take the form of point-like quasi-particles.The quasi-particles can be fused and braided. This structure organizes into a math-ematical structure called a braided fusion category. Moreover, the braiding betweenquasi-particles is always ânon-degenerateâ, in the sense that any quasi-particle mustbraid nontrivially with at least one other particle. This translates into the mathemati-cal condition that the quasi-particles are described by a non-degenerate braided fusioncategory, also known as a modular tensor category.2In general dimension the theory is actually quite involved, but for condensed matter theory thehigh dimensional invertible phases are not relevant.12In recent years this mathematical theory has been extended to higher dimensionswhich yielded a full theory of topological orders in all dimensions. Once again, theguiding principle is that a gapped phase should be determined, up to invertible phases,by its bulk topological defects. The relevant structure is again the fusion and braid-ing properties of these defects. In 3+1D, which is the highest dimension relevant forcondensed matter physics, there can be point-like as well as string-like defects. Thiswas known long ago in the study of superconductivityâthe magnetic fluxes in a 3+1Dsuperconductor are localized in thin-tube-shaped regions, called flux tubes, which areeffectively string-like defects. To describe these higher dimensional defects, the lan-guage of higher category theory is most natural. It was shown that the collection ofparticle-like and string-like defects in a 3+1D gapped phase form a non-degeneratebraided fusion 2-category [58, 140, 56].The above however is not the full story of topological defects of gapped phases.Another important development in recent years is the theory of codimension-1 defects.In the traditional approach to 2+1D topological order, quasi-particles/anyons play thefundamental role. The whole theory of 2+1D topological order is centralized aroundanyons, which is why sometimes it is called the âanyon theoryâ. However line defectsare also possible in a 2+1D gapped phase. The reason that they do not participate inthe classification of gapped phases is that they are determined by the lower dimensionaltopological defects. Once the braiding and fusion of particles are known, the informationabout line defects can be deduced from a procedure called condensation completion [41,25, 138]. Physically, the line defects result from condensation of particles, therefore theydo not play a fundamental role. A basic idea in the categorical-theoretical frameworkof describing gapped phases is that a system of topological defects of spatial dimensionranging from 0 to d is described by a d + 1-category. Therefore the condensationcompletion of a 2+1D topological order is now a fusion 2-category. The 2-category isno longer braided because there is no sense of braiding for the line defects. Howeverthe braiding of anyons is still secretly encoded in this fusion 2-category. If B is abraided fusion category, its condensation completion is denoted as ΣB 3. In general thecondensation completion does not add to or lose any information of the gapped phase,it is simply a procedure of adding all the non-fundamental topological defects into the3It can be defined as the 2-category of separable B-module categories, or equivalently as the 2-category of separable algebras within B.13category. To recover B one can take the âloopingâ, denoted as â¦, and â¦Î£B = B.Although it seems like just a convenient mathematical choice, and in principle we donot need to work with the condensation completion, it will turn out that it is the fusion2-category picture, not the braided fusion 1-category picture, that generalizes to gaplessphases. Similarly, in a 3+1D gapped phase, there can be membrane defects that resultfrom condensation of particles or strings. Condensation completing a 3+1D topologicalorder results in a fusion 3-category. The condensation completed picture is also crucialwhen we discuss SETs.Although it is relatively easy to argue that topological defects are described by fusionhigher-categories, not every fusion higher-category is allowed in a gapped phase. In thebraided fusion formulation, the non-degenerate-braiding condition is very importantand physically originates from the very âgappedâ assumption. In the condensationcompleted formulation, there is a similar non-degeneracy condition that is equivalentto the non-degenerate-braiding condition in the braided version. In the following dVecis the trivial fusion d-category generated by one simple object.Definition 2.1.2: A fusion d-category C is non-degenerate if it satisfies Z1(C) = dVec,where Z1 is the Drinfeld center.The two notions of non-degeneracy are indeed equivalent: ΣB is non-degenerate inthe above sense if and only B is not-degenerate in the braiding-nondegenerate sense.Therefore the classification of topological orders take the following elegant form.Theoremph 2.1.1 ([58, 47]): 4 d+ 1D topological orders are classified, up to stackingwith invertible phases, by non-degenerate fusion d-categories.2.2 Symmetry-Enriched Topological OrdersAnother important aspect of quantum phases is symmetry-enrichment. The introduc-tion of symmetry leads to a new concept of phase. However we must first answer thevery basic question of what is a symmetry. In recent years the meaning of symmetryhas undergone radical changes. To avoid making things technical this early, right nowwe simply say that at the bare minimal a symmetry should be a collection of locally-4The superscript means it is a âphysical theoremââa result that is derived based on physical prin-ciples and physical reasoning.14generated operators that commute with the Hamiltonian. These operators need not beunitary, in order to include the so-called ânon-invertible symmetriesâ. They also neednot act on the entire system, to allow for âhigher-form symmetriesâ. We do requirethese operators to act âtopologicallyâ, meaning that they act on sub-manifolds of thesystem, and their expectation values are invariant under continuous non-crossing de-formation of their locations. This collection of operators actually has extra structurethat makes it into a fusion higher-category.Definition 2.2.1: Let S be a symmetry, two gapped Hamiltonians H1, H2 with sym-metry S are said to be in the same S-enriched gapped phase, if there is a continuouspath of S-symmetric gapped Hamiltonian H(s) such that H(0) = H0 and H(1) = H1.This defines an equivalence relation on the space of S-symmetric gapped Hamiltonians,an S-enriched topological order is then an equivalence class of this equivalence relation.Notice that if two gapped systems have the same SET order, then they must havethe same topological order. Therefore we can focus on one topological order at a time,and classify all SETs with a fixed underlying topological order. This turns out to bea straightfoward representation theory problem. Intuitively, different SETs with thesame underlying topological order differ by how the symmetry act on the topologicalorder. Let C be a fixed underlying d + 1D topological order. We use the same symbolC to denote the fusion d-category of its topological defects(we are working with thecondensation completed description). Consider inserting a symmetry defect Ds for asymmetry generator s â S. Then if we forget about the symmetry S, this defect willbecome a topological defect of the underlying topological order. This defines a map fromS to C, D : S â C, which roughly means how S acts on C. Furthermore the algebraicstructure of S, such as composition of symmetry generators, should be preserved by themap D. Therefore we should naturally expect the symmetry S, when all the algebraicstructure of symmetry generators is taken into account, forms a fusion d-category, andthe map D is simply a functor between fusion d-categories. This suggests the followingresult.Theoremph 2.2.1: Let C be a d + 1D topological order, i.e., a non-degenerate fusiond-category, and S be a symmetry, i.e., a fusion d-category. Then an S-enriched C is afunctor between fusion d-categories D : Sâ C.This a very modern way of stating the classification of SETs. It is elegant, matches15with our intuition, and is general enough to cover almost all generalized symmetries.We first show that this is equivalent to the traditional approach.Example 2.2.1: Traditionally a symmetry is described by a group G, if the symmetryis non-anomalous and the group G is finite, then in the categorical formulation itis described by the d-category dVecG. Traditionally a d + 1D topological order isdescribed by a non-degenerate braided fusion d â 1-category B. In the condensation-completed description the topological order is described by the fusion d-category ΣB.By the above result a G-SET with underlying topological order B should be a functorD : dVecG â ΣB. Such a functor is equivalent to giving a G-crossed extension of B,which is exactly the traditional description of a G-SET with underlying TO B.The result Theorem 2.2.1 says that the classification of SET is simple representationtheory. Recall a representation of a group G is a group homomorphism G â End(V )where V is a vector space. This can be viewed as a functor between âfusion 0-categoriesâ. Representation theory in modern sense simply means the study of functors.Let stress again the physical meaning of Theorem 2.2.1: an SET is determined byhow the symmetry defects map to the topological defects of the underlying topologicalorder. This interpretation will be crucial for our formulation of symmetric-enrichedquantum orders.The simplest SETs orders are the ones with trivial underlying topological order.They are called symmetry-protected topological phases(SPT).Definition 2.2.2: Let S be a symmetry, an S-SPT is an S-SET such that the underlyingtopological order is the trivial one: dVec.The usual cohomology classification of SPTs can be recovered from the classificationof general SETs.Example 2.2.2: Let S = dVecG be a usual non-anomalous G-symmetry describedabove, and C = dVec be the trivial topological order. Then a functor D : dVecG âdVec is determined, up to equivalence, by an element of Hd+1(G,U(1)).2.3 Comments on the Notion of SymmetryThe developments of generalized symmetries in recent years have shed light on ourunderstanding of quantum phases, but it has also caused certain degree of confusion.16It is important to clarify the notion(s) of symmetry here.It is often said that a (2+1D) topological order has 1-form symmetries, and topo-logical orders can be described as 1-form spontaneous symmetry-breaking orders. Thisstatement on its own is not wrong. If a topological order has a nontrivial anyon c, thenany lattice model realization of the order will have a string operator corresponding to cthat commutes with the Hamiltonian. In this sense any lattice model realizing the sametopological order will have the same 1-form symmetries. However the actual form of the1-form symmetries depends on the realization of the topological order. This is differentfrom the notion of symmetry in SETs/SPTs, where the form of the symmetry actionis required to be preserved under deformation. Therefore although a topological orderhas 1-form symmetry, it is not protected by or enriched with any 1-form symmetry.The topological defects of a topological order can all be viewed as generalized sym-metries of the topological order. In this sense a topological order described by a fusiond-category C has symmetry C. But we should keep in mind that topological order isdefined without any notion of symmetry, and symmetries only emerge as a result ofclassification. These emergent symmetries are intrinsic properties of topological orders,which are invariant under generic UV deformation. In the continuum, the differencebetween emergent symmetries of topological orders and imposed symmetries of SETstake the following form: in the continuum a topological order is described by a topolog-ical quantum field theory(TQFT) defined on plain manifolds, but a G-SET is describedby a TQFT defined on manifolds with G-bundles.This picture will be important when we discuss topological skeleton of quantumorders.2.4 Topological Defects in Gapless PhasesWe now move on to the central subject of this dissertation, topological gapless phases.Since the notion of a gapless phase is not defined yet, our discussion will be informaland focused on examples. The examples will guide us towards a proper definition ofgapless phases.The study of topological phenomena in gapless phases is actually much older than ingapped phases. Familiar systems such as superfluids, quantum electrodynamics(QED),conformal field theories, all have non-trivial topological properties in the form of topo-17logical defects. A more modern example is gapless spin liquids.Example 2.4.1 (Superfluids): In a 2+1D superfluid, a continuous symmetry U(1) isspontaneously broken. Since Ï1(U(1)) = Z, there can be topologically nontrivial orderparameter configuration when a point defect is present. These nontrivial configurationsare known as vortexes. They have fusion rule Z. Under the particle-vortex duality,they can also be viewed as the gauge charges of an emergent U(1)â²-gauge field. Thevortexes are very much like anyons in a topological order: they are topological defects,and can only be created in vortex-anti-vortex pairs. However a major difference fromtopological order is that the number of vortex types in a superfluid is infinite(Z). Thisis impossible in any gapped phase. We see an interesting feature of gapless phases: thenumber of topological defects can be infinite.Example 2.4.2 (QED and topological insulator): QED and topological insulator areboth described by U(1)-gauge theories with gapped matter fields. In the IR when allmatters are integrated out, we have an actionSθ =12e2F â§ âF + Î¸16Ï2F â§ F (2.3)where the second term is called the theta-term. The topological defects of the theory areelectric charges and magnetic monopoles. The charges are given by representations ofU(1), which form a group Z. The monopoles are given by Ï1(U(1)) = Z. Together thetopological defects of a 3+1D pure U(1)-gauge theory form the fusion group ZE ÃZM .This is again an infinite group. The theta-term is topologicalâit evaluates to zero ona trivial manifold. It has the effect of changing the statistics between charges andmonopoles, but it does not change the local dynamics of the theory.More generally, in any gauge theory with continuous gauge group G, the gaugecharges will be classified by Rep(G), which is almost a fusion category except thatthere are infinite many simple objects. The monopoles will be classified by Ï1(G), whichcan be finite or infinite. The possibility of infinite many topological defects is uniqueto gapless phases. This also poses a mathematical challenge to an algebraic theoryof topological defects, since the theory of âalmost-fusion categories with infinite manysimple objectsâ is not well-developed. There are also many gapless phases where thetypes of topological defects are finite. The following example is particularly important18for our purposes. 1+1D rational conformal field theories are the most well-understoodclass of gapless phases. There is a rigorous mathematical theory of RCFTs. This allowsus to base our physical intuitions on concreteExample 2.4.3 (CFTs): It was realized long ago that there can be topological defectsin CFTs. The theory of topological defects in CFTs is most well-understood for 1+1Drational CFTs(RCFT). The rational condition is roughly the same as the condition thatthere are finitely many topological defects. A RCFT is determined by two pieces ofdata: the local operator algebra, described by a vertex operator algebra(VOA) V 5; anda fusion category C of its topological defects. The non-topological and the topologicalaspects of a RCFT must be compatible with each other, this condition is [39]ModV â Z(C), (2.4)where the left side is the category of modules over V , which is itself a modular ten-sor category. This is sometimes called the sewing condition, which can be viewed asspecifying how the chiral and anti-chiral modes of the CFT are sewed together. For adiagonal CFT, the topological defects are in one-one correspondence with the primaryoperators, and they are known as the Verlinder lines of the diagonal CFT. For exam-ple, for the Ising CFT, there are three topological defects 1, Ï, f corresponding to thethree primary operators. Their fusion rule is the same as the OPE of the correspondingconformal blocks.It is possible for two different RCFTs have the same local operator algebra: therecan be different fusion categories C,D such that Z(C) = Z(D) = ModV for the sameVOA V . For instance, if a RCFT has a finite group symmetry G, then gauging Gdoes not change the local operator algebra(gauging a finite group is a topological op-eration), but it does change the topological defects in general. This is known as theorbifold construction in CFT literature. For a fixed VOA, different RCFTs are re-lated by âgeneralized gaugingâ, also known as generalized orbifold in CFT literature.Mathematically generalized gauging is described by Morita equivalence.Definition 2.4.1: We say two fusion d-categories C,D are Morita equivalent if thereis an equivalence Mod(C) âMod(D).5More precisely, for a non-chiral CFT the local algebra is a full-field algebra that is a tensor productof a chiral VOA and an anti-chiral VOA.19A famous result is that two fusion 1-categories are Morita equivalent if and only iftheir Drinfeld centers are equivalent. Therefore two RCFTs with the same local operatoralgebra have Morita equivalent topological defects. Morita equivalence should be viewedas the precise mathematical formulation of âgaugingâ or âtopological manipulationâ.The theory of RCFT is a perfect illustration of a general paradigm: a gapless phaseis determined by a local data and a global data. The local data is the structure ofits local operators, including operator contents and OPEs. The global data is thetopological defects and their algebraic relations. This structure is also present in theQED example: the theta term alters the global structure of the theory but has no effecton local dynamics. If the number of topological defects is finite, like in RCFTs, thelanguage of fusion higher category naturally applies. This fusion higher category oftopological defects must be compatible with the local data. In RCFT this is explicitlyrepresented by Eq .(2.4).The topological defects of RCFTs have also been called generalized or non-invertiblesymmetries. However RCFTs should not be viewed as phases protected or enriched withnon-invertible symmetries. This is similar to the situation of topological orders. In factRCFTs can be axiomatized as functorial quantum field theories defined on Riemannsurfaces. This is known as Segalâs approach [96]. Just like a TQFT is determined by itspartition function on all manifolds, a RCFT is determined by its partition function onall Riemann surfaces. In this sense RCFTs are not phases enriched with symmetries.This observation is crucial to our definition of quantum order.Another important observation is that the topological defects of a RCFT do notsatisfy the non-degeneracy condition as those in a topological order. Recall a fusiond-category C is non-degenerate if Z(C) = dVec. However, for a fusion 1-category,Z(C) = Vec if and only if C = Vec, i.e. C is trivial. This can be viewed as a formalderivation of the well-known result that there is no non-trivial topological order in1+1D. Therefore the topological defects in a RCFT generally do not satisfy the non-degeneracy condition.Next we discuss some 2+1D examples. Although lacking a rigorous mathemati-cal formulation, the examples in the following still share a similar structure as 1+1DRCFTs.Example 2.4.4: The Kitaev honeycomb model can be viewed as obtained by gauging20the fermion parity ZF2 of a Majorana fermion. Therefore the system always contains agapped vortex(the fermion parity vortex). Depending on what happens to the Majoranafermion one obtains various phases. The actual model realizes three possibilities.1. If the Majorana fermion forms a trivial(s-wave) superconductor, the ZF2 -gaugedtheory is the toric code topological order.2. If the Majorana fermion forms a p + ip superconductor, the ZF2 -gauged theory isa non-abelian topological order known as the Ising category.3. If the Majorana fermion forms a metal, then the model contains gapless emergentfermion and gapped vortexes. This case is called the gapless Kitaev spin liquid.In all cases, the emergent fermion has well-defined fusion and (self and mutual)braiding.In case 3, the vortex does not have well-defined braiding due to gapless fermions, in thissense the vortex is not topological. In the gapless phase, there is a quasi-particle, theemergent fermion. It has well-defined self and mutual braiding, however the braidingis degenerate: there is nothing topological to braid with the fermion. The emergentfermion is described by the degenerate braided fusion category sVec of Z2-graded finitedimensional vector spaces. The condensation completion of the emergent fermion isΣsVec, a degenerate fusion 2-category. We see that the Kitaev spin liquid is alsodescribed by a local data and a global data. The local data is simply a fermi surface,and the global data is the presence of an emergent fermion, or that the topologicaldefects form ΣsVec. The topological defects form a fusion 2-category that is notpossible in any gapped phases. The above structure is very similar to that of 1+1DRCFTs.The above example can be generalized.Example 2.4.5: The Kitaev gapless spin liquid is a gauged metal. Let us considergauging the finite symmetry of a generic gapless system. Let G be a finite group. Con-sider a gapless system with bosonic G-symmetry, for instance it can be a G-symmetrybreaking phase transition point. Then after gauging G the G-charges become super-selection sectors/topological defects. The G-fluxes can however be non-topological.We have witnessed this in the Kitaev model, where the fermion parity flux becomesnon-topological in the gapless phase. A gauge flux of G is topological if and only ifit only braids with gapped charges. Therefore we need to specify the collection of21gapped/gapless gauge charges. This is determined by how the symmetry G acts inthe original ungauged system. If the symmetry acts faithfully, meaning that all of thesymmetry charges are gapless, then after gauging all the gauge charges will be gapless,which would make all gauge fluxes non-topological. In general the symmetry G maynot act faithfully on the original system. The faithful symmetry action is a quotient ofG, denoted as K = G/N for some normal subgroup N â G. This means the gaplesscharges form the category Rep(G/N), which is a sub-category of the category Rep(G)of all symmetry charges. Therefore a gauge flux of G will be topological if and only if ittakes value in N , i.e., if it acts trivially on charges in Rep(G/N). Thus after gaugingG we have two families of topological defects: all the gauge charges and gauge fluxesof N . These topological defects will form a degenerate braided fusion category as longas N Ì¸= G. Classifying braided fusion categories with this structure will lead to veryinteresting results, see Chapter 3.The topological defects in above example can also be regarded as generalized sym-metries. Should we view them as emergent symmetries like topological defects of topo-logical orders and RCFTs, or should we view them as defects associated with a pro-tecting/enriching symmetry, like symmetry defects of SETs/SPTs? To answer this wemust have a proper definition of phases and symmetry-enriched phases. Notice thatthe original ungauged theory also has topological defects associated with the symmetryG, namely the G-domain walls. Are they more like topological defects of topologicalorders, or more like symmetry defects of SPTs? Let us consider two extreme cases:â¢ If all G-charges are gapped, then all the local operators that transform nontriviallyunder G die out in the IR. This means if we only look at the local properties ofthe system in the IR, we would not be able to infer the symmetry G. We can alsoexplicitly break G without changing the local IR properties of the system.â¢ If all G-charges are gapless, then there are local operators, whose action are non-trivial in the IR, that transform nontrivially under the G-action. This means thesymmetry G can be inferred from the local properties of the system. Also anyattempt to break G explicitly will change the local IR properties such as local cor-relation functions. The symmetry G is now enforced by the IR local properties. Ifwe deform the system in the UV, as long as the IR local properties stay the same,the symmetry G is intact, although its actual form in the UV may be altered.22The symmetry G in the first case is more similar to symmetry of SPT phases, whilethe symmetry G in the second case is more similar to topological defects of topologicalorders or RCFTs. To spell out these similarities more rigorously, we must define thenotion of gapless phases and symmetry-enriched gapless phases properly.Now we believe we have collected enough clues to develop the general theory ofgapless phases and symmetry-enriched gapless phases.2.5 Quantum Order and Symmetry-enriched Quan-tum OrderIn parallel with the classification of gapped phase, there should be a notion of gaplessphase whose definition does involve any symmetry, and a notion of symmetry-enrichedgapless phase whose definition involves explicitly given symmetry. The first notion iscalled a quantum order. It is roughly speaking quantum phase with no UV symmetries.A quantum order may have emergent IR symmetries, but similar to topological defectsof topological orders, they are intrinsic properties of the order, and their UV realizationdepends on the UV realization of the order. On the other hand, the word symmetryin symmetry-enriched quantum order means a UV symmetry whose UV realization isfixed in any order-preserving deformation.Definition 2.5.1: The IR scaling limit of a correlation function is the limitâ¨Ï1(x1) · · Â·Ïn(xn)â©IR := limÎ»â+âÎ»ââ¨Ï1(Î»x1) · · Â·Ïn(Î»xn)â© (2.5)where â is the unique real number such that the limit exists. The IR local operatoralgebra of a system is the collection of IR scaling limits of all local correlation functions.The reason that we define local operator algebra in the IR limit is physically well-motivated: a phase is a IR concept. Taking the IR scaling limit is the same as followingthe renormalization group flow to the extreme IR. Therefore the IR operator algebra ofa system should be the local operator algebra of a scale-invariant quantum field theory.If we further assume that the IR limit of the system is a conformal field theory, thenthe IR operator algebra is simply determined by the set of primary operators and thecoefficients of their three-point functions.23If we deform a system, and we observe that the local IR dynamics stays the same,then we should view the deformed system as in the same phase. This leads to thefollowing definition.Definition 2.5.2: Two Hamiltonians H0, H1 are said to have the same quantum orderif there is a continuous path of Hamiltonian H(s), s â [0, 1] with isomorphic IR localoperator algebra, such that H(0) = H0, H(1) = H1.Notice that in this definition no symmetry of any kind is mentioned. In this sense aquantum order is a phase not protected by or enriched with any symmetry. A specialcase of quantum order is topological order.Example 2.5.1: A topological order is a quantum order with trivial local operatoralgebra. Indeed, demanding the gap to remain open in a deformation is the same asdemanding the local operator algebra to remain trivial.Example 2.5.2: 1+1D CFTs describe quantum orders. Two 1+1D critical systemshave the same quantum order if and only if they are described by the same CFT in theIR. The local operator algbera of a 1+1D CFT is a vertex operator algebra(or a fullfield algebra).A quantum order is defined without specifying any symmetry, but it can have emer-gent symmetries in the IR. As we have seen in the 1+1D CFT example, these emergentsymmetries take the form of topological defects. Unlike topological defects in topolog-ical orders, topological defects in gapless quantum orders can be degenerate. In orderto apply the language of fusion higher-category theory, we have to assume a finitenesscondition on the types of topological defects.Definition 2.5.3: A rational quantum order is a quantum order with finite types oftopological defects.In the following we will assume the rational condition. The topological defectsin a d + 1D rational quantum order will then form a fusion d-category, without thenon-degeneracy condition.Definition 2.5.4: The topological skeleton of a quantum order is the fusion d-categoryof its topological defects. The gravitational anomaly of a topological skeleton C is theDrinfeld center Z(C).The definition of the gravitational anomaly follows [58]. It is the obstruction that24a set of seemly consistent fusion rules cannot be realized by any well defined gappedquantum model in the same dimension.The gravitationally anomaly can either be matched by a higher dimensional topo-logical order Z(C), or a gapless system. In a quantum order, it is the gaplessness thatmatches the gravitational anomaly of the topological skeleton.A quantum order is determined by its local operator algebra and its topologicalskeleton. We have seen this in 1+1D RCFTs. A 1+1D RCFT is determined by aVOA V and a fusion category of topological defects C, such that ModV â Z(C). TheVOA V is the local operator algebra and C is the topological skeleton. In general thereshould be a compatibility relation between the local operator algebra and the topologicalskeleton of a quantum order. Although we are not able to provide this relation(sincethe theory of local operator algebra is not unknown in high dimensions), we do observethat this relation should only depend on the Morita class of the topological skeleton.This is because a Morita equivalence corresponds to a topological manipulation such asgauging a finite symmetry group 6. Such topological manipulation should not changethe local operator algebra up to certain notion of Morita equivalence of local operatoralgebras. Therefore we expect the following.Conjecture 2.5.1: The compatibility relation between the local operator algebra Aand the topological skeleton C takes the formP (A) â Z(C). (2.6)where P (A) is a nondegenerate braided fusion d-category constructed from A.Another motivation for this conjecture is that the Drinfeld center Z(C) measures thegravitational anomaly of the topological skeleton, therefore the compatibility conditionshould be a statement that the local operator algebra matches with this gravitationalanomaly. Recently there have been some developments in formulating local operatoralgebras of quantum field theories rigorously using factorization algebras. It would bevery interesting to test this conjecture using the factorization algebra formulation.Example 2.5.3: The gapless Kitaev spin liquid is a quantum order whose local opera-tor algebra is the bosonized version of that of a fermi surface, and the topological skele-6A Morita equivalence is realized by a generalized gauging, which sums over insertions of a subsetof (potentially non-invertible) symmetry defects [10].25ton is ΣsVec. The gravitationally anomaly of the topological skeleton is Z(ΣsVec),which is a 3+1D topological order with a fermionic particle and a bosonic string.Example 2.5.4: Let us revisit Example 2.4.5. Consider again a system with globalsymmetry G such that only G/N acts faithfully in the IR. Should all of G-defects beviewed as topological skeleton of a quantum order? The answer is clearly no. Consideragain two extreme cases.â¢ IfN = G, then allG charges are gapped, therefore theG-charges do not participatein the IR LOA at all. We can explicitly break the symmetry without affecting theIR LOA. In this case the symmetry is not the topological skeleton of a quantumorder. For instance the system can itself be gapped, then it is simply a G-SET.Symmetry defects in a G-SET are clearly not topological skeleton of a topologicalorder.â¢ If N = 1, then all G charges are gapless, therefore the symmetry G can be inferredfrom the IR LOA. In this case the symmetry defects are determined by the quantumorder, therefore should be identified as topological skeleton of a quantum order.In general the topological skeleton should include the defects of those global symmetriesthat act faithfully in the IR, i.e., dVecG/N . Moreover, it is possible that the faithfulsymmetry G/N is anomalous, therefore in general the topological skeleton contains adVecÏG/N , for some Ï â Hd+2(G/N,U(1)).Then how should we view the symmetry G in the intermediate case(1 âª N âª G)?The only way to make sense of it is to view it as an enforced symmetry like the symmetryin SET/SPT orders. This leads to the notion of symmetry-enriched quantum order.Definition 2.5.5: Let S be a symmetry, two Hamiltonians H0, H1 with symmetry Sare said to be in the same S-enriched quantum order, if there is a continuous pathof S-symmetric Hamiltonian H(s) with isomorphic local operator algebra, such thatH(0) = H0, H(1) = H1.In this definition we have kept the symmetry S general. For example it can be acontinuous Lie group symmetry, or a non-topological rigid symmetry that acts onlyin certain directions. If it is a finite topological symmetry, then it is described by afusion d-category. Following the same logical that led to the classification of SET inSec. 2.2. We expect that symmetry-enriched quantum orders with a fixed underlying26quantum order ae determined by how the symmetry defects map to the topologicaldefects of the quantum order. Namely, with an explicit symmetry action by S, we caninsert symmetry defects of S, then if we forget the symmetry S these defects becometopological defects of the underlying quantum order. This process defines a map fromthe symmetry category to the topological skeleton category that preserves all the fusionstructure.Theoremph 2.5.1: Let S be the fusion d-category of a topological symmetry in d+1D,and Q be a quantum order with topological skeleton Q. An S-enriched Q is determinedby a (linear monoidal) functor F : Sâ Q.Example 2.5.5: A topological order has a topological skeleton ΣB, where B is thenon-degenerate braided fusion dâ 1-category of topological defects of codimension> 2.Then an S-enriched B is a functor F : S â ΣB. As discussed in Sec. 2.2, this agreeswith the usual notion of SETs.Similar to topological orders and symmetry-enriched topological orders, a generalquantum order should be described by a (scale invariant)quantum field theory definedon manifolds with a pseudo-Riemann metric in the continuum, while a G-enrichedquantum order should be described by a quantum field theory defined on manifoldswith both a pseudo-Riemann metric and a G-bundle.It is often said that a topological order has higher-form symmetry. This can beinterpreted as follows.Example 2.5.6: Consider a topological order with topological skeleton ΣB, let D â Bbe any braided fusion sub-category of B, the we have inclusioni : ΣD â ΣB (2.7)In this way the topological order is viewed as ΣD-enriched topological order.The next example is essentially a classification of the so-called âgapless symmetry-protected topological phasesâ(gSPT).Definition 2.5.6: A d+1D (G,K)-gSPT is a G-enriched quantum order whose topo-logical skeleton takes the form dVecÏK .Example 2.5.7: Consider a system with global non-anomalous symmetry G whosefaithful action in the IR is reduced toG/N . As discussed before, the topological skeleton27of such a system contains a dVecÏG/N , the system can be viewed as a quantum orderenriched with G, i.e., a (G,G/N)-gSPT. This is given by a functordVecG â dVecÏG/N (2.8)Such a functor is determined by a group homomorphism Ï : G â G/N togetherwith a trivialization of Ï in G: dη = ÏâÏ. Physically this means the anomaly ofthe IR faithful symmetry G/N is an emergent anomaly that is trivialized in the UV.Notice that for a fixed emergent anomaly Ï, different trivializations of it form a torsorover Hd+1(G,U(1)). Physically this means we can stack a (G,G/N)-gSPT with a G-SPT without changing the underlying quantum order. However as an operation onG-enriched quantum orders, the stacking can be nontrivial. A more careful analysis ofthe classification of gSPTs will be given in Chapter. 3.The next example is an application of our result. As far as we know this concepthas not been introduced in the literature at this level of generality.Definition 2.5.7 (Symmetry-enriched RCFT): Let C = (V , C) be a 1+1D RCFT,where V is a full vertex operator algebra and C is the fusion category of its topologicaldefects. Let S be a fusion category. An S-enriched C is given by a functor F : Sâ C.Example 2.5.8: Consider the critical Ising model. This is a very simple and well-understood model. As a diagonal RCFT, its topological defects 1, Ï, f form the Isingfusion category, with fusion rules(1 is the unit)Ï â f = f â Ï = Ï, Ï â Ï = 1 + f (2.9)Therefore the topological skeleton of the Ising CFT is the Ising category, pretty tauto-logical. We often say Ising CFT has Z2 symmetry, this is clear from the lattice model,but how to understand it on the level of CFT? One formulation is that the Ising VOAhas a Z2-automorphism. On other hand, we should also be able to identify the Z2symmetry in the Ising category. Indeed it is generated by the f -line. We can view Isingas a Z2-enriched quantum order using a functorVecZ2 â Ising (2.10)28that maps the non-identity object of VecZ2 to f . This is simply an inclusion of sub-category. We can replace Z2 by a general symmetry G, and obtain various G-enrichedIsing CFTs. However it is readily seen that a general functor VecG â Ising mustfactor through VecZ2 . In this sense the Z2-symmetry of Ising is the maximal intrinsicinvertible symmetry.In general for a 1+1D RCFT with a topological skeleton C, we can define its maximalinvertible symmetry to be the full subcategory of invertible objects in C. The maximalinvertible symmetry takes the form VecÏK for some finite group K and cocycle Ï âH3(K,U(1)). Any functor from an invertible symmetry category to C must factorthrough VecÏK . Then a G-enriched RCFT is essentially a 1+1D gSPT, determined bya functor VecG â VecÏK .Example 2.5.9: Consider the compact boson at radius R2 = 2, denoted as U(1)2. Weconsider the non-chiral version U(1)2ÃU(1)â2. It is holographically dual to the Chern-Simons theory U(1)2ÃU(1)â2, called the double semion theory. The topological skeletonof U(1)2 is given by the semion modular tensor category which is equivalent to VecÏZ2as a fusion category, with Ï a nontrivial 3-cocycle. Therefore the maximal invertiblesymmetry of the RCFT U(1)2 Ã U(1)â2 is an anomalous Z2 symmetry. A G-enrichedU(1)2 ÃU(1)â2 is therefore a (G,Z2)-gSPT with anomalous Z2. The anomaly must betrivialized in G via a homomorphism Ï : G â Z2. The simplest nontrivial exampleis to take G = Z4 and Ï to be a projection. This gives the so-called Z4-intrinsicallygapless SPT [130, 106, 75].An interesting question is the relation between the VOA formulation and the topo-logical skeleton formulation of invertible symmetry.Conjecture 2.5.2: The group of automorphism of a full vertex algebra V =W âWâis isomorphic to the group of invertible objects in ModW .29Chapter ThreeGapless Symmetry-Protected Topo-logical PhasesIn this chapter we analyze the structure of gapless symmetry-protected topologicalphases(gSPT) more carefully. gSPT is arguably the simplest symmetry-enriched quan-tum orders. By restricting to gSPTs we obtain a more refined and detailed theory.Our approach here is more elementary compared to Chapter 2. In Sec 3.1, we con-sider the partition function of a gapless system with symmetry coupled to backgroundgauge fields, and from it we derive a classification of gSPTs. The result of the clas-sification takes the form of group cohomology, closely resembling the classification of(gapped) SPTs. We the show that the classification reduces to the usual cohomologyclassification of SPTs in the gapped cases. The classification result is the compared toother approaches in literature, and is found to be consistent or equivalent to existingclassification results.In Sec. 3.2 we focus on 2+1D gSPTs and discuss their gauging. Gauging the sym-metry has been a common and powerful method for studying gapped phases with sym-metry. Here we adapt this method to gapless phases and analyze the effect of gaugingthe full symmetry is a gSPT. What we find is a natural generalization of a well-knownresult in the gapped case: gauging a gSPT(a symmetry-enriched quantum order) resultsin an intrinsic quantum order, with nontrivial intrinsic topological defects. A generalconstruction of 2+1D gSPTs is also proposed.In Sec. 3.3 we consider gSPTs with generalized symmetries. For invertible gener-30alized symmetries, we show that the cohomology-style classification still applies. Formore general potentially-noninvertible symmetries, we find that the reasonable gener-alization of the classification is simply the classification of general symmetry-enrichedquantum orders proposed in Chapter 2.3.1 Classification of gSPTA precise definition of gSPT would inevitably require a precise definition of gaplessphases and symmetry-enriched gapless phases. We wish to be more causal here andwork with a less rigorous definition. Roughly speaking, a gSPT is a gapless system withsymmetry that has no âtopological orderâ. Since topological order is a gapped concept,what we mean here is that it has no deconfined quasi-particles or superselection sectors.Using the language of topological skeleton, this means its topological skeleton does notcontain any topological defects of spatial codimension > 1. In 2+1D, this means thereare no anyons in the system. In 3+1D, this means there are no particle-like nor string-like topological defects. It is actually possible to obtain the classification of gSPTsimmediately from this definition together with some facts about fusion higher category.However, let us be elementary here.3.1.1 General considerationConsider a gapless system with symmetry group G(assumed to a bosonic internal sym-metry), if there is no superselection sector, then we can fully integrate out any gappedsector, leaving behind a topological term of background G-gauge fields. The partitionfunction of such a system in the presence of background G-field takes the formZ(AG) = Zgapless(AG)eiStop[AG], (3.1)where Stop[AG] is the topological response of the gapped sector. However, this separationbetween a gapless part and a topological part of the partition function is generally notwell-defined. If the symmetry G acts faithfully on the gapless degrees of freedom, thenthe gapless part Zgapless is only defined up to local counter-terms of AG, which meansthat any topological term Stop[AG] can be absorbed by a redefinition of Zgapless.31The issue here is that the symmetry G may not act faithfully in the IR. In general,for a UV symmetry GUV, its action in the IR is reduced to a faithful action of GIR,which is a quotient of GUV. The symmetry structure defines a group extension1â N â GUV â GIR â 1 (3.2)where N â GUV is the subgroup of GUV that acts trivially on the gapless degrees offreedom.Definition 3.1.1: Let N â G and K := G/N , then a (G,K)-gSPT is a gSPT withsymmetry G whose action in the IR is reduced to K.Notice that what we are trying to classify are not all gapless phases with symmetryand no superselection sector, but really the âsymmetry-enrichment patternsâ of gaplessphases with no superselection sector. Therefore we should expect the classification of(G, 1)-gSPTs to be simply the classification of G-SPTs, even though (G, 1)-gSPTs arestill gapless.3.1.2 Classification via topological response theoryNow let us revisit the structure of partition function of a (G,K)-gSPT. A backgroundgauge field for a (G,K)-gSPT is a pair of gauge fields (AG, AK), then the partitionfunction takes the formZ(AG, AK) = Zgapless(AK)eiStop[AG] (3.3)To classify (G,K)-gSPTs then amounts to classifying the topological response theoryStop[AG]. For this we simply need to determine the conditions Stop[AG] need to satisfy.It is possible that the IR symmetry of a system is anomalous even if the UV sym-metry is anomaly-free. Let us assume that K has an anomaly ÏK . For now let usalso assume this anomaly is captured by group-cohomology. Then the gapless part ofthe partition function Zgapless(AK) is not gauge invariant. The statement that K hasanomaly ÏK is equivalent to the statement that if we introduce a higher dimensional32bulk Nd+1(with âNd+1 =Md being the system), then the partition functionZgapless(AK)eiâ«N ÏK(AK) (3.4)is gauge invariant. Now the full partition function Eq.(3.3) should be gauge invariantat UV level. This means if we lock AK = Ïâ(AG) then the partition function Eq.(3.3)should be gauge invariant with respect to the UV gauge field AG. Combine this withthe fact that Eq.(3.4) is gauged invariant, we see that the combinationeâiâ«N ÏK(AK)eiStop[AG] (3.5)must be gauge invariant when AK |M=âN = Ïâ(AG). Therefore to classify gSPT wesimply need to classify all gauge invariant response theories of the form Eq.(3.5). Wewill speak of this response theory as the topological response theory of a gSPT. It is afully gauge invariant quantity that contains all information of the topological skeletonof the gSPT. The input of such a response theory areâ¢ A d+ 1-dimensional manifold Nd+1 with boundary Md.â¢ A K-bundle AK on Nd+1.â¢ A G-bundle AG on Md, such that AK |M = Ïâ(AG).and the output is a U(1)-number. The input can be summarized as a commutativediagram:Md BGNd+1 BKAGi ÏAK(3.6)Therefore (G,K)-gSPTs are simply gauge invariant U(1)-valued functions of all suchcommutative diagrams. The demand that the topological response theory Eq. (3.5) isâtopologicalâ can be formulated as its value being invariant under a bordism betweentwo such commutative diagrams, therefore a cobordism-style classification of gSPT33should be possible. 1. However for simplicity let us stick to the group-cohomologyframework. It is then useful to discuss the response theory for group-cohomology SPTs.Mathematical Background (Partition function of group-cohomology SPTs): For aG-SPT given by a cohomology element Ï â Hd(G,U(1)), its partition function takesinput a d-dimensional manifold M together with a G-bundle AG : M â BG, andoutputs a U(1)-number.2 The element Ï can be viewed as a map BGâ BnU(1). Thenthe composition Ï â¦ AG : M â BnU(1) is an element of Hd(M,U(1)). If we weredoing De Rham cohomology then we could simply integrate this element over M toget a number. But we are actually doing singular cohomology. In this case integrationof a top-dimension form is defined as follows. If the manifold is oriented then thereis a canonical fundamental class [M ] â Hd(M,Z)(which generates Hd(M,Z) = Z),then using the pairing between homology and cohomology(denoted by â¨â,ââ©) we haveâ¨[M ], Ï â¦ AGâ© â U(1). This is formally written as an integrationâ«MÏ(AG) := â¨[M ], AG â¦ Ïâ© â U(1).Now let us return to the topological response theory of gSPTs. In this case thepartition function Eq. (3.5) is an integration over a manifold with boundary. In thesingular (co-)homology setting integration on manifold with boundary is defined asfollows. An oriented manifold M with boundary âM comes with a standard funda-mental class [M,âM ] â Hd(M,âM ;Z) in the relative homology group. Then takea top-dimension relative cohomology class â¦ â Hd(M,âM ;U(1)), one can form theparing â¨[M,âM ],â¦â© â U(1) which is the integration of â¦ over M . Then mimickingthe partition function of a cohomology-SPT we see that one can obtain a topologicalresponse theory of a gSPT as follows. If we are given a homotopy-commutative diagram1The cobordism classification of SPT is derived as follows: the response theory of a G-SPT takesinput a map M â BG(i.e. a G-bundle on M), and outputs a U(1) number. Demanding the partitionfunction to be topological means the output is bordism invariant, therefore the partition function onlyrelies on the bordism class, i.e. it is a co-bordism element.2For discrete G there is no choice of connection, on any G-bundle there is a unique G-connectionthat is automatically flat. For continuous G, the partition function of an SPT should only depend onthe G-bundle and not the choice of connectionâthis is a necessary condition for the partition function torepresent an SPT, but it is not sufficient. For example, the θ-term of U(1)-gauge field only depends onthe U(1)-bundle, but it does not serve as a topological response theory if time-reversal is not imposed,in which case the θ-angle is continuous and does not represent SPT phases.34BG âBK Bd+1U(1)ÏÏÎ·then by pasting it to the right of the diagram Eq.(3.6) we obtain a diagramMd âNd+1 Bd+1U(1)iÏâ¦AKÎ·which is nothing but an element of the relative cohomology groupHd+1(Nd+1,Md;U(1)),which we can then integrate over N to obtain a U(1)-number. Unpacking this a bit wesee that the topological response theory can written aseiâ«Nd+1Ïd+1(AK)eiâ«MdÎ·(AG) (3.7)with dη = ÏâÏ. The above formal discussion is really just to justify the meaning ofintegration of a non-closed form like η. It is only well-defined when combined with thebulk term Ïd+1(AK).Therefore we see that the topological response theory of a (G,K)-gSPT is deter-mined by an âanomaly trivializationâ relation. The IR anomaly Ï must be trivializedin the UV, and we need to pick a trivialization η: dη = ÏâÏ. The pair (Ï, η) thendefines a topological response theory as Eq.(3.7). One can check topological responsetheory defined in this way is indeed bordism invariant, thus it is truly topological. Nowthere are some obvious equivalences among such trivialization relations.â¢ If we change Ï by a coboundary Ï â Ï + dδ, then we see that the trivializationmust be shifted by η â η+ ÏâÎ´ to satisfy dη = ÏâÏ. The value of response theoryis unaffected by this change. Therefore (Ï, η) â¼ (Ï + dδ, η + ÏâÎ´).â¢ The trivialization η itself can be shifted by a coboundary η â η + dÏ, withoutaffecting the value of the response theory. Therefore (Ï, η) â¼ (Ï, η + dÏ).We are done! The above is clearly some sort of cohomology structure. If you know alittle about homological algebra you may realize this means the pair (Ï, η) is well-definedas an element of Hd+1(C(Ï), U(1)) where C(Ï) is the mapping cone of Ï : Gâ K.35Theoremph 3.1.1: In-cohomology d-dimensional (G,K)-gSPTs are classified byâ¢ IR anomaly: Ï â Zd+1(K,U(1)),â¢ UV trivialization : η â Cd(G,U(1)) such that dη = ÏâÏ,â¢ subject to equivalence relation (Ï, η) â¼ (Ï + dδ, η + ÏâÎ´ + dÏ) for any δ âCd(K,U(1)), Ï â Cdâ1(G,U(1)).This is exactly the definition of cohomology of the mapping cone C(Ï), therefore,d-dimensional (G,K)-gSPTs are classified by Hd+1(C(Ï), U(1)).3.1.3 Consequence of the classificationWe now examine some standard facts about cohomology of mapping cone and theirphysical interpretation.Fact 1: For K = 1 and Ï : G â 1 the trivial map, we have Hd+1(C(Ï), U(1)) =Hd(G,U(1)). Recall that a (G, 1)-gSPT is the same as a G-SPT. This means theclassification of gSPT reduces to the classification of SPT in the gapped limit.Fact 2: There is a long exact sequence· · · ââ Hd(K,U(1)) Ïâââ Hd(G,U(1)) Rââ Hd+1(C(Ï), U(1))Oââ Hd+1(K,U(1)) Ïâââ Hd+1(G,U(1)) ââ · · · (3.8)This contains very rich information. First of all let us rewrite it in physical terms· · · ââ SPTdK Ïâââ SPTdG Rââ gSPTdG,K Oââ SPTd+1KÏâââ SPTd+1G ââ · · · (3.9)â¢ The map R stands for restriction. It takes a d-dimensional G-SPT and turns itinto a d-dimensional gSPT. The resulting gSPT has no emergent anomaly and thetopological response theory is just given by the input G-SPT. The image of R isthe so-called weak gSPTs. The topological features of such gSPTs are restrictionsof the parent SPT.36â¢ The map O stands for obstruction. It gives the emergent anomaly of a (G,K)-gSPT, which can be viewed a d + 1-dimensional K-SPT. The emergent anomalyis the obstruction to trivializing the system via gap opening.â¢ Exactness at SPTdG: a weak (G,K)-gSPT obtained from a parent G-SPT may betrivial even if the parent is nontrivial. This happens exact when the G-SPT is infact the pull-back of some K-SPT(the G-SPT is secretly a K-SPT). This agreeswith our notion of strong equivalence of gapless phases: such a weak gSPT can bemade trivial be stacking with a K-SPT. This means some element is in the kernelof R if and only if it is in the image of Ïâ.â¢ Exactness at gSPTdG,K : It simply says a gSPT is weak(comes from a parent SPT)if and only if it has no IR anomaly. A physical proof of this is straightfoward(noIR anomaly means direct gap opening is possible).â¢ Exactness at SPTd+1K : It simply says the IR anomaly of a gSPT is emergent: itgets trivialized in G. Also, any such trivialization relation can be realized in agSPT.3.1.4 Matching with other classification methodsIn a series of works [131, 128, 129] the authors studied gSPTs in 1+1D and 2+1Dand proposed several classification schemes. We will leave the details of the otherclassification methods and their physical interpretation to the rest of the dissertationand only state their mathematical equivalence here. The goal is to convince the readerthat all the classification methods give the same result. The other classification methodsfocus on the case where Ï : Gâ K is surjective, so there is no emergent symmetry inthe IR. To compare with the other methods we will for now restrict to such cases. Inthis case we also denote N := kerÏâG, called the gapped symmetry of the gSPT.Theorem 3.1.1 (1d, Else-Nayak-style): In [131], the authors obtained a classificationof 1+1D gSPTs using methods adopted from the Else-Nayak classification of SPTs.The result is:â¢ A 1+1D (G,K)-gSPT is classified by pairs (η, ϵ), where η â C2(N,U(1)), ϵ âC1(G,C1(N,U(1))) satisfying certain coherence conditions.37â¢ A careful examination of these coherence conditions reveals that they are nothingbut what are needed to define a G-crossed commutative algebra A with trivialcomponent being Ae = C.It is a fairly standard exercise(for a math student maybe) that such algebras are clas-sified by trivialization relations dη = ÏâÏ modulo the expected equivalence relation.Therefore the classification in [131] agrees with what we obtained here.Theorem 3.1.2 (1d, Holographical): In [131], the authors also showed the Else-Nayak-style classification agrees with the classification of a class of condensable algebras in the2+1D topological order Z[VecG]. Then by transitivity the classification of this class ofcondensable algebras agrees with the classification of gSPTs obtained here. However itcan actually be shown more directly that condensable algebras in Z[VecG] are classifiedby trivialization relations dη = ÏâÏ modulo the expected equivalence relations [137].Theorem 3.1.3 (2d, Categorical): In [128], the author argued 2+1D G-gSPTs areclassified by G-crossed braided fusion categories with unfaithful grading and trivial triv-ial component Ce = Vec. This is motivated by the classification of 2+1D G-SETs,which are G-crossed braided fusion categories with faithful grading. The grading is theflux sectors. For a gSPT not all G-flux sectors are topological, and this leads to the un-faithful grading. The trivial trivial component condition means there is no topologicalorder. It can be deduced from the homotopy theory of G-crossed braided extensionsthat such categories are exactly classified by trivialization relations dη = ÏâÏ.Theorem 3.1.4 (2d, Holographical): The above categorical classification was partlymotivated by expectation from topological holography. Indeed a class of condensablealgebras in the 2-category Z[2VecG] are given by G-crossed braided fusion categoriessatisfying the aforementioned properties [129]. Then by transitivity this class of con-densable algebras have the same classification as the classification of gSPTs obtainedhere.Theorem 3.1.5 (General d, Decorated domain wall): A perhaps more physical ap-proach to the gSPT problem is the decorated domain wall(DDW) approach. This isalso a constructive approach that allows you to construct lattice models from the clas-sification data. Roughly speaking the idea is to start with a K-broken state, and thendecorate the K-defects with G-SPTs. After this is done in a consistent way we in-crease quantum fluctuation of the K-defects until they go critical. Unlike the DDW for38SPTs, we can allow the decoration to give a K-anomaly for a gSPT. Some standardspectral sequence argument shows consistent DDW patterns for gSPT are again givenby trivialization relations dη = ÏâÏ. However to deduce the DDW pattern from thetrivialization is not an easy job.3.2 Gauging of 2+1D gSPTIn the study of ordinary SPTs, gauging the symmetry is a standard method of diagnos-ing the properties of the SPT. For an SPT with finite symmetry group G, its gaugedtheory is a Dijkgraaf-Witten theory with gauge group G. This is a topological order inany dimension(>2). The goal of this section is to develop the theory of gauging gSPTs.We will focus on gSPT with ordinary symmetry in dimension 2 + 1D for simplicity.Generalization to higher dimensions is straightfoward.Generalizing the story of SPT to gSPT, we expect âa gauged gSPT is a (nontriv-ial)quantum orderâ. Our focus here is on the effect of gauging on topological skeleton.The theory of gauging gSPT is mathematically very similar to the theory of gaugingSPT/SET, therefore we briefly review the latter theory here.In 2+1D, the relation between a SPT and the gauged theory are well-understood.Theoremph 3.2.1: Let C be a G-crossed modular tensor category, describing a 2+1Dtopological order enriched with symmetry G. Then the gauged theory is given by theequivariantization of C, denoted as CG.Remark. A G-crossed modular tensor category C has a decompositionC = âgCg, (3.10)where each component Cg describes anyons in flux sector g. It also has a G-action,which maps the flux sectors as g â (Ch) â Cgh. It describes all topological defects in aG-enriched topological order, including the symmetry defects. The equivariantizationCG is a modular tensor category describing the topological order obtained by gaugingG. It always contains a subcategory Rep(G) â CG.Example 3.2.1: A 2+1D G-SPT, viewed as a G-enriched trivial topological order,is described by the G-crossed modular category VecÏG, with Ï â H3(G,U(1)). Thegauging is (VecÏG)G â Z(VecÏG), i.e. the twisted G-gauge theory with twist Ï.39Now let us discuss some physical intuition about 2+1D gSPTs and their gauging.By definition a gSPT has no anyons. The topological skeleton of a 2+1D (G,K)-gSPTis then given by the symmetry defects. A general symmetry flux Fg associated withg â G may not be topological. This happens when some charge Ï â Rep(G) withnontrivial g-action is gapless, Ï(g) Ì¸= 1. If the g-flux Fg can âseeâ a gapless charge,then any fusion and braiding process involving Fg does not yield quantized results. Thegapless charges of a (G,K)-gSPT are those that transform trivially under N âG, thekernel of the RG-map Ï : G â K. Conversely, a flux is invisible to all the gaplesscharges if and only it takes value in N . On the other hand all the symmetry charges ofG are topologicalâ fusion and braiding of G-charges are well-defined in any G-symmetricsystem, more precisely they form the braided fusion category Rep(G). Therefore aftergauging G we have the following gapped topological excitations:â¢ Symmetry charges become gauge charges, they form a Rep(G).â¢ Only N -fluxes become dynamical topological defects.Therefore after gauging G we obtain a gapless system whose topological super-selectionsectors consist of all G-charges and all N -fluxes. The rest of the gauge fluxes are lockedwith the gapless defects of the IR theory, e.g. conformal defects of a K-symmetricCFT. The topological skeleton, viewed as a system of anyons with braiding and fusionrules, has degenerate braiding. Namely some G-charges are invisible to N -fluxes, theseare exactly in the full subcategory Rep(G/N) â Rep(G). Therefore they can not bedefected by braiding with N -fluxes. They can however be detected by braiding withgapless/conformal defects in K-flux sectors.We see that the topological skeleton of a gauged gSPT is described by an anyontheory with degenerate braiding. The degenerate braiding is compensated by gaplessdefects in the system. In fact, the existence of a degenerate topological skeleton pre-dicts the existence of gapless defects, since anyons always need to braid with something,gapped or gapless. A formal statement is that a degenerate anyon theory has a gravita-tional anomaly. Therefore either the system develops âmore anyonsâ to braid with theexisting ones, or the system is gapless. For now, let us turn the above rather physicaldiscussion into precise mathematical statements to complete the our theory of gauginggSPT. We start by picking one classification of gSPT out of the many discussed inSec. 3.1 suitable for describing gauging.40Theoremph 3.2.2: A 2+1D (G,K)-gSPT with N = kerÏ is given by a G-crossedbraided fusion category C such thatâ¢ The grading of C is supported on N .â¢ The trivial component Ce is trivial, Ce = Vec.Remark. Physically, the support of C is the topological flux sectors. The trivial compo-nent Ce is the underlying topological order of the system. By assumption a gSPT hasno topological order sector, thus Ce = Vec. The full data of G-crossed braiding andfusion can be constructed from a trivialization relation dη = ÏâÏ, although the actualexpressions are quite complicated. The details may be found in [128]. The G-crossedcategory describes the fusion, braiding and G-action on N -fluxes.Gauging is simply the process of promoting symmetry defects to dynamical fluxes.The argument leading to âgauging of an SET amounts to equivariantization of theG-crossed categoryâ carries over to gSPT and we have the following.Theoremph 3.2.3: Let C be a G-crossed braided fusion category as above describinga 2+1D (G,K)-gSPT. Then the topological skeleton after gauging G is described bythe equivariantization CG.We now immediately check that mathematical structure of the equivariantizationCG matches with our physical intuition.â¢ The equivariantization CG is a braided fusion category, its braiding is non-degenerateif and only if the support N of C is G. This means the topological skeleton aftergauging is a non-degenerate anyon theory if and only if we started with an SPT.â¢ The transparent sub-category of CG(formally called the Muger center, denoted asZ2), is Z2(CG) â Rep(G/N). This matches with our previous discussion that theG/N -charges braid trivially with all gapped excitations in the gauged gSPT.3.2.1 Construction of 2+1D gSPTThe above analysis can actually be exploited to yield a general construction of 2+1DgSPTs. In fact we will see that all 2+1D gSPTs are constructed in this way. Ourobservation is that gauging finite group a topological manipulation, therefore it is alwaysinvertible. We lose no information by gauging a finite group.41Consider a (G,K)-gSPT with gapped symmetry N = kerÏ. Previously we discussedthe effect of gauging the full symmetry G. Now let us consider a different approach:gauging the gapped symmetry N . After gauging N , the gapped sector now consist ofal N -fluxes and all N -charges. Their braiding is now non-degenerate. Therefore thegapped sector becomes an N -gauge theory enriched with K. The gapless sector hastrivial N -action, therefore gauging simply reduces the symmetry from G to the faithfulsymmetry K. The system is now a simple product:(gSPT)N â CFTK â Z(VecαN)K (3.11)where the subscript denotes the remaining global symmetry, and we used the factthat 2+1D N -gauge theories are described by categories of the form Z(VecαN), α âH3(N,U(1)). For simplicity we also assume the bulk universality class is a CFT. TheK-action on both the CFT and the N -gauge theory can be anomalous, but the anomaliesmust be opposite to each other. We see that the general structure of an N -gaugedgSPT is very simple. The original gSPT can then be obtained by âungauging Nâ, i.e.gauging the 1-form Rep(N)-symmetry.Construction 3.2.1: Any 2+1D (G,K)-gSPT can be constructed as follows. Startwith a CFT S with faithful symmetry K and anomaly ÏK â H4(K,U(1)). Find a 2+1Dfinite gauge theory with K-action and anomaly Ïâ1K , say Z(VecαN). Then(Sâ Z(VecαN))condensing N -chargesâââââââââââââ (G,K)â gSPT (3.12)Notice we did not specify G in the input of the construction. This is because a K-actionon an N -gauge theory defines an extension of K by N by itself. This extension is theG in the output of the construction.3.3 gSPT with generalized symmetriesThe above classification result can be generalized to generalized symmetries straight-forwardly. Generalized symmetries can be roughly divided into two types, invertibleand non-invertible. For invertible symmetries the mathematics involved are essentiallyalgebraic topology and we can remain in the no-category territory. Non-invertible sym-42metry inevitably requires the language of category theory. Therefore we will presentthe generalization of classification of gSPTs in two forms, one using algebraic topologyand the other using category theory.3.3.1 Invertible generalized symmetryA general invertible generalized symmetry contains invertible symmetries of variousform-levels that potentially interact with each other. A standard example is a 2+1Dtopological order with a global ordinary symmetry G. The global symmetry G is a 0-form symmetry while the anyon worldlines are 1-form symmetries. If the G-symmetryacts nontrivial on the topological order, the total symmetry of the system is not a simpleproduct of a 0-form symmetry and a 1-form symmetry. Instead it is described by a 2-group. The formal name for a generalized invertible symmetry with form-level 0 â¼ nâ1is an n-group. Similar to ordinary SPTs, SPTs protected by n-group symmetry have acohomology classification.For any n-group G there is a topological space that entails all the algebraic structureof the n-group, called the classifying space of G and denoted by BG. G can be recoveredfrom the topological space BG by taking the higher fundamental groupoid. So noinformation is lost by passing to the space BG. In fact, the defining property of asymmetry being invertible is that it can be captured by a topological space 3. Then allphysics regarding the invertible symmetry can be described via this topological space.This justifies the earlier claim that the mathematics of invertible symmetry is essentiallyalgebraic topology and no category theory is needed.Theoremph 3.3.1: Let G be a higher group, d-dimensional SPTs protected by G areclassified by Hd(BG, U(1)).This is well-known(see e.g. [118]). Then for gSPT we have direct generalization.Theoremph 3.3.2: Let G,K be higher groups and Ï : G â K be a homomorphism ofhigher groups, d-dimensional (G,K)-gSPTs are classified by Hd+1(C(Ï), U(1)).Unpacking this, it simply says (G,K)-gSPTs are classified by its IR K-anomaly andtrivialization of the anomaly in G, modulo stacking with K-SPTs. Some 2-group gSPTexamples can be found in [129, 7].3The formal statement is that the category of higher groups is equivalent to the category of topo-logical spaces up to homotopy.433.3.2 Non-invertible symmetryThe description of non-invertible symmetries necessarily involves category theory. There-fore to generalize the classification of gSPT to include non-invertible symmetries wemust first reformulate our previous classification in the language of category theory.Let us first consider 1+1D (G,K)-gSPTs. A finite group symmetry G in 1+1D withanomaly ÏG â H3(G,U(1)) is described categorically by the fusion categoryVecÏGG , i.e.,the fusion category of G-graded finite dimensional vector spaces with associator givenby ÏG. The RG flow defines a group homomorphism Ï : G â K. In fact the RG flowdefines more than juts a group homomorphism, it also preserves the associator anddefines a monoidal functor from the UV symmetry category VecG to the IR symmetrycategory VecÏKK , where ÏK is the emergent anomaly of the gSPT. A linear monoidalfunctorF : VecG â VecÏKK (3.13)is determined exactly by a trivialization dη = Ïâ(ÏK). Therefore (G,K)-gSPTs canbe simply classified by functors from the UV symmetry category to the IR symmetrycategory(up to equivalence of symmetry categories). With this understanding we canimmediately generalize the classification to non-invertible symmetry-protected gSPTs.Theoremph 3.3.3: Let SUV be the UV symmetry d-category and SIR be the IR sym-metry d-category. Then a (SUV, SIR)-gSPT is given by a linear monoidal d-functorF : SUV â SIR, modulo the equivalences F â Fâ² if there is a diagramSUV SIRSâ²UV Sâ²IRFf gFâ²that commutes up to invertible higher morphisms, where f, g are isomorphisms.It can be readily verified that the above formulation reproduces the classificationof invertible symmetry gSPTs. Examples of non-invertible symmetry gSPT are mosteasily produced via the SymTFT trick, see Chapter 5.44Chapter FourConstruction of Gapless SPT fromGroup-Cohomology4.1 IntroductionWe say a gSPT is an intrinsically gapless symmetry-protected topological phases (igSPT)if it has nontrivial emergent anomaly. In this chapter, we introduce a systematic methodfor constructing examples of from symmetry-anomalies that are classified by group co-homology [17]. This framework generalizes the structure identified in 1d igSPT modelsintroduced in Ref. [106, 73], to enable the construction of a large variety of igSPT latticemodels in various dimensions. Furthermore, this formalism clarifies the relationship be-tween the topological edge degrees of freedom (DOF), and the emergent IR symmetryanomaly that protects the bulk gaplessness. Using this approach, we construct igSPTmodels for bosonic systems in one-, two-, and three- spatial dimensions (1d, 2d, 3d)with various symmetries.Denoting the number of spatial dimensions as D, our construction takes as inputa group-cohomology anomaly specified by an element of HD+2(GIR, U(1)) for the low-energy symmetry group GIR, and outputs a lattice model of an igSPT in Dd with anenlarged UV symmetry group GUV that realizes this anomalous GIR-symmetry action inthe IR. Specifically, we define a lattice model with GIR-rotors on vertices of the latticeand N rotors on plaquettes, with a microscopic onsite GUV symmetry action. Then,we design a Hamiltonian, Hâ, that locks the N rotors on each plaquette to topological45<latexit sha1_base64="frUL22 +4NQ/Oo1VPcyLAZz02n0U=">AAAE0nicbZNPa9swFMDV1tu67F+7HXc [binary]</latexit >E<latexit sha1_base64="OcDr080R jKrGQExDOVRi5GitVFY=">AAAE13icbZNPb9MwFMC9LcAo/zY4comoJrWHV [binary]</latexit ><latexit sha1_base64="MPKqy3OMHXy/b3hS LrXHOmYjOkA=">[binary]</latexit>N<latexit sha1_base64="K5eHQ3iTFU+0BZ5KvkBWrxIPpBo=">[binary] sxDNzeqnFdoZbV+/QdrrrvM</latexit>self-dual edge<latexit sha1_base64="gvRqyMTF/BWdWF75CpTvyycPib0=">[binary] </latexit>anomalous bulk<latexit sha1_base64="YrB5CZlNSLbq78Hjszv/lbLitQw=">AAAE/nicbZNPb9MwFMC9rcAof7bBkYtFN6kTWtV [binary]</latexit>N ! 1d SPT pump<latexit sha1_base64="Ku54W8CAa4pIKk PVdw3eydPQLf0=">[binary]</latexit>GIR<latexit sha1_base64="dJf+MCFFG1ha9n AeAkn02ZB6M1A=">[binary]</latexit>GUV<latexit sha1_base64="Ku54W8CAa4pIKk PVdw3eydPQLf0=">[binary]</latexit>GIR<latexit sha1_base64="C0DLZM9NAqUAoe2 TBzEpf6BMgko=">[binary] DW+Nb43fixDd3fKnDdoYzV+/Qcecrqc</latexit>(anomalous)Figure 4.1: (Left) N -rotors (spins with open circles) are locked to topological defectsin GIR-rotor (spins with filled circle) symmetry-breaking patterns (e.g. dashed blue linedepicts 1d Ising domain wall) at high-energy (UV) scale â. This imprints an anomalousGIR = GUV/N symmetry action at low energies (IR). (Right) Schematic of a 2d igSPT.The igSPT bulk GIR anomaly forbids a trivially gapped symmetric bulk. In the IR Nsymmetry operations act trivially in the bulk, and pump a lower-d gapped GIR SPTonto the boundary. This pumping action forces the edge to a self-dual critical point,leading to symmetry-protected gapless edge states.defects of the GIR rotors such that, for energy scales âª â (âthe IRâ), the N -rotorsare gapped, and an emergent anomaly is exactly imprinted on the low-energy sector.We show that, if the symmetry GIR remains unbroken and the system does not developfractionalized topological order, then the emergent anomaly protects nontrivial bulkand edge modes. The connection between emergent anomalies and igSPTs was pointedout in the original work on this topic [106], and the structure of group extensions wasexplored in [73]. This chapter extends these concepts to lattice models in dimensionshigher than one, constructs a number of new examples, and elucidates the connectionbetween the emergent anomaly and the igSPT edge physics. Specifically, we give ageneral argument for a bulk-boundary correspondence connecting the emergent GIRanomaly and the structure of the group extension, based on gauging the extending Nsymmetry to form an anomalous GIR symmetry enriched topological order (SET). Inmany cases, the edge states can also be understood via a physically-intuitive mechanismof SPT pumping. Namely, for these cases, the extended symmetry operations N acttrivially in the igSPT bulk, but pump a (D â 1)d GIR-SPT onto the edge. This lower-46dimensional SPT pumping action forces the igSPT edge to reside at a self-plural 1critical point among different (Dâ 1)d GIR-SPT phases, which prevents the edge frombeing in a trivial symmetric state.In 1d, the emergent anomaly forces the igSPT bulk to either be gapless or sponta-neously break symmetry. By contrast, higher dimensions D â¥ 2, an additional possibil-ity arises that the emergent bulk anomaly bulk can be satisfied by certain gappedand symmetric states with anomalous symmetry-enriched topological order (SET).Using techniques introduce in [104, 122], we construct exactly solvable igSPT-typemodels with an emergent anomalous SET ground state. In this context, the monikerâintrinsically gaplessâ is no longer appropriate, and we instead refer to these phasesas quotient-symmetry enriched topological phases (QSETs) following the language ofquotient-symmetry protected topological phases (QSPTs) introduced in [111]. QSETshave an anomalous implementation of the IR symmetry, which is lifted to an anomaly-free symmetry action in the UV, a notion that requires a separation of scales, δ âª â,between the gap, δ, to creating anyonic excitations in the QSET, and the scale â atwhich the anomaly emerges. As for QSPTs, we argue that the QSET orders are sharplydistinguished from ordinary GUV-SETs by an edge phase transition where the gap tothe extending N -DOF closes at the edge. Namely, if the gap to N charges remainsopen, then the anomalous lower-dimensional SPT pumping symmetry ensures that theQSET edge is either gapless or symmetry breaking (D = 2). In 3d, the edge is itself2d, and it is also possible that both the bulk and edge form a QSET order. We con-struct an explicit example of this in Section 4.6, which takes the form of a 3d toric-codetopological order in which gauge magnetic flux lines are decorated with 1d igSPT states.The paper is organized as follows. In Section 4.2, we review the taxonomy of knowntypes of gapless SPTs. In Section 4.3, we review the 1d igSPT models with GUV = Z4symmetry introduced in [106, 73] in the language of the group-cohomology framework,and show how this perspective connects their edge states to the emergent anomaly.Here, we also lay the ground-work for constructing lattice models with fractionalizedanomalous QSET orders in higher dimension. In Section 4.4, we formally generalize thisstructure holds to igSPTs in various dimensions and symmetry classes with emergentanomalies classified by group-cohomology. We then illustrate this general formalism byconstructing lattice models for a 2d time-reversal symmetric igSPT (Section 4.5) and a1The generalization of self-dual to potentially more than two phases.473d Ising igSPT (Section 4.6). We close with a brief discussion of prospects for realiz-ing igSPTs with beyond group cohomology models and as Mott insulators of realisticelectron systems. Since there are few reliable theoretical tools for studying stronglycoupled field theories in 2d and 3d, throughout, we will not attempt to analyze theultimate low-energy (deep-IR) field theory description of the lattice models we con-struct. Rather, we will use the presence of a gapped sector to make sharp topologicalstatements about the emergent anomalous symmetry, and use these to deduce informa-tion about the possible structure of higher-dimensional igSPT bulk and edge modes.Finally, the Appendices contain additional formal details, and construct several addi-tional igSPT model examples in 1d, 2d, and 3d with various combinations of ZN , U(1),and time-reversal symmetries.4.2 Gapless SPTs: Definitions and TaxonomyColloquially speaking, a gapless symmetry protected topological (SPT) state is a scale-invariant gapless system that possesses symmetry protected topological edge states thatcannot be removed without undergoing a phase transition where the bulk changes itsuniversal scaling properties. Previous studies [93, 113, 109, 106, 73] have identified fourdistinct categories of gapless SPTs defined by two binary characteristics:1. whether or not the gapless SPT edge states can be trivialized by stacking it witha gapped SPT with the same symmetry,2. whether or not the edge states are exponentially well-confined to the edge by agapped sector.Gapless SPTs that fail to have the first property are called intrinsically gapless SPTs(igSPTs) [106, 73], and will be the focus of this chapter. Those that fail to have thesecond property have been called âpurely gaplessâ SPTs, and have edge states that arenot exponentially localized to the edge. Rather, their influence decays as a power ofthe distance into the bulk [109]. It remains an open question whether there are gaplessSPTs that are both purely and intrinsically gapless [110].Despite their nom de plume, âgapless SPTsâ are perhaps better thought of as a form48of symmetry-enriched criticality (SEC) [115] 2. Adding a symmetry to gapped systemswith a fixed type of intrinsic topological order leads to distinct symmetry enrichedtopological (SET) phases. Unlike gapped SPT phases, SET phases cannot be smoothlyconnected to a trivial atomic insulator via a gapped path of Hamiltonians, even whenthe protecting symmetry is broken due to the underlying long-range entanglement of theintrinsic topological order. Similarly, for a given universality class of a gapless system,there may be multiple inequivalent implementations of symmetry, which cannot besmoothly interpolated between along a path of Hamiltonians whose ground-states havethe same universality class [115].Gapless SPTs are examples of SECs, in which one cannot locally distinguish betweena âtrivialâ critical point and a symmetry-enriched one from local bulk measurements.Rather, the differences between different gapless SPTs are only evident in non-localbulk probes, or local edge probes. One can make this notion more specific by analogyto gapped SPTs. Different gapped G-SPT ground-states can be connected by a finite-depth local unitary (FDLU), U , that is overall symmetric ([U, g] = 0 âg â G) but whichis not symmetrically generated (U Ì¸= eâiH for any localG-symmetricH). This definitioncannot be directly ported to the gapless setting as even different instances of a gaplessstate with the same universal scaling properties, e.g. two instances of a conformal fieldtheory (CFT) perturbed by different irrelevant operators, cannot be connected by anFDLU. However, we can generalize the notion of symmetric FDLU-(in)equivalence bydefining i) that two ground-states are in the same universality class if they flow tothe same RG fixed point after applying an overall symmetric FDLU, ii) ground-stateswith the same universality class are distinct gapless SPT classes if they cannot beconnected in this way by any symmetrically-generated FDLU. An immediate corollaryof this definition is that local scaling operators in distinct gapless SPTs of the sametype of criticality have the same symmetry properties as conjugating a local operatorwith definite symmetry quantum number with an overall-symmetric U preserves itssymmetry quantum number. However, the symmetry properties of non-local scalingoperators, such as a disorder operator that inserts a domain wall, may change under2Unlike gapped phases, which are stable to generic perturbations, gapless states may have oneor more relevant perturbations that change their universality class. We will remain agnostic aboutthe number of relevant perturbations, i.e. whether we are discussing a stable gapless phase or (morecommonly) a fine-tuned critical or multi-critical point, and use the terms âcriticalâ and âgaplessâ assynonyms.49such an overall symmetric FDLU, leading to distinct classes of gapless SPTs. Thesenotions can be made more precise for 1 + 1d conformal field theories (CFTs) [115], forwhich the data specifying a universality class is well understood and characterized bythe spectrum and fusion rules for primary scaling operators.4.3 Low dimensional igSPTsGiven the formal nature of our constructions, we begin by warming up with an (almosttrivial) example of how emergent anomalies can arise in a 0d system. We then, reviewthe GUV = Z4 symmetric 1d igSPT with emergent GIR = Z2 anomaly previouslyconstructed in [106], in a language that is amenable to generalization to other symmetrygroups and dimensions.4.3.1 Warmup: Emergent anomalies in 0dTo see a simple example of how an IR anomaly can emerge from an anomaly-freeUV system, consider the 0d spin-1/2 edge state of a Haldane/AKLT spin-chain [43, 1]with spin-rotation symmetry (GIR = SO(3)). While this 1d G-SPT phase is typicallydiscussed as a spin-1 chain, in any physical realization, it arises only as an emergentdescription of spin-1/2 electrons in a Mott phase, so that only spin-1 DOF are activebelow the Mott gap, â. The spin-1/2 DOF transform under a larger group GUV =SU(2): rotations that add up to 2Ïn around any spin-axis, which are identity operationsin GIR, give a Berry phase of (â1)n with n â N = Z2. Formally GUV = SU(2) is a(central) extension of GIR = SO(3) by the normal subgroup N = Z2 â GUV. The spin-1/2 representation of SO(3) is a projective representation characterized by a nontrivialelement of the second group-cohomology Ï2(g, h) â H2(SO(3), U(1)) = Z2, where theZ2 group structure indicates that two spins-1/2 form an ordinary linear representationof SO(3). Namely, for GIR, h â SO(3), the spin-1/2 representation, Rg, Rh satisfymultiplication rule RgRh = Ï2(g, h)Rg·h where an explicit representation of Ï2 is theBerry phase for sweeping the spin around a loop starting from up in the z-direction,rotating by g then h, then (gh)â1.One can realize the system with a spin-1/2 ground-space trivially in a single spinfulfermion site, simply by considering a single-site Hubbard model: H = âµcâ ÏcÏ+ U2 n(nâ501) with onsite repulsion U and chemical potential 0 < µ < U/2 so that there is a singleoccupied electron in the ground-space. This system has a (Mott) gapped sector witheven fermion parity consisting of the empty and doubly-occupied states with energyâ = min(µ, U â 2µ), and a doubly-degenerate ground-space consisting of the singly-occupied spin-up or down states. In this low-energy ground-space there is an emergentSO(3) anomaly: the states transform projectively under SO(3) as an ordinary spin-1/2representation of SU(2). This anomaly is stable: it is protected by the Mott gap tofermion excitations, and can only be removed by closing the gap separating the fermionparity even and odd ground-spaces.While this example may seem merely an overly complex re-interpretation of a com-pletely trivial single-site problem, it lays the ground-work for more complicated higherdimensional examples. Specifically, it suggests that ingredients for realizing an emer-gent anomaly are: i) a (central) group extension that lifts the anomaly, ii) an interactionterm that locks the extending degrees of freedom away at high energies in such a waythat imprints the anomaly on the low-energy subspace.4.3.2 Review of 1d igSPT with Z4 symmetryWith this nearly-trivial 0d example in hand, we next review the constructions of [106, 73]for 1d igSPTs. Our goal will be to adapt the notation such that generalizations tohigher dimension and other symmetry groups become obvious. We also reveal addi-tional structure about the connection of igSPT edge states, and the group extension,via a symmetry under pumping lower dimensional GIR SPTs onto the boundary. Ad-ditionally, we introduce various notions of gauging the symmetry that serve as usefultools for characterizing the igSPT topology, and constructing QSET phases in higherdimensions.In this section, we focus on the example where the low-energy IR symmetry groupis GIR = Z2 and the UV symmetry group is GUV = Z4. Additional examples for otherGIR are detailed in Appendix 4.8 and summarized in Table 4.1. The emergent GIR = Z2anomaly is the same one that protects the edge of the 2d G-SPT constructed by Levinand Gu [71]. However, realized as a pure 1d igSPT with extended GUV = Z4 symmetry,we can interrogate the 0d ends of an open chain with this bulk anomaly.Ref. [106] proposed an elegant physical model realizing this igSPT phase in terms51of an extended Ising-Hubbard model with discrete Ising symmetry corresponding toÏ-rotations around a specific (x) spin-axis. The electronic degrees of freedom form aspin-1/2 projective representation of this group for which a 2Ï rotation results in a(â1) phase, and hence only a 4Ï rotation is trivial corresponding to symmetry groupGUV = Z4. Denoting Z4 = {0, 1, 2, 3}, the fermion parity, (â1)Nf where Nf is thenumber of fermions, forms a normal sub-group N = {0, 2} = ZF2 . The igSPT phasecorresponds to a Mott insulator, in which the fermion excitations have an energy gap,so that the only IR degrees of freedom are bosonic spins that transform under a quotientgroup GIR = GUV/N = {0, [1]} = Z2 where [. . . ] denotes the equivalence class under thequotient. The IR igSPT phase has an emergent anomaly of GIR, corresponding to theanomaly of the edge of a 2d gapped Z2-SPT [71]. While the extension of the spin-modelby fermionic spinor DOF is natural for physical realizations, Ref. [73] showed that thesame IR theory can arise from a purely bosonic Z4 spin chain. We will follow the latterall-boson approach, since it meshes nicely with the group cohomology formalism, butwill comment on cases where the igSPT might alternatively arise from a Mott insulatorof fermions. While all the main points of this 1d Z4 igSPT were previously explainedin [106, 73], we give additional arguments that clarify the structure of edge states,and use a notation and framework that readily generalizes to other dimensions andsymmetries.4.3.3 Anomalous Ising spin-chainConsider a spin-1/2 chain with on-site Z2 spin DOF gi â GIR = Z2 = {0, 1}. Specif-ically, we can define the standard Pauli operator Ïzi = (â1)gi . Further, define theconjugate operator as gËi â {0, 1}, via Ïxi = (â1)gËi . To start consider an infinite chainor with periodic boundary conditions (PBCs). We will then consider the effect of open-boundary conditions and the edge. An anomalous non-onsite symmetry action is:UA1 =âiÏ3(gi â giâ1,âgi â 1, 1)Uos1â¡âi(â1)gi(giâgiâ1)âi(â1)gËi(4.1)52<latexit sha1_base64="RyRDj4WSaVk477OgGl VHQCCetH0=">[binary] </latexit>"<latexit sha1_base64="h5qD5ek+sbEKwJDGyNHIIYPW xF0=">[binary]</latexit>â 1/L <latexit sha1_base64="QPDjsMjfJl3wRDj0Z7z11UzguQU="> [binary] Rz9aX1s/W79av8vQ1ZUq5zlaWK0//wF059CM</latexit>â eL/â <latexit sha1_base64="U1ZI4pI1/r0uVXpV9I1cCANA3bg= ">[binary]</latexit>Edge<latexit sha1_base64="EIxNBaGbGROY3pTZpMkucZN c7lU=">[binary] VurlR57xFK8v58x+wj8VD</latexit>Bulk<latexit sha1_base64="wW0amJGBSygk5PqSHbkPmvKcgI8=">AAAFIXicbVRN [binary]</latexit>| ""i± | ##i<latexit sha1_base64="gVDQyCeCGFxsiCcCarGZ9XKLKJU=">AAAFD3icbZTPb9MwFMe9rcAYvzY4IiS [binary]</late xit>Bulk: Luttinger LiquidFigure 4.2: Schematic depiction of the structure of 1d GUV = Z4 intrinsically gap-less SPT as originally explained in [106]. The bulk has an emergent anomalous Z2symmetry, which protects a gapless Luttinger liquid state (gray arcs schematically de-note power-law range correlations) with finite size gap â¼ 1/L for chain length L or re-quires spontaneous symmetry breaking (SSB). With open boundaries, there is a twofoldground-space degeneracy with much smaller finite-size splitting â¼ eâL/Î¾. Crucially, theedge state space does not have a local tensor product structure, or equivalently, thereis long-range entanglement between the edge modes in the exact ground-states (longdashed blue line).where Ï3(g, h, k) = (â1)ghk, and the second factor in UA1 is just the ordinary on-siteaction of GIR = Z2 on each site which maps |giâ© â |gi + g mod 2â©. Physically, thenon-onsite phases, (â1)gi(giâgiâ1) means that each domain wall (DW) between Ïzi =â,Ïziâ1 =â carries a Z2 charge (gives a phase of (â1) in UA1 ) 3. Roughly-speaking, thissymmetry action is anomalous because it causes non-trivial (semionic) statistics forIsing domain walls (DWs) [71], which intuitively presents an obstacle for reaching atrivial gapped, symmetric state from a spontaneous symmetry broken one by âcon-densingâ DW defects, as the non-onsite factors lead to destructive interference in theDW dynamics preventing them from condensing [51].4.3.4 Onsiteing the symmetryFollowing [106, 73], this GIR = Z2-anomaly can emerge as the low-energy sector of anigSPT with an enlarged symmetry GUV = Z4 which is a central extension of GIR byN = Z2. Introduce auxiliary qubit DOF with Pauli operators Ïxi = (â1)nËi , Ï zi = (â1)ni3This phase can be more symmetrically apportioned such that âââ and âââ DWs each contributea factor of i, which is related by a finite-depth unitary or equivalently redefining Ï3 by a coboundary.However, we prefer that the non-onsite phases take values in the IR symmetry group (GIR = Z2).53with ordinary on-site (extended) symmetry generated by:Uos1 =âi(â1)ginËi(â1)gËi . (4.2)Then, energetically lock nËi to (gi â giâ1) via a Hamiltonian:Hâ = âââiÎ´nËi,giâgiâ1 = âââi12(1 + Ïxi Ïzi Ïziâ1), (4.3)such that the symmetry action restricted to the ground-space of Hâ is equivalent tothe anomalous symmetry action:Uos1 âIRâi(â1)gi(giâgiâ1)âi(â1)gËi = UA1 (4.4)We will denote this as Uos1 âIRUA1 . To determine the structure of the UV symmetrygroup GUV implemented by (4.2), note that (Uos1 )2 =âi(â1)nËi , and (Uos1 )4 = 1, fromwhich we see that the extended group structure is GUV = Z4.Transforming into the IR space â An alternate perspective on the emergent anomalyis obtained by starting with the on-site Z4 symmetry action (4.2), and performing alocal unitary transformation:V =âi(â1)ni(giâgiâ1) (4.5)that maps Hâ to a trivial N -paramagnet:V â HâV = âââiÎ´nËi,0 (4.6)and converts the onsite symmetry to the anomalous one:V â Uos1 V =âi(â1)gi(giâgiâ1)âi(â1)ginËiâi(â1)gËi (4.7)54which coincides with the anomalous symmetry transformation, (4.1) if we restrict tothe ground-space of Hâ where nËi = 0.Zero-correlation length (ZCL) IR Hamiltonians â We can write down an idealizedmodel for the emergent anomaly by adding generic local interactions: H = Hâ +HIR, where HIR is symmetric, and commutes with Hâ, i.e. exactly preserves the IRsubspace with emergent anomaly. For such Hamiltonians, the emergent IR symmetryoperation is precisely (4.1), for which the non-onsite phases are strictly local, actingonly on neighboring sets of three sites. We will refer to such models as having zero-correlation length (ZCL), with the understanding that this refers to the spatial range ofthe symmetry action (and, as we will see shortly, the localization length of igSPT edgestates) but not to the correlation length of local correlation functions which is infinitein a gapless system.A general prescription to construct a ZCL HIR is to take any local interactioninvolving only GIR-rotors, conjugate it by V to generate an interaction term thatcommutes with Hâ, but may not respect symmetry. Then explicitly symmetrize theterm by superposing it with its symmetry conjugate (assuming this sum does not van-ish). For example, starting with a simple transverse field term H1 = âJâi(â1)gËi =âhâi Ïxi , conjugation by V gives: V â H1V = âJâi Ïxi=1Ïxi Ïxi which commutes withHâ but is not symmetric. Then, we can add its symmetry conjugate, to obtainHIR = Vâ H1V + Uâ 1Vâ H1V U1:HIR = âJâi(Ïxiâ1Ïxi + Ïyiâ1Ïyi)Ïxi (4.8)which satisfies the fixed-point properties. This Hamiltonian can further be exactlysolved by fermionization, and results in a gapless Luttinger liquid with topological edgestates [73]. We mention in passing that an alternative prescription for constructingfixed-point HIRâs would be to consider including a generic local symmetry-preservingterm H1 with coefficient âª â, but which may not commute with Hâ. Then, perform(degenerate) higher-order perturbation theory with H1 to approximately compute itsinteraction projected into the low-energy subspace.Clearly there are many different possible options for HIR, resulting in a potentiallyrich phase diagram in the deep IR. This phase diagram is constrained by the emer-55gent anomaly, which prevents trivial (symmetric and non-fractionalized) gapped states.Rather than confronting the (generally hard) task of working out this phase diagram forspecific choices of HIR, we focus instead on deducing sharp topological features arisingfrom the emergent anomaly.4.3.5 Edge states via lower-d SPT pumpingTo explore the edge of this Ising igSPT, restrict the system to an open chain of Z4 rotors,with sites i = 1 . . . L, and microscopic (UV) symmetry action (4.2). Further, restrictHâto this open chain by simply omitting term that spill past the boundaries, and add theedge term ââÎ´n1,0 to remove the dangling N rotor that does not interact with any GIRdomain walls on the links. The numerical analysis and analytic arguments of [106, 73]show that the Z4 igSPT has gapless DOF corresponding to operators localized to theedge of the igSPT chain with open boundaries. Specifically, in a length-L chain, thebulk topological Luttinger liquid has a finite size gap δB â¼ 1/L. Within this gap, thereis a near-exact two-fold degeneracy when the system has open boundary conditions,with the two near-ground-states being split by an amount δE â¼ eâL/Î¾, where ξ isproportional to the inverse of the energy gap â to the UV N DOFs.Here we give a general argument that clarifies this structure, and which readilygeneralizes to other examples with various symmetries and dimensions. Specifically,consider the anomalous symmetry action UAg . In the ground-space (IR) of Hâ, theextending ni spins are locked to DWs of the gi spins, and we can write a description ofthe action just in this IR sector:Uos1 âIRLâi=2(â1)gi(giâgiâ1)Lâi=1(â1)gËi (4.9)In the IR symmetry group GIR = Z2 = {0, 1} the group operation is 1 + 1 = 0. Onemight therefore expect that Ug=2 would yield the identity in the IR. Instead, one finds:Uos2 = (Uos1 )2 âIRLâi=2(â1)giâgiâ1 = (â1)gLâg1 = Ïz1ÏzL. (4.10)We can interpret U2 as a unitary that pumps a 0d GIR-SPT onto the edges of the56chain. 0d G-SPTs are classified by representations of GIR on U(1) â i.e. different 0dSPTs correspond to different possibly symmetry quantum numbers or âchargesâ of theground-state (which cannot be changed without closing the gap to an excited statewith a different charge). For Z2 symmetry there are two possible symmetry charges,±1, and U2 flips the local symmetry charge of each edge, toggling this 0d SPT invariant.Specifically, the local action of U2 on one end of the chain, say x = 1, is (â1)g1 whichanticommutes with the GIR-symmetry generatorâi(â1)gËi . As a result, the edge istuned to a degenerate critical point where there are two degenerate edge configurations.We now connect this lower-d SPT pumping symmetry to the ground-space degener-acy of the open igSPT chain, and also to the non-local string order identified in [106].By assumption, UGUV=0,1,2,3 are symmetries of the full bulk and edge Hamiltonian.Moreover, since in the IR U2 acts nontrivially only on the edge of the chain,(U2)[1,L] = (â1)g1âgL = Ïz1ÏzL, one can define the restriction to one end of the chainby U2|1,L := Ïz1,L. This local action of the pumping actions must commute with any lo-cal Hamiltonian thatâs invariant under the GUV symmetry. Furthermore, each of themanti-commutes with the symmetry generator U1. This non-commuting algebra of sym-metry acitons {(U2)|i=1, U1} = 0 immediately leads to an exact ground state degeneracythat is at least 2-fold. Moreover, this degeneracy is directly associated with edge-localoperators and does not arise with periodic boundary conditions.We refer to the edge-local version of the pumping operators as topological edgemodes. However, a key point that distinguishes them from ordinary topological edgestates of gapped systems is that the degenerate ground-space does not have a tensor-product structure. Specifically, unlike for the Haldance/AKLT chain [43, 1], the de-generacy cannot be decomposed into a local âspin-1/2â living at each end of the chain(which would result in 4-fold GS degeneracy). Formally, the distinction is that theedge pumping modes, (U2)i=1,L share a common global conjugate operator U1, whoseaction cannot be localized to one or the other edge because of the gapless bulk degreesof freedom (in contrast to the Haldane/AKLT chain where there are independent edgemode Sx and Sz operators for each end of the chain). We note that a similar non-tensor product structure edge degeneracy was observed at the critical point betweena Haldane chain (gapped Z22-SPT) and a reduced symmetry magnet gapped with onlya single Z2 symmetry [93]. The fractional quantum dimension of the edges harkensthat of Majorana zero modes at the end of a topological superconductor. However, a57more apt analogy is that of (boundary) spontaneous symmetry breaking (SSB) since:the true ground-state of a finite-size chain is a Schrodinger-cat/GHZ like state of op-posite edge magnetizations which has long-range entanglement that can be collapsedby a local measurement of the edge spin or which can be removed by an infinitesimalboundary field. Ordinarily, a 0d system could not spontaneously break symmetry onits own, however, here the 0d edge is stabilized by the 1d bulk, despite that the bulkitself does not break symmetry. An equivalent interpretation of these edge modes is asdifferent symmetry-breaking conformal boundary conditions on the bulk CFT [93, 27].Effect of finite correlation length â Next consider moving away from the ZCL limit,whereHIR is symmetric, but does not commute withHâ. The allowed emergent anoma-lies are discrete and cannot be continuously altered by GIR-symmetric perturbationsthat do not close the gap to theN charge degrees of freedom. Namely, consider perturb-ing the Hamiltonian to: H = Hâ + λV , where λ is a dimensionless coupling constantand V is GUV-symmetric. Then, the stability of the gapped N degrees of freedom tolocal perturbations [13] implies there is a critical λc > 0, such that for λ < λc the N -charged DOF remain gap, and the emergent anomaly remains stable. In this regime,one can define an IR Hilbert space by adiabatically continuing the ground-space ofHâ(s) = Hâ + λ(s)V with s â [0, 1], λ(0) = 0, and λ(1) = λ < λc. The resultingadiabatic evolution is accomplished by a GUV-symmetric finite-depth local unitary cir-cuit (FDLU): F = limtfâ«V/â2 T {eâiâ« tf0 Hâ(t/tf )dt}. Properties in the FDLU-transformedframe, F , then differ from those in the lab-frame by exponentially-well-localized sym-metric dressing [45]. In particular, the IR action of symmetry is equivalent to the idealnearest-neighbor anomalous action, UAg , above, up to this FDLU transformation, Forexample, in the perturbed model, the pumping operation F â U21F is no longer preciselylocalized to the edge, but has exponentially decaying tails into the bulk that decay withdistance x as eâx/Î¾ where the correlation length ξ is bounded by the Lieb-Robinson dis-tance for F .From this we deduce that, away from the ZCL limit, we can define quasi-locallydressed operators W1,L = Fâ Ïz1,LF that approximately commute with H to accuracyeâL/Î¾, and this edge-state degeneracy is split by this exponentially small amount, buthowever, remains sharply distinct from the bulk finite-size gap â¼ 1/L for Lâ« ξ.58The N -charge gap does not close at the edge â Since the SPT pumping operation isimplemented by an N -symmetry operation, and ends up acting nontrivially at the edgeof the igSPT, one might be tempted to conclude that the gap to the N charged DOFcloses at the edge. We emphasize that this is not the case. Specifically, the unitary V in(4.5) explicitly transforms Hâ to a paramagnet that fully gaps all the N charges. In theZCL limit, one can readily check that there are no local N -charged operator that actswithin this 2-fold degenerate ground-space associated with these edge model (a featurethat is preserved up to local dressing by F away from the ZCL limit). Physically, theeffect of the N symmetry transformation on the IR theory with emergent anomalousGIR symmetry, arises due to a non-trivial entanglement between N charges and GIRcharges that is locked in by Hâ at UV scale â. However, it costs finite energy, â, tounlock an N rotor.Connection to string order â We note that the lower-d SPT pumping operation isclosely connected to the nonlocal string order identified by TVV in [106]. In TVVâsIsing/Hubbard model exhibited a string order Sziâi<x<j(â1)NxSzj where Ni was thefermion number on site i and Szi was the electron spin operator at site i. In thatexample, the group extension that lifted the anomaly was to extend a GIR = Z2 x-axisspin rotation symmetry by fermion parity N = ZF2 . Hence we can understand thestructure of the string operator as follows: the bulk of the stringâi<x<j(â1)nx wherenx is the number of fermions at position x, is simply the N = ZF2 symmetry generatorrestricted to an interval (i, j). The Sz operators terminating the ends of the stringadd a symmetry charge, i.e. add a 0d SPT onto the end of the string. This cancelsthe 0d SPT pumping action of the symmetry operation, such that the string operatorhas a finite expectation value for an arbitrarily long string. In principle, this picturecoupled with the SPT-pumping perspective enables one to identify non-local membraneâorder-parametersâ that detect higher dimensional igSPTs which have an analogousform of acting with an N symmetry operation restricted to Dd region, multiplied by(D â 1)d boundary operators that add an appropriate GIR SPT to their boundaries.A subtlety (common to all higher-d analogs of non-local string order), is that, awayfrom the ZCL, long-range membrane order is characterized by exponential decay ofmembrane correlations with the perimeter/surface-area of the membrane-boundary (anO(1) constant suppression for 1d string order parameters) rather than with the bulk59membrane area (similar to the perimeter vs. area law for diagnosing confinement in 2dgauge theories). For this reason, we do not pursue this non-local order perspective inhigher-dimensions.4.3.6 Gauging the symmetryGauging the symmetry often provides non-perturbative insights into its topology. Gaug-ing a G-symmetry can either be done by exploring the response of the system to anexternal, background G-gauge field, or by promoting a global G symmetry to a localgauge redundancy coupled to dynamical gauge fields. For our purposes, it will be mostuseful to consider gauging the central extension, N , that lifts the anomaly, which re-sults in an anomalous GIR-SET. While this gauging perspective will be most useful in2d and 3d, we use the 1d version as a warmup.Coupling to a background gauge N -gauge field â For a gapped system, the responsetheory to an external gauge field is a local topological quantum field theory (TQFT).For an igSPT, the presence of gapless bulk modes can lead to non-local responses toa general GUV-gauge field. To avoid this non-locality, one must be selective in whatgauge-fields or gauge configurations we couple into the system. Since the N DOF aregapped in the IR, gauging only the normal sub-group N of the extended symmetryGUV results in a local response theory. As explained in [104], gauging the extendingsymmetry N results in a theory with an anomalous GIR = GUV/N symmetry. Here,we explore the relation between this and the igSPT edge states.To explicitly gauge the lattice model, introduce a non-dynamical (backgrond) Z2gauge link variable Ai,i+1 â N = {0, 1} associated with link (i, i + 1), and conjugategauge electric field Ei,i+1 â NÌ = {0, 1} satisfying {(â1)Ai,j , (â1)Ei,j} = 0, and enforcethe Gaussâ law gauge constraints: nËi = Ei,i+1 âEiâ1,i, which generate the gauge trans-formations: ni â ni + Ïi and Ai,j â Ai,j + Ïj â Ïi where Ïi â {0, 1}. We note thatthe following discussion is equivalent to the discussion of the effect of symmetry-twistedboundary conditions on 1d igSPTs in [73].The unitary V in (4.5), which converts the anomalous symmetry to an on-site one,is not gauge invariant as written. However, we can remedy this by minimally couplingni to Ai,i+1. With periodic boundary conditions (PBCs) note that:âi(â1)ni(giâgiâ1) =â(â1)gi(ni+1âni). Hence, we can minimally couple (ni+1 â ni)â (ni+1 â ni âAi,i+1) to60make this operator gauge invariant resulting in:V =Lâ1âi=1(â1)gi(ni+1âniâAi,i+1) (4.11)where the script font indicates that the N symmetry subgroup has been gauged. Thistransformation effects Vâ HâV = âââi Î´nËi,0 and (with PBCs):Vâ U1V âIR(â1)âi Ai,i+1âi(â1)gi(giâgiâ1)âi(â1)gËi (4.12)We see that the U1 symmetry eigenvalues ±1 are toggled by an N -gauge flux (â1)âA =(â1). Hence, the N gauge flux carries a GIR-symmetry charge. This is consistent withour above discussion of the edge states â since inserting an N symmetry flux with PBCsis equivalent to cutting open the chain, acting locally on one end with U21 = W (whichwe have seen changes the U1 charge), and then gluing the chain back together. Hence,the structure of the igSPT edge states can also be deduced by gauging the extendingsymmetry, N .Fractionalizing the anomalous GIR-symmetry â In higher dimensions, it is possiblethat the bulk igSPT anomaly is satisfied by a gapped symmetric, but topologicallyordered state. Following [122], we can view this topological order as arising from frac-tionalizing GIR-rotors into GUV DOF, and projecting out the unphysical fractionalstates by coupling the N -subgroup of the GUV rotors to a dynamical N gauge field.Here, it will be crucial to assume a hierarchy of scales between the UV gap, â that im-prints the anomaly, and the (assumed much smaller) energy gap, δ, to the fractionalizedexcitations: δ âª â.Specifically, consider a microscopic lattice model with GUV symmetry, imposing Hâso that in the IR at scales âª â the symmetry action is effectively the anomalous onegiven by (4.1) acting on GIR-rotors. Then, fractionalize each GIR-rotor into GUV rotor,with onsite basis |νi, giâ© with νi â Ng = Z2, but where the νi DOF is coupled to adynamical Ng-gauge field, ai,i+1 (we use lower- and upper-case letters to distinguishemergent and external/background gauge fields). We caution that one should distin-guish the gapped ni DOF from the fractionalized ν DOF: the latter are not gauge61invariant, i.e. are not microscopic excitations, but may only emerge as fractionalizedIR excitations if a is in a deconfined phase (similar to spinons in a more typical partonor slave-particle description of electronic models). For this purpose, we include a GIRsubscript on Ng, to distinguish the gauge group from the global N symmetry acting onthe gapped UV sector.In higher dimensions, this dynamical gauge theory could emerge from a parton de-scription where a microscopic boson operator B with GIR charge (â1), is fractionalizedinto a pair of 12-GIR charged operators: B = b2 (i.e. Ug : b â igb acts as a Z4 symme-try on the partons). In 1d, any such gauge theory will generically become confined ifthere are any quantum fluctuations in the gauge field. Nevertheless, we find it useful tostudy a fine-tuned fluctuation-less gauge model as preparation for higher-dimensionalexamples.The gauge invariant sector satisfies a Gauss law Î½Ëi = ei,i+1 â eiâ1,i where Î½Ë and eare conjugate to ν and a respectively. This Gaussâ law generates gauge transformationsνi â νi + Ïi and ai,i+1 â ai,i+1 + Ïi+1 â Ïi for any local gauge transformation Ïi.Further, the IR symmetry action on the fractionalized degrees of freedom is:U frac1 =Lâi=1(â1)gi(giâgiâ1)Lâi=1(â1)giÎ½Ëi(â1)gËi (4.13)As for the external gauge response, the finite-depth unitary V (with n â ν andAâ a) simplifies the symmetry action to be almost on-site:Vâ U frac1 V âIR(â1)Î½LââLâ1i=1 ai,i+1âÎ½1âi(â1)giÎ½Ëi(â1)gËi . (4.14)This differs from an ordinary onsite Z4 symmetry only by a gauge-string with chargedends that terminate on the edge. As a result, one can write add a trivial Z4 bulk-62paramagnet Hamiltonian in the V-transformed frame:Vâ HZ4-PMV = âÎ´Lâ1âi=2Î´Î½Ëi,0Î´gËi,0= âÎ´Lâ1âi=2â(Î½i,gi),(Î½â²i,gâ²i)âZ414|(Î½i, gi)â©â¨(Î½ â²i, gâ²i)| (4.15)that gaps out all the bulk DOF (except the total gauge flux). In additional to thetopological edge degeneracy, this leaves an additional accidental edge degeneracy thatcan be partially removed by further adding:Hedge = âV(Î´âi=1,L(â1)gi+Î½Ëi)Vâ , (4.16)which commutes with U frac1 and locks gi1, L = Î½Ë1,L. Then the remaining DOF aresimply Î½Ë1,L which naively leaves a fourfold GSD. However, total gauge charge vanishes,âi Î½Ëi = 0, for any physical states, which (considering the bulk ground-state of HZ4-PMhas no charge) reduces the physical edge-GSD of gauge invariant states to two (for fixedgauge-flux sector). Again, the pumping operation pumps 0d GIR-SPTs onto the edge:(Vâ U frac1 V)2 = (U frac1 )2 = (â1)gLâg1 , so that, just as for the gapless Luttinger-Liquidbulk, there is no local symmetric perturbation one can further reduce this two folddegeneracy since it is protected by a non-trivial local anti-commutation between U1and (the edge restriction of) U2.In addition there is a pair global topological supers-election sectors labeled by totalgauge fluxâi ai,i+1 = 0, 1. As mentioned above adding quantum fluctuations in thegauge field would immediately lead to a confined theory in 1d. However, this construc-tion will prove useful in producing lattice models of bona-fide anomalous topologicalorders in higher-dimensional examples.634.4 Constructing igSPTs from group cohomologydataWe next formally show how this 1d example generalizes for emergent anomalies classifiedby group cohomology. To begin let us fix some notation.4.4.1 NotationGeneral groups â For equations involving a general, unspecified group, G we willuse multiplicative notation for group operations: (Â·) : G Ã G 7â G, denoted by g · h(we will occasionally omit the · for brevity). We denote the representation of symmetryelement g â G on quantum states by Ug.Abelian groups â For specific examples, we focus on groups that are either Abelianor semidirect products of Abelian groups and time-reversal. For Abelian groups we willtypically use additive notation for the group operation + : GÃG 7â G, for example wewrite ZN = {0, 1, . . . N â1} with group operation addition modulo N , and denote U(1)elements by a phase α â [0, 2Ï). For direct product of cyclic groups, such as ZN1ÃZN2 ,denote N12 = GCD(N1, N2) where GCD denotes the greatest-common divisor.For Abelian groups, define the âvorticity indicatorsâ v as follows: for g, h â ZN , 0 â¤g, h â¤ Nâ1 define vN(g, h) to be 1 if g+h â¥ N and 0 otherwise, where, here, we regardg, h as integers and and + as integer-addition (i.e. not modulo N). Similarly for U(1)define v2Ï(Î±, β) to be 1 if α + β â¥ 2Ï, and 0 otherwise. When clear from context, wewill drop the subscripts on v. This function will frequently appear in cocycles, whereit can physically be interpreted as detecting vortices on triangular plaquettes.We denote the Pontryagin dual of a group G by GÌ â¡ H1(G,U(1)) = Hom(G,U(1)),which consists of linear representations of G on U(1), i.e. gË â GÌ is a homomorphismgË : G 7â U(1), satisfying property gË(g1) · gË(g2) = gË(g1 Â·g2) âg1,2 â G. For Abelian groupsthe Pontryagin dual is given by the group Fourier transform. All the examples we willconsider involve the relations: ZÌN = ZN , ZÌ = U(1). As an example, when G = ZN , wehave GÌ = ZN , and an element gË â GÌ is the homormphism: gË(g) = e2ÏigËg/N .64Figure 4.3: The models are defined on a simplicial complex: a generalized triangulationwith orientation s = Â± for each D-simplex (link in 1d, triangle in 2d, tetrahedron in 3d,etc...). The on-site Hilbert space for the anomalous model consists of |ga â Gâ© for eachvertex a. The anomalous GIR-symmetry action is specified by an element [ÏD+2] of thecohomology groupHD+2(G,U(1)), it differs from the trivial onsite action by a product ofphase factors given by the cocycle ÏD+2 and field configuration on each simplex. E.g. in2d the phase factor associated with the yellow triangle is Ï4(gâ11 g2, gâ12 g3, gâ13 gâ1, g)â1,the overall inverse is due to the clock-wise orientation of the triangle.Cup Product â We will make use of the cup product of two co-cycles, defined asfollow. For Ïm â Hm(G,M) and ηn â Hn(G,N), the cup product of them is an elementof Hm+n(G,M âZ N) defined as:Ïm âª ηn(g1, · · · , gm+n)â¡ Ïm(g1, · · · , gm)âZ η(gm+1, · · · , gm+n). (4.17)We will only be concerned with the case where M , N are Pontryagin dual of eachother: M = NÌ , in which case we can identify the tensor product M âZ N as U(1) byidentifying the elementmâZn asm(n) â U(1), this identification will be made implicitlythroughout this paper. For example when M = NÌ = ZN , both Ïm and ηn are integers65â {0, 1, . . . N â 1}, and accroding to our definition we have: Ïm âª ηn(g1 . . . gm+n) =exp[2ÏiNÏm(g1, . . . gn)ηn(gm+1, . . . gm+n)]. Therefore Ïm âª ηn is a m + n cocycle withU(1) coefficients.G-rotor models â We will consider quantum lattice models with onsite Hilbert spacebuilt from quantum rotors, i.e. Hi = Span{|giâ©, gi â G} for each site i in the lattice.With a slight abuse of notation, we define the operator g which is diagonal in the g-basiswith eigenvalue g, and the dual operators gË whose exponential raise the eigenvalues ofg, for example: ï£±ï£²ï£³e2ÏiNjgË|gâ© = |g + jâ©, G = ZN ,eiÎ±gË|gâ© = |g + Î±â©, G = U(1). (4.18)We will refer to such DOF as G-rotors. Note the eigenvalues of the dual operatorstake values in the Pontryagin dual group, thus the notation here is consistent with thenotation for the Pontryagin dual.4.4.2 Review: Anomalous lattice models from cohomology dataWe will exploit several constructions from the group cohomology (partial) classificationof bosonic SPT phases in (D+1)d with symmetry G. Refs. [17, 19] provided a generallattice model for the anomalousDd surface of a (D+1)d gapped SPT with symmetry G.These models are G-rotor models on a lattice with simplicial structure, i.e. a suitabletriangulation of space with sites indexed by a â Z, and a G-rotor |gaâ© at each site a.The order of site labels induces an orientation to edges, which is conventionally drawnas arrows pointing from smaller to larger site number. We will focus on cases where thesimplicial structure forms a regular lattice: a chain of sites in 1d, a triangular lattice in2d, or cubic lattice of face-sharing tetrahedra in 3d. We label elementary D-simplices(links in 1d, triangles in 2d, or tetrahedrons in 3d) by letters i, j, k · · · , and verticesby a, b, c · · · . The D + 1 vertices of a fixed simplex i are labeled as i1, i2, · · · , iD+1.Each simplex is assigned an orientation si = Â±. For example, in 2d with the simplicialstructure shown in Fig.4.3, upwards pointing triangles have s = 1 and downwardspointing triangles have s = â1.66An anomalous symmetry action is locally generated but not strictly onsite. It differsfrom the ordinary onsite G-symmetry action |gaâ© â |ggaâ© by non-onsite phases thatdepend on topological defect configurations on each simplex i. Specifically, for g â G,the anomalous symmetry can be chosen (up to a symmetric finite-depth local unitarytransformation) to act as [17, 19]:UAg |{ga}â© =âi[ÏD+2(gâ1i1gi2 , . . . , gâ1iDgiD+1 , gâ1iD+1gâ1, g)]si |{gga}â© (4.19)where ÏD+2 â ZD+2(G,U(1)) is a representative co-cycle for the anomaly, the productis over all simplices. For group elements that act anti-unitarily (i.e. time-reversal), thephases above should additionally be complex conjugated.The general form of this equation is that the ordinary onsite symmetry actionga â gga is modified by phases, ÏD+2, that depend on symmetry domain wall con-figurations gâ1iÎ± giα+1 on the links (iαiα+1) of each simplex i. Physically, these phasesgive rise to destructive quantum interference between different domain wall rearrange-ments [51], that prevent the system from reaching a gapped symmetric state (whichwould be a quantum superposition of all possible domain wall configurations). Groupcohomology implicitly assumes both a tensor product structure (i.e. the anomaly isin the symmetry action not the Hilbert space) and that all topological defects in sym-metry breaking order are gappable. These assumptions are known to breakdown forselected âbeyond-group-cohomologyâ SPTs that are characterized by symmetry defectsthat carry ungappable chiral modes or unpaired Majorana zero modes. In the followingwe focus on the group-cohomology anomalies, and comment only briefly at the endabout possible generalizations to beyond cohmology igSPTs.4.4.3 Lifting the anomalyStarting from an anomalous GIR-rotor model, we now show a general prescription forobtaining an lattice model with an extended and anomaly-free onsite symmetry GUV inthe UV, and emergent GIR-anomaly in the IR. The idea will be to introduce additionalN -rotors for some appropriate Abelian group N , ni (with dual operators nËi), on each67plaquette, such that the central extension:0â N Î¹âªâââ GUV GUV/Nâââââ GIR â 0 (4.20)lifts the GIR anomaly in GUV, meaning that a nontrivial cocycle [Ï] â HD+2(GIR, U(1))becomes trivial when viewed as a cocycle in HD+2(GUV, U(1)). Extensions of of GIR byN are classified by H2(G,N ): denoting elements of the extended GUV symmetry groupby (n â N , g â G), the extended group corresponding to an element e2 â H2(G,N ) hasgroup operation: (n1, g1) · (n2, g2) = (n1 + n2 + e2(g1, g2), g1 · g2), and the associativityof this group operation, γ1 · (γ2 · γ3) = (γ1 · γ2) · γ3 for γi â GUV, is guaranteed by thecocycle condition de2 = 1.It was shown that group-cohomology anomalies can be lifted by such a centralextension [104, 122] including a constructive proof in [104]. Moreover, it was shownin [104] that existence of such a group extension guarantees the D + 2 cocycle ÏD+2that specifies the Dd GIR-anomaly has a representative in the decomposed form:ÏD+2 = bD âª e2, (4.21)with bD â ZD(G, NÌ ), e2 â Z2(G,N ). We will show below that this decomposition im-plies a bulk-boundary correspondence for igSPTs tying the emergent group-cohomologyanomaly to the presence of SPT edge states. In general, bD will characterize the symme-try transformation ofN gauge fluxes when one gauges the extendingN symmetry whichwe will argue guarantees non-trivial igSPT edge states. In many cases, bD will manifestas a lower-dimensional SPT pumping symmetry that generalizes the 1d GUV = Z4 casediscussed above, and gives a physically transparent mechanism for igSPT edge states.Second, since bD has coefficients that are dual to those of e2, referring to the discus-sion in Section 4.4.1 above, their cup product [bDâªe2] can be identified with an elementofHD+2(G,U(1)) by âfeedingâ the e2(gD+1, gD+2) â N output into bD(g1, . . . gD) â NÌ =Hom(N , U(1)). As a concrete example, the 1d GUV = Z4 igSPT discussed above cocy-cle Ï3(g, h, k) = (â1)ghk which we can decompose into b1 âª e2 with b1(a) = a â NÌ = Z2and e2(g, h) = g · h â N = Z2.684.4.4 Extending and onsiteing the symmetryThe decomposition structure permits us to define an ordinary onsite GUV symmetryaction that can be reduced to the anomalous GIR action in the IR. We introduce anN -rotor ni â N for each simplex i. We define a single unit cell of the model as follows:associate each simplex, i, with the vertex ai â i that has the largest site number in thatsimplex ni. This results in groups of D simplices for each vertex. Define a single unitcell of the lattice model to consist of associated D N -rotors and one GIR rotor. Then,we may define the GUV symmetry action as:Uosg |{nËi}, {gi}â© =âinËi âZ e2(gâ1iD+1gâ1, g)|{nËi}, {gga}â©. (4.22)which is a tensor products of local unitaries acting only on a single unit-cell of thelattice model, i.e. is an on-site symmetry action. See Fig.4.4 for an illustration of 1dand 2d lattices.Then, to imprint the IR anomaly, we introduce an interaction:Hâ = âââiÎ´nËi,bsiD (gâ1i1gi2,...,gâ1iD giD+1 ), (4.23)In the ground-space of Hâ, nËi gets locked to bsiD(gâ1i1gi2 , . . . , gâ1iDgiD+1), which we denoteas: nËi âIRbD, and we obtain the low energy anomalous actionUosg |{ga}â© âIRâibsiD(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iD giD+1)âZ e2(gâ1iD+1gâ1, g)|{gga}â© (4.24)=âiÏsiD+2(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iD+1gâ1, g)|{gga}â© (4.25)= UAg |{ga}â©. (4.26)The result is that, at energy scales âª â (henceforth referred to as âthe IRâ), theextending N -rotors are gapped out by Hâ. Importantly, it is possible to fully gap allthe N rotors in this way, even in open geometries with boundaries. Specifically, withopen boundaries one needs to add additional terms âââiâS Î´nËi,0 where 0 is the identityelement of NÌ and S is the set of boundary sites that are not the terminal site of some69Figure 4.4: The N rotors nËi live on the sites of the dual lattice(white circles), orequivalently, the center of every simplex of the simplicial lattice. An N rotor in asimplex is grouped with the vertex of that simplex with largest site label to form a unitcell. The GUV symmetry action becomes onsite with this definition of a unit cell. Thedashed lines in the figure indicate the locking of N rotors to GIR rotors in the IR.complete simplex, which fully gaps all N -rotor DOF not involved in Hâ. Importantly,this gapping is done in such a way as imposes a non-trivial entanglement between theGIR and NÌ rotors that imprints an anomalous GIR-symmetry action on the GIR degreesof freedom in the IR.The unitaries U osg do not yet form a group, because the multiplication is not closed:Uosg1Uosg2=(âinËi âZ e2(g1, g2))Uosg1g2 . (4.27)To obtain a closed group, we enlarge our set of symmetry actions by defining a symmetryaction for each pair (n, g), n â N , g â G asUos(n,g) :=(âinËi âZ n)Uosg . (4.28)70The original symmetry actions (4.22) are identified with Uos0,g. Itâs easy to check thatthe Hamiltonian (4.23) is symmetric under Uos(n,g) actions. The multiplication rule isnowUos(n1,g1)Uos(n2,g2)= Uos(n1+n2+e2(g1,g2),g1g2), (4.29)which is the group law of the extended group GUV. In conclusion, we have constructeda lattice model with onsite GUV-symmetry in the UV, and emergent GIR = GUV/N -anomaly in the IR.4.4.5 Edge states: perspective from gauging NHow is the emergent anomaly related to the presence of SPT-edge states of the igSPT?To answer this question, consider the gedanken experiment of gauging the extendingN symmetry of the igSPT model described above.General picture â Gauging N leaves an anomalous GIR symmetry enriched topolog-ical (SET) order. Due to this anomaly there must be no way to confine this SET orderwithout either closing the gap to the N gauge charges (i.e. driving a phase transitionout of the igSPT phase), or breaking the GIR symmetry. In particular, it must not bepossible to condense N -flux excitations to result in a trivial gapped, confined, and GIRsymmetric theory. In general there can be three different types of obstacles to condens-ing the N flux excitations. i) The N -flux may carry gapless modes, for example withgaplessness protected by a GIR anomaly. In this case, their condensation would resultin a gapless state. ii) The N flux may carry a fractional GIR charge which cannot bescreened by any local excitations. Here, condensing the N flux would necessarily breakthe GIR symmetry. iii) In 2d and 3d the N flux could also have non-trivial self-statistics,such that it could not be directly condensed.In our construction, the N flux is always a boson, but may have non-trivial GIRsymmetry properties, so only possibilities i) and ii) are realized in our models. We notethat N -fluxes are co-dimension two [(D â 2)-dimensional] objects: instantons in 1d,point-particles in 2d, and line or loop excitations in 3d. In Appendix 4.11, we show thatthe GIR symmetry properties of fluxes are characterized by [bD] â HD(GIR, NÌ) whicharises in the decomposition of the anomaly cocycle ÏD+2 = bDâªe2. Gapless fluxes [case71i) above] corresponds to models where [bD âZ n] â HD(GIR, U(1)) is non-trivial forsome n â N . In this case, we will show below that the n-flux carries the gapless edgestates of a GIR-SPT, and [bDâZn] â HD(GIR, U(1)) characterizes the anomaly of thoseSPT edge states. Fractional GIR-charged fluxes [case ii) above] correspond to situationswhere [bDâZn] = [0] is trivial ân â N , so that there is no obstruction to having a gappedN -flux, but where [bD] â HD(GIR,N ) is non-trivial. For example H2(Z2, U(1)) = 0(there are no projective representations of Z2) whereas H2(Z2,Z2) = Z2 (it is possibleto define a fractional half-charge for a Z2-gauge flux). We note that for an igSPT withnontrival emergent anomaly, the cocycles bD, e2 that constitute the anomaly cocycleÏD+2 must be both nontrival. For instance, if bD is trivial, i.e. bD = dcDâ1, thend(cDâ1 âª e2) = dcDâ1 âª e2+ (â1)Dâ1cDâ1 âª de2 = bD âª e2 = ÏD+2, making ÏD+2 trivial.Therefore given an igSPT with nontrivial emergent anomaly, it is guaranteed that thefluxes and charges will have nontrivial GIR symmetry properties.The non-trivial properties of N -gauge fluxes are directly related to the local actionof symmetry on the edge of the igSPT. Namely, consider an interface between a trivialN gauge theory, âthe vaccumâ (with no emergent anomalies) and the N -gauged igSPT(with emergent GIR anomaly). Then the N flux transforms trivially in the vacuum. Ifwe drag theN flux into the system, it must change itsGIR symmetry action to [bD] Ì¸= [0].In order that the overall system obey an ordinary linear representation of GIR, theremust be a compensating codimension two excitation that transforms as â[bD]. Sincethe fluxâs symmetry property changes immediately upon passing through the vacuumâigSPT interface, this compensating excitation must reside on the igSPT edge. Hence,the igSPT faces the same obstacle to forming a trivial, GIR-symmetric, confined stateas the bulk vortex. Pulling this behavior back to the ungauged theory (e.g. by fixingto a flat N -gauge configuration), this implies an obstacle to forming a trivial GUV-symmetric edge of the igSPT â establishing a bulk-boundary-correspondence betweenthe emergent anomaly and the edge states.Special Cases: Edge states from SPT pumping symmetry â Having elucidated a gen-eral bulk-boundary correspondence from the N -gauging arguments presented above, wenow restrict to the special case where [bDâZn] is a non-trivial element of HD(GIR, U(1))for some n â N . In these cases, it is possible to understand the igSPT edge states from aphysically intuitive picture of an emergent lower-dimensional SPT-pumping symmetry.72Specifically, consider the extending symmetry operation(s):Uos(n,0) =âinËi âZ nâIRâibsD(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iD giD+1)âZ n. (4.30)Since bD(g1, . . . gD) â NÌ , for fixed n â N , [bD(â¦, . . . , â¦)[n]] â HD(GIR, U(1)) can beidentified with a (Dâ1)d GIR-SPT. This suggests a close relation betwteen the decom-position Ï = b âª e and the SPT pumping action noted in the the 1d example. Indeed,in Appendix 4.10, we show that (4.30) is precisely a unitary that converts a trivialproduct state of the edge (with symmetry action U trivial) into an edge GIR-SPT (withsymmetry action given in (4.78)). More generally starting from an SPT with invariant[ν] â HD(GIR, U(1)), the pumping operation Uosn converts this into a state with SPTinvariant [ν + bD âZ [n]].Since these pumping operations are symmetries of any GUV-symmetric Hamilto-nian, any symmetry-respecting edge must be invariant under toggling its GIR-SPTinvariant. This SPT-pumping symmetry forbids the igSPT edge from being triviallygapped without breaking symmetry. Heuristically, the SPT pumping symmetry forcesthe edge of the igSPT to sit at a self-plural4 (multi)critical point between the differentpossible (D â 1)d GIR-SPT phases. When this self-plural critical point is a continuousphase transition, this results in symmetry-protected gapless edges. If this represents adiscontinuous, first order, transition, then this results in an edge that spontaneouslybreaks the N -symmetry. A third possibility that arises only in D â¥ 3 is that thisself-dual point could be satisfied by a gapped-symmetry SET order (we construct anexample in Sec. 4.6). Formally, the SPT-pumping symmetry forces the edge to havean anomaly that has an LSM-type obstruction to forming a trivial, gapped, symmetricstate. Specifically, the igSPT edge with symmetry GUV is the same as the anomalousedge of a gapped Dd SPT with a different symmetry GIR ÃN in which N symmetry-breaking domain walls are decorated [19] by (D â 1)d GIR SPTs with cocycle bD. Weremark that a related construction was used in [107] to establish anomaly constraintson self-dual transitions between SPT and trivial phases.4generalizing the notion of self-dual, self-trial, etc... to arbitrary number73We emphasize the fact that the GUV-igSPT edge states have the same anomalyas a GIR Ã N gapped SPT does not imply that there is a gapped GUV-SPT withthe same edge states since GUV Ì¸= GIR Ã N for any non-trivial extension. As weprove in Appendix. 4.12, and illustrate through specific examples below, the non-trivialextension required to lift the anomaly always forms an insurmountable obstacle tocreating a gapped SPT with the igSPT edge properties â i.e. the igSPT models withemergent group-cohomology anomalies are indeed intrinsically gapless.We also remark that the SPT pumping mechanism for igSPT edge states can bedirectly related to the anomaly-cancellation argument used by TVV [106] to deduce theexistence of edge states. In Appendix 4.13, we explain how the edge-SPT-pumping pic-ture reproduces the edge-anomaly-cancellation picture while revealing additional struc-ture about the edge-anomaly.4.4.6 Stability and bulk phase diagramSo far, we have not fully specified the bulk igSPT Hamiltonian, which is necessary to describe aparticular igSPT state, or explore the bulk and boundary phase diagram of the system with theemergent anomaly. To remedy this we should introduce a local GUV symmetric Hamiltonian,H = HIR +Hâ such that: i) HIR is GUV symmetric, and ii) HIR is sufficiently weaker thanHâ that it does not close the gap that imprints the IR anomaly. As argued in the 1d igSPTdiscussion above, the latter assumption implies that the anomalous IR symmetry action isadiabatically connected, by a GUV-symmetric finite-depth local unitary (FDLU) circuit, tothe ideal zero-correlation length one in (4.19), and similarly for the lower-dimensional SPTpumping action of N -symmetries. Hence, under these conditions, the edge states will bestable, and are at most spread out by virtual quantum dressing within the Lieb-Robinsonlengthscale of this FDLU. Therefore, as is common in gapped topological systems, for genericigSPTs, the edge states will not be confined strictly to the edge of the system, but will have anexponentially decaying envelope â¼ eâx/Î¾ with distance x from the edge, where ξ is the (finite)correlation length induced by the gap to N -rotors. Throughout most of the remainder of thepaper, we will not attempt to solve the (generally hard) problem of deciding what happensfor a particular HIR, but rather will deduce sharp, topological anomaly based constraints onthe possible outcomes.Examples â Table 4.1 compiles a summary of igSPTs in various dimensions and with var-ious symmetry groups obtained from this general framework. In the remainder of the main74Symmetry group extensionN â GUV â GIRAnomaly(sub)groupHD+2(G,U(1))RepresentativeCocycleÏD+2(g, h, k, . . . )SPT-pumping/Symmetryfractionalization1d igSPTsZN â ZN2 â ZN ZN e 2ÏiN g·vN (h,k) UN pumps a ZN rep.Zâ Râ U(1) Z eig·v2Ï(h,k) U2Ï pumps a U(1) rep.ZN12 â ZN1 Ã ZN2N12 âZN1 Ã ZN2 ZN12 e2ÏiN12g1·vN2 (h2,k2) U(0,N2) pumps a ZN1 rep.ZN123 âZN2 â (ZN1 Ã ZN3 Ã ZN123)âZN1 Ã ZN2 Ã ZN3ZN123 e2ÏiN123g1h2k3 [U(0,0,N2),U(0,0,N3)]g pumps aZN1 rep.Zâ Râ ZT2 â U(1)â ZT2 Z2 eig1h2k2 T 2 pumps a U(1) rep.2d igSPTs(Z2)boson â ZT4 â ZT2 Z2 (â1)ghkl T 2 pumps a Haldane chain.ZN12 â ZN1 Ã ZN2N12 âZN1 Ã ZN2 Z2N12e2ÏiN12g1h2vN2 (k2,l2) U(0,N2) pumps a1d ZN1 Ã ZN2-SPTZN12 â ZN1 â ZN2N12 âZN1 Ã ZN2 Z2N12e2ÏiN12g1h1k1l2 edge ZN1 domain walls havefractional ZN1-chargesZ2 â R4ÏZ Ã ZT2 â R2ÏZ Ã ZT2 Z22{(â1)v2Ï(g1,h1)v2Ï(k1,l1)(â1)g2h2v2Ï(k1,l1)U2Ï pumps a 1d U(1)ÃZT2 -SPT3d igSPTsZN â ZN2 â ZN ZN e 2ÏiN g·vN (h,k)·vN (l,m) UN pumps a 2d ZN -SPTTable 4.1: Topological data for igSPTs in 1d, 2d, and 3d with Abelian symmetriesand/or time-reversal. An anomalous IR GIR symmetry emerges from a UV symmetryGUV symmetry which is a central extension of GIR by N , where the N DOF are gappedout. The data is listed only for ârootâ phases (from which other phases can be triviallyobtained by stacking). For product groups we only list mixed anomalies that do notfollow from previous table entries. E.g. for the third row (1d GIR = ZN1 Ã ZN2) wedo not repeat the pure ZN1 or ZN2 anomalies described in the first row). For productgroups G1 Ã G2 â G3 . . . we denote elements by (g1, g2, g3, . . . ). The explicit cocylesÏD+2 â ZD+2(G,U(1)) represent the anomalies. For on-site symmetries, we can readoff the the cocycle from the higher-d SPT response theory by replacing Ai â gi anddAâ v(g, h). We list the SPT pumping or the symmetry fractionalization that protectthe non-trivial edge for each igSPT. For unitaries U, V , [U, V ]g = UV Uâ V â denotes thegroup commutator.75text we focus on a single representative example in 2d and 3d respectively. In higher dimen-sions, it is possible to have stable QSET phases consistent with the emergent GIR anomalies.Straightforward generalizations of the gauging-by-fractionalization procedure outlined for 1dabove yield general formulas for solvable lattice models of these QSETs. Rather than dis-playing these results for general unspecified symmetry groups and dimensions, which followsclosely the results of [122] and results in somewhat unwieldy notation we instead apply thisframework to specific examples 2d and 3d in the following sections.4.5 A 2d Time-Reversal Symmetric igSPTWe now apply this general framework to construct a 2d time-reversal symmetry igSPT, whichhas IR symmetry GIR = ZT2 where the T superscript indicates that the symmetry operation isrepresented as an antiunitary operator. Following the group-cohomology cook book, one canconstruct an anomalous action of ZT2 on a lattice model with spin-1/2 DOF Ïza = (â1)ga onsite a of a 2d triangular lattice, Σ2, with a simplicial structure. There is a single anomalous2d action of time-reversal symmetry, H4(ZT2 , UT (1)) = Z2 which can be implemented as:T A =âi(â1)gi1i2gi2i3gi3âa(â1)gËaK (4.31)where we have defined the shorthand gi1i2 = gi1 â gi2 , K denotes global complex conjugationin the ga eigenbasis. We can read from this anomalous action N = NÌ = Z2, b1(g) = g ande2(g1, g2) = g1g2. To onsite this symmetry, we introduce additional NÌ = Z2 rotors, nËi â {0, 1}on each triangle, with onsite symmetry action:T os =âi(â1)nËigi3âa(â1)gËaK. (4.32)where gË, n denote the conjugate operators to g, nË respectively, and K is now global complexconjugation in the g, n-eigenbasis. To imprint the anomaly in the IR, we then energeticallylock nËi âIRgi1i2gi2i3 on each triangle i via the Hamiltonian:Hâ = âââiÎ´nËi,gi1i2gi2i3 . (4.33)In the ground-space of Hâ, we then have an emergent time-reversal anomaly: T os âIRT A.Similar to 1d, we see that the extension which lifts the anomaly is N = Z2, and GUV =ZT4 = {1, T , T 2, T 3} where T 2n+1 are antiunitary, T 2n are unitary, and T 4 = 1, the extending76group N is the normal subgroup {1, T 2}. This clearly yields an intrinsically gapless SPT,as there are no gapped bosonic SPT phases with ZT4 symmetry in 2d. Noting that, micro-scopically, T 2 = (â1)âi nËi , we see that physically, we have to extending the Hilbert space toinclude Kramers doublet bosons 5.For completeness, we note a unitary V that maps the on-site symmetry to the anomalousone, and equivalently maps Hâ to a simple paramagnet for the extending spins, n, Vâ HâV =âââi Î´nËi,0:V =âi(â1)gi1i2gi2i3ni . (4.34)This mapping makes it manifest that Hâ (augmented with âââi Î´nËi,0 terms for incompleteboundary simplices) fully gaps out the extending DOF, even in the presence of a boundary.As in 1d, this means that the N -rotors are fully gapped in both the bulk and the edge, andthere are no local N -charged operators (i.e. Kramers doublet boson operators) that createlow energy excitations.4.5.1 Edge statesAs in 1d, we can expose the structure of the edge states by considering a composition ofsymmetry operations that produces an N -element microscopically, and would act triviallyon a system with non-anomalous IR ZT2 -symmetry. Here there is a single pumping operatorgiven by T 2 which gives a unitary pumping operation:T 2 âIRâiâÎ£2(â1)gi1i2gi2i3 =âiââÎ£2(â1)gi1gi1i2 . (4.35)where âÎ£2 is the boundary (edge) of Σ2.Based on the 1d examples, we now expect that this pumping operation, T 2 should tog-gle the 1d ZT2 -SPT invariant of âÎ£2. In 1d, there is a single nontrivial ZT2 -SPT: the Hal-dane/AKLT phase [43, 1] which is characterized by Kramers doublet edge states. This canbe explicitly confirmed by verifying that (â1)gijgjk = Ï2(gij , gjk) is a non-trivial 2 cocycle inH2(Z2, UT (1)). Alternatively, note that restricting the boundary string operator to an open1d chain x â {1, 2, . . . L} gives: T 2[0,L] âIRâLi=2(â1)gi(giâgiâ1). Conjugating an ordinary on-site5We will later argue that extending by Kramers doublet fermions (such that T 2 = (â1)NF whereNFis the fermion number) would not lift this anomaly, but rather would instead lift a more complicatedbeyond-cohomology ZT2 anomaly.77TR symmetry on this chain: T (0) =âLi=1(â1)gËiK, by T 2 gives:(T 2[1,L])â T (0)T 2[1,L] = (â1)gLâg1âi(â1)gËiK (4.36)and the local action of this twisted time-reversal on site i = 1 is: (â1)g1(â1)gË1K, which squaresto T 2edge = (â1), which is the signature of anomalous edge action of a Haldane chain. Fromthese considerations, we confirm that acting on the 2d igSPT with T 2 pumps a Haldane/AKLTchain onto it s 1d edge.The symmetry of the system under the SPT-pumping action of T 2 forces the edge toreside at a self-dual point between the trivial and Haldane phases. As explained in [107], thisself-duality enforces an anomaly constraint that forbids the edge from being trivially gappedand symmetric. Specifically, if we write down a 1d family of edge Hamiltonians:Hedge(λ) = â12âiââÎ£2[(1â Î»)Ïziâ1Ïxi Ïzi+1 + (1 + Î»)Ïxi](4.37)with â1 â¤ λ â¤ 1. which realizes a trivial paramagnet for λ > 0, the Haldane/AKLT phasefor λ < 0, and a gapless CFT with central charge c = 1 at the self-dual point λ = 0. Thepumping symmetry maps λ â âÎ», and forces the edge to sit at the gapless point if thissymmetry is intact. We note that another possibility (which does not arise for this particularedge Hamiltonian) is that there could be a first order phase transition between the trivial andSPT states. This would correspond to spontaneously breaking the N -symmetry of the system,since it would result in a degenerate pair of ground-states for the edge that are interchangedby N .While Hedge in Eq. 4.37 with λ = 1 operates in the IR subspace, and is symmetric underT 2IR, it is not invariant under TIR due to the non-onsite phases in this anomalous time-reversalsymmetry operator which depend on the bulk spin configuration as well as the edge. Hence, inthe true igSPT the Luttinger liquid described by Eq. 4.37 can potentially become intertwinedwith the bulk critical modes in a manner that is hard to calculate in general. Below, we willdiscuss two instances where this bulk-edge coupling can be studied in a controlled fashion,and the gapless self-dual Luttinger liquid described by Eq. 4.37 with λ = 1/2 dynamicallydecouples from bulk modes.784.5.2 On the bulk phase diagramOur construction of the time-reversal symmetric 2d igSPT so far simply engineers a gappedsector that protects an emergent IR anomaly. However, Hâ still has an extensively degenerateground-space. To create a bona fide model, we should then add generic local time-reversalsymmetry interactions, HIR, that do not close the â-gap. As in 1d, any terms in HIR that donot commute with Hâ can be adiabatically eliminated, so we may restrict our attention tointeractions that commute term-by term the terms of Hâ. We may follow either of the twoprocedures that were outlined for the 1d example for generating ZCLHIR models. In this case,the resulting models involve complex several body interactions: generically, in 2d, single siteterms map to seven-body interactions under V involving products of operators over a site andits nearest neighbors on the surrounding hexagon of the triangular lattice). Moreover, thereare few non-numerical tools for studying 2d interacting lattice models. Therefore, insteadof confronting the difficult task of mapping out the phase diagram for a particular family ofHIR, we instead use the emergent anomaly data to deduce general anomaly constraints onthe possible bulk and boundary phases that might arise.Symmetry breaking bulk â A trivial option for the bulk is that it could be gapped and spon-taneous symmetry breaking. An example Hamiltonian for this would be a nearest neighborIsing ferromagnetic interaction: HIR = âJââ¨ijâ©(â1)gËi+gËj where â¨ijâ© denote nearest neigh-boring sites.Gapless bulk â One possible fate of the igSPT bulk that is consistent with the emergentanomaly is a symmetric gapless state. This problem has been previously studied in the con-text of 3d bosonic ZT2 -SPT surface physics [117], and one candidate gapless state is an SO(5)1Wess-Zumino-Witten model [11], which has various lower-symmetry and dual fermionic in-carnations [119]. A closely related field theory also appears in descriptions of deconfinedquantum critical points [99] between antiferromagnet and valence bond solids, or betweenquantum spin-Hall states and superconductors in fermionic systems [70] (though both ofthose involve different emergent anomalous symmetries at the critical point). The fate ofthe interaction between gapless Luttinger liquid edges and bulk DQCP modes was recentlystudied in a large-N limit [83], where the edge and bulk naturally decouple. Understandingthe ultimate fate of these 2d DQCP-type field theories and their interplay with edge modesremains an outstanding challenge. For example, it remains an open question whether theseDQCPs are continuous or weakly first order (which in the igSPT context with an emergentanomaly would signal spontaneous symmetry breaking).794.5.3 Quotient-symmetry-enriched topological (QSET) orderFor 1d igSPTs the only ground-states consistent with the emergent anomaly are either gaplessor symmetry breaking. In 2d, there is an additional possibility that the bulk of the igSPT isgapped and symmetric, but with an anomalous symmetry enriched topological order (SET).For the 2d ZT2 in-cohomology that emerges in this model, a known anomalous SET is a Z2(toric-code) topological order with anyons {1, e,m, f = e Ãm} such that e and m are bothKramers doublets under time-reversal, denoted as eTmT [117, 120].In this section, we construct an explicit lattice model in which eTmT SET order arisesin the bulk of the ZT4 igSPT model using the âgauging by fractionalizationâ procedure intro-duced for the 1d igSPT. Analyzing the edge of this model shows that the edge-SPT-pumpingsymmetry and associated self-dual edge gapless states survive when the bulk becomes gappedto form the anomalous SET order. Here, some care is needed to properly define what onemeans by an anomalous SET, since the anomaly is lifted deep in the UV. To see this, notethat extending the symmetry to ZT4 by adding microscopic Kramers bosons, bT clearly lifts theeTmT anomaly in the UV, as the SET can be trivially confined to a symmetric paramagnetby condensing the composite bT ÃmT (or bT ÃeT ) 6. However, if there is a hierarchy of scales,such that the gap, â, to bT excitations greatly exceeds the gap δ to the anyonic excitations,we can meaningfully talk about an emergent anomaly even with a gapped bulk. Borrowing aterm from an analogous situation for SPTs [111], we will call such a state a quotient-symmetryenriched topolgoical order (QSET). The sharp distinction between a QSET and a GUV-SETsis that the QSET will have symmetry-protected edge states that cannot be removed withoutundergoing an edge phase transition in which the gap, â, to N -DOF closes.Following the 1d approach described above, we can construct a lattice model of this QSETby fractionalizing the on-site gi â Z2 DOF into (νa â {0, 1}, ga) â Z4 DOF, and couple thefractional n-DOF to an emergent Z2 lattice gauge field, Aa,b so that the physical (gaugeinvariant) degrees of freedom are still GIR = Z2 rotors. The anomalous time-reversal isimplemented asT fracIRâi(â1)gi1i2gi2i3gi3âa(â1)gaÎ½Ëa(â1)gËaK (4.38)To interpret this expression it is useful to redraw the triangular lattice as shown in Fig. 4.5.Here, for each 2-simplex (i1i2i3), we color the i1i2 bond blue and the i2i3 bond dashed red.6We note that, extending the symmetry by adding local Kramers doublet fermions, cT , would notlift the eTmT anomaly, but rather would lift a different beyond-cohomology anomaly which supportsa representative efTmfT SET in which the e and m particles are both Kramers doublets under time-reversal and also have fermionic statistics.80Geometric picture of factorization property => DDW pictureFigure 4.5: As described in the text, the triangulated simplicial complex (left) can beredrawn as an inter-penetrating pair of dual square lattices (right), and the anomaloussymmetry can be interpreted as decorating domain walls on the blue lattice with 1digSPT phases on the dashed red sublattice.The blue and red bonds form a pair of (skewed) square lattices. We can straighten theplaquettes and shift the red square lattice relative to the blue one so that the red and bluesquare lattices are dual to each other. Here, we must remember that the purple-circledsites should be regarded as the same site. Then, we can interpret the non-onsite part ofthe symmetry as follows: along TR DWs (gab = 1) on the blue sub-lattice, the non-onsiteaspect of the symmetry action is essentially the same as that of the 1d igSPT model withGIR = Z2 described above. We can make this decorated domain wall (DDW) picture preciseby temporarily extending the IR symmetry group to ZT2 ÃZ2 = {1, T }Ã{1, g}, with ZT2 rotorsliving on the vertices of the blue lattice and Z2 rotors living on the vertices of the dashed redlattice, and decorating ZT2 DWs with 1d GIR = Z2 (GUV = Z4) igSPTs. Then, we can breakthe enlarged symmetry back down to the diagonal subgroup generated by (T , g), by lockingthe Z2 and ZT2 rotors together in pairs as indicated by the purple ovals.With this DDW picture, we can readily extend the other 1d constructions, to obtain gauge-invariant a unitary V that transforms this fractionalized symmetry into an almost on-site one(up to edge terms):V =âi(â1)gi1i2gi2 (Î½i2i3âai2i3 ). (4.39)which performs the same transformation as the 1d Z4 igSPT along blue DWs. We can write81the transformed symmetry schematically as:T â² = Vâ T fracV=âDW(â1)â«DW DAÎ½âa(â1)gaÎ½Ëa(â1)gËaK (4.40)whereDW denotes the blue domain walls, and â«DW DAν â¡â(ab)âGUV (νb â νa âAa,b) denotesthe lattice version of a Wilson line. These Wilson lines terminate only on the boundary, andso there are no ν operators in the bulk, only gauge connections Aa,b.Using V we can write down a Hamiltonian that fully gaps the bulk:Vâ HV = âÎ´âaâÎ£o2(â1)Î½Ëa+gËa â Î´âi(â1)Ai1i2+Ai2i3+Ai3i1 (4.41)where the first term makes a trivially-gapped paramagnet of the bulk matter (Σo2 denotes theinterior of the space) and the second term produces a gap for gauge flux excitations.This produces a lattice model of a Z2 (toric-code) QSET order. Denote the topologicalsuper-selection sectors of the Z4-rotor excitations [(Î½Ëa, gËa) Ì¸= (0, 0)] as e and the gauge fluxas m. The Z4 rotors transform under GUV = ZT4 , and hence the e particles are Kramersdoublets.Deducing the QSET anomalous symmetry properties from the cohomology data â We candeduce the symmetry properties ofm from the properties of the pumping symmetry operationsthrough the following argument applied to the case where Σ2 is an open disk (illustrated inFig. 4.6). Sincem is a Ï-flux for ν-DOF, braiding anm flux around a ν excitation is equivalentto acting locally on that excitation by T 2. Hence, creating a pair of m particles in the QSETbulk dragging them around a loop Î and re-annihilating them is equivalent to locally actingwith T 2 in the interior of Î, which we have seen pumps a Haldane chain onto Î due to theanomalous symmetry action. Thus, if we create and separate a pair of mâs, each m mustbehave like the Kramers-doublet edge of a Haldane chain. This shows that the resultingQSET bulk is an anomalous eTmT SET.The symmetry properties of m can also be directly confirmed by examining the action ofT â² on a gauge flux. With PBCs, the Wilson loops are all closed, and the modified symmetry,T â² differs from the on-site ZT4 symmetry by a phase of (â1) whenever the blue DWs enclose aflux â and we can choose a gauge where we consider the interior of the blue DW to be regionswith gi = 1. Suppose we have a gauge flux (m particle) in an initially gi = 0 domain. Thenacting with T flips gi : 0 â 1. Acting again with T then yields a (â1) phase since the fluxsits in an 1 domain. This shows that the local action of T 2 on m-particles gives a (â1) phase.82Edge SPT pumping argument for QSET flux properties<latexit sha1_base64="n+Jchhn35KEjs 1IWzNqMlYnOCFk=">[binary]</latex it>T 2 <latexit sha1_base64="3dsezDBBvPwXDJG+pqzDzGOcUlo=">AAAE0nicbZNPb9MwF [binary]</latexit>m<latexit sha1_base64="3dsezDBBvPwXD JG+pqzDzGOcUlo=">[binary]</latexit>m<latexit sha1_base64="3dsezDBBvPwXD JG+pqzDzGOcUlo=">[binary]</latexit>mFigure 4.6: constructed through the symmetry fractionalization approach can bededuced by the SPT-pumping symmetry action. Consider a 2d disk geometry where thebulk is a gapped time-reversal symmetric QSET. (Left) Creating a pair of gauge fluxes(m particles) and dragging them out of the sample is equivalent to acting locally on theshaded red region by T 2, which by the SPT-pumping property of the bulk anomaly addsa Haldane/AKLT chain with spin-1/2 (Kramerâs doublet) edge states to the boundary.(Right) Considering a similar process where only one m particle is dragged out of thesystem, and noting that projective time-reversal DOF can only be created in pairs, wededuce that the m particle in the bulk must carry a Kramerâs doublet. This argumentcan be readily generalized to various other symmetry groups.83Edge states of the QSET â The SPT-pumping symmetry prevents the edge of the QSETfrom being trivially gapped. We can deduce the structure of the QSET edge states from thefractionalized lattice model construction. This follows analogously to the 1d case discussedabove: wherever a blue DW intersects the boundary one can lock gi to Î½Ëi, and applying thetrivial paramagnetic term on other sites without blue DWs:VHedgeVâ = âÎ´â(i1i2i3)â©âÎ£2[(â1)gi1i2 (gi3+Î½Ëi3 ) + (â1)(1âgi1i2 )(1âÎ½Ëi1 )(1âgËi1 )]. (4.42)The first term in this equation is the same as the Hedge for the 1d igSPT from the prior sectionwherever a blue DW intersects the edge (gab = 1), and the second term simply implementsa trivial paramagnet away from the DWs (gab = 0). Eliminating the ν variables to satisfythis edge Hamiltonian, we obtain a 1d closed chain of free GIR-rotors with the anomalousSPT-pumping symmetry described above in (4.31). Consequently, a valid symmetric edgeHamiltonian is (4.37) tuned to the self-dual point (λ = 0) where it realizes a c = 1 CFT. Incontrast to the complicated case of a gapless bulk and edge, this edge CFT is obviously stableto symmetric coupling to the gapped QSET bulk.UV interpretation of the QSET â The QSET phase is sharply separated from an ordinaryGUV-SET by an edge phase transition where the edge states are lost by closing the N gap.One can ask what the resulting GUV-SET is in this case. For the time-reversal symmetricmodel we have been considering, the GUV-SET is trivial, since we can freely relabel the eand m excitations by binding them to microscopic Kramers-doublet bosons to make themKramers singlets. In the GUV = Z2ÃZ4 example studied in the appendix, the resulting GUV-SET is actually a non-trivial one where the m particle carries a projective representation ofGUV. Whether the resulting GUV-SET is trivial or nontrivial seems to depend on how far thesymmetry is extended to lift the GIR anomaly. Namely, we could lift the anomalous GIR = Z22anomaly in this example to GUV = D4, (the dihedral group with four elements), which is notthe minimal lift, but would result in the QSET order being trivial if the N -gap is closed atits edge.4.6 A 3d Ising igSPTBy now the pattern is clear, and we can readily construct a 3d igSPT model with GIR =Z2,N = Z2, GUV = Z4 that exhibits the surface anomaly of a notional 4d GIR-SPT bulk, but84when realized as an igSPT, the anomaly emerges from a gapped N -sector. We note that,as for the 1d example, the extending degrees of freedom could either be bosons (e.g. spins)or fermions. The latter is perhaps more relevant for physical realizations, as the GIR = Z2symmetry could arise as an Ising spin rotation symmetry such as Ï rotations around a fixedspin-axis, say x: Rx(Ï) for a (possibly doped) Mott insulator of spin-1/2 electrons, whichwould transform under a GUV = ZF4 extension with Rx(Ï)2 = (â1)NF .An explicit 5 cocycle for this state is:Ï5(g, h, k, l,m) = (â1)ghklm (4.43)The igSPT lattice model consists of nË â NÌ/g â G-rotors sitting on sites/simplicies of a 3dsimplicial complex (which, can actually be simplified to a 3d cubic lattice following the DDWpicture described previously) with onsite GUV = Z4 symmetry generated by:Uos1 =âi(â1)gi4 nËiâa(â1)gËa , (4.44)anomaly imprinting HamiltonianHâ = âââiÎ´nËi,gi1i2gi2i3gi3i4 , (4.45)which gives rise to an emergent anomalous GIR = Z2 symmetry in the IR generated by:UA1 =âi(â1)gi1i2gi2i3gi3i4gi4âa(â1)gËa , (4.46)which is related to the original on-site symmetry by the corresponding anomaly-lift unitary:V =âi(â1)gi1i2gi2i3gi3i4ni . (4.47)We note that just as in 2d, we can view this model as a DDW construction with a largerZ32 = {1, a}Ã{1, b}Ã{1, c} symmetry group, which we then break down its diagonal subgroupgenerated by abc. As illustrated above in 2d, in this extended DDW picture, we can againpull apart the simplicial complex into three inter-penetrating sub-lattices, one each for a,b,clinks, with the b sublattice residing on the dual lattice of a and c dual to b. This phase canthen be viewed as a DDW phase in which we decorate the 1d intersection of a and b DWswith 1d GUV = Z4 igSPTs. The original GUV = Z4 igSPT with just a single symmetry factorcan then be recovered by locking the a, b, and c rotors together at each site of the original85lattice.4.6.1 Edge phase(s)The extended group structure, N = Z2 follows from noting that(UA1 )2 =âiââÎ£3(â1)gi1i2gi2i3gi3pumps a 2d Z2 SPT onto the surface, which has a H3(Z2, U(1)) = Z2 classification. Thistunes the surface to a self-dual point between a Z2 SPT (Levin Gu phase [71]) and a trivialZ2 symmetry paramagnet.Spontaneous symmetry breaking edge â A particular instance of (a gauged version) of a 2dself-dual Hamiltonian satisfying this pumping symmetry was studied numerically in [87] whichfound that it realized a first order phase transition, which in the present context correspondsto spontaneously breaking of the GUV = Z4 symmetry. It remains an open question howstable a putative gapless symmetric edge termination would be, which would correspond toasking whether there was a continuous (multi)critical point between the Levin-Gu phase andtrivial PM.QSET edge and bulk â A new possibility that arises in 3d is that the 2d edge of an igSPTcan also exhibit anomalous QSET order that can only arise at the edge of a system withan emergent 3d anomalous symmetry. We focus on the simplest theoretical instance whereboth the 3d bulk and 2d edge have QSET order. An explicit lattice model can be readilyconstructed as in 1d and 2d, but we instead simply deduce a candidate QSET from the generalprinciples identified in the previous sections.The QSET order can have the structure of an emergent N gauge theory, in this case a3d toric code topological order with e particles, and vortex-line like m string excitations. Inour fractionalization construction, the e particle carries a half-charge of GIR, i.e. transformsunder GUV = Z4. In 2d, the string operator whose ends create m particles acted locally likethe lower-d SPT pumping operation. Here, the 3d analog of this is that the 2d membraneoperator whose edges create m line excitations carries a 2d Z2 gapped SPT. The edge of thismembrane, which is the m line therefore carries the (gapless or SSB) edge of this 2d SPT. Inan open geometry, for example the 3-ball pictured in Fig. 4.7, the m line excitations can alsointersect the edge of the system. From our study of 1d igSPTs we know that terminating theanomalous 1d edge of a 2d SPT requires the N extending DOF in the UV and results in aglobal 2-fold ground-space degeneracy of the m line when it intersects the systemâs edge. A86<latexit sha1_base64="3dsezDBBvPwXDJ G+pqzDzGOcUlo=">[binary] 9bPxam21uVz1u0dlq//gNKyLL1</latexit>m<latexit sha1_base64="uI0olJo/2kruGr QGViRysBqC6d8=">[binary] fWj4Xp9lbl8xatndav/ydYsu0=</latexit>e3d QSET for G=Z2Figure 4.7: A possible fate of the 3d GUV = Z4 igSPT model is that the emergentanomaly is satisfied by a fully-gapped anomalous Z2 symmetry enriched 3d Toric codewith point-like e excitations that carry Z4 charge and string-like m excitations thatcarry 1d Z4 igSPTs. The intersection of the m line and the surface hosts the edge zeromode of the 1d igSPT.succinct summary of these properties is that the anomalous feature of the 3d QSET is thatthe m line excitations carry 1d Z4 igSPTs.4.7 DiscussionOur group cohomology based constructions and framework, build on previous (predominately1d) examples [106, 73], to give a clear picture of how emergent anomalous symmetries leadto igSPT and QSET states, with SPT edge states protected by SPT-pumping symmetries.A natural question for future exploration is whether or not igSPTs can be constructed withother types of anomalies, including i) fermionic anomalies, i.e. anomalies involving fermionparity which is not quite a symmetry because it cannot be broken by any local operator andalso cannot be sensibly extended [106] to a larger group, and ii) beyond group-cohomology87anomalies that involve gravitational or mixed-symmetry/gravitational anomalies. To facilitatepotential experimental realizations, it would also be interesting to generate examples wherethe extended symmetry group was naturally realized by spin-1/2 electrons, which can imprintthe anomalous IR symmetry on the low-energy spin-model of a Mott- or Cooper-pair insulator.As we argue next, this naturally leads one to consider beyond cohomology anomalies.4.7.1 Selected boson igSPT candidates with beyond group-cohomology anomaliesBefore closing we briefly comment on some candidate 2d igSPTs with emergent anomalies be-yond group cohomology, highlighting ones that could arise in electronic systems (i.e. emergingfrom unit charged, spin-1/2, fermions). We focus on systems with physically relevant sym-metries: spin-component-conservation and time-reversal GIR = U(1)ÃZT2 , that are robust todisorder. In a system of electrons, bosonic excitations are composites of an even number ofelectrons, and will have charge that is an even multiple of the electron charge, for examplespins have charge 0 and Cooper pairs charge 2.In 2d, anomalous GIR = U(1) Ã ZT2 symmetry actions are governed by three differentZ2-topological invariants. These are conveniently expressed in terms of the properties of ananomalous toric-code-like SETs with anyons {1, e,m, f = eÃm}. We can distinguish differentSET phases by the symmetry properties of these anyons. Following standard notation [117,120], we denote a half-integer charge by a C subscript (note that a half-integer boson chargecorresponds to odd number of units of electron charge), a time-reversal Kramers-doublet bya T subscript, and fermionic self-statistics by an F subscript. The 7 non-trivial anomalousGIR-SETs with can be generated by taking combinations of the three root phases eCmC ,eTmT , and eFmF , where the symmetry properties of the fermion follow from f = e Ã m.Anomalies involving only C or T are captured by the group-cohomology classification and areincluded in our results. Anomalies involving eFmF are beyond the group-cohomology as theyinvolve a mixed time-reversal/graviational anomaly. Namely, any pure 2d bosonic realizationof eFmF must have chiral central charge ±4, which is half of the minimal âintegerâ chiralphase of bosons (the E8 state [79]), and hence breaks time-reversal symmetry.On general grounds, one can lift the eFmF anomaly by introducing a (charge-neutral)local fermion excitation, c, in the system. This can be seen in either of two ways. First,working directly in the topological order, we can relabel eF and mF by binding them to alocal c fermion, resulting in bosons with trivial quantum numbers that can be condensedto trivially confine the SET order. Second, fermions have invertible chiral phases with anyinteger Hall conductance, which can be used to cancel the chiral central charge ±4 of the882d realization. Hence, we expect that one can obtain an emergent eFmF starting from asystem of charge-neutral fermions, and driving these neutral-fermions into a gapped statethat imprints the anomaly on the IR boson DOF.A potentially more interesting example is the eCTFmCTF anomaly (i.e. with all three rootanomalies). In this case, the anomalous properties can be lifted by introducing fermions withunit charge (i.e. half a boson charge) and that are Kramers doublets (spin-1/2) under time-reversal â i.e. ordinary electrons. A concrete, albeit schematic construction of a 2d QSETorder an emergent eCTFmCTF anomaly can be obtained by the slab construction of [106].Start from a thin slab of a 3d boson GIR-SPT with this surface anomaly, in which we view thebosons as emerging from a Mott insulator of electrons. Then, close the electron gap on thebottom surface to bind electrons to themCTF particles, and condense the resulting bound statewhich is now a boson with trivial quantum numbers. This confines the DOF on the bottomsurface, but leaves the anomalous eCTFmCTF SET order on the top surface. An interestingquestion, that we leave for future work, is to investigate the edge of this 2d igSPT/QSET, i.e.the side-interface between the top and bottom surfaces in this slab construction. While wehave cited the slab construction as a proof-of-principle, the same phase should be realizable ina strictly 2d lattice model of electrons, and it would be interesting to see whether this couldbe accomplished with realistic interactions.4.7.2 Relation to half-filled topological flat bandsAnother context in which beyond cohomology anomalies emerge in the IR is half-filled topo-logical flat bands with a non-local particle hole (CT ) symmetry, including the 1/2 filled Lan-dau level of ordinary electrons [101, 121, 85], and various multi-component generalizationsthereof [90, 100]. Here, the role of Hâ is played by the orbital magnetic field, which breaksthe space into Landau levels (LLs). Half-filling the lowest LL results in an anomalous CTsymmetry that is emergent, in the sense that it is only operative within the LL and doesnot extend to the entire state Hilbert space. This CT symmetry realizes the same anomalyas the surface of a 3d U(1) Ã CT symmetric topological insulator. This was notably used toargue [101] and numerically demonstrate [42] that CT -symmetric composite Fermi surface hasa Ï Berry phase.In this context, the particle-hole transformation is unusual in that it involves both per-forming both a CT transformation and then adding filled LLs. The latter operation producesa non-standard notion of CT symmetry at the edge, and for example is compatible with non-vanishing thermal and/or electrical Hall conductance that is half of the allowed amount forinteger states. The non-locality of this symmetry follows from the Wannier obstruction to89forming a local tensor product structure to the lowest LL, which is also a key aspect of thebeyond-group-cohomology anomalies that emerge.If we make peace with the non-local symmetry, we might regard these examples as thefirst known examples of igSPTs. In fact, this half-filled flat band construction provides anexample of a system with an emergent eFmF anomaly discussed above. Namely, consider afour-flavor half-filled Landau levels of fermions with CT symmetry, which realizes the sameanomaly as a four Dirac cone surface of a 3d topological insulator in class AIII 7. In particular,DWs between opposite CT -breaking domans in this surface carry chiral central charge 8 â thehallmark of the eFmF anomaly. Then, consider inducing CT -symmetry interactions that gapout the fermion excitations (for example introduce fermion pairing and disorder the resultingsuperconductor through vortex proliferation without closing the gap to unpaired fermions asdetailed in [84]). The combination of the orbital field and symmetrically-gapping interactionsresults in a low-energy boson-only model with an emergent anomalous anti-unitary ZCT2 sym-metry that exhibits an eFmF anomaly at low energies. We remark that this example waspreviously discussed as a realization of an eTmT anomaly in [90]. If we ignore the hierarchyof scales â, the distinction between the eTmT and eFmF anomaly is lost in a system withmicroscopic Kramers-doublet fermions. Interpreted as an igSPT, this distinction becomesimportant so long as we demand that the fermion gap remains open.It remains an open question whether one can realize such beyond-cohomology anomalieswith a local symmetry, (e.g. one that enforces vanishing thermal Hall conductance).4.8 Additional 1d igSPT examplesIn this section we develop a few other examples of 1d igSPTs. Data for these as well asadditional examples are listed in Table. 4.1.4.8.1 GIR = ZNWe represent elements of ZN by operators gi = {0, 1, . . . N â 1}, and conjugate operators gËisatisfying [gËi, gi] = 1. These define the ZN rotor operators: e2ÏiNg and their dual operators e2ÏiNgËsatisfying:[e2ÏiNgË, e2ÏiNg]g= e2ÏiN where [a, b]g = abaâ1bâ1 is the group commutator. These arethe usual representations for a ZN quantum clock model, that are an N -level generalizationof the Pauli algebra for a qubit.7This system naturally has another U(1) symmetry (charge conservation) which will not play animportant role in the following discussion.90The anomaly classification isH2(ZN , U(1)) = ZN , with an explicit root cocycle: Ï3(a, b, c) =e2ÏiNa·vN (b,c) (this cocycle generates all the other anomalies). The corresponding anomaloussymmetry action isUAg =â(ij)e2ÏiNgi,jvN (âgjâg,g)âie2ÏiNggËi (4.48)=â(ij)e2ÏiNgi,jvN (gj ,g)âie2ÏiNggËi (4.49)We can read from the cocycle that N = NÌ = ZN and b1(a) = a, e2(a, b) = vN (a, b). Thereforewe can onsite this symmetry by extending the group with N = ZN rotors, ni, and lockingnËjIRgi,j on each link (ij) with onsite symmetryUos(n,g) =âie2ÏiNnËi(n+v(gi,g))e2ÏiNggËi , (4.50)which satisfies theGUV = ZN2 symmetry multiplication rule: Uos(n,g)Uos(nâ²,gâ²) = Uos(n+nâ²+vN (g,gâ²),g+gâ²).In the IR the SPT pumping operations are generated byU(1,0)IRâ(ij)e2ÏiNgi,j = e2ÏiN(gLâg1)which pumps the root class of 0d GIR = ZN -SPT onto the edge(endpoints) of the chain.4.8.2 GIR = U(1)2d SPTs with GIR = U(1) are classified by an even-integer Hall conductance Ïxy = 2n withn â Z. Unlike their fermionic counterparts these bosonic integer quantum Hall (bIQH) [97]states have vanishing chiral central charge i.e. vanishing thermal Hall conductance, since eachcharged chiral edge mode is accompanied by a neutral counter-propagating one. The result isan SPT phase since absent U(1) symmetry one could backscatter charge modes into neutralones gapping the edge.Realizing an igSPT with the same edge anomaly would be quite striking, as its bulk wouldhave a quantized conductance just as for the SPT edge. Unfortunately, our construction hassome pathalogical features when applied to this case. First, one needs to introduce continuousU(1) rotors eigi with gi â [0, 2Ï) and gi â gi + 2Ï. This is often a price that one must payto get exactly solvable models for bIQH-type states. Ordinarily one can hope that the sametopological phase could arise in a finite-dimensional truncation of these rotors, e.g. to spin-S91spins with maximum Sz = Â±S. Here, there is a more serious obstruction.Namely it turns out the minimal extending group that can lift the anomaly is Z. Oneway to see this is that the classification of 0d U(1) SPTs that can get pumped onto the edgehave H1(U(1), U(1)) = Rep (U(1)) = Z classification. In our construction, this lower-d SPTclassification dictates the structure of the symmetry extension.We can also verify by explicit construction. A cocycle for the root phase with Ïxy = 2 is:Ï3(a, b, c) = eia·v2Ï(b,c). The anomalous symmetry implementation is thenUAg =â(ij)eigi,jv(gj ,g)âieiggËi (4.51)where gËi is the conjugate âangular momentumâ to the rotor phase gi satisfying [gËi, gi] = 1,which has Z eigenvalues. We read from the cocycle that N = Z, NÌ = ZÌ = U(1) andb1(g) = g, e2(g1, g2) = v2Ï(g1, g2). To onsite this symmetry we can introduce additionalNÌ = U(1) rotors, nËi, and lock nËjIRgi,j on each link (i, j), and defining the on-site symmetryUos(n,g) =âieinËi(n+v2Ï(gi,g))eiggËi (4.52)One can verify that the resulting extending symmetry is actually the entire real line R.Namely, U(1) = R/2ÏZ, and the content of the on-site symmetry operator is that eachtime the symmetry shifts the phase gi past 2Ï it increments nËi by one unit, so that we canidentify each (n â N, g â R/2ÏZ) â GUV with 2Ïn + g â R, and the on-site symmetry issimply UxUy = Ux+y for x, y â R.While mathematically possible, we regard these models as unphysical. For example, whileit is perfectly possible to consider 1d array of quantum beads on loops of wire, faithfullyrealizing the R symmetry in such a model would require having no kinetic energy for the Nbeads, as the symmetry under cycling the GIR-rotors from g â g + 2Ï would increase theN -rotorâs angular momentum, which must commute with the Hamiltonian.4.8.3 GIR =â3I=1 ZNIAn explicit cocycle for the mixed anomaly is Ï3(a, b, c) = e2ÏiN123a1b2c3corresponding to anoma-lous symmetryUAgâ =â(ij)e2ÏiN123g1i,j(âg2jâg2)g3âie2Ïiâ3I=1gËIi gIiNI . (4.53)92We can onsite the symmetry by introducing NÌ = ZN123 rotors nËjIR(g1i,j) and define the UVactions to beUos(n;gâ) =âie2ÏiN123nËi(n+(âg2iâg2)g3)âie2Ïiâ3I=1gËIi gIiNI . (4.54)which obeys a group multiplication rule:(n, gâ) Â·(m, hâ)=(n+m+ g2h3, gâ + hâ)(4.55)We see the extended group is ZN2 â (ZN1 Ã ZN3 Ã ZN123), where the action of ZN2 on ZN1 ÃZN3 Ã ZN123 is:g2 â¦ (g1, g3, n) := (g1, g3, g2g3 + n) (4.56)The pumping actions are generated byUos(1,0)IRâ(ij)e2ÏiN123g1i,j = e2ÏiN123(g1Lâg11) (4.57)which pumps a 1d ZN1 SPT onto the edge.4.8.4 GIR = U(1)â ZT21d anomalies of GIR = U(1) â ZT2 are classified by H3(U(1) â ZT2 , U(1)) = Z2. As for theU(1) example above, 0d SPTs are classified by H1(G,U(1)) = Z = N , so one expects thatthis example requires an unphysical Z extension of the symmetry to lift the anomaly. This isconfirmed by direct computation from an explicit cocycle: Ï3(g, h, k) = eig1h2k2 . We can readfrom this cocycle that N = Z, NÌ = R, making the model unphysical as for the U(1) example.4.9 Other 2d igSPTs4.9.1 GIR = Z22In 2d, GIR = Z22 has H4(Z22, U(1)) = Z2ÃZ2, which can be generated by combinations of thefollowing two representative cocycles:Ï4(g, h, k, l) â {(â1)g1h2k2l2 , (â1)g1h1k1l2} (4.58)93In both cases, the anomalies can be lifted by N = Z2 extensions. However, the extensiongroup structure and edge physics is different. For the first cocycle the anomaly can be lifted byextending to GUV = Z2ÃZ4, producing an igSPT with edge states protected by a 1d Haldanechain (Z22-SPT) pumping symmetry. In the second case, the extension is to GUV = Z4 â Z2,and the edge states are not protected by any SPT-pumping acitonigSPT with SPT pumping symmetry â Let us first consider the anomalous GIR symmetrywith representative cocycle Ï4(g, h, k, l) = (â1)g1h2k2l2 , which can be written as Ï4 = b2 âª e2with b2(g, h) = (â1)g1h2 , which is a non-trivial element of H2(Z22, U(1)), corresponding tothe cocycle of a Haldane/AKLT spin chain. According to the general construction above, weexpect the resulting igSPT has edge states protected by a Haldane SPT pumping symmetry,in close relation with the time-reversal symmetric 2d igSPT example discussed in the maintext. This phase can also be understood via a decorated domain wall construction.Denote the IR symmetry group as ZA2 ÃZB2 where A,B label the two different Z2 factorsfor clarity. This igSPT phase can be thought of as a DDW state where ZA2 DWs are decoratedwith 1d ZB4 igSPTs.The symmetry can be onsited by introducing NÌ = Z2 rotors that are locked to nËiIRs(i1i2i3)g1i1i2g2i2i3for each triangle i where we remind that si = ±1 for up and down triangles respectively, andwriting onsite symmetry:Uos(n;gâ) =âi(â1)nËi(n+v(g2i3 ,g2))âa(â1)âI=1,2 gËIagI(4.59)which obeys the group multiplication of GUV = Z2 Ã Z4.The 1d edge of this 2d igSPT phase carry edge states protected by an SPT pumpingsymmetry U2(0,1)IRâ(ij)ââÎ£2(â1)g1i,jg2j which pumps a 1d Z22-SPT onto the edge.We can deduce a possible symmetry gapped QSET phase for this igSPT by the fraction-alization procedure outlined in the main text. This results in a Z2 gauge theory with a eparticle having a unit of Z4 charge, i.e. half of the elementary B charge in the IR symmetryfactor ZB2 , and an m particle that carries the projective representation of Z22 that is foundat the end 1d SPT pumped by U2(0,1). We refer to this state as eCBmP where CB denotes ahalf-B-charge, and P denotes a projective Z22 representation.The structure above can be readily generalized to products of two cyclic groups, GIR =ZN1 Ã ZN2 , with the relevant data listed in Table 4.1. These results follow readily frominterpreting these phases in a DDW construction where ZN1 DWs carry 1d ZN2N12 igSPTs.igSPT without SPT pumping symmetry â Now let us turn to the other representative co-cycle Ï4(g, h, k, l) = (â1)g1h1k1l2 = b2 âª e2 with b2(g, h) = (â1)g1h1 and e2(g, h) = (â1)g1h2 .94We see that the b2 cocycle involves only the first Z2 symmetry factor. Since there are no1d SPT phases protected by a single Z2 symmetry, when realized as an emergent anomaly,Ï4, will not have a lower-dimensional SPT pumping symmetry, in contrast to all the previ-ously constructed examples. Nevertheless, we will find that the igSPT constructed from thisanomaly via our construction has SPT edge states, which are protected by a long-range stringorder where the condensed string has half-charge of the first Z2 symmetry factor.Following the general procedure outlined in the main text, define the anomalous IR sym-metry action associated with this anomaly cocycle as:UAα =âi(â1)g1i1i2g1i2i3g1i3âa(â1)gË1a+gË2aUÎ² =âa(â1)gË1a . (4.60)where for future convenience we have chosen to generate ZA2 Ã ZB2 = {1, a} Ã {1, b} by {(α =a+ b), β = b}. This symmetry can be on-sited by defining an N = Z2 rotor on each plaquettedefining the on-site symmetry action:Uo.s.α =âi(â1)nËig1i3âa(â1)gË1a+gË2a ,Uo.s.β =âa(â1)gË1aUo.s.n = U2α =âi(â1)nËi , (4.61)which satisfy: U2α =âi(â1)nËi , U4α = 1, U2β = 1, U â βUαUβ = Uâ1Î± , corresponding to UVsymmetry: GUV = Zα4 â Zβ2 . Then, lock nËiIRg1i1i2g1i2i3 with Hamiltonian:Hâ = âââiÎ´nËi,g1i1i2g1i2i3. (4.62)Note that, in the IR, the N symmetry acts non-trivially only on the boundary of thesystem. Defining the system on an open spatial 2-manifold, Σ2:Un = U2Î±IRâiââÎ£2(â1)aiâ1,iai (4.63)where we denote g1i as ai. Restricting this symmetry to an open region R = 1 < i â¤ Lof the boundary, one obtains (Un)|R =â1<iâ¤L(â1)aiâ1,iai , which transforms under the GIR95symmetry as: Uβ : (Un)|R 7â (â1)a1+aL(Un)|R. Examining this expression, one finds thatwhile the bulk the (Un)|R string operator is invariant under GIR, its ends transform as if thereis a local â1/2-chargeâ of Za2. Namely, acting again with Uβ on the string end transformsthe end near site 1 from (â1)a1 7â (â1)a1+1, and the transformation of the bulk string addsanother factor of (â1)a1 , leaving a local phase of (â1) indicating the the string ends locallyhave U2β = (â1).A more precise version of this argument involves gauging the N symmetry, and the ac-tion of GIR on the ends of the restricted N symmetry action (Un)|R translate to the GIRtransformation properties of the N -flux, which are well-defined modulo redefining the trans-formation by a NÌ = Z2 magnetic gauge transformation, i.e. correspond to an element of aprojective symmetry action (PSG): H2(ZA2 , NÌ ) = Z2. The non-trivial element of the PSGcorresponds to the flux having a half-charge of a, which we expect to arise due to the aboveargument. We can verify this by confirming that the emergent anomaly cocycle decomposesas Ï4 = b2 âª e2 where b2(g, h) = (â1)g1h1 is the non-trivial element of the PSG for N -fluxes,and e2 = (â1)g1h2 specifies the group-extension that lifts the anomaly.As weâve shwon in general in Section 4.4.5 of the main text, the non-trivial transformationrules for the N flux ensure that the igSPT edge cannot be symmetric and gapped. Specifically,the edge symmetry action of Un and Uβ has the same anomaly as a gapped SPT phase withZβ2 âZn2 symmetry with cocycle Ï3(g, h, k) = (â1)g1h1k2 . However, as with previous examples,there is no gapped SPT with this edge anomaly when the N = Z2 is the normal subgroup ofa larger Zα4 symmetry, as follows from the general proof in Appendix 4.12.One can explicitly construct a symmetric gapless edge of this model as follows. Definespin Pauli operators: Ïzi = (â1)g1i , Ïxi = (â1)gË1i for each site of the edge. Then, examine theaction of Un on these operators: Un : Ïxi 7â âÏziâ1Ïxi Ïzi+1. I.e. the Un-symmetry forbids atrivial paramagnetic interaction for the edge spins, but does allow a critical edge Hamiltoniansuch as:Hedge = âKâi(Ïxi â Ïziâ1Ïxi Ïzi+1), (4.64)which by maps via a standard Jordan-Wigner transformation to a pair of decoupled criticalIsing models in the even and odd sub-lattice. In fact, this Hamiltonian has an enlargedemergent U(1) symmetry generated by NË =âj NËj with NËj = (â1)j 14(1â Ïzj)(1 + Ïzj+1)=(â1)jaj,j+1aj+1 measuring the number of ââ domain walls, and the UN symmetry correspondsto (â1)NË . As for the other cases described in the main text, this Luttinger liquid edgeHamiltonian needs to be symmetrized over GIR, which will potentially tangle it up with bulkmodes in a manner that is not precisely understood in general, but in select cases (e.g. the96bulk is a gapped QSET) this edge Luttinger liquid can remain decoupled from the bulk modes.4.9.2 GIR = U(1)Ã ZT2In this section, we denote: U(1) rotor states by αi â [0, 2Ï), the phases of U(1) symmetryrotations by Greek letters without site subscripts, and Z2 time-reversal rotor states by Ïi â{0, 1}, in order to clearly distinguish the role of each in formulas. Moreover, we denote theconjugate angular momentum to the rotor phase as âi, with [âi, αi] = 1.This group is physically relevant for spin systems with a conserved spin-component, saySz, which is odd under time-reversal (indicated by the direct product with T ). The 3dgroup-cohomology anomaly classification for GIR is Z32. One of these Z2 factors is the puretime-reversal anomaly discussed above. The remaining two Z2 factors correspond to mixedU(1), T anomalies, and can be thought of as being generated by two different root phases andtheir combinations.Following [120], we can label the anomalies by a representative anomalous SET Z2-gauge-theory (toric code topological order), labeled by specifying whether or not the Z2 gauge chargee and flux m have a half-integer charge (C-subscript) and/or transform as a Kramers doubletunder T (T subscript). The pure T -anomaly is labeled by eTmT , and the mixed anomaliescan then be labeled by different stacking combinations of the two root phases eCmC andeCmT .A 2d igSPT with eCmC anomaly â The eCmC anomaly arises at the surface of a 3dGIR-SPT discussed extensively in [117]. Its gapless states are closely related to field theoriesof deconfined quantum critical points [119]. The physics of this anomalous state can beunderstood through a DDW perspective in which time-reversal DWs are decorated with bosoninteger quantum Hall (bIQH) states with Hall conductance 2, i.e. where the time-reversalbreaking DW in the anomalous theory carries gapless edge modes of the root 2d U(1) SPT(analogous to how magnetic DWs in the surface of a 3d electron GIR-topological insulatorcarry electron IQH edge modes [98]). An equivalent statement is that the bulk responsetheory of the 3d boson SPT to a background U(1) electromagnetic field has a quantizedmagnetoelectric response:Z3d[A] = eiÎ¸8Ï2â«dAdA (4.65)Physically this response function is invariant under θ â Î¸+4Ï (twice the periodi for fermionsystems). Since T : dAdA â âdAdA, this periodicity leaves θ = 0, 2Ï as the distinct time-reversal invariant possibilities, with the latter corresponding to a non-trivial SPT.97An explicit cocycle To construct an explicit cocycle for this phase, let us warm-up witha lower-dimensional example. In 1d the axion response theory is Z1d = ei Î¸4Ïâ«dA, with time-reversal symmetric allowing for θ = 0, 2Ï corresponding to trivial and SPT phases respectively.The 1d SPT phase has edge states that transform under a projective representation of U(1)ÃZT2 defined by: U2ÏâÎ±UÎ± = (â1), corresponding to effectively having half-charge. A suitablecocycle which implements this projective representation is:Ï2[(Î±1, Ï1), (α2, Ï2)] = ei Î¸2v2Ï(Î±1,Î±2) (4.66)with θ = 2Ï, since UαUβ = Ï2UÎ±+Î². To check this expression, we note that time-reversalrestricts θ to be a multiple of 2Ï. Namely, since Ï does not appear in Ï2, T Uα = UαT inthis projective representation. Therefore, we must have T U2ÏâÎ±UÎ±T â = U2ÏâÎ±UÎ± = eiÎ¸/4Ï.The left hand side evaluates to T eiÎ¸/4ÏT â = eâiÎ¸/4Ï. Comparing these requires θ to be aninteger multiple of 2Ï. The cocycle is obviously trivial for θ an integer multiple of 4Ï, since itevaluates to 1 for all arguments. Finally, we note that, while time reversal does not directlyplay a role in this cocycle (there is no explicit Ï dependence), it is crucial to quantizing θ(as for the field theory). Namely, with just a U(1) symmetry alone, the cocycle is a pureboundary for θ /â 4ÏZ: ei Î¸4Ï v2Ï(Î±1,Î±2) = [dÏ1](Î±1, α2) with Ï1(Î±) = ei Î¸4Ï [α]. However, dÏ1 Ì¸= Ï2when the group is extended to include time-reversal. Namely, for GIR = U(1)ÃZT2 : denotinggi = (αi, Ïi) â U(1)Ã Z2, the coboundary condition is modified to:df(g1, g2) =[f(g2)]1â2Ï1 f(g1)f(g1g2)(4.67)and we no longer have dÏ1 = Ï2 for arguments with non-vanishing Ï1. Note, however, thatfor θ â 4ÏZ, Ï2 = 1 for all arguments and the cocycle is explicitly trivial.From this 1d picture, we can analogously construct a 3-cocycle for the 2d anomaly:Ï3[(Î±1, Ï1), . . . , (α4, Ï4)] = eiÎ¸4Ïv2Ï(Î±1,Î±2)v2Ï(Î±3,Î±4). (4.68)Again time reversal quantizes θ to be an integer multiple of 2Ï, and θ = 4Ï is trivial becauseÏ3 evaluates to 1 for all arguments in that case.igSPT from a U(1) rotor model Since time-reversal group elements do not explicitlyappear in the cocycle, we can actually construct the 2d igSPT purely from a U(1) rotorswith no Ising spins for the ZT2 subgroup. This cocycle defines the anomalous U(1) symmetry98action:UAα =âi(â1)v2Ï(Î±i1i2 ,αi2i3 )v2Ï(Î±k,Î±)âaeiÎ±âa (4.69)We can onsite this symmetry by introducing a N = Z2 extension with onsite symmetry:Uos(n,α) =âi(â1)nËi(n+v2Ï(Î±i3 ,Î±))âaeiÎ±âa , (4.70)and energetically locking nËiIRv2Ï(Î±i1i2 , αi2i3) on each triangle i with:Hâ = âââiÎ´nËi,v2Ï(Î±i1i2 ,αi2i3 ). (4.71)We can identify each element of the extended symmetry group (n â Z2, α â U(1) = R/2ÏZ) âGUV, as an element of U(1) with half the elementary charge, i.e. GUV = R/4ÏZ. To see thismap: (n, g) = 2Ïn+ g mod 4Ï, and notice that the onsite group operation is simply ordinaryaddition in R/4ÏZ: U(n,g)U(m,h) = U(n+m+v2Ï(g,h),g+h). Physically, this corresponds to liftingthe anomaly by adding local bosons with half of the charge of the UV degrees of freedom.Put another way, we can view this igSPT as arising in a boson-paired Mott insulator with agap to unpaired bosons. We note in passing that, just as for the igSPT with eTmT anomaly,we expect that the eCmC anomaly cannot be lifted by viewing the bosons as Cooper pairs offermionic electrons with half of the bosonic charge. This would instead be a natural candidatefor lifting the beyond group cohomology eCFmCF anomaly.Edge states The N group is generated by Uos(1,0). In the IR, this operation pumps a 1dU(1)Ã ZT2 SPT:Uos(1,0)IRâ(ijk)âÎ£2(â1)v2Ï(Î±ij ,Î±jk)=â(ijk)âÎ£2dU(1)(ei2[Î±])ijk=â(ij)ââÎ£2(â1)v2Ï(gij ,gj) (4.72)where dU(1) denotes the coboundary operator for the U(1) symmetry subgroup, and [α] denotesα mod 2Ï, and in the second line we have used that v2Ï(Î±, β) =12Ï ([α] + [Î²]â [α+ β]). Inlight of our discussion above, we see that this pumps the 1d U(1)ÃZT2 SPT with mixed U(1)99and T anomaly onto the boundary.QSET Order To obtain a QSET order here, we can follow the fractionalization proceduredescribed in main text. Here, we need to fractionalize the physical boson unit charge B intoa pair of 1/2-charge spinons B = b2 bound together in the UV by an emergent N = Z2 gaugefield. As we have seen in previous examples, in the deconfined phase of this gauge theory, theanomaly is âliftedâ, from the point of view of the fractional excitations, and the gauge flux (mparticles) carry the projective representation of U(1)ÃT that arise at the 1d G-SPT pumpedby U2ÏâÎ±UÎ±. In this case, the projective representation is characterized by a half charge ofthe U(1), and non-Kramers T 2, i.e. the resulting QSET order is eCmC , as expected from theoutset.2d igSPT with eCmT anomaly â The other nontrivial U(1)ÃZT2 anomaly can be realizedby cocycle: Ï4[(Î±1, Ï1), . . . (α4, Ï4)] = (â1)Ï1Ï2v2Ï(Î±3,Î±4), corresponding to anomalous U(1)-symmetry:UAα =âi(â1)Ïi1i2Ïi2i3v2Ï(Î±i3 ,Î±)âaeiÎ±âa , (4.73)and ordinary T = âi(â1)ÏËiK. This can be on-sited by introducing N = Z2 rotors andlocking nËiIRÏi1i2Ïi2i3 on each triangle i, and defining onsite symmetry action as in (4.70) âagain resulting in an extension of U(1) to include half-charges, i.e. extending from R/2ÏZ toR/4ÏZ.The edge states are protected by the SPT-pumping operation:Uos(1,0)IRâiâÎ£2(â1)Ïi1i2Ïi2i3=â(ij)ââÎ£2(â1)ÏijÏj , (4.74)which we recognize (from the discussion of the GUV = ZT4 igSPT in the main text) pumps aHaldane/AKLT chain onto the boundary. Note sharing similar edge states, the R/4ÏZ â ZT22d igSPT is distinct from 2d ZT4 one which has a pure time reversal anomaly. The ZT4 igSPTinvolves a pure time-reversal anomaly, and the SPT pumping is implemented by T 2. Here,instead we have a mixed U(1) and T anomaly, in which it is the U(1) symmetry that pumpsthe lower-d ZT2 SPT.Following the pattern of previous examples a possible QSET order that comes from thefractionalization procedure is eCmT , as expected from our label of the anomaly.1004.10 The SPT pumping action of NIn this appendix, we demonstrate that the N symmetry actions in (4.30) act only on the(Dâ 1)d edge of the igSPT and pump (Dâ 1)d SPTs onto the edge. Recall bD is a D cocyclewith coefficient NÌ , the cocycle condition reads:1 = âbD(g1, · · · , gD+1) = bD(g2, g3, · · · , gD+1) · bD(g1g2, g3, · · · , gD+1)â1 · · Â·Â· bD(g1, g2, · · · , gDgD+1)(â1)D · bD(g1, g2, · · · , gD)(â1)D+1 (4.75)Replacing gi â gâ1il gil+1 for I = 1, 2, · · · , D, and gD+1 â gâ1iD+1 , we obtain the relation:1 = bD(gâ1i2gi3 , gâ1i3gi4 , · · · , gâ1iD giD+1 , gâ1iD+1) · bD(gâ1i1 gi3 , gâ1i3 gi4 , · · · , gâ1iD giD+1 , gâ1iD+1)â1 · · Â·Â· bD(gâ1i1 gi2 , gâ1i2 gi3 , · · · , gâ1iDâ1giD , gâ1iD )(â1)D · bD(gâ1i1 gi2 , gâ1i2 gi3 , · · · , gâ1iDâ1giD , gâ1iD giD+1)(â1)D+1(4.76)The action (4.30) involves the last term, therefore we can use the above relation to replaceit by the other D terms, each of which depends only on a face of the D-complex, i.e. thefirst term doesnât depend on gi1 , the second doesnât depend on gi2 , etc. Replacing each termof (4.30) in this way, we obtain an expression in which each term corresponds to a face of aD-simplex. Each face in the bulk receives contributions from its two adjacent simplices andcancel, hence we are left with only an edge action:Uos(n,0) =â(i1,·Â·Â· ,iD)ââÎ£bsD(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iDâ1giD , gâ1iD )âZ n, (4.77)where the product is now only over (Dâ 1) simplices on the edge âÎ£ of the spatial lattice Σ.These N actions pump (Dâ 1)d GIR-SPTs onto the edge of the igSPT. Conjugating a trivialsymmetry action U trivialg |{ga}â© = |{gga}â© by the operator Uos(n,0) results in:Uos(n,0)Utrivialg Uosâ (n,0)|{ga}â© =â(i1,·Â·Â· ,iD)ââÎ£bsD(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iDâ1giD , gâ1iD gâ1)bsD(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iDâ1giD , gâ1iD )âZ n|{gga}â©.(4.78)This is precisely the symmetry action of a (D â 1)d GIR-SPT with anomaly specified by theD cocycle [bD âZ n] â HD(GIR, U(1)) [17]. To see this, consider restricting the action of(4.78) to an open sub-region A of the edge. Then, applying the cocycle condition ((4.75))with gl â gâ1il gil+1 , l = 1, · · · , D â 1 and gD â gâ1iD gâ1, gD+1 â g, the above expression can101be further reduced to an action with phase factor supported only on the (D â 2)d boundaryof A:(Uos(n,0)Utrivialg Uosâ (n,0))A|{ga}â© =â(i1,·Â·Â· ,iDâ1)ââAbsD(gâ1i1gi2 , gâ1i2gi3 , · · · , gâ1iDâ1gâ1, g)âZ n|{gga}â©.(4.79)Comparing with (4.19) we see the action is the standard anomalous symmetry action onthe edge of a (D â 1)d GIR-SPT with anomaly specified by the D-cocycle [bD âZ n] âHD(GIR, U(1))4.11 Gauging the extending symmetryTo gauge N explicitly in our lattice models, we introduce an |N |-dimensional Hilbert spaceon each link of the dual lattice with basis |aij â Nâ©. We define the gauge field operator bynË âZ Aij |aijâ© = nË âZ aij |aijâ© and the electric field operator Eij by Eij âZ n|aijâ© = |aij + nâ©.Note in general the operators Aij , Eij are not well defined since they are N , NÌ -valued whichare only well defined modulo |N |, only the U(1)-valued vertex operators nË âZ Aij , Eij âZ nare well-defined. Gauge transformations at i is generated byâ¦i = nËiâjâr(i)Eij , (4.80)here r(i) stands for sites adjacent to i on the dual lattice. The gauge invariant subspace isthen obtained by demanding Gaussâs law â¦i = 1 on every site.Now we consider excitations in the gauged model. In the deconfined phase of a GIR-symmetric N gauge theory, topological excitations include fluxes and charges, fluxes are N -valued objects and charges are NÌ -valued. A charge is a site on the dual lattice with nonzeronËi. Charge creation is generated by the gauge invariant string operatorWe = ni1 âAi1i2 âAi2i3 â · · · âAiLâ1iL â niL , (4.81)nË âZ W creates a charge +nË at site i1 and a charge ânË at site iL. An elementary fluxis an elementary(smallest) hexagon on the dual lattice with non-zero value of Wilson loopâ(ij)âh(a)Aij where h(a) stands for the elementary hexagon surrounding the site a of thesimplicial lattice. A pair of flux/anti-flux can be created by acting on the vacuum with astring of electric field operators. Denote the path of the string operator by C, then flux102Figure 4.8: Schematic of the effect of gauging the N symmetry in the igSPT latticemodels models (shown in 2d). Recall GIR-rotors live on the vertices of the simpliciallattice(black dots) while N rotors live on the vertices of the dual lattice(white dots).The N gauge fields then live on the links of the dual lattice(green links). An electricstring operator along the dashed blue path is a product of electric field operator overlinks that cross the path(thickened black links), it creates a pair of flux/anti-flux at itsend, shown as the blue hexagons. A gauge field string(Wilson line) is plotted in redand creates a pair of charge/anti-charge at its end, shown as the red dots.creation is generated byWm =â(pq)â¥CEpq. (4.82)Wm âZ n generates a flux n and an anti-flux ân on the two hexagons at the end of the pathC, the product is over all links of the dual lattice that intersect the path C. The two typesof excitation as well as their corresponding string operators are shown in Fig. 4.8.4.11.1 Review: symmetry fractionalization within the groupcohomology frameworkBefore we proceed to analyze the symmetry properties of excitations in the N -gauged igSPTmodels, we briefly review the group cohomology (partial) classification of symmetry enriched103topological orders (SETs). Since we are interested in studying the result of gauging thecentral extension N , we restrict our attention to Abelian topological orders enriched by ageneral (possibly non-Abelian) global symmetry G.Symmetry fractionalization in 2d â In 2d, all excitations are point-like. Local operatorscannot create topologically non-trivial anyon excitations. Instead, an isolated anyon, a, canonly be created by non-local operators, for example by a string operator that creates an aanyon and its anti-particle, a¯ and moves the a¯ far away. Therefore while the overall symmetryaction on the system must form an ordinary linear representation, Ug, with UgUh = Ughâg, h âG, the action of symmetry restricted to a region with an anyon a, Uag , may act projectively:UagUah = Ïa(g, h)Uagh where Ïa(g, h) â U(1). This is analogous to the projective action at theedge of a 1d G-SPT, with one key difference: the projective phases are discrete and constrainedby the fusion rules of the theory. For example, given an anyon a such that N copies of a fuseto the identity: aN = (aà aà . . . a)ï¸¸ ︷︷ ï¸¸NÃ= 1, it is possible to create N a-excitations with a localoperator. Since all local operators must transform under an ordinary linear representation ofG, this requires [Ïa(g, h)]N be trivial, i.e. is ZN valued rather than U(1) valued. Let us nowformalize this structure.For a general Abelian topological order, the set of anyon excitations A together with thefusion rules Ã : AÃA â A form an Abelian group. The porjective phases Ïa can be regardedas a map: G Ã G Ã A â U(1), (g1, g2, a) 7â Ïa(g1, g2). Consistency with the fusion rulesrequires ÏaÏb = Ïab which allows us to view Ï as a map: G Ã G â AÌ = Hom(A,U(1)), i.e.a 2-cochain with AË-coefficients. The associativity of the unitaries Uag requires the cochain Ïbe closed: Ï â Z2(G, AÌ).The symmetry restriction Uag is not uniquely defined, but rather can be modified byrelabeling by Uag â βagUag , which redefines Ï â dÎ²Ï, where dβ â B2(G, AÌ). Thus, equivalenceclasses of symmetry fractionalization are classified by [Ï] â H2(G, AÌ) = Z2(G, AÌ)/B2(G, AÌ).As an example, in an N gauged theory the fusion rules form the group N ÃNÌ , where thegroup N is generated by gauge fluxes and NÌ is generated by gauged charges. The symmetryfractionalization pattern is therefore classified by H2(G, NÌ Ã N ) = H2(G, NÌ )ÃH2(G,N ).Symmetry fractionalization in 3d â In 3d Abelian topological orders, which may be re-garded as N gauge theories, there are both point-like gauge-charge excitations, and loop-likegauge flux excitations. In an N gauge theory, gauge charges have fusion group NÌ and gaugefluxes have fusion group N . As in 2d, symmetry properties of point-like excitations are classi-fied by H2(G, NÌ ), where N is the fusion group of point-like excitations. For loop excitations,there are two cases for their symmetry properties. First, loops can carry an overall fractional104charge. Second, loops may carry symmetry-protected gapless modes 8. Without loss of gen-erality, we ignore the first possibility because one can always shrink the loop to a point toproduce a fractional-charged point-excitation, or equivalently, the fractional charge can be re-moved by relabeling excitations by binding a point excitation to the loop. For the second case,we follow closely the formalism of [51] for studying 1d symmetry-protected gapless modes atthe edge of a 2d SPT. Specifically, in-cohomology symmetry-obstructions to gapping flux-linesarise from non-trivial fusion symbols for G-domain walls along an flux line. Denote the fluxline anyon type as a, and label domain walls by the element g â G that relates the changein the symmetry breaking configuration across the domain wall. Then, the fusion symbols,Ïa(g, h, k) â U(1) denote overall phase difference resulting from fusing domain walls g, h, kwithin an a-flux line in different orders [(gh)k vs. g(hk)]. For the 1d edges of 2d G-SPTsequivalence classes of Ï are characterized by H3(G,U(1)) [51]. Here, an important differencefor the SET rather than the SPT, is that Ï depends on the type of flux-line, a and is requiredto be consistent with the fusion rules. Following the arguments for 2d above, this implies thatequivalence classes of Ï for G-SETs are characterized by H3(G, AÌ) where A is the dual of thefusion group for loop excitations.As an example, in a 3d G-enriched N -gauge theory, fusion rules of gauged charges formthe group NÌ while fusion rules of flux loops form the group N . Apply the general classificationwe see the symmetry fractionalization pattern is given by the group H2(G,N )ÃH3(G, NÌ ).4.11.2 Symmetry fractionalization of the N -gauged igSPTLet us first focus on 2d. To determine the symmetry fractionalization pattern one looks forrestriction of the symmetry action Ug to a subregion M that only contains a single charge orflux. The GIR action is given by (4.22)Ug|{nËi}, {ga}â© =âinËi âZ e2(gâ1i3 gâ1, g)|{nËi}, {ga}â© (4.83)Now consider creating a pair of charge nË and anti-charge ânË using the gauge field string (4.81)Now since only two sites i1, iL host non-zero nË, we can restrict the symmetry action to i1 bysettingU e,i1g |{nËi}, {ga}â© = nËâZ e2(gâ1(i1)3gâ1, g)|{nËi}, {ga}â©, (4.84)8It is also possible that loop excitations carry ungappable chiral modes, however these cases arebeyond the group cohomology formalism and will not be relevant for the in-cohomology igSPT exampleshere.105which satisfies U egUeh = nËâZ e2(g, h)U egh showing that chargs carry fractionalized transforma-tion given by [e2] â H2(GIR,N ).In order to see the symmetry fractionalization of a gauge flux, itâs useful to do a changeof basis that transforms the IR space into simple nËi paramagnet as we did for the 1d IsingigSPT in 4.3.4. Before gauging, this is accomplished by the unitary:V =âibsiD({giÎ±})âZ ni (4.85)where bD({giα}) stands for bD(gi1i2 , gi2i3 , . . . , giDâ1iD).V transforms the Hamiltonian toV â HâV = âââibâsD ({giÎ±})âZ niï£«ï£­âjÎ´nËj ,bsD({gjÎ±})ï£¶ï£¸ bsD({giÎ±})âZ ni = âââjÎ´nËj+bsD,bsD= âââjÎ´nËj ,0 (4.86)Such change of basis allows us to identify the symmetry fractionalization pattern for fluxes,to illustrate this, let us focus on the case of two special dimension.After gauging N we need to make sure the unitary V used for changing basis is gaugeinvariant. To achieve this we first rewrite V in a form that is convenient for gauging. Recallthe cocycle condition for b2,1 = db2(g, h, k) =b2(h, k)b2(g, hk)b2(gh, k)b2(g, h)(4.87)replace g â gâ1i1 gi2 , hâ gâ1i2 gi3 , k â gâ1i3 , we have:b2(gâ1i1gi2 , gâ1i2gi3) =â(ab)âââibsiab2 (gâ1a gb, gâ1b ), (4.88)here âi is the simplex i and ââi are the three edges of it. siab = ±1 depends on the directionof the edge ab being compatible with the orientation of the triangle i or not. With the help106of this identity we can writeV =âibsi2 ({giÎ±})âZ ni =âiâ(ab)âââibsi·siab2 (gâ1a gb, gâ1b )âZ ni=â(ab),(ij)â¥(ab)b2(gâ1a gb, gâ1b )âZ (ni â nj) (4.89)the last line is obtained by noticing that in the product in the second last expression, eachedge (ab) of the simplicial lattice receives contributions from its two adjacent triangles i, jwith (ij)â¥(ab), and the exponets sisiab in these two terms are always opposite of each other.Therefore the result is a product over directed links (ab), and for each fixed (ab) the corre-sponding (ij) link on the dual lattice is perpendicular to (ab) with i to the left and j to rightof (ab).With this new form we can make V gauge invariant by replacing ni â nj by the gaugeinvariant quantity ni â nj +Aji,VË â¡â(ab),(ij)â¥(ab)b2(gâ1a ab, gâ1b )âZ (ni â nj +Aji) (4.90)= Vâ(ab),(ij)â¥(ab)b2(gâ1a ab, gâ1b )âZ Aji (4.91)The Hamiltonian is transformed by VË to the trivial paramagnet one as before. Next wecalculate the transformed GIR-symmetry action. We do it by examining the action on the107Figure 4.9: The terms inâ(ab),(ij)â¥(ab)b2(gâ1a gâ1,g)b2(gâ1b gâ1,g) âZ aji that involve the vertex a arelinks (abk) shown in the figure, and the dual links (ij) that are perpendicular to (abk)form blue hexagon that encircles the site a. Notice when the arrow on a link abk ispointing away from a, the term is b2(gâ1a gâ1, g)[aji], while when the arrow is pointingtowards a, the term is bâ12 (gâ1a gâ1, g)âZ aji = b2(gâ1a gâ1, g)âZ aij, either way the gaugefield aij that appears is pointing in the counter-clock direction, therefore the productof all 6 terms that involve the vertex a is simply b2(gâ1a gâ1, g)âZ (flux of h(a)), wherewe h(a) stands for the hexagon on the dual lattice that encircles site a.basis |{nËi}, {ga}, aijâ©,VË â UgVË |{nËi}, {ga}, {aji}â© = VË â Ugâ(ab),ijâ¥abb2(gâ1a gb, gâ1b )âZ aij |{nËi + b2({giα})}, {ga}, {aij}â©(4.92)= VË â âi(nËi + b2({giÎ±}))âZ e2(gâ1i3 gâ1, g)â(ab),ijâ¥abb2(gâ1a gb, gâ1b )âZ aji|{nËi + b2({giα})}, {gga}, {aij}â©(4.93)=(âi(nËi + b2({giÎ±}))âZ e2(gâ1i3 gâ1, g))ï£«ï£­ â(ab),ijâ¥abb2(gâ1a gb, gâ1b )b2(gâ1a gb, gâ1b gâ1)âZ ajiï£¶ï£¸ |{nËi}, {gga}, {aij}â©(4.94)IRï£«ï£­ â(ab),ijâ¥abb2(gâ1a gb, gâ1b )b2(gâ1a gb, gâ1b gâ1)âZ ajiï£¶ï£¸(âib2({giÎ±})âZ e2(gâ1i3 gâ1, g))|{gga}, {aij}â©(4.95)=â(ab),ijâ¥abb2(gâ1a gb, gâ1b )b2(gâ1a gb, gâ1b gâ1)âZ ajiUAg |{ga}, {aij}â© (4.96)108where UAg is the anomalous symmetry action defined in (4.19). To get to the second lastline notice in the VË transformed frame the IR space corresponds to nËi = 0. Now to furthersimplify the expression, we apply the cocycle condition db2(g, h, k) = 1 with g = gâ1a gb, h =gâ1b gâ1, k = g and obtain1 =b2(gâ1b gâ1, g)b2(gâ1a gb, gâ1b )b2(gâ1a gâ1, g)b2(gâ1a gb, gâ1b gâ1)(4.97)apply it to (4.96) we obtainVË â UgVË IRâ(ab),(ij)â¥(ab)b2(gâ1a gâ1, g)b2(gâ1b gâ1, g)âZ ajiUAg (4.98)Now we can change the product from being over links to being over sites, for each fixed site athere is one term b2(gâ1a gâ1, g)[aji] from each edge ij of the dual lattce the encircles the thesite a, as shown in Fig. 4.9. Therefore we haveUËg = VËâ UgVË IRâab2(gâ1a gâ1, g)âZï£®ï£° â(ij)âh(a)aijï£¹ï£»UAg =âab2(gâ1a gâ1, g)âZ (flux of h(a))UAg(4.99)where h(a) stands for the hexagon in the dual lattice that surrounds the site a.Now it is clear how to localize the symmetry UËg to a single flux. If we start with a zeroflux and GIR-symmetric vacuum and create a pair of flux +n and ân at site a and b using astring of electric field operators, then we can localize the symmetry action UËg to a region Mthat contains only the +n flux by setting UËmg = b2(gâ1a gâ1, g) âZ nUA,Mg , here UA,Mg standsfor the restriction of the anomalous action to the region M . Now itâs clear the restrictedsymmetry action UËmg forms a projective representation UËmg UËmh = b2(g, h)âZ nUËmgh, confirmingthat the symmetry properties of fluxes are determined by the cocycle b2.An analogous derivation to that above shows that in 3d the local symmetry action ofGIR on N -flux lines is governed by the cocycle b3 â H3(G,N ) appearing in the emergent-anomaly cocycle decomposition Ï5 = b3 âª e2. Physically, we can interpret b3 as the N -valuedfusion-symbol for fusing the 0d endpoints of 1d GIR domain wall defects within an N -fluxline [51].1094.12 Proof of Intrinsicness of igSPT with group co-homology anomaliesIn this appendix we show that the gapless SPTs in our construction are in fact intrinsic,i.e. that their edge states can not be realized as edge states of gapped SPTs with the samesymmetry, GUV. This can be straightforwardly verified for the examples in the main text:for GUV = Z4 in 1d and 3d, and GUV = ZT4 in 2d where there are no non-trivial gappedSPTs. In other cases, such as GUV = Z2 Ã Z4, GIR = Z22 in 2d, there is not only a single 2digSPT but also various 2d gapped SPTs with anomalies in Z2, Z4 and one with mixed Z2,Z4anomaly. However one can check explicitly that none of these gapped SPTs have the sameSPT pumping property of the igSPT, which has the property that acting twice with the Z4generator in the igSPT pumps a 1d GIR = Z22 SPT onto the boundary. In a gapped system,consistency would then require that each GIR-DW carry âhalfâ of the minimal allowed 1dGIR = Z22 SPT. However, this is not possible in a gapped system.In this appendix, we give a general argument that such obstructions to realizing theSPT pumping behavior in a gapped system are intrinsic, i.e that an igSPT with emergentgroup cohomology anomaly is intrinsically gapless. We proceed by assuming the converse andderiving a contradiction. Let Σ be the Dd bulk of a igSPT, âÎ£ be the (D â 1)d edge. TheigSPT edge is characterized by two key features: i) an SPT pumping symmetry: Un, n â Npumps a lower dimensional GIR SPT, and ii) an N -gap: there are no gapless N DOF inthe bulk or at the edge. Suppose now that the same two characteristics were possible witha gapped GUV-symmetric bulk. Then there would be an action of GUV on âÎ£, denoted asVγ , γ â GUV, that contains the pumping actions: Vn = Un, ân â N of the form:Vγ |{gi}â© = â¦(Î³, {gi})|{p(GUV)gi}â© (4.100)where p is the projection GUVpââ G and â¦(Î³, {gi}) is a general phase factor that depends onthe spin configuration as well as the group element γ. Crucially, if the bulk is gapped, theglobal action of VÎ³âGUV on the entire boundary must form an ordinary linear representationof GUV. We will now show that this fail to be the case for systems with SPT pumpingsymmetries. For n â N , Vn is a {gi}-dependent phase â¦(n, {gi}) since p(n) = 1G. Fornontrivial extensions GUV Ì¸= GÃN and there exist γ1, γ2 /â N , but n â¡ γ1γ2 â N . Since Nis central, n = Î³â11 nγ1 = γ2γ1. Hence for an ordinary linear representation we should have:Vγ1Vγ2 = Vγ1γ2 = Vγ2γ1 = Vγ2Vγ1 . Instead, considering the action of these operations on âÎ£,110one finds:Vγ1Vγ2 |{gi}â© = â¦(Î³1, {p(Î³2)gi})â¦(Î³2, {gi})|{gi}â©, (4.101)Vγ2Vγ1 |{gi}â© = â¦(Î³2, {p(Î³1)gi})â¦(Î³1, {gi})|{gi}â©, (4.102)Vn=γ1·γ2 |{gi}â© = Un({gi})|{gi}â©. (4.103)Denote by a GIR-rotor configuration with a domain wall between g and h configurations byDW(g, h) (we will not need to specify the detailed form of the DW configuration just thatwe have a region A such that giâA = g and giâAc = h where Ac denotes the complementof A). Let {gi} = DW(1G, p(γ1)), which is a nontrival domain wall since γ1 /â N . Noticep(γ2) Â·DW(1G, p(γ1)) = DW(p(γ2), 1G), thus from (4.101)=(4.103) we haveâ¦(Î³1,DW(p(Î³2), 1G))â¦(Î³2,DW(1G, p(γ1)))= Un(DW(1G, p(γ1))) (4.104)Next set {gi} = DW(p(γ2), 1G), then p(γ1) · DW(p(γ2), 1G) = DW(1G, p(γ1)), therefore from(4.102)=(4.103), we haveâ¦(Î³2,DW(1G, p(Î³1)))â¦(Î³1,DW(p(Î³2), 1G))= Un(DW(p(γ2), 1G)) (4.105)Notice (4.104) and (4.105) have the same LHS,while the RHS are in general different, becausethe phase factor Un takes the form:Un({gi}) =â(i1,·Â·Â· ,iD)bsD(gâ1i1gi2 , · · · , gâ1iDâ1giD , gâ1iD )âZ n, (4.106)which is not a function of domain wall classes. That is to say, although DW(p(γ2), 1G) andDW(1G, p(γ1)) are equivalent domain walls in the sense that the DW(1G, p(γ1)) = p(γ1) ·DW(p(γ2), 1G), the last argument of bsD in Un is gâ1iD, which is not a function of domainwall classes. Because of this, Un in (4.106), gives different phases depending on the order ofapplication of γ1 and γ2. This raises a contradiction with the assumption that the bulk isgapped since the edge-restriction of symmetry on the edge of a gapped bulk would necessarilyhave commuting action of γ1,2. As a simple example, consider 2d igSPT with GUV = Z2ÃZ4and GIR = Z22. The sole non-trivial SPT pumping operation is U = (â1)(g1i+1âg1i )g2i+1 , and theg2i+1 factor is what makes U not a function of domain wall classes. Apply the constructionto this example by setting γ1 = γ2 = (1, 1) â Z2 Ã Z4, which satisfy γ1 + γ2 = (0, 2) â N .111Then the phase factor W receives a single contribution in the configuration DW((0, 0), (1, 1)):(â1)1·1 = â1, while it is one in the configuration DW((1, 1), (0, 0)).4.13 Relating anomaly cancellation and SPT pump-ing pictures of igSPT edge statesWhile we have focused on the SPT pumping as the origin of igSPT edge states, TVV [106]instead deduce the existence of edge states from an anomaly cancellation argument. Here, werecount this argument and relate these two mechanisms.First recall the anomaly cancellation argument. Consider gauging the GUV symmetry bycoupling the igSPT on a (D+1)d spacetime manifold XD+1 to a background GUV-gauge fieldAGUV = (AN , AGIR), where AN is an N gauge field that acts only on the gapped sector andAGIR a GIR-gauge field that acts on the IR DOF. Then, consider integrating out the gappedN -DOF, which results in the partition function Z[AGUV ] = ZIR[AGIR ]eiâ«XD+2Ï(AGUV )whereÏD+2 is the anomaly cocycle, and XD+2 is a Wess-Zumino-Witten (WZW) type extensionof spacetime whose boundary is the physical spacetime: XD+1 = âXD+2, and ZIR is thepartition function for the gapless IR DOF, which couple only to the IR part of the gaugefield. When considered as a function of g â GIR, ÏD+2 is a non-trivial cocycle, and thereis no way to define the above action on a spatial manifold with a boundary, âXD+1 Ì¸= â .Rather, in this case, the anomalous GIR-action can only arise as the boundary of a gappedGIR-SPT on XD+2. However, the group extension lifts the anomaly such that interpreted asa GUV-cocycles one writes ÏD+2 = âÎ±D+1 [104], and with the extended DOF it is possible todefine the action on an open spacetime as: Z = ZIR[AG]eiâ«XD+1Î±(AGUV ). As argued in [106],α cannot be invariant under N gauge transformations: AN â AN + dÏN , otherwise onewould have Ï = dα as GIR-cocycles, and there would be no emergent GIR-anomaly. Rather,N gauge transforms change α by an exact but nonzero form: Î±â α+ dλ(AGUV , ÏN ), whichresults in a transformation of the partition function: Z â eiâ«âXDÎ»Z. From this, one deducesthat additional gapless edge states are required to cure this lack of gauge invariance.We now show that the SPT pumping picture reproduces the anomaly cancellation pictureas well as clarifying the physical properties of the gapless edge states that produce the cancel-lation. Consider instead the the decomposition ÏD+2 = bD âª e2 given by the anomaly-liftingextension. Due to the non-trivial group extension structure, the merger of two AGIR fluxes withflux values gi1, g2 with 0 Ì¸= e2(g1, g2) â N , acts as a source for AN : âAN = e2(AGIR) [104].This gives a solution to the anomaly lifting equation ÏD+2 = âÎ±D+1 with αD+1 = bD âªAN 9,9Note that from Ï â Hâ(GIR,M) = Hâ(BGIR,M) and gauge field AGIR : X â BGIR, we can112from which we can deduce explicitly the lack of gauge invariance of the partition function Zon open manifolds and see the connection to our SPT pumping picture. Specifically: consideran N gauge transformation AN â AN +dÏN where ÏN (x â âX) = n â N is constant on theboundary. This changes the partition function by: Z â eiâ«âXDbD(AGIR )âZnZ. The prefactorof this change is precisely the partition function for the (Dâ 1)d SPT pumped by the n â Nsymmetry. From this, we see that the SPT-pumping picture reproduces the anomaly cancel-lation argument, but additionally reveals a further substructure: α = bâªAN for the solutionto the the anomaly lifting equation âÎ± = Ï, and gives a simple physical interpretation interms of SPT pumping for the relevant class of igSPTs.4.14 Perspective from Gauge-Response Theory fromGapped SPTsThe emergent GIR-symmetry anomaly underlying the topological feature of a Dd igSPT arisesfrom an anomaly-free UV symmetry extension GUV. A more conventional context for thisanomaly is to arise at the Dd surface of a (D+1)d gapped GIR-SPT with no extended DOF.A useful way to characterize gapped GIR-SPTs is to consider their response to a backgroundGIR-gauge field, which will be described by a TQFT. In this section, we review some TQFTsfor various GIR-SPTs relevant to the igSPT examples we study, recall the construction ofRef. [124] which enables one to deduce explicit cocycles from these TQFTs. We then arguethat the structure of these TQFTs and corresponding cocycles implies the decompositionform ÏD+2 = bD âª e2 needed for our igSPT construction, and show how the lower-d SPT edgepumping symmetry that protects the igSPT edge states arises from a dimensional reductionprocedure from the TQFT describing the higher-d GIR-SPT with the same surface anomaly.In this section we consider only Abelian, unitary symmetries.Topological terms in discrete gauge theoriesWe follow the notation of Ref. [124], where we define a ZN gauge field A such that itsholonomies take valuesâ®A â 2ÏN and its gauge transformations satisfyâ®Î´A = 0 whereâ®isshort-hand for a discrete lattice sum around a closed loop. We abbreviateN12 = GCD(N1, N2)where GCD denotes the greatest common divisor.define an actionâ«XÏ[A] by the pull back Ï[A](x) â¡ Ï (A(x)) where we use the same symbol for bothinterpretations of Ï.113We define the response field theory as the partition function for the system in the presenceof a background (nondynamical) gauge field A. This is well defined as long as the systemhas a unique gapped ground-state, which requires that we consider an SPT with periodicboundary conditions. This gapped requirement may fail in the presence of certain gauge fluxconfigurations, which must then be excluded.For example, in 1d the GIR = ZN1 Ã ZN2 SPTs, which can be interpreted as ZN1 domainwalls (DWs) being decorated by p â N12 charges of the ZN2 symmetry. As a result, theconfigurations with dA1 Ì¸= 0, i.e. with spacetime instantons where the A1 flux changes, inducea change in the ZN2 charge of the system causing a zero of the partition function, treâÎ²H[A]since the trace forces the starting and ending states to have the same charge. Hence, weonly expect to get a well-defined partition function if A1,2 are flat, i.e. dA1,2 = 0. Thecorresponding gauge response theory is [124]:Z[AI=1,2] = eipN1N2N12â«M2A1A2(4.107)where AI=1,2 is a ZNI gauge field,M2 is a (closed) 1+1d spacetime, and we have suppressed â§products between each term for simplicity. This term is gauge invariant under AI â AI+dgIwith I = 1, 2 only if we demand that the gauge fields are flat: dA1,2 = 0. The restrictionson gauge configurations depend on dimensionality and topological term in a way detailed inRef. [124].Deducing Cocycles from Response TheoryIn general, for finitely generated Abelian symmetry groups, GIR =âQI=1 ZNI the topologicalterms will have Lagriangian proportional to products of k factors of AI distinct I and â(Dâk)/2â factors of dAJ . For example in 2d, one could have either mixed Chern-Simons termstype terms 2d, L2d â¼ AIdAJ , or discrete terms such as A1A2A3. In 3d one can have either L =A1A2dA3 or A1A2A3A4. In 4d (which is relevant to 3d IGSPTs) there are various possibilitiessuch as L â¼ AdAdA,AAAdA,AAAAA with various combinations of flavor indices.As shown in Ref. [124], from these topological terms, one can readily read off an explicitcocycle, ÏD+2 for the corresponding SPT, by taking eiL[A] and replacing each factor of AI â2ÏNIgI and each factor of dAI by 2ÏvNI (g, h) where we remind that the the vorticity functionis defined by vN (a, b) = 1 if a + b > N (with addition taken in Z i.e. not modulo N) and 0114otherwise. For example for GIR = ZN1 Ã ZN2 in 3d,Z[A] = eiN1N2(2Ï)2N12â«A1A2dA2âÏ4(g, h, k, l) = e2ÏiN12g1h2vN2 (k2,l2). (4.108)We note that, the decomposition ÏD+2 = bD âª e2 follows naturally from the exterior productstructure of the topological terms in the response theory in this case.igSPT edge-SPT pumping symmetry from dimensional reduc-tionWe also note that there seems to be a rather general prescription to deduce the (D â 1)dSPT pumping symmetry of a Dd igSPT with emergent anomalous GIR symmetry, from theresponse theory for the (D + 1)d gapped SPT whose surface has the same anomaly.We illustrate this first with a D = 1 GIR = ZN example, with Z[A] = ei2Ïâ«M2AdA, withcorresponding cocycle Ï3(g, h, k) = e2ÏiNgvN (h,k). We say that the 0d SPT pumping symme-try arose from just the first factor e2Ïig/N , which we could obtain by composing symmetrytransformations (N â j) and j which set the vN argument to 1 in the cocycle term.Acting with a symmetry j is like considering tr[UjeâÎ²H] which is equivalent to a gaugeconfiguration with a fluxâ®Ï A =2ÏjN in the time cycle of the partition function. Composingsymmetries that add up to N is then like performing a large gauge transformation aroundthe time-cycle. In the igSPT context, we would like to explore the effect of applying thistransformation only to the anomalous 1d surface of the 2d SPT (in fact, we would like to gofurther and ask about the effect on the 0d edges of the 1d igSPT which do not exist whenrealized as the surface of a 2d SPT).Considering the 2d SPT response theory on a three-torus with coordinates (Ï, x, y), wecan emulate the effect of the SPT-pumping transformation restricted to the y = 0 line at timeÏ = 0 by taking A = A1d(x) + AF with dAF = â2ÏdÏdyÎ´(Ï)Î´(y), which inserts a 2Ï flux inthe Ï -cycle of the torus only at y = 0. Inserting this form into the response theory results in a1d partition function Z1d[A1d] = eiâ«Ax1ddx, which we can identify as the response field theoryfor the (0 + 1)d GIR-SPT that gets pumped onto the igSPT boundary by the N symmetry.This dimensional reduction procedure works for all of the examples listed in Table 4.1.Selected additional examples are summarized in Table. 4.2 below.115igSPTResponseTheory ZD[A]Reduction AnsaztA = AD + AFReduced ResponseTheory ZDâ2[A]1d ZN ei2Ïâ«M2AdAdAF = 2ÏÎ´(Ï)Î´(y)dÏdy eiâ«Ax1dx2d ZN1 Ã ZN2 eiN1N2(2Ï)2N12â«A1A2dA2 dAF = 2ÏÎ´(Ï)Î´(x3)dÏdx3 eiN1N22ÏN12â«M2A1A23d ZN ei(2Ï)2â«M5AdAdAdAF = 2ÏÎ´(Ï)Î´(x4)dÏdx4 ei2Ïâ«AdATable 4.2:116Chapter FiveClassification of 1+1D gSPT viaSymTFT5.1 Background: gapless SPTs, and SymTFT5.1.1 Gapless SPTs: GeneralitiesGapless SPTs are examples of symmetry-enriched quantum orders, in which one cannot locallydistinguish between a âtrivialâ critical point and a symmetry-enriched one from local bulkmeasurements. Rather, the differences between different gapless SPTs are only evident in non-local bulk probes, or local edge probes. One can make this notion more specific by analogyto gapped SPTs. Different gapped G-SPT ground-states can be connected by a finite-depthlocal unitary (FDLU), U , that is overall symmetric ([U, g] = 0 âg â G) but which is notsymmetrically generated (U Ì¸= eâiH for any local G-symmetric H). This definition cannot bedirectly ported to the gapless setting as even different instances of a gapless state with the sameuniversal scaling properties, e.g. two instances of a conformal field theory (CFT) perturbed bydifferent irrelevant operators, cannot be connected by an FDLU. However, we can generalizethe notion of symmetric FDLU-(in)equivalence by defining i) that two ground-states are inthe same universality class if they flow to the same RG fixed point after applying an overallsymmetric FDLU, ii) ground-states with the same universality class are distinct gapless SPTclasses if they cannot be connected in this way by any symmetrically-generated FDLU. Animmediate corollary of this definition is that local scaling operators in distinct gapless SPTs ofthe same type of criticality have the same symmetry properties as conjugating a local operatorwith definite symmetry quantum number with an overall-symmetric U preserves its symmetry117quantum number. However, the symmetry properties of non-local scaling operators, such asa disorder operator that inserts a domain wall, may change under such an overall symmetricFDLU, leading to distinct classes of gapless SPTs. These notions can be made more precisefor 1+1D conformal field theories (CFTs) [116], for which the data specifying a universalityclass is well understood and characterized by the spectrum and fusion rules for primary scalingoperators.Example of a 1+1D igSPT: â A simple igSPT with full symmetry Î = Z4 was constructedin studied in [74, 130]. We will use this example extensively throughout this paper to illustratethe formal results. A lattice Hamiltonian realizing the Z4-igSPT chain is given by [74]HZ4âigSPT = H0 +HâH0 = âgâi(Ï ziâ1/2Ïxi Ïzi+1/2 + Ïyiâ1/2Ïxi Ïyi+1/2)Hâ = ââiÏziâ1Ïxiâ1/2Ïzi (5.1)with two species of spins per unit cell represented by Pauli operators ÏÎ±, ÏÎ±. The system hasa Z4 symmetry generated byUs =âiÏxi eiÏ4(1âÏxi+1/2) (5.2)Below the energy scale, â (taken to be very large), Ïxiâ1/2 is locked to Ïziâ1Ïzi , and thesymmetry effectively reduces to an anomalous G = Z2 symmetry:Us ââiÏxi eiÏ4(1âÏzi Ïzi+1) (5.3)Viewed as a Z2 symmetry, this symmetry action is anomalous, and has the same anomaly asthe edge of a 2+1D SPT phase (the Levin-Gu phase [72]). This emergent anomaly preventsthe system from realizing a symmetry-preserving short-range entangled state. Instead, theonly options are to spontaneously break the symmetry, or form a gapless state. The abovechoice of H0 realizes the latter option, reducing at low energies to the critical spin-chain:H0 â ââiÏxi â Ïziâ1Ïxi Ïzi+1. (5.4)Further, in an open chain, one finds that acting twice with the low energy symmetryoperator âpumpsâ a G = Z2 charge of the low energy symmetry group onto each end of the118boundary: U2s = Ïz1ÏzL, where 1, L are the first and last sites in the chain respectively [74, 130].This symmetry-charge pumping, locally anticommutes with the Us symmetry, protecting atwo-fold ground-state degeneracy, split only exponentially-weakly in system size, L by finitesize effects (in contrast to polynomial scaling with 1/L for gap to bulk excitations). Awayfrom this idealized fixed point, this ground-state degeneracy is robust, so long as symmetryremains intact, and the gap to the Ï z particles remains open.Formally, the structure of the emergent anomaly is captured by the short-exact sequenceof groups:1â A jââ Î pââ Gâ 1 (5.5)which defines a group extension of G by A to Î. Microscopically (in the UV), the system hasfull symmetry Î, but excitations carrying charges of the normal subgroup A â´ Î are gappedout by Hâ. Therefore, below the energy scale of this gap, the symmetry is effectively reducedto the quotient G = Î/A. We call A,G the gapped symmetry and the gapless symmetryrespectively. The gapless symmetry G can be anomalous and the anomaly is representedby a cocycle Ï â Hd+2(G,U(1)). In the above example, A,G = Z2, and Î = Z4, andÏ(a, b, c) = (â1)abc is the anomaly of the edge of a 2+1D Levin-Gu phase [72]. Since thefull symmetry Î is not anomalous, the emergent anomaly must be âliftedâ by the gappedsector. Formally, this implies that ,upon pull-back by the projection p, the cocycle becomesa coboundary: pâ(Ï) = dα. Moreover, the system can be put in a Î-symmetric gappedphase. However, the anomaly in the IR has the consequence that one must first close the gapto the A-degrees of freedom before tuning the system to a gapped symmetric state. Thusthe nontrivial topological properties of the system are protected by the A-gap, despite thepresence of gapless G-charged excitations.5.1.2 The Symmetry topological field theory (SymTFT) con-ceptSymTFT is a method of studying a given symmetry via a higher dimensional topologicalorder. SymTFT has been proven powerful in classifying SPTs protected by generalized sym-metry, identifying anomalies of non-invertible symmetries, revealing dualities between differ-ent phases, and beyond [86, 38, 39, 3, 50, 37, 49, 139, 5, 77, 2, 78, 20]. We here review theconstruction of SymTFT for ordinary internal symmetries (invertible, 0-form symmetries).For a system with a symmetry group G, the SymTFT is a G-gauge theory in one higherdimension. This correspondence can be motivated in two different ways: First, there is a119relation between global gauge transformations, and a boundary global symmetry, as we nowreview. Though the SymTFT correspondence has only been established for discrete groupsG, consider for a moment the more familiar U(1) gauge theory on a space Σ, with electricfield E, and charge density Ï. By Gaussâ law, â · E = Ï, a global gauge transformation,reduces to a U(1) symmetry transformation that acts only on the boundary, âÎ£ of the system:Uθ = eiÎ¸â«Î£Ï = eiÎ¸â« â·E = ei â«âÎ£ E . This boundary symmetry of a gauge theory is also calledasymptotic symmetry [44, 92, 102, 103] and plays a crucial role in the recently proposedâHiggs=SPTâ correspondence [105, 112]. This relation between bulk gauge symmetry andedge global symmetry also hold for discrete symmetries. Thus, a G-symmetric system canalways be embedded into the boundary of a G gauge theory in one higher dimension. We notethat, if the G symmetry action is anomalous, the corresponding bulk gauge theory would bea twisted gauge theory, also known as the Dijkgraaf-Witten theory with a nontrivial cocycle.A second complementary perspective, is that one can recover a G-symmetric system ind + 1D from a dimensional reduction procedure by considering a thin slab of (d + 1) + 1D(twisted) G gauge theory, with appropriate open boundary conditions in the thin direction,and periodic boundary conditions in the other directions. We illustrate this dimensionalreduction for the simplest case of a 2+1D Z2 gauge theory (toric code topological order)with anyon types e,m, f = e Ã m. Here, consider the top boundary of the slab to be afixed reference gapped state. Gapped boundaries of topological orders correspond to anyoncondensations that fully confine the topological order [54, 53]. Here, we consider the referencetop boundary state as an e-condensed boundary. Define a global Z2 symmetry operation asnucleating a pair of m particles, and dragging them around the periodic cycle. Gappedstates of this quasi-1d system correspond to different gapped boundaries for the bottomof the slab. The m condensed boundary is symmetric under the above-defined symmetry,since it can absorb the m line that generates the symmetry. By contrast, the e condensedbottom boundary corresponds to a spontaneous symmetry broken state, which has a dualsymmetry under pulling an e particle from the top-boundary condensate and moving it tothe bottom boundary condensate. This dual symmetry anti-commutes with the original Z2symmetry, due to the mutual semionic statistics of the e and m particles, leading to a two-fold degenerate ground-space. This structure of the global Z2 m-symmetry and dual Z2e-symmetry, matches that of the lattice spin-1/2 Ising model with global Z2 symmetry gm =âi Ïxi respected by symmetric (paramagnetic) short range entangled phases, and dual domainwall conservation symmetry: ge =âi Ïzi Ïzi+1 respected in symmetry broken phases: these twosymmetries anticommute when restricted to overlapping finite intervals, and were referred toas âcategorical symmetriesâ or symmetry topological order (SymTO) in [55, 16, 15, 46, 14].The Ising critical point separating the two phases also corresponds to the phase transition120between the e and m condensed bottom boundary [77, 15].In 1+1D systems, this structure generalizes to any finite internal unitary symmetries. Inthe broken phase of such a symmetry, a dual symmetry appears that encodes the conser-vation of domain wall excitations. The original symmetry and the dual symmetry togetherform the categorical symmetry of the system and can be uniformly described by a 2+1Dtopological order. As we have seen in the toric code example, the symmetry and the dualsymmetry correspond to anyon string operators of gauge flux and gauge charge respectively.The statistics of anyons in the 2+1D topological order then encodes the commutation relationsbetween the symmetry and the dual symmetry. The thin-slab construction is a way to makethe categorical symmetry manifest: if the 1+1D system has a symmetry G with anomalyÏ â H3[G,U(1)], then one can always put the system on the boundary of the twisted gaugetheory (a.k.a twisted quantum double) DÏ(G). The original/dual symmetry is then realizedby dragging fluxes/charges around the periodic direction. To specify the symmetry and thedual symmetry, one fixes the top boundary to be one that condenses all the charges. Differentphases of the 1+1D system are therefore realized by choosing different boundary conditionson the bottom boundary.This thin-slab construction, which can be viewed as a concrete realization of the SymTFTidea, translates the question of classifying phases of matter with symmetry in 1+1D into thequestion of classifying boundaries of 2+1D topological orders. The latter question is well-understood: Boundaries of 2+1D topological orders are completely determined by anyonsthat condense at the boundary[54, 60, 65, 67, 53]. For abelian topological orders, an anyoncondensation is specified by a group of self and mutual bosons A. After condensing A theanyons that have non-trivial braiding with A become confined and the topological order isreduced to one whose excitations only include the deconfined anyons that braid trivially withA. The theory of anyon condensation for generic topological orders is more involved andwe defer the discussion to section 5.4. In general we can devide anyon condensations intotwo families: If the condensation completely confines the theory and no anyons other thanvacuum stay deconfined, then we call the condensation fully-confining, otherwise we call thecondensation partially-confining. In the boundary/anyon condensation dictionary of 2+1Dtopological orders [60, 65, 67], gapped boundaries corresond to full-confining condensationsand gapless boundaries corresond to partially-confining condensations. It is then clear that inthe thin-slab construction, gapped/gapless phases of the 1+1D system is obtained by choosingfully/partially confining condensations on the bottom boundary. Moreover, if on the bottomboundary any of the gauge charges is condensed, then the effective 1+1D system is invariantunder a dual symmetry implying the original symmetry is at least partially broken. Thereforein order to describe a fully symmetric 1+1D system the bottom boundary must condense no121Figure 5.1: The thin slab hosts toric code in its bulk. On the top reference boundarywe fix the e-condensed boundary condition. The horizontal m-line becomes a globalsymmetry for the effective 1+1D system which we identify as the generator of a Z2symmetry. In the left figure we consider condensing m on the physical boundary. Inthis case the horizontal m-string can be absorbed by the physical boundary and actsas identify. The effective 1+1D system is therefore gapped and symmetric. In the rightfigure we consider condensing e on the physical boundary. In this case the verticale-string acts as the order paramater of the symmetry and has non-zero expectationvalue, signaling SSB. Equivalently, the system is now invariant under the dual symmetryrepresented by a horizontal e-string.charges. Motivated by this relation between charge condensation and symmetry breaking, wecall a condensation symmetric if it does not condense any gauge charges.The SymTFT is known to be capable of characterizing gapped phases. We have seen in theZ2 symmetry example that the dual toric code has three boundary conditions corresondingto exactly three different phases of the Z2 symmetry. In 1+1D there are also nontrivialSPT phases that have edge modes protected by symmetry. The SymTFT is also capable ofdescribing edge modes of SPTs, as we discuss now. To illustrate the principle we consider thesymmetry Z2ÃZ2 that supports a non-trivial SPT known as the cluster chain. The SymTFTin this case is the quantum double of Z2 Ã Z2, which can be viewed as two copies of toriccode. We label the anyons in each copy by ei,mi, fi, i = 1, 2. Since the cluster chain is agapped symmetric system, it should be dual to a symmetric, fully-confining condensation ofD(Z2 Ã Z2). Consider the option ASPT = â¨e1m2, e2m1â© that satisfies all the conditions. Wethen consider a thin-slab construction where on the top boundary we condense all chargese1, e2, and on the bottom boundary there is an interface between ASPT condensation andAm = â¨m1,m2â© condensation. The set up is illustrated in Fig. 5.2. As shown in Fig 5.2,the symmetry of the system, in this case a horizontal m1 or m2 string, can be localized tothe edge of the SPT and the localized symmetry actions form a projective representation ofZ2 Ã Z2. Therefore the condensation ASPT corresponds to a gapped Z2 Ã Z2 symmetric 1+1system with non-trivial edge modes, i.e. the cluster state.122Figure 5.2: 1.The condensation {e1m2, e2m1} describes the SPT state protected byZA2 ÃZB2 . The m1-line operator is the generator of ZA2 . 2. we pull out an e2-string fromthe e2-condensation at the top boundary, which acts as identity on the ground state.3.We multiply the e2-string and the m1 string to obtain an H-shape operator. 4. Themiddle part of the H-shape operator is absorbed by the bottom m1e2-condensation,leaving an operator that is supported at the left and right edge of the SPT. Similardeformation can be perfomed for ZB2 action. Therefore the symmetry actions of theSPT can be localized to the edge in the ground space. 5. The localization of ZA2action and ZB2 action to the right edge of the SPT is shown. Since the two operatorsintersect at one point, the mutual Ï statistics between e2m1 and m1 indicates that thetwo operators anti-commute. Therefore the edge symmetry actions form a projectiverepresentation of ZA2 Ã ZB2 .1235.2 SymTFT for GSPTS?SymTFT has been applied widely to analyze gapped phases. As we have seen, it is able tocharacterize both symmetric and symmetry-breaking phases. In the symmetric case, SymTFThas been shown known to be able to reproduce the classification of SPTs [139]. Given thiscapacity, one might wonder if SymTFT could shed light on gapless phases. A preliminaryexploration of this idea was performed in [15], where the authors conjectured that SymTFTis capable of classifying 1+1D gapless phases with symmetry. To elucidate, the thin slabconstruction indicates that a symmetric gapless 1+1D system corresponds to a symmetric,partially-confining condensation on the bottom boundary. Thus, itâs plausible that all sym-metric, partially-confining condensations of a 2+1D quantum double D(G) represent theclassification for all possible 1+1D G-symmetric gapless systems. Furthermore, a local low-energy equivalence principle (LLEP) was introduced in [15]. If the topological order obtainedby condensing A in the topological order D(G) is represented as D(G)/A, this principlesuggests that the local low-energy properties of a 1+1D system dual to a condensation A ofD(G), are equivalent to those of a 1+1D system dual to the trivial condensation 1 of D(G)/A.Using LLEP, we can derive the local low-energy properties of 1+1D systems dual to a givenpartially-confining condensation of the SymTFT. As an example, by condensing the doubleflux m2 in D(Z4), we obtain a toric code topological order D(Z2) formed by e2 and m. Thus,the condensation m2 corresponds to a 1+1D system exhibiting local low-energy propertiesidentical to the Ising critical point.However, the LLEP does not fully clarify the correspondence between 1+1D phases andanyon condensation of 2+1D quantum doubles. For example, there could be multiple conden-sations giving rise to equivalent post-condensation topological order, then LLEP suggests thatthese condensations describe 1+1D systems with identical local low energy properties. How-ever there are non-local properties of 1+1D systems that can be used to distinguish differentphases, the LLEP does not clarify how these non-local properties are described in the anyoncondensation picture. As a more concrete example, all SPT phases associated with a givensymmetry have the same local low energy properties: they are adiabatically connected to atrivial product state and there is no local order parameter that can distinguish one SPT fromanother. However different SPT phases are inequivalent quantum phases if symmetry is tobe preserved and they can distinguished by non-local string order parameters. Therefore theLLEP must only be part of a more complete dictionary. In the gapped phases/fully-confiningcondensation case this is well-understood: different symmetric full-confining condensationscorrespond to different 1+1D SPT phases. However it is still unclear what is the complete1+1D correspondence for a partially-confining condensation other than what the LLEP pro-124Figure 5.3: The slab describes an effective 1+1D system that we identify as the Z4-igSPT. On the top boundary we condense the charge e, on the bottom boundary wecondense e2m2. The H-shaped operator in blue represents the string order parameter ofthe Z4-igSPT. It is a decorated Us2 action, with ends charged under Us, i.e. a horizontalm-string.vides.On other hand, gSPTs are gapless systems with symmetry, therefore one might wonderhow does gSPT fit into the proposed SymTFT description of gapless systems? Specifically,are there symmetric partially-confining condensations that are dual to gSPTs? And if thereare, how does anyon condensation describe the properties of a gSPT, such as the symmetryextension structure, emergent anomaly, and edge modes? We address these questions in therest of this paper and establish a complete framework for describing gSPTs by SymTFT. Wewill see that symmetric partially-confining condensations are exactly dual to gSPTs, therebysolidify the SymTFT for gapless systems conjecture. To illustrate, let us examine the Z4-igSPT as a case study.5.2.1 SymTFT for the Z4-igSPTThe Z4-igSPT is a gapless system with a non-anomalous symmetry Z41. Thus we expect thatit is dual to a symmetric, partially-confining condensation of D(Z4). A condensation thatsatisfies these conditions is A = â¨e2m2â©. Let us analyze this condensation. After condensinge2m2, the deconfined anyons are generated by e2, em. e2 and em both have order 2 aftercondensing e2m2: (e2)2 = 1 trivially, and (em)2 is the condensed anyon, which is identified asvacuum. Thus we have in total 4 inequivalent anyons after the condensation: 1, e2, em, e3m.Interestingly, both em and e3m are not bosons: em is a semion with θem = i and e3m isan anti-semion with θe3m = âi. We conclude that the post-condensation topological order1The ZG2 anomaly of this igSPT is only emergent and the full system has no Z4 anomaly.125Figure 5.4: The slab contains a segment of igSPT with two interfaces to the trivialsymmetric gapped phase (vacuum). The ZA2 action can be localized to the interfaces,similar to the localization of symmetry actions of SPTs. The localized A-symmetryaction intersects the G-symmetry action (m-string) at one point. Since e2 and m hasmutual Ï statistics, the two operators anti-commute, giving rise to a 2-fold GSD.is the twisted quantum double DÏ(Z2), also known as the double semion theory. The twistÏ â Z3[Z2, U(1)] is the Levin-Gu anomaly.By invoking the LLEP, we see that the e2m2 condensation describes a 1+1D Z4 symmetricgapless system whose local low energy properties are the same as that of a system with Z2symmetry and the Levin-Gu anomaly. This means the dual 1+1D system has a low energysector with an effective anomalous Z2 symmetry. This is exactly the structure of the Z4-igSPT,therefore it is tempting to identify the condensation A = {1, e2m2} with the Z4-igSPT. Howdoes one describe properties of the Z4-igSPT other than the emergent anomaly in the SymTFTlanguage? Using the thin-slab construction, we show below that all physical properties of theZ4-igSPT can be extracted from the e2m2 condensation.String order parameter from SymTFT â An important feature of a non-trivial gSPT is theexistence of non-trivial string order parameters. The charges carried by ends of these stringorder parameters can be used to distinguish different gSPT phases. In the Z4-igSPT example,this can be seen by looking at the following string operator: Sa,b := Ïzaâbâ1i=a eiÏ2(1âÏxi+1/2)Ïzb .In the low energy space below â, we have â¨Sa,bâ© = â¨ÏzaÏzaÏzbÏzb â© = 1. Therefore this stringoperator acquires non-zero vacuum expectation value. Notice the end point of this stringoperator is charged under the Us symmetry: UsSââ,bU â s = âSââ,b. Now let us use thethin-slab construction with the e2m2-condensation to reproduce this result. Following thestandard prescription, we put the Ae = {1, e, e2, e3} condensation on the top boundary andthe AigSPT = {1, e2m2} condensation on the bottom boundary. As we show in figure Fig. 5.3,a non-local H-shaped operator can then be constructed in the thin-slab. In the middle of thisoperator is an m2-string, while near the end of it the m2 string joins an e2 string from the126top boundary to form an e2m2 string which is then absorbed by the bottom boundary. Thestructure of this operator mimics that of the operator Sa,b constructed before. Specifically, theend of this operator anti-commutes with an m string that intersects it, see Fig 5.3. Since thisoperator does not create any excitations, it acts as identity on the physical state representedby the thin-slab. Therefore it is a non-local string order parameter that has non-zero vacuumexpectation value, with ends charged under the Us symmetry. We see that by utilizing thethin-slab construction, one can reconstruct the string order parameters of a gSPT from itsdual anyon condensation.Edge modes from SymTFT â Another important property of non-trivial gSPTs is the edgemodes that appear when the system is put on an open chain. To observe the edge modesof the Z4-igSPT in the thin-slab construction, consider a setup where the igSPT thin-slabis put in adjacent with slabs that represent the vacuum. The vacuum is represented as aall-flux-condensed boundary. This can be seen from, for example, the fact that condensing allfluxes on the bottom boundary gives a symmetric gapped 1+1D system with no non-trivialstring order parameter. Therefore in this setup we have on the bottom boundary a segment ofe2m2-condensation that is adjacent tom-condensations. As we show in Fig. 5.4, the symmetryaction by ZA2 , i.e. a horizontal m2-string, can be localized to the interfaces between the igSPTand the vacuum, and the localized ZA2 symmetry action anti-commutes with the symmetryaction by ZG2 , i.e. a horizontal m-string. This is exactly the algebra of symmetries of the Z4igSPT: The edge of the gapped symmetry UËs2 is charged under the gapless symmetry Us.Stability of the igSPT from SymTFT â Starting with a 1+1D phase dual to a condensationA, going to an adjacent phase amounts to adding anyons to the list A or deleting some anyonsfrom A [15]. This relation constrains the phase diagram of the dual 1+1D system. Let usnow list all possible condensations of the Z4 Toric code and their relations, shown in Fig. 5.5.In this diagram, we connect two condensations if one is a subset of the other. It is clearfrom the diagram that the Z4-igSPT is stable: although the system has no Z4-anomaly, inorder to go to the gapped Z4-symmetric phases(Z4 paramagnet) the system has to either gothrough the Z4-critical point where the gap to the A-dof is closed, or open up a gap for theG-dof by breaking the ZG2 symmetry and then restore the symmetry by condensing domianwalls of ZG2 and going through a ZG2 -critical point. From the anyon condensation perspective,the anyon m can not be added to the condensation {1, e2m2} because it braids non-triviallywith e2m2, this indicates a direct transition from Z4-igSPT to Z4-paramagnet is impossible.This is an instance of âanomaly protected gaplessnessâ that appears in all igSPTs, where theexistence of an emergent anomaly gives obstruction to directly gapping out the system whilepreserving the symmetry. We see that the relation between different anyon condensation1271 â¨e2â© â¨eâ©â¨e2m2â©â¨m2â© â¨e2,m2â©â¨mâ©ZÎ4 â criticalZÎ4 Broken to ZA2ZA2 â criticalZÎ4 â BrokenZ4 â igSPTZG2 â CriticalZA2 â ParamagnetZÎ4 Broken to ZA2ZA2 â ParamagnetZ4 â ParamagnetFigure 5.5: In the top diagram vertices are condensations of D(Z4), and there is anarrow between two condensations if one is a subset of the other, the arrow is pointingin the direction of the larger set. The condensations in red are fully-confining. In thebottom diagram the vertices are replaced by the corresponding 1+1D phases, the phasesin red are gapped. The Z4-igSPT is only adjacent to the Z4-Ising critical point and thepartially symmetry broken phase, therefore the gaplessness of the igSPT is protected.128patterns provides a SymTFT description of this mechanism.The Z4-igSPT example demonstrates that the SymTFT is capable of providing compre-hensive characterization of 1+1D gSPTs. Therefore it is plausible that every Î-gSPT is dualto certain symmetric, partially-confining condensation of the quantum double D(Î). Howeverin order to fully establish such a duality it is crucial to understand the classification of twotypes of objects in two categories: on one hand the classification of gSPTs in 1+1D, and onthe other hand the classification of symmetric partially confining condensations of quantumdoubles. This will be the goal of the rest of this chapter. After carefully examining the struc-ture of 1+1D gSPTs, we provide a classification of 1+1D gSPTs protected by internal unitaryfinite symmetry. The classification itself turns out to provide us some surprising insights intothe structure of 1+1D gSPTs. For example, we find that there is an edge/bulk (0+1D/1+1D)correspondence for 1+1D gSPTs in a strict sense to be defined. After having a completecharacterization of 1+1D gSPTs, we review the established theory of anyon condensationand discuss the structure of symmetric partially confining condensations of 2+1D quantumdoubles as well as its connection to 1+1D gSPTs. A 1-1 correspondence between the two willby then become evident.5.3 Structure of 1+1D bosonic gSPTsIn order to establish formally the connection between 1+1D gSPTs and their SymTFT dual,it is crucial to have a complete understanding of the structure of 1+1D gSPTs. This is thetask of this section.5.3.1 Algebra of symmetry and consistency conditionsLet us consider a general 1+1D gSPT with a gapped sector and a gapless sector having thestructure of a symmetry extension1â A jââ Î pââ Gâ 1. (5.6)Here A â´ Î is a normal subgroup of Î, not necessarily Abelian. Being a normal subgroup,we have Î³AÎ³â1 = A for any γ â Î, and this defines a Î-action on A. We ask what extra databesides the group extension is needed to specify a gSPT. From our experience with SPTs,we know that the low energy physics of an SPT is completely specified by the anomaloussymmetry action on the boundary. For a 1+1D SPT, the algebra of edge symmetry action,namely the projective representation of edge symmetry actions, is the origin of the SPT edge129modes. Motivated by this, we look at symmetry actions and their algebra in a 1+1D gSPT,this algebra will similarly determine the edge modes of the gSPT. First of all the UV A-symmetry actions act trivially in the bulk in the low energy sector, since by definition there isno A-charges in the gapless sector in the bulk. Therefore on a closed chain we expect to haveU IRa = 1. In the following we will drop the IR superscript and it is understood that operatorsare restricted to the gapless low energy subspace. However on an open chain the A-action canstill act non-trivially on the edge of the chain, this is known as symmetry fractionalizationprinciple for symmetries acting on gapped degrees of freedom [89, 108, 114]. Therefore anA-symmetry action Ua reduces to an edge action Ua = ULa â URa below the gap to A-dof.Similar to the edge symmetry action of SPTs, the localized edge actions UËa := ULa only needto form a projective representation of the gapped symmetry group A: UËaUËb = η(a, b)UËab,where η â Z2[A,U(1)] is a 2-cocycle of A. We also need to consider the interplay betweenthe localized edge action UËa and the non-localized G-action. Consider a general γ â Î action,and the conjugation UÎ³UËaUâ1Î³ , the group law states that UÎ³UaUâ1Î³ = UÎ³aÎ³â1 , however usingthe decomposition Ua = ULa â URa , we haveUγULa Uâ1Î³ â UγURa Uâ1Î³ = ULÎ³aÎ³â1 â URÎ³aÎ³â1 . (5.7)Notice that UγUL,Ra Uâ1Î³ is an operator supported on the left/right edge: away from the edgeUL,Ra acts as identity and UγUL,Ra Uâ1Î³ = UÎ³Uâ1Î³ = 1. Therefore both sides of (5.7) decomposeinto tensor products of operators acting on separate Hilbert spaces, thus each factor must beequal up to a phase:UÎ³UËaUâ1Î³ = ϵ(a, Î³)UËÎ³aÎ³â1 (5.8)The phase ϵ(a, γ) can be thought of as measuring the charges of the fractionalized UËa actionunder Uγ . The data (η, ϵ) naturally satisfies a set of consistency conditions. For example,UÎ³(UËaUËb)Uâ1Î³ should be equal to (UÎ³UËaUâ1Î³ ) · (UÎ³UËbUâ1Î³ ), which leads to an equationη(a, b)ϵ(ab, γ) = Î·(Î³aÎ³â1, Î³bÎ³â1)Ïµ(a, γ)ϵ(b, γ). (5.9)We perform careful analysis of consistency conditions in appendix 5.6 and summarize themhere:A 1+1D Î-gSPT is defined by (Aâ Î, η, ϵ), where A is a central subgroup of Î, and η, ϵ130Figure 5.6: 1. UV symmetry actions Ua, a â A reduce to edge actions in the gaplesssector, and the restriction of them to one of the endpoints make up a projective repre-sentation of A given by a 2-cocycle η. 2. The restriction of an A-symmetry action toone of the endpoints fails to satisfy the commutation relation with another symmetryaction.satisfy the following conditions:η â Z2[A,U(1)], ϵ : AÃ Îâ U(1)ϵ(a, γ)ϵ(b, γ)ϵ(ab, γ)=η(a, b)Î·(Î³aÎ³â1, Î³bÎ³â1)(5.10)Ïµ(a, γ1γ2) = Ïµ(Î³2aÎ³â12 , γ1)ϵ(a, γ2) (5.11)ϵ(a, b) =η(b, a)Î·(babâ1, b), âa, b â A (5.12)Moreover, the decomposition Ua = ULa â URa only defines the localized A-action UL,Ra up toa phase. Redefining ULa â α(a)ULa , URa â Î±â(a)URa does not change the overall symmetryaction Ua. This will have the effect of modifying the 2-cocycle η by a coboundary, and willalso affect the ϵ factor by ϵ(a, γ) â ϵ(a, Î³)Î±(Î³aÎ³â1)Î±(a) . Since the choice of the phase factor α isartifitial, two solutions to the consistency conditions related by such a transformation shouldbe viewed as defining the same gSPT.131Two sets of data (η, ϵ), (Î·â², Ïµâ²) define the same gSPT, if they are related byÎ·â²(a, b) = η(a, b)Î±(a)Î±(b)Î±(ab)(5.13)Ïµâ²(a, γ) = ϵ(a, Î³)Î±(Î³aÎ³â1)Î±(a). (5.14)We will refer to this equivalence relation as gauge equivalence, and a transformation of a pair(η, ϵ) by the equivalence relation as a gauge transformation.The set of equivalent classes ofsolutions (η, ϵ) to the consistency conditions is our classification of 1+1D bosonic gSPT withunitary symmetry.Finiteness of the classification. It is not immediately clear that the classification aboveis finite or not, although in many cases it can be verified by hand that the solutions are finitemodulo gauge transformation. However we will show that gSPTs are in 1-1 correspondencewith symmetric partially confining condensations of 2+1D quantum doubles, and the latteralways has a finite classification for any finite gauge group. Thus the SymTFT dual willprovide a definite answer to the finiteness question, at least for finite Î. Thus there are onlyfinite many gSPTs in 1+1D for any finite symmetry group Î. A seemingly contradiction isthat we can always stack a gSPT with a CFT that transforms trivially under the symmetrywithout changing the data (η, ϵ). The resolution is that we should view the classification askinematic and does not determine what CFT lives in the bulk. That is, it classifies possibleÎ-symmetry enriching patterns of all G-symmetric gapless states. Alternatively, one can takethe bulk phase as an extra data that needs to be specified besides (η, ϵ). However the data(η, ϵ) does put constraints on the possible gapless state. For example, the gapless sector needsto have nontrivial G-symmetry, since if G-symmetry acts trivially on the gapless sector, itwould have been identified as a gapped symmetry. Also, as we will show soon, the pair (η, ϵ)determines the emergent anomaly of the bulk, which also needs to be matched by the gaplesssector.Group structure of the classification. There is an obvious group structure of theclassification. Namely since the consistency conditions are all multiplicative in η, ϵ, the prod-uct of two solutions is again a solution. The solution η = 1, ϵ = 1 serves as group identity, andthe inverse of an element (η, ϵ) is (Î·â1, Ïµâ1). It is tempting to interpret the group structureas the physical operation of stacking two gSPTs together, however there is a subtlty. Forexample, if we stack a gSPT with a CFTA bulk and another with a CFTB bulk, then thestacked system will have CFTAÃCFTB bulk. Therefore the stacking operation only defines132a group structure modulo the bulk state. This is tight to the fact that the classification isreally a classification of symmetry enriching patterns.Invariants of gSPTs. The data η, ϵ provides some discrete invariants for a gSPT whichwill be useful in practice. Consider a given a â A, let Ca(Î) be the centralizer of a in Î,i.e. the subgroup consisting of elements that commute with a. Then we have UcUËaUâ c =ϵ(a, c)UËa, âc â Ca(Î), and the factor ϵ(a, c) is now gauge invariant: a gauge transformationchanges it to Ïµâ²(a, c) = ϵ(a, c)Î±(caaâ1)Î±(a) = ϵ(a, c). Therefore ϵ restricted to {a}ÃCa(Î) providesan invariant for any a â A. This is nothing but the charge carried by UËa under Ca(Î). Ingeneral UËa does not have a well-defined Î-charge, since a Uγ conjugation maps it to UËÎ³aÎ³â1up to a phase factor. Restricting to Ca(Î) gives a well-defined charge of UËa. However, weshould stress that in general these invariants do not determine the gSPT completely. Themethod of using charges of edge symmetry actions as invariants also applies to 1+1D SPTs,and it is known that there are examples where such charges do not fully determine the SPT(although the minimal counter-example requires a symmetry group of order 128) [88]. Putthe counter-example as the gapped sector will give a counter-example for gSPTs.5.3.2 Emergent anomaly of gSPT and a bulk-edge (1+1D/0+1-D) correspondenceWe next show that the data (η, ϵ) determines the emergent anomaly of the gSPT. Since thedata (η, ϵ) are directly related to the edge modes of the gSPT, the result that they determinethe emergent anomaly can then be viewed as an edge-to-bulk map for gSPTs: the edge modesdetermine the emergent anomaly. Whatâs interesting is that the converse is also (partially)true: the emergent anomaly determines (η, ϵ) up to stacking with a Î-SPT. Therefore thereis a bulk-edge correspondence2 for gSPTs in the following sense: the edge modes determinethe bulk emergent anomaly, and the bulk emergent anomaly determines the edge modes upto stacking with an SPT.There are more than one way to detect the anomaly of a symmetry. For example anomalyamounts to obstruction to consistently coupling the system to a background gauge field, equiv-alently anomaly is the obstruction to consistently inserting a defect network of the symmetry.A practical way of computing the anomaly when the symmetry action is explicitly knownis provided by [35], where anomaly is formulated as the obstruction to implementing thesymmetry on a system with boundary. Here we review this approach briefly.2not to be confused with the holographic duality between 1+1D gSPTs and 2+1D quantum doubles.133For a symmetry action Ug, g â G acting on a 1+1D system with open boundary condition,if we compare UgUh and Ugh, they may fail to be equal up to operators supported on the edge:UgUh = â¦(g, h)Ugh, (5.15)where â¦(g, h) is a local unitary operator supported on the edge: â¦(g, h) = â¦L(g, h)ââ¦R(g, h).From the definition (5.15) the operators â¦ must satisfy a condition from the associativity ofthe Ug operators,(UgUh)Uk = Ug(UhUk)â â¦(g, h)â¦(gh, k) = Ugâ¦(h, k)Uâ1g â¦(g, hk). (5.16)However, since the operator â¦ splits into tensor product of â¦L and â¦R, the restriction of itto one of the edges only need to satisfy the consistency condition up to a phase:Ï3(g, h, k)â¦L(g, h)â¦L(gh, k) = Ugâ¦L(h, k)Uâ1g â¦L(g, hk). (5.17)The phase Ï3 can be shown to be a 3-cocycle and represents the anomaly of the symmetryaction Ug.We now show that the data η, ϵ suffice to determine the emergent anomaly of a gSPT.The group extension (5.6) defines a 2-cocycle of G with A coefficients: e2 â Z2[G,A], with Gacting on A by conjugation. The role of this 2-cocycle is to specify the group law of Î fromthat of G and A. One can write elements of Î as pairs γ = (a, g), a â A, g â G. The grouplaw of Î is then given by (a, g) · (b, h) = (a · gbgâ1 · e2(g, h), gh). Now with this understandingof group structure of Î, one can compute the composition of symmetry actions by G:UgUh = U(e2(gh),gh) = Ue2(g,h)Ugh. (5.18)The operator Ue2(g,h) when restricted to the gapless sector acts only non-trivially on the edge,therefore the G-action(when restricted to the gapless sector) does take the form of (5.15),with â¦(g, h) = Ue2(g,h) and â¦L(g, h) = UËe2(g,h). Now we can calculate the anomaly according134to equation (5.17):â¦L(g, h)â¦L(gh, k) = UËe2(g,h)UËe2(gh,k)= η(e2(g, h), e2(gh, k))UËe2(g,h)e2(gh,k), (5.19)Ugâ¦L(h, k)Uâ1g â¦L(g, hk) = UgUËe2(h,k)Uâ1g UËe2(g,hk)= ϵ(e2(h, k), g)UËge2(h,k)gâ1UËe2(g,hk)= ϵ(e2(h, k), g)η(ge2(h, k)gâ1, e2(g, hk))UËge2(h,k)gâ1e2(g,hk) (5.20)The cocycle condition on e2 isge2(h, k)gâ1e2(g, hk) = e2(g, h)e2(gh, k), (5.21)therefore the last lines of (5.19) and (5.20) only differ by a phase factor, which is the anomaly3-cocycle of the G-symmetry action:Ï3(g, h, k) =η(ge2(h, k)gâ1, e2(g, hk))η(e2(g, h), e2(gh, k))ϵ(e2(h, k), g) (5.22)Trivial vs. nontrivial extension. From the expression (5.22) it is clear that if thegroup extension of a gSPT is trivial, then e2 = 1 and the emergent anomaly vanishes. Thusin order to have an emergent anomaly the group extension must be non-trivial, i.e. the fullsymmetry group Î can not be a direct product or semi-direct product of the gapped symmetryA and the gapless symmetry G.gSPTs of SPT-pump type. Any solution to the consistency conditions (5.10)-(5.12)defines a gSPT and determines the emergent anomaly through (5.22). Consider a special casewhere η = 1 and the group extension is central. In this case elements of A commute with allelements of Î: Î³aÎ³â1 = a, and the consistency conditions on ϵ reduce toϵ(a, γ)ϵ(b, γ) = ϵ(ab, γ) (5.23)ϵ(a, γ1γ2) = ϵ(a, γ1)ϵ(a, γ2) (5.24)ϵ(a, b) = 1 âa, b â A. (5.25)The first two conditions state that ϵ is now linear in both factors, and the last conditionstates that the ϵ is well-defined on the quotient AÃ Î/A = AÃG. Viewed from the original135definition, UÎ³UËaUâ1Î³ = ϵ(a, Î³)UËa, these conditions imply that now every a â A action carries awell-defined charge under G. The anomaly according to (5.22) is Ï3(g, h, k) = ϵ(e2(h, k), g) =ϵ âª e2(g, h, k), and the edge modes of the gSPT can be intuitively understood as A-actionspumping G-charges (which are 0+1-D SPTs of G). Such gSPTs have been studied in [130]and the SPT-pump mechanism was also used to understand igSPTs in higher dimensions.Bulk-edge (1+1D/0+1-D) correspondence for gSPT Interestingly the emergentanomaly determines the data (η, ϵ) up to stacking with Î-SPTs. This means up to Î-SPTs thephysics of a Î-gSPT is completely determined by its emergent anomaly. This has an immediateconsequence that the two definitions of igSPT are equivalent: 1. A gSPT is intrinsic if theedge modes can not be trivialized by stacking with an SPT. 2. A gSPT is intrinsic if it hasa nontrivial emergent anomaly. We provide a comprehensive derivation of these statementsin appendix 5.8. The idea of the derivation can be summarized as follows. Denote the groupextension as1â A jââ Î pââ Gâ 1 (5.26)Since the anomaly is trivialized upon pull-back by p, we have on the cochain level pâ(Ï) = d(α),where α â C2[Î, U(1)] is a 2-cochain of Î. The composition of maps p â¦ i is trivial since thesequence 5.26 is exact: pâ¦j = 0. This implies the composition of pull-backs (pâ¦j)â = jââ¦pâ = 0is also trivial. Therfore 0 = jâ â¦ pâ(Ï) = jâ(dÎ±) = d(jâÎ±), meaning that jâÎ± is a 2-cocycle ofA representing a 1+1D A-SPT. This is to be identified with the SPT of the gapped sector,i.e. η = jâÎ±. ϵ can be found by considering the value α((a, 1), (1, g)). Since the trivializingcochain α is not unique, this procedure only determines η, ϵ up to stacking with Î-SPTs. Insection 5.5.3 we will address the question of stacking gSPTs with SPTs.5.4 Condensation of quantum double and gSPTThe Z4-igSPT example suggests that symmetric partially-confining condensations in the 2+1Dquantum double D(Î) correspond to Î-gSPTs. Now that we have a classification of 1+1DgSPTs, we are in position to establish this correspondence in a rigorous manner. We firstbriefly summarize the results.1365.4.1 Summary of resultsThe SymTFT for a symmetry Î is the quantum double D(Î), also known as the gauge theorywith gauge group Î. Anyons of D(Î) are gauge charges, gauge fluxes and composites of chargeand fluxes (dyons). An anyon condensation amounts to specifying a set of condensed anyonsand also a fusion channel for fusing any two condensed anyons. For quantum doubles, ananyon condensation can be specified as follows. One can first separate the condensation intotwo steps. In the first step we only condense pure charges, which will lead to gauge symmetrybreaking from Î to some subgroup H < Î. This is dual to a global symmetry breaking in theSymTFT setup, therefore for describing symmetric 1+1D systems one can neglect this stepand set H = Î. In the second step we only condense fluxes and/or dyons. In general a dyonhas a flux sector and a charge sector. The contents of the condensed dyons and their fusionchannels are then specified by 3 objects: a normal subgroup A < Î and two phase factorsϵ : A Ã Î â U(1), η : Î Ã Î â U(1). A specifies the flux sectors of the condensed dyons, ϵdetermines the charge sector of each condensed dyon, and η determines the fusion phase offusing two condensed dyons. In general fluxes take value in Î. Then the condensed dyons musthave flux sectors taking values in the normal subgroup A. For a given dyon with flux sectora â A, ϵ(a, γ) then measures the charge sector. It can be thought of as the phase that is pickedup by braiding a flux of value γ around the dyon with flux sector a. By collecting the phasesϵ(a, γ) resulting from braiding with all the fluxes one can then determine the charge sector ofthe dyon. Finally η(a, b) is the fusion phase of fusing two condensed dyons with flux sectorsa â A and b â A. One immediately sees that the set of data needed to specify a symmetriccondensation is identical to that is needed to specify a 1+1D Î-gSPT. It can be shown that inorder for the data A, ϵ, η to define a consistent boson condensation, they must satisfy certainconsistency conditions. These turn out to be identical to the consistency conditions of 1+1DgSPTs (5.10)-(5.12). Moreover, a condensation gives rise to a new topological order. It canbe shown that the condensation specified by the data A, ϵ, η is fully-confining if and only ifA = Î. And when A Ì¸= Î, the post-condensation topological order is in general a twistedquantum double of the quotient group G := Î/A: DÏ(G), and the twist Ï turns out to begiven by the same formula as (5.22). This then establishes a 1-1 correspondence betweensymmetric partially-confining condensations of D(Î) and Î-gSPTs.In the rest of this section we review the formal theory of anyon condensation and discussthe structure of anyon condensation in 2+1D quantum doubles.1375.4.2 Review of condensation in 2+1D topological ordersWe first review aspects of UMTC relevant to our discussion and then discuss the descriptionof anyon condensation in this language. A general 2+1D topological order is described bya unitary modular tensor category(UMTC) C. Objects of C correspond to anyons of thetopological order. A UMTC is equipped with a tensor product of pairs of objects â, whichdescribes the fusion of anyons, an associator αX,Y,Z : (XâY )âZ â Xâ (Y âZ) that relatesdifferent orders of fusing three anyons, and a braiding cX,Y : X â Y â Y âX that describesthe process of exchanging two anyons. The data αX,Y,Z , cX,Y satisfy certain consistencyconditions, including the pentagon equation of the associators and the hexagon equation thatrelates the associator with the braiding [52, 12, 62].A condensation amounts to bringing a large number of anyons in a region together to forma new phase from C. This physical process corresponds to obtaining a new UMTC from C byspecifying a condensable algebra A of C [54]. A condensable algebra describes the contents ofcondensed anyons, if several simple anyons ai are condensed, we can view A as the compositeanyon A = a1+a2+Â· · · . After condensing A, we obtain a new phase D, with a new fusion ruleâD. In the new phase A is identified as the vacuum, A = 1D. The vacuum of the new phasemust fuse trivially with itself: AâD A = A. Since A is also an anyon of the original phase,and fusion channels of the new phase D come from those of the original phase C, this meanswhen viewed as an anyon of C, there must be a nontrivial fusion channel µ : A âC A â A.This makes A an algebra in C. There are extra conditions on the algebra due to the physicalrequirements that the condensed anyons should be self and mutual bosons. These conditionsmake A a so called connected separable commutative algebra, or a condensable algebra forsimplicity. Now consider an excitation M that lives in the post-condensation phase D, thefusion M âD A is equal to M since now A is the vacuum: M âD A = M âD 1D = M . Onthe other hand both M and A are also anyons(possibly composite ones) of the original phase,and fusion channels of D come from that of the origin phase C. Therefore when viewed asanyons of C, there should be a nontrivial fusion channel νM : M âC A â M . This makesM a module over A. For example, in the toric code the composite anyon m+ f is a moduleover 1 + e, therefore is an excitation after condensing 1 + e. Deconfined anyons in the newphase must braid trivially with vacuum, then when viewed as anyons of the original phase,this means the diagram138M âC A MAâCM M âC AνMcM,AcA,MνMcommutes for any deconfined anyon represented by a module M . This condition makes M aso called local module over A. Thus the deconfined excitations in the new phase are given bylocal modules over A in the original phase. The category of local modules over A, ClocA has afull UMTC structure induced from C [54], therefore describes a valid topological order.5.4.3 Condensation of 2+1D quantum doubleLet us now focus on a specific kind of topological order of interests to us, the 2+1D quantumdoubles. The quantum double D(Î) is a gauge theory with gauge group Î. The anyons inD(Î) can generally be identified as gauge charges, gauge fluxes, and composites of chargeand flux. Gauge fluxes are in general labelled by conjugation classes of the group Î, and thecharges are labelled by irreducible representations of the group. A unified way of describinganyons of D(Î) is to identify an anyon of D(Î) as a Î-graded vector space V = âÎ³âÎVÎ³ ,together with a Î-action, such that Î³(VÎ³â²) = VÎ³Î³â²Î³â1 . In this way a charge corresponds to thecase where the graded vector space is only supported at the identity element: V = V1Î , thenthe anyon is completely determined by a representation of the group Î, recovering the factthat charges are labelled by representations of Î. On the other hand a flux corresponds to thecase where the vector space is supported on at least one nontrivial element while the Î-actionis trivial. The vector space has to be supported on conjugation classes of Î to be invariantunder the Î-action, therefore a flux is labelled by a conjugation class of Î. A generic bondstate of charge and flux then corresponds to a graded vector space with nontrivial support aswell as nontrivial Î-action.Condensable algebras of D(Î) have been classified in [23]. A condensation in D(Î) canalways be done in two steps. In the first step, only charges are condensed, reducing D(Î) toD(H), for some subgroup H < Î. This step can be thought of as gauge symmetry breakingfrom Î to H. The condensable algebra for this step is given by the algebra of functionsover the coset Î/H: C(Î/H). As an element of D(Î), C(Î/H) is a graded vector spacesupported only at the identify 1Î, with the Î-action defined by (γ · f)([Î³â²]) := f([Î³â²Î³â1]) forany function f on the coset Î/H. Clearly this defines a representation of Î and correspondsto a condensation of charges. Local modules over C(Î/H) can be shown to form a UMTCequivalent to the quantum double D(H). In the second step no charges are condensed, and139the condensable algebra is determined as follows:Theorem 5.4.1 ([23], Thm. 3.5.3): A condensable algebra of D(H) that does not condenseany charges are determined by η â H2(H,U(1)), a normal subgroup A âH, and a functionϵ : AÃH â U(1), such that the following conditions are met:ϵ(a, h)ϵ(b, h)ϵ(ab, h)=η(a, b)Î·(hahâ1, hbhâ1)(5.27)Ïµ(a, h1h2) = Ïµ(h2ahâ12 , h1)ϵ(a, h2) (5.28)ϵ(a, b) =η(b, a)Î·(babâ1, b), âa, b â A (5.29)The algebraA[A, η, ϵ] determined by these data can be described as follows. FirstA[A, η, ϵ]should be an (composite) anyon of D(H), i.e. an H-graded vector space with an H-action. Asa graded vector space, A[A, η, ϵ] = âaâACa is supported on the subgroup A, let the basis of Cabe ea. The H-action is defined by h(ea) := ϵ(a, h)ehahâ1 . The algebra structure on A[A, η, ϵ]is defined on the basis ea by ea · eb := η(a, b)eab. Requiring the algebra A[A, η, ϵ] to be well-defined and condensable then leads to the consistency condition on ϵ and η (5.27)-(5.29). Itwas also shown in [23] that the condensation is fully-confining if and only if A = H, in whichcase the function η determines ϵ through (5.29) and the condensation is determined solely bya subgroup H < Î and a 2-cocycle of H, which we recall is the same as the classification of1+1D gapped Î-phases, i.e. an SSB from Î to H, and an SPT of H. On the other hand,if no charge is condensed and the condensation is not fully-confining, meaning that H = Î,and A is a proper subgroup of G, the data becomes exactly the same as what determines anigSPT: a group extension 1 â A â Î â G := Î/A â 1, a 2-cocycle of A, and a functionϵ : AÃH â U(1). And the conditions (5.27)-(5.29) are exactly the same as the consistencyconditions in (5.10)-(5.12). We conclude that there is a 1-1 correspondence between symmetricpartially-confining condensations of D(Î) and 1+1D Î-gSPTs.Moreover, the post-condensation topological order is in general a twisted quantum doublegiven by the following:([24], Thm. 2.17) The category of local modules over A(A, η, ϵ) is equivalent to D(G :=Î/A)Ï as a UMTC, with the twist Ï given by:Ï3(g, h, k) = ϵ(e2(g, h), kâ1)Ã η(e2(g, hk), e2(h, k))η(e2(g, hk)e2(h, k), kâ1e2(g, h)â1k) (5.30)Although appears to be different from (5.22), it can be shown that (5.30) is equivalent to the140emergent anomaly of gSPT (5.22) as cocycles. We show the equivalence between the two inappendix 5.9. In summary, there is a 1d-2d correspondence between gSPTs and symmetricpartially-confining condensations of quantum doubles:A gSPT defined by the data (A â Î, η, ϵ) is described in SymTFT by the condensablealgebra A(A â Î, η, ϵ) of D(Î). If the post-condensation topological order is D(Î)locA â¼=DÏ(G), then the emergent anomaly of the gSPT is equal to the twist Ï â H3[G,U(1)].It fact it is readily seen that the symmetry of a gSPT has the structure of a Î-graded vectorspace with Î-action. The localized edge actions UËa span a graded vector space supported onA, with a Î-action given by the conjugation UÎ³UËaUâ γ = ϵ(a, Î³)UËÎ³aÎ³â1 . Therefore the symmetryalgebra of a Î-gSPT naturally defines an object of D(Î). The product of operators: UËaUËb =η(a, b)UËab endows the object with an algebra structure, and the consistency conditions(5.10)-(5.12) make sure the algebra is condensable.5.5 Special cases of gSPTs and their SymTFT de-scriptionHaving established the formal relation between anyon condensation and gSPT, let us nowlook at several typical classes of gSPTs and how they are described by the SymTFT asapplications of this formalism. There are two families of gSPTs that have been studied insystematic manners. One is the gSPTs of SPT-pump type, another is the week gSPTs thatcan thought of as transition from SPT to spontaneous symmetry breaking. We discuss howthey are incorporated into our general classification in terms of the data (η, ϵ) as well as theirSymTFT description.5.5.1 igSPT of SPT-pump typeA class of igSPTs whose edge modes can be explained by the mechanism of SPT-pump wasstudied in [130]. In terms of the data η, ϵ, these gSPTs are solutions with η = 0, and whoseemergent anomaly is given by Ï3 = ϵ âª e2. In this section we show how gSPTs of SPT-pump type are described by the SymTFT. For simplicity let us consider the case where Î isAbelian. The quantum double D(Î) that is dual to the igSPT is an Abelian gauge theorywhose excitations are charges, fluxes and their bond states. In general fluxes are labelled byelements of the gauge group Î, while charges are labelled by irreducible representations ofthe group, or in the abelian case the dual group ÎÌ := H1[Î, U(1)]. Therefore we denote the141Figure 5.7: On the top reference boundary we fix the charge-condensed boundarycondition. On the physical boundary we consider a condensation determined by theSPT-pump factor ϵ where fluxes of A are binded with charges Ïµ(a,â) and then con-densed. The resulting slab has the expected SPT-pump edge modes: an a â A actionis localized to the edge of the igSPT, and its charge under a g â G action is ϵ(a, g).fluxes as mγ , γ â Î and the charges as eÎ³Ë , Î³Ë â ÎÌ. Both charges and fluxes are self-bosons,and the mutual statistics between a charge and a flux is given by the pairing between Î andÎÌ: SeÎ³Ë ,mγ = Î³Ë(Î³) â U(1). From our understanding of igSPTs of SPT-pump type, we willbe able to reverse engineer the corresponding anyon condensation. If the igSPT has a groupextension of the form1â Aâ Îâ Gâ 1, (5.31)then we expect an action by a â A to pump a charge Ïµ(a,â). In the Z4-igSPT example, them2 anyon is binded with e2 before condensation, which is essentially what is pumped by theZA2 -action. Therefore in order for a-action to pump a charge Ïµ(a,â) â GÌ 3, we expect thebond states maeϵ(a) to be condensed, where we defined ϵ(a) := Ïµ(a,â) which is an element ofGÌ, corresponding to a gauge charge. It is readily seen that anyons maeϵ(a) can be condensedsimultaneously: the quantum spin is θ(maeϵ(a)) = Sma,eϵ(a) = ϵ(a, a) = 1 since Ïµ|AÃA = 1,the mutual statistics between maeϵ(a) and mbeϵ(b) is Smaeϵ(a),mbeϵ(b) = ϵ(a, b)ϵ(b, a) = 1. Bythe same diagramatic reasoning that leads to edge modes of the Z4-igSPT, we see that thecondensation A = {maeϵ(a), a â A} makes the A-actions localized to the edge while theG-actions are not localizable. The charges binded with ma give the expected SPT-pumpstructure. Therefore this condensation should be dual to the Î-igSPT of SPT-pump type.Since we also know that these igSPTs have emergent anomalies given by Ï3 = Ïµâªe2, we arrive3The phase ϵ(a, γ) in general is defined on AÃÎ, but when η = 0, the consistency condition (5.12)states that Ïµ|AÃA = 1. ϵ is now also linear in both variables by (5.10)-(5.11), therefore ϵ is now anelement of H1[AÃG,U(1)] = H1[A, GÌ]142at the following conclusion:Let A < Î be a subgroup, G := Î/A, e2 is the extension class associated with 1â AâÎ â G â 1. For any ϵ â H1[A, GÌ], condensing A = {maeϵ(a), a â A} in the D(Î)quantum double results in the topological order DÏ(G = Î/A), with Ï = ϵ âª e2.We see that via the SymTFT/igSPT duality and by invoking our knowledge about igSPTs in1+1D, we have inferred information regarding condensations in 2+1D quantum doubles. SeeFig. 5.7 for an illustration of the thin slab construction of igSPTs of SPT-pump type.Similar to the Z4-igSPT, igSPTs of SPT-pump types are all stable due to their emergentanomalies. From the anyon condensation perspective, this means it is impossible to furthercondense anyons to arrive at a fully-confining condensation that does not condense charges.This can be seen from that fact that the post-condensation topological order is a twistedquantum double, and it is well known that in a twisted quantum double there is no fully-confining condensation that does not condense charges [139]. Therefore in order to arrive ata gapped symmetric phase one has to either condense charges, corresponding to breaking thesymmetry, or âuncondenseâ some anyons in A, which corresponds to closing the gap to Adegrees of freedom. This anyon condensation picture matches with our expectation from the1+1D analysis: as the gap to A-dof is perturbatively stable, closing the gap to A degrees offreedom will require a phase transition out of the igSPT phase. On other hand the anomalyof the gapless sector forbids opening a gap directly without breaking the symmetry.5.5.2 Weak gSPTsThere is also a family of gSPTs named weak gSPTs that are transitions between SPT phasesand symmetry breaking phases, these have been studied in [116, 74]. Here we discuss somegeneralities of such gSPTs and show how they fall into our classification in terms of the pair(η, ϵ) as well as their SymTFT description. The common mechanism for such gSPTs is thatthe full symmetry has the form GA Ã GB, and on one side of the transition is an SPT ofdecorated domain wall(DDW) type, where domain walls of GA are decorated with chargesof GB and vice versa. On the other side of the transition the symmetry is broken down toGA, and system is a GB-paramagnet. The transition between the two is thus in the sameuniversality class as a usual GB-SSB transition, which is further enriched by GAÃGB to forma gSPT. The GA symmetry is intact in both the SPT and the GB-SSB phases, therefore actsas a gapped symmetry at the transition point. Thus GA-action still localizes to edge actions atthe transition, and the restriction to one edge carries charges of GB from the DDW structureof the SPT phase. This leads to edge modes and makes the transition a nontrivial gSPT.143On the other hand the GB-action, which acts on gapless dof, no longer fractionalizes and theedge modes of the gSPT are reduced from that of the nearby GA Ã GB-SPT. As the groupextension is trivial, there is no emergent anomaly. The gSPT can be gapped out directly whilepreserving symmetry-it is adjacent to a gapped SPT by construction. Thus such gSPTs arenot stable, and the edge modes of them are nothing but residue of SPT edge modes, hence thename weak gSPTs. Nevertheless, the symmetry enriching pattern and nontrivial edge modesmake it distinct from a usual GB-SSB transition with a trivial GA-action. In particular, thecharge carried by edge GA-action provides a discrete invariant that can not be altered withoutchanging the bulk universality class. To go to a trivial GB-SSB transition the system has toeither open up a gap or go through a multi-critical point with a different bulk universalityclass.Examples of weak gSPTs include the Haldane chain to SSB transition, which has an IsingCFT bulk and a 2-fold GSD on an opne chain, and similarly Z3ÃZ3-SPT to SSB transitions,which have bulk three-state Potts CFT with central charge c = 4/5 and 3-fold GSD on anopen chain [116]. Numerical results on the phase diagrams of these models have been obtainedin [116], in either case it was confirmed that the weak gSPT is separated from a trivial one bya multi-critical point. We will later provide a rather simple derivation of the phase diagramusing SymTFT.Let us now see how the weak gSPTs fall into our classification in terms of η, ϵ. A weakgSPT has a symmetry group GAÃGB, and on one side of the transition is in an SPT of DDWtype. In the cohomology classification this means the SPT is described by a cocycle β2 âH1[GA, H1[GB, U(1)]], which is a subgroup of the full cohomology group H2[GA ÃGB, U(1)]by the Ku¨nneth formula. This means β2 = β2(gA, hB) is linear in both factors and describesthe charges carried by edge GA and GB actions viaUgB UËgAUâ gB = β2(gA, gB)UËgA (5.32)UgAUËgBUâ gA = β2(gA, gB)â1UËgB (5.33)At the transition point the subgroup GB becomes gapless and the second equation above islost. However we still have the first equation from which we see that it corresponds to a gSPTwith η2 = 1, ϵ(a = gA, γ = (hA, hB)) = β2(gA, hB). 4We next discuss the SymTFT description of weak gSPTs. We take Î = Z2 Ã Z2 as an4We note that a weak gSPT can be defined more broadly as any SPT to (partial)SSB transition. Letβ2 â H2[Î, U(1)] represent a Î-SPT. If one drives the SPT to the Î broken down to Aâ´ Î transition,then the transition is a gSPT with extension 1 â A iââ Î â G â 1 and η = iâ(Î²2), ϵ(a, γ) =Î²2(Î³,a)Î²2(Î³aÎ³â1,Î³).144example, and generalization to generic weak SPTs is straightforward. Since the full symmetryis Z2 Ã Z2, we search for symmetric partially-confining condensations that do not condensecharges in D(Z2 Ã Z2), i.e. two copies of toric code. One choice is the A = {e1m2} conden-sation. The post-condensation topological order is generated by e2m1 and e1, both of whichare bosons and these is a mutual Ï statistics between them. Thus the post-condensationtopological order is equivalent to the toric code, D(Z22)/A â¼= D(Z2), and the condensationdescribes a 1+1D gSPT with a trivial group extension 1 â ZA2 â ZA2 Ã ZB2 â ZB2 â 1.Using the thin slab construction, it is readily seen that ZA2 -action is localized to the edge andthe edge action is charged under the ZB2 -action. In terms of the pair (η, ϵ), this means wehave η = 1, ϵ(a, bâ) = (â1)ab1 . Since the extension is trivial, there is no emergent anomaly.Nevertheless, the nontrivial ϵ leads to a 2-fold GSD on an open chain. Starting with thecondensation e1m2, further condensing e2m1 is allowed and will result in the condensationâ¨e1m2, e2m1â©, which is dual to the cluster chain. On the other hand further condensing e1or m2 will result in the condensation â¨e1,m2â©, which is dual to a ZA2 -SSB/ZB2 -paramagnetphase. Therefore the condensation â¨e1m2â© describes the transition from the cluster chain tothe ZA2 -SSB/ZB2 -paramagnet phase. The trivial gSPT with the same bulk universality classis described by the condensation â¨m2â©, which corresponds to the ZA2 Ã ZB2 -paramagnet toZA2 -SSB/ZB2 -paramagnet transition. From the SymTF it is clear that the two gSPTs are notconnected if symmetry is preserved. Both gSPTs are in the Ising-CFT universality class, butin order to go from the e1m2 condensation to the m2 condensation one has to either condensecharges, corresonding to breaking the symmetry, or uncondense e1m2, which will lead to the{1} condensation that is dual to a ZA2 ÃZB2 -multi-critical point. The same phase diagram hasbeen obtained in [116] via DMRG.5.5.3 Stacking gSPT with SPTOne can always stack a gSPT with an SPT to obtain a new gSPT. We discuss some generalitiesof this kind of construction in this section.An example â Consider the topological order D(ZA4 ÃZB4 ), which is two copies of Z4-Toriccodes. Denote the charges and fluxes in each copy as e1,2 andm1,2. Consider the condensationA = â¨m21e21e22,m2e1â©. The post-condensation topological order consists of the deconfinedanyons m1e1e32,m1e31e32 and m22, which are semion, anti-semion and boson respectively. Weconclude that the condensation leaves a twisted quantum double DÏ(Z2). Let us considerthe slab construction with the condensation A on the physical boundary. Following the samearguement as in Fig. 5.2, one can see that the Za2 subgroup of Za4 and the full Zb4 subgroup canbe localized to the edge. This corresponds to a gapped symmetry A = Za2ÃZb4, giving a group145extension is 1â Za2ÃZb4 â Za4ÃZb4 â Z2 â 1. However now the edge Z2 action and edge Z4action of A anti-commute. This property is similar to the symmetry action of a non-trivialZ2 Ã Z4-SPT. We conclude that this condensation corresponds to a gSPT with a non-trivialη â H2[Z2 Ã Z4, U(1)] â¼= Z2. Also, the action by both Za2 and Zb4 of A on the edge anti-commute with the G-action, i.e. a horizontal m1-string. Therefore this condensation/gSPTalso has a non-trivial ϵ factor.The condensation and the thin slab construction provide a complete description of thedual gSPT: the group extension, the edge modes and the emergent anomaly. As a consistencycheck, one can verify that Î·(âa, bâ) = (â1)a1b2 , Ïµ(âa, Î³â) = (â1)a1Î³1+a1Î³2ia2Î³1 is a solution to theconsistency conditions on (η, ϵ), and by inspection this solution gives a gSPT with the samestructure as that is given by the condensation A = â¨m21e21e22,m2e1â©.Notably this igSPT can be viewed as obtained by stacking the Z4-igSPT with a Z4 Ã Z4-SPT. In other words the A-SPT given by η can be cancelled by stacking it with a Î-SPT. Inthis case Î = Z4ÃZ4-SPTs have a Z4 classification. Take the generating phase, the pull-backof this Î-SPT to the subgroup A = Z2 ÃZ4 is exactly the non-trivial Z2 ÃZ4-SPT, thereforecancels the A-SPT of this igSPT upon stacking. The SPT+igSPT will still have the symmetryextension 1â Za2 ÃZb4 â Za4 ÃZb4 â Z2 â 1, but now the second factor Zb4 of Î plays no roleand the system is equivalent to the Za4-igSPT with an extra trivial Zb4-action.General consideration â Let us now consider the effect on the data (η, ϵ) of stacking withan SPT specified by Ï2 â H2[Î, U(1)]. For simplicity we assume all groups are abelian here,generalization to non-abelian groups is straightforward. The stacked system has symmetryÎSPT Ã ÎgSPT , but we are only interested in the diagonal symmetry Î and can assume thesymmetry is broken down to this diagonal subgroup by perturbations. Now the symmetryextension is unchanged by the stacking, because still only the A-subgroup acts trivially onthe gapless sector, and the quotient G = Î/A acts nontrivially on the gapless sector comingfrom the gSPT. The A-action is now modified by the Î-SPT, namely η is now modified byjâ(Ï2), where j is the inclusion Ajââ Î and jâ is the pull-back of Ï2 to A. Recall the factorϵ is the measurement of the charge of the edge A-actions under Î. An SPT also defines sucha factor: the edge of an a â A action carries charge Ï2(Î³,a)Ï2(a,Î³) under γ. Therefore the factor ϵis modified by iaÏ(Î³), where we used the slant product iaÏ(Î³) :=Ï2(Î³,a)Ï2(a,Î³). Notice the set ofdata (jâ(Ï2), iaÏ2(Î³)) satisfies the consistency conditions (5.10)-(5.12). The product of twosolutions is again a solution, therefore the data (Î·jâ(Ï2), ϵ(a, Î³)iaÏ2(Î³)) is a valid solutionand corresponds to the gSPT obtained by stacking the (η, ϵ)-gSPT with the Ï2-SPT. It canalso be shown by direct computation that two gSPTs differ by stacking with an SPT have thesame emergent anomaly.146In the Za4 Ã Zb4-gSPT example studied earlier, one can verify that choosing the Ï2 corre-sponding to the generating element of H2[Z4 Ã Z4, U(1)] will lead to a pull-back jâ(Ï2) thatis equal to the generating element of H2[Z2 Ã Z4, U(1)], therefore trivializing the η factor ofthe Za4 Ã Zb4-gSPT upon stacking, leaving an igSPT of SPT-pump type.5.6 Consistency conditions on the data η, ϵFirst of all we can compute UÎ³UËaUËbUâ1Î³ in two ways,UÎ³UËaUËbUâ1Î³ = η(a, b)UÎ³UËabUâ1Î³ = η(a, b)ϵ(ab, Î³)UËÎ³abÎ³â1 , (5.34)we also haveUÎ³UËaUËbUâ1Î³ = UÎ³UËaUâ1Î³ UÎ³UËbUâ1Î³ (5.35)= ϵ(a, γ)ϵ(b, Î³)UËÎ³aÎ³â1UËÎ³bÎ³â1= ϵ(a, γ)ϵ(b, Î³)Î·(Î³aÎ³â1, Î³bÎ³â1)UËab, (5.36)we concludeη(a, b)ϵ(ab, γ) = Î·(Î³aÎ³â1, Î³bÎ³â1)Ïµ(a, γ)ϵ(b, γ) (5.37)Secondly we can computeUÎ³1UÎ³2UËaUâ1Î³2 Uâ1Î³2= ϵ(a, Î³2)UÎ³1UËÎ³2aÎ³â12 Uâ1Î³1 (5.38)= ϵ(a, Î³2)Ïµ(Î³2aÎ³â12 , Î³1)UËÎ³1Î³2aÎ³â12 Î³â11 . (5.39)On the other handUÎ³1UÎ³2UËaUâ1Î³2 Uâ1Î³2= UÎ³1Î³2UËaUÎ³1Î³2 = ϵ(a, Î³1Î³2)UËÎ³1Î³2aÎ³â12 Î³â11 . (5.40)We conclude that ϵ satisfies:ϵ(a, γ1γ2) = Ïµ(Î³2aÎ³â12 , γ1)ϵ(a, γ2). (5.41)147Lastly, there are two ways of computing UÎ³UËaUâ1Î³ when γ = b is an element of A:UbUËaUâ1b = UËbUËaUËâ1b = η(b, a)UËbaUËâ1b= η(b, a)η(ba, bâ1)UËbabâ1=Î·(b, a)Î·(babâ1, b)UËbabâ1 , (5.42)where we used η(ba, bâ1) = 1/Î·(babâ1, a) which comes from cocycle condition on η (we assumecocycles are normalized so that η(1, a) = η(a, aâ1) = 1). Since we also haveUbUËaUâ1b = ϵ(a, b)UËbabâ1 , (5.43)we conclude that ϵ(a, b) = Î·(b,a)Î·(babâ1,b) , âa, b â A.5.7 Connection to the LHS spectral sequenceHere we show that the consistency conditions (5.10)-(5.12) as well as the emergent anomaly (5.22)have the structure of the LHS spectral sequence. We focus on computation aspects of the LHSsequence and work with cochain-level expressions. We also assume the group A is abelian inthis appendix.Consider a 3-cocycle of G: Ï3 â Z3[G,U(1)] that is trivialized via a group extension1â A jââ Î pââ Gâ 1. (5.44)This means that there is some 2-cochain of Î: α â C2[Î, U(1)], such that dα((a1, g1), (a2, g2), (a3, g3)) =Ï(g1, g2, g3). Here we have written elements of Î as pairs (a, g), a â A,G â G. We also denotethe action of G on A by ga := gagâ1.The group extension (5.44) specifies an extension classe2 â Z2[G,A]. By modifying with coboundaries one can always decompose the 2-cochain αinto the following form [125]:α((a, g), (b, h))= F 0,2(g¯h¯a,h¯ b)F 0,2(g¯h¯e2(g, h),gh ah¯b)F 1,1(g¯a, h)F 2,0(g, h), (5.45)where F 0,2 â C2[A,U(1)], F 1,1 â C1[A, C1[G,U(1)]], F 2,0 â C2[G,U(1)]. Requiring that148dα((a, g), (b, h), (c, k)) only depends on g, h, k results in the following conditions:δ0F0,2 = 1, (5.46)δ1F0,2δ0F1,1 = 1, (5.47)δ2F0,2δ1F1,1δ0F0,2 = 1. (5.48)Here the differentials δi are defined as follows. δ0 is the cochain differential with respect toA: δ0 = dA, δ1 is the cochain differential with respect to G: δ1 = dG, andδ2F0,2(a, g, h) :=F 0,2(a, e2(g, h))F 0,2(e2(g, h), a). (5.49)Once these conditions are met, the trivializing relation Ï(g, h, k) = dα((a, g), (b, h), (c, k))becomesÏ(g, h, k) = δ3F0,2δ2F1,1δ1F2,0 (5.50)where δ3, δ2 are defined byδ3F0,2(g, h, k) :=F 0,2(e2(g, hk),g e2(h, k))F 0,2(e2(gh, k), e2(g, h))(5.51)δ2F1,1(g, h, k) := F 1,1(e2(g, h), k) (5.52)Since δ1 = dG, we see that on the cocycle level we can neglect F2,0 and Ï â¼= δ3F 0,2δ2F 1,1 asG-cocycles. Now we can identify F 0,2(a, b) = η(b, a) and F 1,2 = ϵ(a, g). The condition that ηis a 2-cocycle of A is the same as δ0F0,2 = 1. The consistency condition (5.10) isϵ(a, g)ϵ(b, g)ϵ(ab, g)=η(a, b)η(ga,g b)(5.53)which is the same as dAϵdGη = 1, or δ0F0,2δ1F1,1 = 1. Lastly using the condition (5.11),ϵ(ha, g)ϵ(a, h) = ϵ(a, ghe2(g, h)) = ϵ(a, gh)η(a, e2(g, h))η(e2(g, h), a)(5.54)which is the same as δ1F1,1δ2F0,2 = 1. The emergent anomaly (5.22) can be readily seen tobe the same as δ3F0,2δ2F1,1. Since now dÎÎ± = δ3F0,2δ2F1,1, this means the anomaly (5.22)is the differential of the 2-cochain α, therefore is trivialized by the group extension.1495.8 The bulk-edge correspondence for 1+1D gSPTsA 1+1D gSPT is defined by a group extension (5.44) and a pair (η, ϵ). The emergent anomalyis determined by (η, ϵ) via (5.22). This can be viewed as an edge to bulk map: the edge modesdetermines the bulk emergent anomaly. To show the converse the question is equivalent tofinding η, ϵ for a given emergent anomaly. To start with, one finds a trivializing cochain: dα =Ï, α â C2[Î, U(1)]. From the decomposition (5.45) we see that α((a, 1), (b, 1)) = F 0,2(a, b).Formally this means the pull back of α by j is a 2-cocycle of A: djâ(Î±) = jâdÎ± = jâpâÏ =(pâ¦ j)âÏ = 1, where we used (pâ¦ j)â = 1 because the sequence (5.44) is exact. Similarly, otherterms in the decomposition F 1,1, F 0,2 can be extracted from α by the Lyndonâs algorithm [80].For exampleϵ(a, g) = F 1,1(a, g) = iaα(g) :=α((a, 1), (1, g))α((1, g), (a, 1)). (5.55)Therefore ϵ, η can be reconstructed once the trivializing cochain α is found. However, thechoice of the trivializing cochain is not unique: if dα = dÎ±â² = Ï, then d(Î±Î±â²â1) = 1. Thus anytwo Î-cochains α, Î±â² that differ by a Î-cocycle give the same emergent anomaly Ï. There-fore the data η, ϵ can only be determined from Ï up to a 2-cocycle of Î. Consider anothertrivializing cochain Î³â² = γδ, where δ â Z2[Î, U(1)]. Then the data Î·â², Ïµâ² associated with Î³â² isÎ·â² = jâ(Î±â²) = jâÎ± · jâÎ´ = η · jâÎ´, (5.56)Ïµâ²(a, g) = iaÎ±â²(g) = iaα(g) · iaδ(g) = ϵ(a, g)iaδ(g) (5.57)Recall from section 5.5.3 that stacking a gSPT defined by (η, ϵ) with an SPT defined byδ â Z2[Î, U(1)] results in a new gSPT with Î·â² = η · jâÎ´, Ïµâ²(a, g) = ϵ(a, g) · iaδ(g). Comparingwith (5.56)-(5.57), we see that the new data Î·â², Ïµâ² associated with the new trivializing cochainÎ±â² differs from the original data η, ϵ associated with α by stacking with a Î-SPT given by δ.Therefore we conclude that for a given emergent anomaly Ï, the gSPT is only determinedby Ï up to stacking with Î-SPTs. The fact that stacking with SPTs does not change theemergent anomaly may not very surprising, given that SPTs do not have bulk anomalies,what is surprising is that all gSPTs with the same emergent anomaly are obtained in thisway. Therefore the gSPT bulk-edge correspondence can stated as follows: The edge modesof a gSPT determine the bulk emergent anomaly, and the bulk emergent anomaly determinesthe edge modes of the gSPT up to stacking with SPTs.Now with the help of the bulk-edge correspondence we can prove that the two definitionsof igSPT are equivalent: 1. A gSPT is intrinsic if the edge modes can not be trivialized by150stacking with an SPT. 2. A gSPT is intrinsic if it has a nontrivial emergent anomaly. 1â2: if the emergent anomaly is trivial: Ï = 1, then η0 = ϵ0 = 1 is clearly a solution to theconsistency conditions. By the bulk-edge correspondence any other solution with the sameÏ = 1 must differ from η0 = ϵ0 = 1 by stacking with a Î-SPT. Therefore if the gSPT hasnon-trivial edge modes, η Ì¸= 1 and/or ϵ Ì¸= 1, they can be eliminated by stacking with a Î-SPT.2 â 1: If the edge modes of a gSPT can be trivialized by stacking with a Î-SPT, then itmeans the data η, ϵ differs from the tivial one η0 = 1, ϵ0 = 1 by stacking with a Î-SPT. Sincestacking with Î-SPTs does not change the emergent anomaly, the data η, ϵ gives the sameemergent anomaly as that is given by the trivial one, Ï = 1. â¡5.9 Equivalence between the post-condensation twistand gSPT emergent anomalyIn the computation of emergent anomaly, we defined UgUh = â¦(g, h)Ugh. Alternatively onecan define UgUh = Ughâ¦â²(g, h). Similarly â¦â²(g, h) = â¦â²L(g, h) â â¦â²R(g, h) The associativitycondition on â¦â² isâ¦â²(gh, k)Uâ1k â¦â²(g, h)Uk = â¦â²(g, hk)â¦â²(h, k) (5.58)Restrict to â¦â²L, the associativity condition is then only satisfied up to a phase,Ïâ²(g, h, k)â¦â²L(gh, k)Uâ1k â¦â²L(g, h)Uk = â¦â²L(g, hk)â¦â²L(h, k) (5.59)The â¦â²(g, h) is related to â¦(g, h) via â¦â²(g, h) = U â ghâ¦(g, h)Ugh. Using this relation one canshow that Ïâ² = Ï:â¦â²L(gh, k)Uâ1k â¦â²L(g, h)Uk= Uâ1ghkâ¦L(gh, k)UghkUâ1k Uâ1gh â¦L(g, h)UghUk= Uâ1ghkâ¦L(gh, k)â¦L(g, h)Ughk (5.60)andâ¦â²L(g, hk)â¦â²L(h, k) = Uâ1ghkâ¦L(g, hk)UghkUâ1hk â¦L(h, k)Uhk= Uâ1ghk(â¦L(g, hk)Ugâ¦L(h, k)Uâ1g )Ughk. (5.61)151Comparing with (5.17), one sees that Ïâ² = Ï. Next notice that â¦â²L(g, h) = U â ghâ¦(g, h)LUgh =U â ghUËe2(g,h)Ugh = ϵ(e2(g, h), gh)UËe2(g,h). Therefore replacing â¦â²L(g, h) by UËe2(g,h) in (5.59) willonly affect the cocycle Ï by a coboundary (since every â¦â²L(g, h) factor only gets modified bya phase factor ϵ(e2(g, h), gh)). Therefore we can take â¦â²L(g, h) = UËe2(g,h) in (5.59). Write(5.59) asÏ(g, h, k)UËe2(gh,k) = UËe2(g,hk)UËe2(h,k)Uâ1k (UËe2(g,h))â1Uk. (5.62)Use UËaUËb = η(a, b)UËab, together with UgUËaUâ1g = ϵ(a, g)UËgagâ1 , we arrive at the expressionÏ(g, h, k) = ϵ(e2(g, h), kâ1)Ã η(e2(g, hk), e2(h, k))η(e2(g, hk)e2(h, k), kâ1e2(g, h)â1k) (5.63)which is the twist in (5.30). Therefore the twist (5.30) and the anomaly (5.22) are equivalentas cocycles of G.152Chapter Six2+1D Phases with Symmetries andSymTFT6.1 Introduction6.1.1 Gapped 2+1-D phasesWe study gapped 2+1-D phases with bosonic fusion 2-categorical symmetries via the sandwichconstruction. We formulate the interval compactification of the sandwich as a relative tensorproduct, which allows us to freely compute properties of the obtained 2+1-D gapped phase.A class of gapped boundaries of the SymTOis called minimal gapped boundaries, whichphysically correspond to those gapped boundaries with no intrinsic topological order. For thecase where the symmetry and the physical boundaries are both minimal gapped boundaries,we give a complete description of the obtained 2+1-D gapped phase using tools from higherlinear algebra, including the underlying topological order and the symmetry action. We findthat the obtained 2+1-D gapped phase generally has multiple vacuua, and each vacuua hostsa 2+1-D Dijkgraaf-Witten theory. We also study concrete examples where the SymTOisZ1[2VecZ2 ], Z1[2VecS3 ] or Z1[2VecS4 ]. The first two cases have been discussed recentlyin [8, 9], although through different methods. The case Z1[2VecS4 ] is previously unknownand contains some new features. In particular, we findâ¢ It admits a non-invertible minimal symmetry boundary that is neither a 2-group symme-try 2VecÏG nor a 2-representation symmetry 2Rep(G). In this sense it is intrinsicallynon-invertible.153â¢ This symmetry admits a partial symmetry-broken phase with two vacuua, with onevacuum having the S3-gauge theory and the other having the Z2-gauge theory.â¢ This symmetry admits two families of SPT phases that are not gauge-related to any singlefamily of invertible SPTs. These SPTs are in this sense intrinsically non-invertibleSPTs.6.1.2 Gapless 2+1-D phasesTo date, little research has focused on topological holography for 2+1-D gapless phases,and even in the case of invertible symmetries, a systematic understanding remains elusive.This is largely attributable to the complexity of condensable algebras in 3+1-D topologicalorders. In this chapter we take on this challenge and establish a rigorous duality betweencondensable algebras in the SymTOZ1[2VecG] and 2+1-D G-gSPTs. To achieve this, weneed to understand both sides of the duality systematically.The condensable algebras1 in Z1[2VecG] are described by G-crossed braided fusion cate-gories. However we do not expect all of them to correspond to 2+1-D G-gSPTs. From theclub-sandwich construction, certain condensable algebra will induce (2+1-D)topological orderon the domain wallM. Such a club-sandwich describes a gapless 2+1-D system whose gappedsector has nontrivial topological order. This leads to the intriguing concept of âgapless sym-metry enriched topological orderâ(gSET). Naturally, we define a 2+1-D gSPT to be a specialkind of gSET whose gapped sector has no topological order. This is completely in parallel withthe gapped SET/SPT relation. This class of condensable algebras in Z1[2VecG] that we ex-pect to be dual to 2+1-D G-gSPT are called minimal condensable algebras 2. Mathematically,such a condensable algebra is a G-crossed braided fusion category B whose trivial componentBe is trivial. Equivalently, we may say B is a G-crossed extension of Vec. However, thegrading of B is generally not faithful but only supported on a normal subgroup N âG. Thegrading of the condensable algebra is faithful precisely when the algebra is Lagrangian. Theclass of minimal condensable algebras supported on N is then denoted as GCrExt(Vec, N),which stands for âG-crossed extensions of Vec with support Nâ. This is the key subject onthe SymTOside: Elements of it are presumably dual to 2+1-D G-gSPTs. We reveal a spectralsequence structure of minimal condensable algebras:Theorem 6.1.1: Elements in GCrExt(Vec, N) are parameterized by Ï â H3[N,CÃ] suchthat d2Ï = 0, d3Ï = 0 together with a solution (γ, p)3. Denote the corresponding condensable1More precisely condensable E2-algebras, which describe condensation of strings.2The Lagrangian case exactly corresponds to the minimal gapped boundaries, hence the name.3Each condition diÏ = 0 means certain equation defined on the E2-page has a solution. See154algebra as Vec(Ï,Î³,p)N , the condensed phase isMod0Vec(Ï,Î³,p)N(Z1[2VecG]) â Z1[2Vecd4ÏG/N ].Here di are the differentials in the LyndonâHochschildâSerre(LHS) spectral sequence, andby d4Ï we mean a representative of it in H4[G/N,CÃ] computed using the solution (γ, p).Furthermore, GCrExt(Vec, N) is naturally an abelian group and we prove an exactsequence relating it to ordinary group cohomology.Theorem 6.1.2: There is an exact sequenceH3[G/N,CÃ] pâ3ââ H3[G,CÃ] Rââ GCrExt(Vec, N) O4âââ H4[G/N,CÃ] pâ4ââ H4[G,CÃ].Here pâ3, pâ4 are induced by projection p : Gâ G/N .If we accept that minimal condensable algebras are dual to G-gSPTs(which we will provelater), the above sequence can be written in physical terms asSPTG/N3D â SPTG3D â gSPT(G,N)3D â SPTG/N4D â SPTG4D.This in fact has simple physical interpretation. Take the right half of the sequence asan example. For a G-gSPT, the IR symmetry G/N may be anomalous, this anomaly is theimage in SPTG/N4D . However this anomaly is an emergent one, meaning that it trivializeswhen pulled back to G. Conversely, for every such emergent anomaly, one can construct aG-gSPT a` la Wang-Wen-Witten [123]. This means the right half of the sequence is exact. Thephysical meaning of the rest part of the sequence is also straightfoward and will be explainedin Sec. 6.8.This exact sequence allows us to compute the classification G-gSPTs. The exactness ofthe sequence implies 4gSPTG,N3D = H3[G,CÃ]/im(pâ3)ï¸¸ ︷︷ ︸weak gSPTÃ ker(pâ4)ï¸¸ ︷︷ ︸emergent anomaly(6.1)The first factor comes from H3[G,CÃ] and physically corresponds those gSPTs that areobtained by tuning a gapped G-SPT to a G/N -critical point. Historically, these were theearliest known examples of gSPTs [94, 76] and were often referred to as weak or non-intrinsic.The reason for the quotient by im(pâ3) is that, if a G-SPT comes from a G/N -SPT, then as itSec. 6.8.2 for more details.4This is an equality as sets.155G N gSPTG,N3D igSPT?Z4 Z2 Z4 ÃZpq Zp Zp(p,q) ÃZa2 Ã Zb4 Zb2 (Z2 Ã Z4)Ã Z2âZan1ÃZbpq Zbp(Zp(p,q) Ã Z(p,n1))ÃZ(p,,q,n1)iff (p, q, n1) Ì¸= 1S3 Z3 Z3 ÃD4 Z2 Z4 Ã Z22âTable 6.1: Classification of 2+1-D gSPT with exact symmetry G and gapped symmetryN and whether the symmetry allows for igSPT.is tuned to a G/N -critical point, all the topological defects from the G/N -SPT are lost andit can not be distinguished from a trivial gSPT. The second factor comes from H4[G/N,CÃ],it captures precisely the emergent anomalies of G-gSPTs.Readers familiar with the decorated domain wall(DDW) construction of SPTs may havenoticed the similarity between the spectral sequence structure of GCrExt(Vec, N) and thestructure of 2+1-D G-SPT. Indeed, in the standard DDW construction for a 2+1-D G-SPTwith N â G a normal subgroup, a decoration pattern is determined by a Ï â H3[N,CÃ]such that d2Ï = 0, d3Ï = 0, d4Ï = 0 together with a solution. However, as we argue inSec. 6.9, for a 2+1D G-gSPT we no longer demand d4Ï = 0. This is because the quotientG/N acts faithfully on the gapless degrees of freedom and is allowed to be anomalous. Thed4Ï obstruction is in fact exactly the emergent anomaly of the gSPT. Therefore, the structureof GCrExt(Vec, N) matches precisely with the structure of 2+1-D G-gSPTs with gappedsymmetry N . Furthermore, the twist of the condensed phase, d4Ï, is exactly the emergentanomaly of the 2+1-D G-gSPT. This establishes a rigorous correspondence between mini-mal condensable algebras in the SymTOZ1[2VecG] and 2+1-D G-gSPTs, which naturallycategorifies the results in [131]. We summarize it as the following physical theorem.Theoremph 6.1.1: A 2+1D G-gSPT with gapped symmetry N and domain wall decorationdata Ï, γ, p corresponds to the minimal condensable algebraVec(Ï,Î³,p)N in the SymTOZ1[2VecG].The emergent anomaly of the G-gSPT is equal to the twist of the condensed phase.156Using the formula Eq. (6.1), we compute the classification of G-gSPTs for several G andN . The results are summarized in Table. 6.1.Finally, we exploit the SymTOframework to construct 2+1D gapless phases with general-ized symmetries. Using the club-sandwich construction, we construct 2+1-D igSPTs with 2-group or non-invertible symmetries, and emergent anomalous 2-group symmetry 2VecÏZ[0]2 ÃZ[1]2.6.2 Bosonic fusion 2-categories and their SymTFTThe full definition of fusion 2-category can be found in [25]. Fusion 2-categories come in twotypes, called bosonic and fermionic. For any fusion 2-category C, its looping â¦C is defined asâ¦C := HomC(1,1), (6.2)where 1 is the monoidal unit of C. This is always a braided fusion category. We may thenexamine its Mu¨ger center Z2(â¦C), which is a symmetric fusion category. We say the fusion2-category C is bosonic if Z2â¦C is Tannakian(isomorphic to Rep(G)), and fermionic if Z2â¦Cis super-Tannakian(isomorphic to Rep(G, z) for some super group (G, z)). In this chapter,we are only concerned with bosonic fusion 2-categories. In 2+1-D, a general finite internalsymmetry is described by a fusion 2-category. The SymTOof a fusion 2-category is given byits E1-center(also known as the Drinfeld center):Z1[C] := EndCâC(C), (6.3)which is a non-degenerate braided fusion 2-category, i.e. a 3+1D topological order. Someexamples of fusion 2-categories that will appear In this chapter are listed below.Example 6.2.1: â¢ For any braided fusion category B, we denote by ΣB :=Mod(B) the2-category of finite semi-simple left B-module 2-categories. This is naturally a fusion2-category with monoidal product given by relative tensor product over B [25].â¢ For any 2-group G, the fusion 2-category of G-graded finite semi-simple categories is de-noted as 2VecG . One can also twist the monoidal structure by any class Ï â H4[BG,CÃ],the resulting fusion 2-category is denoted as 2VecÏG .â¢ For any 2-group G, the fusion 2-category of 2-representations of G is denoted as 2Rep(G).For any bosonic fusion 2-category C, its E1-center must be a Dijkgraaf-Witten theory,i.e. a (potentially twisted) finite gauge theory, Z1[2VecÏG] [30]. The complete braided fusion2-category structure of Z1[2VecÏG] was first studied in [57].157Fusion 2-categories have been classified [31]. We summarize here the classification in thebosonic case.Fact 3: Bosonic fusion 2-categories are parameterized byâ¢ A finite group G.â¢ A class Ï â H4[G,U(1)].â¢ A subgroup H < G up to conjugation.â¢ A non-degenerate braided fusion category B with H-action Ï.â¢ A class Ï â C3[H,U(1)] such that dÏ = O4(Ï)Ï|â1H , where O4(Ï) is the anomaly of theH-action Ï.The corresponding fusion 2-category will be denoted as C[G,H, Ï, C, Ï]. Among these data,G, Ï determines the SymTO: Z1[C[G,H, Ï, C, Ï]] â Z1[2VecÏG]. In other words, any fusion2-category of the form C[G,H, Ï, C, Ï] can be realized as a gapped boundary of Z1[2VecÏG].Below we discuss how the examples discussed before fit into this classification.Example 6.2.2: Let B be a braided fusion category. ΣB is a bosonic fusion 2-categoryif and only if Z2B is Tannakian. Say Z2B = Rep(G), then the de-equivariantization BGis a non-degenerate braided fusion category with a natural G-action. Then we have ΣB âC[G,G, Ï,BG, Ï], where Ï is cohomologous to the O4âanomaly of the G-action on BG and Ïis an arbitrary cochain such that dÏ = O4Ïâ1.Example 6.2.3: Let G = A[0]âH [1] be a split finite 2-group, then 2VecG â C[AÌâH, AÌ, 1,Vec, 1]and 2Rep(G) â C[AÌâH,H, 0,Vec, 0], where AÌ is the Pontryagin dual of A. The general(non-split) case can be found in [34].6.3 Topological holography for 2+1-D gapped phasesIn this section we discuss generalities of topological holography for 2+1-D gapped phases withfusion 2-categorical symmetries. In the gapped scenario, most constructions can be rigorouslydefined and computed. We begin by reviewing the definition of 2+1D gapped phases withsymmetry. We then formulate the sandwich construction in terms of relative tensor product.When the symmetry boundary and physical boundary are Morita equivalent, the sandwichadmits a particularly simple description as a 3-representation of the symmetry 2-category.1586.3.1 Gapped 2+1-D phases with symmetryA bosonic gapped 2+1-D phase is determined by its topological order up to stacking with theinvertible E8-state5. We recall the classification of 2+1-D topological orders:Fact 4 ([47]): A 2+1-D topological order(with possible local degeneracy) is a multi-fusion2-category T such that Z1[T] = 2Vec.Physically, T describes all the topological defects of the topological order and their fusion.The condition Z1[T] = 2Vec is an anomaly-free condition: the topological order can berealized by a lattice model in pure 2+1-D. If there is no local degeneracy, then T is fusion,and the anomaly-free condition implies that T = ΣB for some non-degenerate braided fusioncategory B [47].We have now defined symmetries and gapped phases in 2+1-D. The definition of a 2+1-Dgapped phase with symmetry is as simple as it gets.Definition 6.3.1: A 2+1-D gapped phase T with symmetry C is a (linear monoidal 2-)functor:F : Câ T. (6.4)To see the correctness of this definition, we recall some familiar examples.Example 6.3.1: 1. If the underlying topological order is trivial: C = 2Vec, and the sym-metry is a a finite group 0-form symmetry 2VecG, then a functor F : 2VecG â 2Vec isexactly defined(up to equivalence) by a class Ï â H3[G,CÃ]. We recover the cohomologyclassification of 2+1-D G-SPTs.2. More generally, if the symmetry is a non-anomalous 2-group symmetry C = 2VecG , thena linear monoidal functor F : 2VecG â 2Vec is the same as a homotopy class of mapBG â B3CÃ, i.e. a class in H3[G,CÃ]. We recover the cohomology classification of2+1-D G-SPTs.3. If the underlying topological order has no local GSD, i.e. takes the form ΣB, and thesymmetry is a finite group 0-form symmetry 2VecG, then a functor F : 2VecG â ΣB isthe same as a homotopy class of mapsF : BGâ BPic(B) (6.5)5In the rest of this chapter, we will neglect the subtly caused by the invertible E8-state and viewgapped phase and topological order as synonymous.159where Pic(B) is the Picard 3-group of B 6. Such maps are in 1-1 correspondence withG-crossed extensions of B. This is exactly the classification of G-symmetry enrichedtopological orders(SET) in 2+1-D.4. More generally, if the finite group 0-form symmetry is anomalous, then a functor F :2VecÏG â ΣB describes a 2+1-D SET with an anomalous G-symmetry.The examples considered above all have invertible symmetries. In general it is diffi-cult to construct 2+1-D topological orders with non-invertible symmetries systematically.SymTOprovides an efficient method for achieving this.6.3.2 The sandwich constructionTo construct phases with symmetry C via SymTO, we start with the 3+1D topological orderZ1[C], and put it on a sandwich geometry with two boundaries, dubbed symmetry boundary,Bphys, and physical boundary Bphys. The topological order Z1[C] has a canonical gappedboundary condition, where the topological operators form exactly C. This is chosen as theboundary condition at Bsym. Upon interval compactification, the sandwich reduces to a 2+1-D system with symmetry C. We may informally write this 2+1-D system asBphys â Z1[C] Bsym = Bphys â Z1[C] C, (6.6)and the symmetry C simply acts by mapping into the second factor:F : C1â idCâââââ Bphys â Z1[C] C. (6.7)However we have not given Eq. (6.6) a precise mathematical meaning. For constructinggapped 2+1-D phases with symmetry C, we should consider Bphys to be gapped as well. Inthis case, we may adopt the following description of gapped boundaries of 3+1D topologicalorders [59, 60]: a gapped boundary of Z1[C] is defined by a fusion 2-category D together witha braided equivalence Ï : Z1[C] â Z1[D]. This makes the gapped boundary D into a so-called E1-fusion module 2-category over Z1[C]. Then the expression Eq. (6.6) is precisely therelative tensor product between left and right E1-fusion module 2-categories over Z1[C] 7 [63].Physically, the topological defects in the bulk can act on both the left and the right boundary.After interval compactification, these actions should be identified.The dimension reduction of a gapped sandwich is particularly simple if Bsym and Bphys6See Appendix A.3 for definition.7See Appendix A.1 for definition.160happen to be Morita equivalent 8. In this case, we may writeBphys = CâM := EndC(M) (6.8)for some (right)C-module 2-category M, and CâM is the dual fusion 2-category with respectto M. Physically M describes a gapped domain wall between Bphys and Bsym = C. Then wehaveBphys â Z1[C] Bsym = EndC(M)â EndCâC(C) EndC(C)= End(Mâ C C) = End(M). (6.9)The symmetry C then acts on End(M) according to the C-module structure on M:F : Câ End(M), a 7â ââ a. (6.10)Therefore the symmetry C simply acts on the 3-vector space M by endomorphism. Thisis a direct 3-categorification of 0+1D phases with symmetry, which are vector spaces withsymmetry acting by endomorphism. This is the most general form of a non-chiral 2+1-Dtopological order with symmetry.The interval compactification Eq. (6.9) in fact directly follows from the bulk-boundaryrelation of 2+1-D topological orders: if the physical boundary is Morita equivalent to thesymmetry boundary, then we can form a gapped domain wall M between them, see Fig. 6.1,then upon interval compactification the sandwich becomes the 2+1-D bulk of the 1+1-Dboundary M. By the bulk-boundary relation in 2+1-D [59, 63], the sandwich must be thecenter of the boundary: Z0(M) = End(M). We will study several examples of gapped phasesobtained through this construction in the next section.6.4 2+1-D Gapped phases with generalized symme-tries from SymTOIn this section we apply the SymTOsandwich construction to study 2+1-D gapped phaseswith generalized symmetries. By our discussion in Sec. 6.2, possible gapped boundaries ofZ1[2VecÏG] are fusion 2-categories of the form C[G,H, Ï, C, Ï]. Here we focus on the caseC = Vec, and we will call the corresponding gapped boundary a minimal gapped boundary.Physically, such gapped boundaries have no intrinsic topological order. As we shall see,8Morita theory of fusion 2-categories is reviewed in Appendix A.1161BphysCâMBsymCZ1[C] âMCâM CZ1[C] âMZ0(M) = End(M)Figure 6.1: If the physical and symmetry boundaries happen to be Morita equivalent,then we can bend them to make a a gapped domain wall M. Therefore the dimensionreduction of the sandwich is the bulk of the domain wall M. By âbulk=center ofboundaryâ, we see that the dimension reduction of the sandwich must be Z0(M) =End(M). The symmetry C then acts on End(M) according to the module 2-categorystructure on M.minimal gapped boundaries already give a large class of interesting 2+1-D gapped phases.We derive the general structure of a sandwich construction with minimal gapped symmetryand physical boundaries, including the underlying topological order and the symmetry action.We then study several concrete examples.6.4.1 Sandwiches with minimal gapped boundariesFor a minimal gapped boundary of Z1[2VecÏG], we will neglect the trivial factor C = Vec anddenote the fusion 2-category of topological defects on the boundary as C[G,H, Ï, Ï]. Suchfusion 2-categories are called group-theoretical fusion 2-categories in [34]. They cover a widerange of fusion 2-categories, including 2VecG , 2Rep(G) for any 2-group G. By the resultsin [34], C[G,H, Ï, Ï] is Morita equivalent to 2VecÏG. Specifically, we haveC[G,H, Ï, Ï] â BimodVecÏHâVecÏH (2VecÏG). (6.11)where VecÏH is viewed as a condensable E1-algebra in 2VecÏG9. A more explicit descriptionof this fusion 2-category is [34]C[G,H, Ï, Ï] = â[g]âHG/HÎ£VecÎ¾gHâ©gH , (6.12)9For definition of condensable E1-algebra, see Appendix A.1.162where HG/H is the set of double cosets and ξg is a 3-cocyle constructed from Ï and Ï. Inthe following we will need the more general case, [34]BimodVecÏHâVecÏK (2VecÏG) = â[g]âHG/KÎ£VecÎ¾gHâ©gK , (6.13)ξg(h1, h2, h3) :=Ï(h1, h2, h3)Ï(gâ1hâ13 g, gâ1hâ12 g, gâ1hâ11 g)Ã Ï(h3, hâ13 g, gâ1hâ12 g, gâ1hâ11 g)Ï(g, gâ1hâ13 g, gâ1hâ12 g, gâ1hâ11 g)Ï(h2, h3, hâ13 hâ12 g, gâ1hâ11 g)Ï(h1, h2, h3, hâ13 hâ12 g).Denote the 2-category of modules over VecÏH in 2VecÏG as MH,Ï := ModVecÏH (2VecÏG),we haveEnd2VecÏG(MH,Ï) = BimodVecÏHâVecÏH (2VecÏG). (6.14)Now consider a sandwich with a Z1[2VecÏG] bulk, a symmetry boundary C[G,H, Ï, Ï] anda physical boundary C[G,K, Ï, Ï]. Then the interval compactification of the sandwich can becomputed as follows,T = Bsym â SymTO Bphys= C[G,H, Ï, Ï]â Z1[2VecÏG] C[G,K, Ï, Ï]= End2VecÏG(MH,Ï)â Z1[2VecÏG] End2VecÏG(MK,Ï). (6.15)Using Eq. (7.9) and then Eq. (7.3), this is equal toT = End(MH,Ï â 2VecÏG MK,Ï) (6.16)= End(BimodVecÏHâVecÏK (2VecÏG)) (6.17)Now using Eq. (6.13), this is equal toT = End( â[g]âHG/KÎ£VecÎ¾gHâ©gK)= â[g1],[g2]Func[Î£VecÎ¾g1Hâ©g1K ,Î£VecÎ¾g2Hâ©g2K ]= â[g1],[g2]BimodVecÎ¾g1Hâ©g1KâVecÎ¾g2Hâ©g2K(2Vec) (6.18)From this, we can read off the structure of the underlying topological order of the sandwich:â¢ In general it has multiple vacuua. This happens when the double coset HG/K contains163more than one element, which means HK Ì¸= G. Physically, the local vacuum degeneracycomes from spontaneous breaking of 0-form symmetries. The number of vacuua is equalto the cardinality of the double cosetGSD = |HG/K|. (6.19)â¢ Each vacuum has topological orderT[g],[g] := BimodVecÎ¾gHâ©gKâVecÎ¾gHâ©gK(2Vec) = Î£Z1[VecÎ¾gHâ©gK ], (6.20)i.e. a 2+1-D H â© gK-gauge theory with twist ξg. Since there is no intrinsic topologicalorder on the symmetry or the physical boundary, the topological order of the sandwichcomes from spontaneous breaking of (possibly non-invertible)1-form symmetries.â¢ The 2-category of domain walls between vacuum [g] and vacuum [h] isT[g],[h] := BimodVecÎ¾gHâ©gKâVecÎ¾hHâ©hK(2Vec) (6.21)This is exactly the 2-category of ALL domain walls between Z1[VecÎ¾gHâ©gK ] and Z1[VecÎ¾hHâ©hK ] [53].â¢ The fusion rule of domain walls is given by taking relative tensor product: forM[g],[h] âT[g],[h], M[h],[k] â T[h],[k], we haveM[g],[h] âM[h],[k] :=M[g],[h] â VecÎ¾hHâ©hKM[h],[k]. (6.22)The symmetry category is C = BimodVecÏHâVecÏH (2VecÏG), the symmetry action functorF : Câ T is given by relative tensor product: For any symmetry generator D â C,F(D) = D â VecÏH â â End(BimodVecÏHâVecÏK (2VecÏG)). (6.23)6.4.2 Anomaly of fusion 2-categorical symmetryThe above analysis allows us to determine whether a general fusion 2-categorical symmetryis anomalous. Recall that a fusion 2-categorical symmetry C is called non-anomalous if thereis exists a trivial gapped phase with symmetry C. This means there is a fiber 2-functorCâ 2Vec. (6.24)Using the sandwich construction, this means there exists a gapped physical boundary of164Z1[C], such that the sandwich is a trivial 2+1-D topological order with no local degeneracy.If the symmetry boundary itself has intrinsic 2+1-D topological order, then the sandwichmust have nontrivial topological order. Therefore for C to be non-anomalous, the symmetryboundary itself must be a minimal gapped boundary. Similarly the physical boundary mustbe a minimal gapped boundary for the sandwich to have trivial topological order. Then byour analysis before, the sandwich is a trivial topological order if and only if1. GSD = |HG/K| = 1, meaning that G = HK.2. The topological order Z1[VecÎ¾eHâ©K ] is trivial, meaning that H â©K = {e}.This leads to the following.Corollary 6.4.1: A bosonic fusion 2-categorical symmetry C[G,H,B, Ï, Ï] is non-anomalousif and only B = Vec, and there exists K < G such that H â©K = {e} and HK = G.This result is known in mathematics literature [34].Example 6.4.1: Let G = Z4, H = Z2, then the symmetry C[Z4,Z2, 0,Vec, 0] is anomalous.In fact it is equivalent to the anomalous 2-group symmetry2VecÏZ[0]2 ÃZ[1]2 (6.25)with a nontrivial class Ï â H4[Z[0]2 Ã Z[1]2 ,CÃ] [34].6.4.3 Example: Gapped phases from Z1[2VecZ2]Topological holography with SymTOZ1[2VecZ2 ] has been studied in [8]. Although this ex-ample is relatively simple, it serves to illustrate the general strategy.For Z1[2VecZ2 ], there are three minimal gapped boundaries:â¢ One with H = {1}. This is often called the rough boundary, denoted as Be.â¢ One with H = Z2 and Ï = 0. This boundary condenses the flux m. This is often calledthe smooth boundary, denoted as Bm.â¢ One with H = Z2 and a nontrivial twist Ï â H3[Z2, U(1)]. This boundary also condensesthe flux, but the endpoint of a flux is a semion [140]. This is called the twisted smoothboundary, denoted as BÏm.Bsym = Be, Bphys = Be â This corresponds to H = K = {1}. The double cosetHZ2/K = {[1], [g]}. By Eq. (6.12), the symmetry category isC = â1,gÎ£Vec = 2VecZ2 , (6.26)165a 0-form Z2-symmetry. By Eq. (6.18), the underlying topological order of the sandwich isT = End(â1,gÎ£Vec) = â i,j=1,gΣVec. (6.27)In other words, there are two vacuua, each is in a trivial gapped phase. In the symmetrycategory 2VecZ2 , there is only one nontrivial symmetry generator, denoted as Dg, with Dg âDg = 1. By Eq. (6.23), it is mapped to the operatorF(Dg) = Vecg â â =(0 VecVec 0)â End(â1,gÎ£Vec). (6.28)We have F(Dg)(Vec1) = Vecg,F(Dg)(Vecg) = Vec1. In other words, the two vacuua areexchanged under the action of the Z2-symmetry. We conclude that this sandwich describes aZ2-SSB gapped phase.Bsym = Bm, Bphys = Bm â This corresponds toH = K = Z2, the double cosetHG/K ={[1]} and H â©K = Z2. The symmetry boundary is described as follows.â¢ By Eq. (6.12), the symmetry category isC = ΣVecZ2 â ΣRep(Z2). (6.29)i.e. a Z2 1-form symmetry.â¢ It has two simple objects(Rep(Z2)-module categories),10DZ2 := Rep(Z2), D0 := Vec. (6.30)â¢ The fusion rule isRep(Z2)â Rep(Z2) Rep(Z2) = Rep(Z2),Rep(Z2)â Rep(Z2) Vec = VecVecâ Rep(Z2) Vec = VecâVec. (6.31)By Eq. (6.18), the underlying topological order of the sandwich isT = End(ΣRep(Z2)) = Î£(Rep(Z2)â Rep(Z2)) = ΣZ1[VecZ2 ], (6.32)10Generally an indecomposable Rep(G)-module category is labeled by a subgroup H < G and a2-cocyle Ï of H. We denote such a module category by DH,Ï.166Mm,m Mm,e Me,m Me,eFigure 6.2: A solid line represents a smooth boundary and a dashed line represents arough boundary. There is also an invertible domain wall corresponding to a non-trivialbraided auto-equivalence. As module category over Rep(Z2), the rough boundary isVec and the smooth boundary is Rep(Z2).i.e. a toric code.Before we describe the symmetry action on the toric code, let us recall the domain wallsin a toric code, their description as Rep(Z2)âRep(Z2) bimodule categories, and their fusionrules.â¢ A trivial domain wall M1, the bimodule category is Rep(Z2).â¢ Four domain walls corresponding to combinations of smooth/rough boundaries. SeeFig. 6.2. They correspond to the bimodule categories Mi,j = Ni â Nj , with i, j = e,mand Ne = Vec, Nm = Rep(Z2). They have fusion ruleMi,j âMk,l = (δj,k + 1)Mi,l (6.33)â¢ An automorphism domain wall MÏ. This domain wall will not enter our discussionbelow.By Eq. (6.23), the nontrivial symmetry generator D0 acts asF(D0) = Vecâ Rep(Z2) â â End(ΣRep(Z2)) â Î£(Rep(Z2)â Rep(Z2)) (6.34)The isomorphism Func[ΣA,ΣB] â Î£(Aopâ B) sends f to f(A). Therefore as aRep(Z2)âRep(Z2) bimodule category we haveF(D0) = Vecâ Rep(Z2) Rep(Z2) = Vec =Me,e. (6.35)We see that the symmetry generator D0 acts as the rough-rough domain wall of the toriccode, also known as the Cheshire string. This domain wall is the condensation descendant of167the gauge charge. The symmetry action is simply an embeddingF : Î£Rep(Z2)â ΣZ1(VecZ2) (6.36)induced by the embedding ofRep(Z2)â Z1[VecZ2 ]. In other words, the 1-form Z2-symmetryacts as the e-particle of the toric code. The 1-form symmetry is broken since there is an orderparameterâ the m-particle.6.4.4 Example: Gapped phases from Z1[2VecS3]We now consider a SymTOZ1[2VecS3 ], where S3 = Z3 â Z2 = â¨r, s|r3 = s2 = 1, srs = r2â©.This example is particularly interesting for the following reasons.â¢ It admits a symmetry boundary describing a non-invertible fusion 2-category symmetry2Rep(Z[1]3 â Z[0]2 ).â¢ This non-invertible symmetry admits a symmetry-broken phase where different vacuuahave distinct topological orders.â¢ This non-invertible symmetry itself is non-anomalous and admits SPT phases.There are two non-trivial subgroups(up to conjugation): Z3 = â¨râ© and Z2 = â¨sâ©. LetC[S3, H, Ï = 0, Ï = 0] be the symmetry boundary, and C[S3,K, Ï = 0, Ï = 0] be the physicalboundary. WithH,K being either the Z3 or the Z2 subgroup of S3, we obtain four sandwiches.In the following we study these four cases.H = K = Z3 â We have HG/K = {[1], [s]} and H â©K = Z3. The symmetry category isdescribed as follows.â¢ By Eq. (6.12), the underlying 2-category isC = Î£VecZ3,[1]âÎ£VecZ3,[s] â Î£Rep(Z3)[1]âÎ£Rep(Z3)[s]. (6.37)â¢ There are two connected components, each isomorphic to ΣRep(Z3). This means thesymmetry is necessarily a 2-group symmetry [34]. Notice that conjugating with s ex-changes the two non-trivial irreducible representations of Z3, therefore this is a 2-groupsymmetryC = 2VecZ[1]3 âZ[0]2. (6.38)where the 0-form Z2-symmetry acts nontrivially on the 1-form Z3-symmetry11.11This also follows from the general case Example. 6.2.3168â¢ There are four simple objects, two in each connected component, denoted as DZ3[1] âRep(Z3),D0[1] â Vec,DZ3[s] â Rep(Z3),D0[s] â Vec. The unit is DZ3[1]. The fusion rulesareD0[1] âD0[1] = 3D0[1],D0[1] âDZ3[s] = D0[s],D0[1] âD0[s] = 3D0[s],DZ3[s] âDZ3[s] = DZ3[1],DZ3[s] âD0[s] = D0[1],D0[s] âD0[s] = 3D0[1].(6.39)By Eq. (6.18), the underlying topological order of the sandwich isT = End(C) (6.40)= âi,j=[1],[s]Î£(Rep(Z3)â Rep(Z3)). (6.41)In other words, we have two vacuua, each supporting a Z3-toric code. Next we analyzethe symmetry action on this topological order.Consider the symmetry generator D0[1], we haveF(D0[1]) = D0[1] ââ â End(Î£Rep(Z3)[1]âÎ£Rep(Z3)[s]) (6.42)Since the isomorphism Func[ΣA,ΣB] â Σ(Aop â B) sends f to f(A), to find the matrixelements F(D0[1])i,j , i, j = [1], [s], we only need to compute the action of F(D0[1]) on theâbasisâ Rep(Z3)[1] â DZ3[1],Rep(Z3)[s] â DZ3[s] . Using the fusion rules Eq. (6.39), we haveF(D0[1])(DZ3[1]) = D0[1] = Vec, (6.43)F(D0[1])(DZ3[s]) = D0[s] = Vec. (6.44)Therefore we haveF(D0[1]) =(Vec 00 Vec)â End(Î£Rep(Z3)âÎ£Rep(Z3)). (6.45)Similar to the Z2-toric code, the Rep(Z3)âRep(Z3)-bimodule category Vec corresponds169to the rough-rough domain wall in the Z3-gauge theory. Therefore the 1-form Z3-symmetryacts as the e-particle of the Z3-gauge theory in each vacuum. Next we consider the symmetrygenerator DZ3[s] , using fusion rules of C, it is not hard to deduceF(DZ3[s]) =(0 Rep(Z3)Rep(Z3) 0)â End(Î£Rep(Z3)âÎ£Rep(Z3)). (6.46)The Rep(Z3)âRep(Z3)-bimodule category Rep(Z3) is the trivial domain wall betweentwo Z1[Rep(Z3)]. Therefore DZ3[s] simply exchanges the two vacuuaâ the 0-form Z2-symmetryis broken.H = K = Z2 â We have HG/K = {[1], [r]} and H â©K = Z2. Notice Z2 is not normalin S3 and H â© [r]K = {1}. The symmetry category is described as follows.â¢ By Eq (6.12), the underlying 2-category isC = ΣVecZ2 âÎ£Vec = Î£Rep(Z2)[1]âÎ£Vec[r]. (6.47)â¢ There are two connected components, however they are not isomorphic. This means thesymmetry is not a 2-group symmetry.â¢ There are three simple objects, two in ΣRep(Z2)[1] and one in ΣVec[r], denoted asDZ2 â Rep(Z2),D0 â Vec,W â Vec. DZ2 is the unit, and fusion rules of other objectsare [6, 34]D0 âD0 = 2D0,D0 âW = 2W,W âW =W âD0.(6.48)which we recognize is the fusion 2-category 2Rep(Z[1]3 â Z[0]2 )12.The underlying topological order of the sandwich isT = End(Î£Rep(Z2)âÎ£Vec). (6.49)There are two vacuua, one has the toric code topological order, the other has trivialtopological order. Next we analyze the action of 2Rep(Z[1]3 â Z[0]2 ) on this topological order.12This follows from the general case Example 6.2.3.170Consider the symmetry generator, DZ2 , it actsF(D0) = D0 ââ. (6.50)Using the fusion rules Eq. (6.48), we obtainF(D0) =(Rep(Z2)VecRep(Z2) 00 Vec(VecâVec)Vec,)â End(Î£Rep(Z2)âÎ£Vec), (6.51)where we added subscripts to express the module category structure. We see that in the toriccode sector DZ2 acts as the Cheshire string, while in the trivial phase sector it creates a 1+1Dphase with two GSD, 21. Next we consider the symmetry generatorW, using the fusion rulesEq. (6.48), we obtainF(W) =(0 Rep(Z2)VecVecVecVecRep(Z2) VecVecVec)â End(Î£Rep(Z2)âÎ£Vec). (6.52)One can directly verify, using relative tensor product of module categories, that the fusionrules Eq. (6.48) are preserved under F.H = Z3,K = Z2 â Since HK = S3 and H â© K = 1, we see from Eq. (6.6), that theunderlying topological order of the sandwich is trivial(2Vec). This means the sandwich isnow a 2+1-D SPT with 2-group symmetry 2VecZ[1]3 âZ[0]2. There are in fact two SPTs, as thephysical boundary can be twisted by a class Ï â H3[Z2, U(1)].Remark. Since these two SPTs differ by the twist Ï â H3[K,CÃ] on the physical boundary,they can be viewed as gauged version of ordinary(group-like) SPTs. In other words, if wechange the symmetry boundary to 2VecG, they would become different SPTs of the group-likesymmetry 2VecK . In this sense they are not âintrinsically non-invertible SPTsâ.H = Z2,K = Z3 â Since HK = S3 and H â© K = 1, we see from Eq. (6.6) that theunderlying topological order of the sandwich is trivial. Therefore the sandwich describes a2+1-D SPT with non-invertible symmetry 2Rep(Z[1]3 â Z[0]2 ). There are in fact three SPTs,since the physical boundary can be twisted by a class Ï â H3[Z3, U(1)].Similar to the previous case, these SPTs are not intrinsically non-invertible SPTs.1716.4.5 Example: Gapped phases from Z1[2VecS4]We now consider a SymTOZ1[2VecS4 ], with S4 = â¨(1, 2), (1, 2, 3, 4)â© be the permutation groupof four elements. This case is particularly interesting for the following reasons.â¢ There exists a symmetry boundary that defines a non-invertible symmetry C[S4, S3, 0, 0]that is neither a 2-group symmetry 2VecG nor a 2-representation symmetry 2Rep(G).In this sense this symmetry is â intrinsically non-invertibleâ.â¢ The symmetry C[S4, S3, 0, 0] admits a partial symmetry-broken phase with two vacuua,with one vacuum having the S3-gauge theory and the other having the Z2-gauge theory.â¢ The symmetry C[S4, S3, 0, 0] admits two families of SPT phases that are not related bytwisting the physical boundary by cocycles as in the previous examples. Therefore it isimpossible to gauge-relate these SPTs to a family of invertible SPTs with fixed symmetrygroup. These two families of SPTs are therefore âintrinsically non-invertible SPTsâ. Forinstance, the gapless edge modes between these two families of SPTs should be distinctfrom the edge modes of any invertible SPTs.H = K = S3 â We take H = K = S3 = â¨(1, 2), (1, 2, 3)â©. There are two double cosets,HG/K = {[e], [(3, 4)]}. We have H â© (3,4)K = {e, (1, 2)} â Z2. The symmetry category isdescribed as follows.â¢ By Eq. (6.12), the underlying 2-category isC = ΣRep(S3)e â ΣRep(Z2)[(3,4)] (6.53)â¢ Since S4 is not a semi-direct product, by Example 6.2.3, this is not a 2-group symmetrynor a 2-representation symmetry.â¢ There are in total six simple objects, four in ΣRep(S3)e, two in ΣRep(Z2)[(3,4)]. Letg := (3, 4), the simple objects are denoted asDS3e ,DZ2e ,DZ3e ,D0e ,DZ2g ,D0g . (6.54)â¢ The fusion rules are 13The underlying topological order is, by Eq (6.18),T = End(Î£Rep(S3)â ΣRep(Z2)) (6.55)13This can be derived by decomposing (H,K)-bisets into unions of double cosets.172DS3e DZ2e DZ3e D0e DZ2g D0gDS3e DS3e DZ2e DZ3e D0e DZ2g D0gDZ2e 3DZ2e D0e 3D0e 3DZ2g 3D0gDZ3e 2DZ3e 2D0e D0g 2D0gD0e 6D0e 3D0g 6D0gDZ2g DZ2e âD0g D0e â 2D0gD0g 2D0e â 4D0gTable 6.2: Fusion rules of objects in C[S4, S3, 0, 0]There are two vacuua, one has the S3-gauge theory and the other has the toric code. Thematrix elements of the symmetry generators can be deduced from the fusion rules in Table.6.2similar to the previous examples.H = S3, K = Z4 â Here the symmetry boundary is as before, C = C[S4, S3, 0, 0]. We takeK = Z4 = â¨(1, 2, 3, 4)â©. Since H â©K = {e} and HK = S4. From our discussion in Sec. 6.4.1,we see this this choice leads to an SPT of C[S4, S3, 0, 0].In fact there are four SPTs for this choice, since the physical boundary can be twisted bya class Ï â H3[Z4,CÃ].H = S3, K = Z2ÃZ2 â We takeK = Z2ÃZ2 = â¨(1, 2), (3, 4)â©. Again we haveHâ©K = {e}and HK = S4. This define another family of SPTs of C[S4, S3, 0, 0].There are eight SPTs, since the physical boundary can be twisted by a class Ï â H3[Z2ÃZ2,CÃ] â Z32.Remark. The above two families of SPTs are not gauge related to any single family of invertibleSPTs. More precisely, if we change the symmetry boundary to be 2VecS4 , then these twofamilies are mapped to gapped phases with different unbroken symmetry groups, one withZ4, the other with Z2 Ã Z2. Therefore these two families of SPTs can not be obtained bygauging a single family of invertible SPTs. It would be particularly interesting to explorethe gapless edge modes between them, which should be distinct from the edge modes of any2+1D invertible SPTs.1736.5 Topological holography for 2+1-D gapless phasesWe now turn to the study of topological holography for 2+1D gapless phases. A completemathematical theory for gapless phases and their interplay with generalized symmetries re-mains an open challenge. Within the scope of this paper, we show that the SymTOclub-sandwich construction determines the topological skeleton of gapless phases.The concept of topological skeleton was introduced in a series of papers [68, 69]. Thespirit of [68, 69] is that all the physical observables of a gapless phase can be divided intotwo pieces:â¢ The information of local observables such as the OPE of local fields. This is called thelocal quantum symmetry of the gapless phase. In 1+1D, this can be the chiral algebraor full field algebra of a CFT.â¢ The information of all the topological defects. This is called the topological skeleton ofthe gapless phase.It is expected that the topological skeleton of a gapless phase has a higher categoricalstructure. These two pieces of information are not independent and must be compatiblein certain ways. Studies on 1+1D phases with symmetry have shown that the topologicalskeleton of a phase(gapped or gapless) is intimately related to condensable algebras in theSymTO [131, 61]. This is best illustrated through the club-sandwich construction 14.6.5.1 The club-sandwich constructionThe basic setup is summarized in Fig. 6.3. The bulk is divided into regions, separated bya gapped domain wall M. To the right of this wall hosts the SymTOZ1[C] with its rightboundary being the symmetry boundary C. The left region is obtained through a condensablealgebra A in Z1[C] and is denoted as Z1[C]0A. The left boundary of the left region is a gaplessboundary of Z1[C]0A. This gapless boundary should be irreducible in some sense: all operatorsof Z1[C]0A should act nontrivially on this gapless boundary, otherwise we can further reduceZ1[C]0A by a condensation.This construction describes the topological skeleton of a 2+1D phase as follows [68, 69, 61].â¢ The SymTOZ1[C] describes the topological skeletons of operators, including both localand non-local operators symmetric under C.14In [68, 69, 61], this is called the âtopological wick rotationâ principle.174BphysXlocM BsymZ1[C]0A Z1[C] C âBphys BIRsymZ1[CIR] CIRFigure 6.3: The club-sandwich separates the topological skeleton from the local quan-tum symmetry of a gapless phase. The left boundary encodes all the local data of thegapless phase such as OPE of local fields. The domain wall M encodes the topologicalskeleton of the gapless phase, including the topological skeletons of states and opera-tors. By fusing the domain wall M with the symmetry boundary Bsym, we obtain anordinary sandwich that describes the IR symmetry of the gapless phase. In particular,the new symmetry boundary describes the IR symmetry of the gapless phase, denotedas BIRsym.â¢ The domain wall M describes the topological skeletons of states of the 2+1D phase. Atopological skeleton of state is a subspace of the Hilbert space that is invariant underthe action of local symmetric operators.â¢ The operators from Z1[C] can move to the domain wall and act on objects on the wall.This corresponds to changing the topological skeleton of states by acting with topologicalskeleton of operators.â¢ The operators from Z1[C] that act trivially on the wall M are the topological skele-tons of operators that leave the vacuum sector invariant. These operators are thereforecondensed on the wall.If we shrink the Z1[C] region and fuse the domain wall M with the symmetry boundaryBsym, we obtain an effective ordinary sandwich. See Fig. 6.3. The new symmetry boundarydescribes the effective symmetry of the 2+1-D phase that acts nontrivially in the IR, calledthe IR symmetry, denoted by CIR. It is given by a relative tensor productCIR =Mâ Z1[C] C. (6.56)If we consider a topological defect on the original symmetry boundary C, then after shrink-ing the Z1[C] region it will become a topological defect on the new symmetry boundary CIR.175This process defines a functorR : Câ CIR, (6.57)which describes how the exact symmetry C reduces to the IR symmetry CIR as we zoom out.6.6 Higher condensation theory for 3+1D topolog-ical ordersThe crucial input in the club-sandwich construction is a condensable algebra in the SymTO.Therefore the theory of condensation is critical to the topological holography paradigm forgapless phases. Here we review the part of this elegant theory [63] relevant for this chap-ter, namely E2-condensation in 3+1D topological orders. More details can be found in Ap-pendix A.2.At the microscopic level, a condensation of topological defect amounts to turning on somecommuting projector Hamiltonian to trivialize certain topological defects. At the macroscopiclevel, this process is described by algebras in the category of topological defects [40, 63]. In3+1D there are point-like as well as string-like defects, therefore we naturally have two typesof condensable algebras in 3+1D. To describe condensation of strings, one considers the so-called condensable E2-algebras15. The collection of all condensable E2-algebras in a 3+1Dtopological order C will be denoted as AlgcE2(C).After condensing an algebra A â AlgcE2(C), we have the following physical picture.MD CThe objects of interests are the condensed phase D and the domain wall M between theoriginal phase and the condensed phase. These objects are described by category of modulesover A.15A particle condensation is described by a so-called E3-condensable algebra. Particle condensationcan always be reduced to string condensation, see discussion near the end of this section.176Theoremph 6.6.1: Let C be a 3+1D topological order, i.e. a non-degenerate braided fusion2-category. Then a string condensation is specified by a condensable E2-algebra A â AlgcE2(C),and we have:â¢ The condensed phase is a new topological order D, described by the braided fusion2-category of local modules over A: D =Mod0A(C).â¢ The domain wall between the old phase and the new phase is described by the fusion2-category of modules over A: M =ModA(C).â¢ Moving defects in C onto the wall defines a functor from C to M, given byX 7â X âA (6.58)â¢ Conversely, any defect on the wallM can be moved into C, given by the forgetful functorF :ModA(C)â C (6.59)â¢ Defects in D can move onto the wall by the inclusion Mod0A(C)âModA(C), and thencan move freely into C via the forgetful functor (6.59).â¢ By folding, M is a gapped boundary of C â Dop. By the bulk-boundary relation oftopological orders we haveZ1[ModA(C)] â Câ Mod0A(C)op (6.60)A special case of string condensation is when the condensed phase is trivial, the corre-sponding condensable algebra in this case is called Lagrangian.Definition 6.6.1: A condensable E2-algebra is called Lagrangian if Mod0A(C) = 2Vec.For a Lagrangian condensation, the domain wallM becomes a gapped boundary of C. Thebulk-boundary relation of topological order states all gapped boundaries arise in this way.Theoremph 6.6.2: Let M be a gapped boundary of C, then there is a Lagrangian algebra Ain C such thatM =ModA(C), (6.61)and we haveZ1[ModA(C)] â C. (6.62)177Sequential condensation Consider starting with a 3+1D topological order C, performa string condensation A1 to obtain phase D, and then perform another string condensationA2 in D to obtain phase E. This two-step condensation is in fact equivalent to a singlecondensation in C, described as follows.Definition 6.6.2: Let A â AlgcE2(C), a condensable E2-algebra over A is a condensablealgebra B together with a map between condensable E2-algebras: f : Aâ B.Theorem 6.6.1: Let A â AlgcE2(C), then a condensable algebra B in Mod0A(C) is the sameas a condensable algebra in C over A, and we haveMod0B(Mod0A(C)) âMod0B(C). (6.63)All the above statements are natural generalization of the anyon condensation theory to3+1D. However, the structure of condensable algebras is more complicated in 3+1D, thus itis desirable to have systematical methods for constructing condensable algebras, as well asfor computing the condensed phase Mod0A(C)). In the next section, we focus on the caseC = Z1[2VecG], for which more explicit description of condensable E2-algebras can be given.We also describe how to compute the condensed phase for a general string condensation inZ1[2VecG].Remark. We briefly comment on particle condensation in 3+1D here. For a 3+1D topologicalorder C, particles live in â¦C, a symmetric fusion category. A particle condensation is definedby a condensable E3-algebra in â¦C, A â AlgcE3(â¦C). A particle condensation can be treatedas a string condensation: we can first condense the particles along one direction, obtaining theso-called condensation descendant of the particles. Then condensing particles is the same ascondensing this descendant string. Mathematically, the descendant string is ΣA â AlgcE2(C).For example, the Cheshire string 1c in the 3+1D toric code is the condensation descendantof the gauge charge: 1c = Σ(1+ e). Condensing the particle e is then the same as condensingthe string 1c.6.7 String condensation in Z1[2VecG]In this section we review the structure of condensable algebras in Z1[2VecG] [137]. We alsoprovide a systematic method for determining the condensed phase for any condensable algebrain Z1[2VecG].1786.7.1 Condensable E2-algebras in Z1[2VecG]The full structure of the braided fusion 2-category Z1[2VecG] can be found in [57, 128]. Wenote that it can be formulated as the 2-category of G-crossed finite semi-simple 1-categories.From this formulation, it follows from the definition of condensable E2-algebras Def. A.2.1,Def A.2.2 that we have the following fact.Fact 5: A condensable E2-algebra in Z1[2VecG] is a G-crossed braided multi-fusion categorysuch that(C1) The G-action permutes the simple summands of the monoidal unit transitively.This follows directly from the definition of condensable E2-algebras and the descriptionof the braided fusion 2-category Z1[2VecG]. Roughly speaking, the underlying object ofthe algebra is by definition a G-crossed finite semi-simple category, then the algebra struc-ture(product, unit, unitors, associator) equips it with a G-crossed monoidal structure. Thebraiding in Def. A.2.1 further equips it with a G-crossed braiding. Demanding the algebra tobe separable amounts to demanding it to be multi-fusion. Finally, condition (C1) is equivalentto the connectivity condition in Def. A.2.2.A physical interpretation of the multi-fusion structure of the condensable algebra is asfollows. If we take the SymTOsandwich construction and put a condensable algebra at thephysical boundary, and fix the symmetry boundary to be 2VecG, then we obtain a 2+1-Dphase with a global symmetry G. This phase breaks the symmetry G exactly when the con-densable algebra at the physical boundary is multi-fusion, in which case the simple summandsof the monoidal unit correspond to the vacuua of the phase, and they are permuted by theG-action. This G-set of simple summands must be irreducible and is equal to the coset G/Hfor some subgroup H < G, which is the unbroken subgroup of the 2+1-D phase.Let A be a condensable algebra in Z1[2VecG], the full sub-category generated by the sim-ple summands of the monoidal unit ofAmust be equivalent to a functor category Func[G/H,Vec]for some subgroup H < G. This sub-category is by itself a G-crossed braided multi-fusioncategory satisfying (C1), i.e. a condensable algebra in Z1[2VecG]. Thus we have inclusion ofcondensable algebras Func[G/H,Vec] â A. By the sequential condensation theorem 6.6.1,condensing A can then be divided into two steps, where in the first step we condense thesub-algebra Func[G/H,Vec]. This step is rather simple(proof can be found in, e.g. [63]):Mod0Func[G/H,Vec](Z1[2VecG]) â Z1[2VecH ]. (6.64)The second step is then to viewed A as a condensable algebra in Z1[2VecH ]. By results179in [137], viewed as a condensable algebra in Z1[2VecH ], A is now a H-crossed braided fusioncategory. Thus we have the following equivalent characterization of condensable algebras inZ1[2VecÏG].Fact 6: Condensable E2-algebras in Z1[2VecG] are labeled by pairs (H,B), where H < G isa subgroup, and B is a H-crossed braided fusion category.We say the condensable algebra is strongly connected if H = G in the above description.In this case, it is described by a G-crossed braided fusion category. We also stress here thathere the H-crossed braided fusion category B does not need to have faithful grading. In thedecompositionB = âhBh, (6.65)we do not demand that all components are non-zero. Also, the trivial component Be,being a braided fusion category, does not have to be non-degenerate. In fact, these twoconditions(faithful grading and non-degenerate trivial component) are met exactly when thecondensable algebra is Lagrangian [127, 137].Theorem 6.7.1: Let (H,B) â AlgcE2(Z1[2VecÏG]) be described above. Then it is Lagrangianiff the grading of B is faithful and Be is non-degenerate.6.7.2 Computing the condensed phaseFor a non-Lagrangian condensable algebra, we are interested in the condensed phase. In theSymTOconstruction, the condensed phase describes the low energy symmetry of the corre-sponding gapless phase. When the SymTOis Z1[2VecG], there is a rather complete answer.There is a standard Morita equivalence: 2VecG â¼ ΣRep(G), where ΣRep(G) is thefusion 2-category of Rep(G)-module categories. This leads to an equivalence between theirDrinfeld centers:Φ : Z1[2VecG] â Z1[ΣRep(G)] (6.66)Consequentially there is an equivalence between condensation in Z1[2VecG] and conden-sation in Z1[ΣRep(G)]. The description of condensable E2-algebras in Z1[ΣRep(G)] andtheir condensed phase are rather simple [32]:Fact 7: A condensable E2-algebra in Z1[ΣRep(G)] is a braided fusion category B together180with a braided functor from Rep(G). The condensed phase isMod0B(Z1[ΣRep(G)]) â Z1[ΣB]. (6.67)Let A be a strongly-connected condensable algebra in Z1[2VecG], i.e. a G-crossed braidedfusion category. Under the equivalence Φ, it is mapped to a condensable algebra in Z1[ΣRep(G)],i.e. a braided fusion category with a braided functor fromRep(G). This is given by the equiv-ariantization of A, Φ(A) = AG [32]. Therefore we haveMod0A(Z1[2VecG]) â Z1[ΣAG]. (6.68)Eq. (6.68) provides an answer to the condensation problem. However, it is not an idealanswer. For instance, in many cases the Mu¨ger center Z2AG is Tannakian, then we knowthat Z1[ΣAG] must take the form Z1[2VecÏK ] for some K and some Ï â H4[K,U(1)], and wewould want to determine K and Ï from the condensable algebra A. In other words, if weknow that the condensed phase is a Dijkgraaf-Witten theory, we wish to determine the gaugegroup and the twist. Using the classification of fusion 2-categories developed in [31], this canbe done as follows.Lemma 6.7.1: Let B be a braided fusion category with a Tannakian Mu¨ger center: Z2B =Rep(K). Then the de-equivariantization BK is a non-degenerate braided fusion categorycarrying a natural K-action. Let Ï â H4[K,U(1)] be the O4-obstruction of this K-action 16.Then there is an equivalenceZ1[ΣB] â Z1[2VecÏK ] (6.69)Therefore, we obtain the following practical algorithm for computing the condensed phase.Determining the condensed phase. Let A â AlgcE2(Z1[2VecG]) be a condensablealgebra such that Z2Ae is Tannakian. Then the condensed phase is a finite gauge theory,Mod0A(Z1[2VecG]) = Z1[2VecÏK ], (6.70)where K is determined from Z2(AG) = Rep(K), and Ï â H4[K,U(1)] is the O4âobstructionof the K-action on (AG)K . Some useful results about computing O4âobstruction of braidedfusion category with group action can be found in [22].16See appendix A.3 for the definition of the O4-obstruction.181The case Ae is non-degenerate. In the examples studied In this chapter, the trivialcomponent of the condensable algebra Ae will always be non-degenerate. In this case we maygive a more concrete description for the condensed phase.Let A â AlgcE2(Z1[2VecG]) be a strongly connected condensable algebra such that Ae isnon-degenerate. Then the support of the A must be a normal subgroup N â G. By resultsin [26], we have Z2(AG) = Rep(G/N). Therefore in this case we haveMod0A(Z1[2VecG]) = Z1[2VecÏG/N ] (6.71)where Ï is the O4âanomaly of the G/N -action on (AG)G/N â AN . In other words, ifthe support of A is N , then we may view it as an N -crossed braided fusion category, andequivariantize with respect to N . The result AN now carries an action by G/N , and itsO4-obstruction is exactly the twist of the condensed phase.6.8 Minimal condensable algebras: Cohomology methodFrom the general principle of topological holography, we expect a non-Lagrangian condensablealgebra in the SymTOto describe the topological skeleton of a gapless 2+1-D phase withsymmetry. However, not all such condensable algebras should be considered âgapless SPTâ.Similar to the gapped case, we should expect there to be a notion of âgapless symmetryenriched topological orderâ(gSET) in 2+1-D, and gSPT should be a special kind of gSET,where the gapped sector has trivial topological order.In this section, we introduce the class of condensable algebras in Z1[2VecG] that we expectto correspond to 2+1-D G-SPTs/gSPTs based on the picture of club-sandwich. We call themminimal condensable algebras. The definition is:Definition 6.8.1: Recall a condensable E2-algebra in Z1[2VecG] is labeled as (H,B), whereB is an H-crossed braided fusion category. We say the condensable E2-algebra is minimal ifBe = Vec.In the club-sandwich picture Fig. 6.3, the braided fusion category Be describes the intrinsic2+1-D topological order on the domain wall M. If M has nontrivial topological order, thenthe gapped sector of the corresponding gapless phase has nontrivial topological order and weconsider such gapless phase as âgapless symmetry-enriched topological orderâ(gSET).One of the main results of this paper is an exact duality between minimal condensablealgebras and 2+1-D G-gSPTs. To establish this correspondence, it is crucial to understand themathematical structure of minimal condensable algebras, which is the goal of this section. Wegive a cohomological characterization and classification of minimal condensable algebras using182the theory of crossed extension of braided fusion categories and the LHS spectral sequence.We also provide a method for computing the condensed phase of any minimal condensablealgebra via spectral sequence.6.8.1 Structure of minimal condensable algebrasWe only need to focus on the case H = G , the general case can be dealt with using sequentialcondensation theorem 6.6.1. In this case, the condensable algebra is a G-crossed braidedfusion category B such that the trivial component is trivial Be = Vec. Equivalently B is aG-crossed extension of Vec. It is well-known that if the grading of the G-crossed extensionis faithful, then such extensions are classified by the group cohomology H3[G,CÃ]. Howeverwe see from Theorem 6.7.1 that this is only true when the condensable algebra is Lagrangian.For a general condensable algebra, the grading of B is supported on a normal subgroup NâG.We denote the set of equivalent classes of minimal condensable algebras supported on Nas GCrExt(Vec, N), which stands for âG-crossed extensions of Vec supported on Nâ. Westart with the following observation.Lemma 6.8.1: A G-crossed braided fusion category B with non-degenerate Be and supportN âG is equivalent to its trivial component Be plus the following data:1. A faithful N -crossed extension BÃe .2. A G/N action on (BÃe )N :Φ : G/N â Autbr((BÃe )N ) (6.72)such that G/N acts on Rep(N) â (BÃe )N as the canonical action induced by the actionof G/N on N .Proof. For every G-crossed braided fusion category B supported on N âG, by restricting toN -action we an view B as an N -crossed braided fusion category with faithful grading. It isstandard that if B has a G-action, then BN has a natural G/N -action that acts on Rep(N)canonically [26, 22].Conversely, if the trivial component Be and the above data are given, theG-crossed braidedfusion supported on N can be recovered as follows. Using the G/N -action on (BÃe )N , we mayform the equivariantization ((BÃe )N )G/N . Since Rep(N) â (BÃe )N , we haveRep(N)G/N â Rep(G) â ((BÃe )N )G/N . (6.73)183Thus ((BÃe )N )G/N contains Rep(G). We may then de-equivariantize with respect to G toobtain (((BÃe )N )G/N )G. This is a G-crossed braided fusion category. Since Z2(((BÃe )N )G/N ) =Rep(G/N), the support of (((BÃe )N )G/N )G is exactly N .Specializing to the case Be = Vec, a faithful N -crossed extension is determined by a classÏ â H3[N,CÃ], and the correspondingN -crossed extension isVecÏN . TheN -equivariantizationof VecÏN is know: (VecÏN )N â Z1[VecÏN ]. We obtain the following characterization of minimalcondensable algebras in Z1[2VecG].Corollary 6.8.1: An element B â GCrExt(Vec, N) is determined by1. A class Ï â H3[N,CÃ].2. A homomorphism of 2-groupsΦ : G/N â Autbr(Z1[VecÏN ]) (6.74)such that G/N acts on Rep(N) â Z1[VecÏN ] canonically.6.8.2 The LHS spectral sequenceWe now explain how the above characterization of GCrExt(Vec, N) may be reformulated interms of the LHS spectral sequence. We have Autbr(Z1[VecÏN ]) â BrPic(VecÏN ). Thereforethe 2-group morphism Φ can be viewed asΦ : G/N â BrPic(VecÏN ). (6.75)We understand that a morphism of 3-groupsΦ : G/N â BrPic(VecÏN ) (6.76)corresponds to a G/N -extension of VecÏN . As explained in [36], such an extension amountsto a group extension of G/N by N together with an extension of Ï to a cocycle in H3[G,CÃ].In our case, the group extension is fixed to be 1 â N â G â G/N â 1. Therefore therest of information in the 3-group morphism is just an extension of Ï from a N -cocyle to aG-cocycle. From the theory of spectral sequence, we know that such an extension exists ifand only if d2Ï = 0 in E2,22 , d3Ï = 0 in E3,13 , and d4Ï = 0 in E4,04 . When these conditionsare met, an extension of Ï may be constructed step-by-step as follows. In the first step,we examine d2Ï â H2[G/N,H2[N,CÃ]]. The condition d2Ï = 0 means certain equationinvolving Ï has a solution. We need to choose a solution γ, then Ï, γ together defines a class184iÎ³,Ï â E3,12 = H3[G/N,H1[N,CÃ]]. This element iÎ³,Ï is in the kernel of d2, and its image inE3,13 is d3Ï. The condition d3Ï = 0 means that we can choose γ so that iÎ³,Ï = 0 in E3,12 .We make such a choice. Then iÎ³,Ï = 0 means certain equation involving Ï, γ has solutionand we choose a solution p. From Ï, γ, p, we can construct a class jÏ,Î³,p â E4,02 = H4[G,CÃ].The class jÏ,Î³,p is in the kernel of d2, d3, and its image in E4,04 would be d4Ï. The conditiond4Ï = 0 means that for certain choice of γ, p we can make jÏ,Î³,p = 0 in E4,02 . We make sucha choice. Then jÏ,Î³,p = 0 means certain equation has solution, and we choose a solution β.Together the triple (γ, p, β) gives a desired extension of Ï to G. Cochain level expression ofthe classes (γ, p, β) and the equations they satisfy may be found in [126].The obstruction classes d3Ï, d4Ï correspond to the obstruction of lifting Φ to a 2-grouphomomorphism and to a 3-group homomorphism, often called the O3-obstruction and theO4-obstruction. In the description of GCrExt(Vec, N), Corollary 6.8.1, Φ is only a 2-grouphomomorphism, and the obstruction of lifting it to a 3-group homomorphism is exactly thetwist of the condensed phase by Eq. (6.71). Therefore we have the following.Theorem 6.8.1: Elements in GCrExt(Vec, N) are parameterized by Ï â H3[N,CÃ] suchthat d2Ï = 0, d3Ï = 0 together with a solution to these equations (γ, p). Denote the corre-sponding condensable algebra as Vec(Ï,Î³,p)N , the condensed phase isMod0Vec(Ï,Î³,p)N(Z1[2VecG]) â Z1[2Vecd4ÏG/N ]. (6.77)Here by d4Ï we mean a representative of it in H4[G/N,CÃ] computed using the solution(γ, p).6.8.3 Group structure of minimal condensable algebras and anexact sequenceThe set of equivalent classes of minimal condensable algebras with supportN , GCrExt(Vec, N),actually has a natural group structure. Consider two elements inGCrExt(Vec, N),Vec(Ï,Î³,p)N ,Vec(Ïâ²,Î³â²,pâ²)N .Take the diagonal components of Vec(Ï,Î³,p)N â Vec(Ïâ²,Î³â²,pâ²)N , we obtain another element ofGCrExt(Vec, N). This defines a product â¦ on GCrExt(Vec, N), which is clearly associa-tive and commutative. In terms of the data Ï, γ, p we simply have Vec(Ï,Î³,p)N â¦Vec(Ïâ²,Î³â²,pâ²)N =Vec(Ï+Ï,Î³+Î³â²,p+pâ²)N . Therefore GCrExt(Vec, N) is naturally an abelian group. We prove anexact sequence relating GCrExt(Vec, N) to ordinary group cohomology, which enables us tocompute GCrExt(Vec, N) as a set.There is a natural group homomorphism from GCrExt(Vec, N) to H4[G/N,CÃ] givenby taking the obstruction class of the 2-group morphism Φ. This obstruction class is exactly185the obstruction jÏ,Î³,p of lifting Ï to G. Since jÏ,Î³,p is linear in Ï, γ, p, this defines a grouphomomorphismO4 : GCrExt(Vec, N)â H4[G/N,CÃ]. (6.78)We claim the image of this map exactly contain those elements that trivialize inH4[G,CÃ].Indeed, for any element in GCrExt(Vec, N), the 2-group morphism Φ gives us âalmost anextension of VecÏNâ: VecαG, except that the associator αg,h,k : (VgâVh)âVk â Vgâ (VhâVk)may fail to satisfy the pentagon equation. The O4-obstruction jÏ,Î³,p exactly measures thefailure of the pentagon equation. Namely jÏ,Î³,p([g], [h], [k], [l]) = dα(g, h, k, l). Therefore theO4-obstruction is always trivialized in H4[G,CÃ]. Conversely, take any Ï â H4[G/N,CÃ]such that pâ(Ï) = dα. Then α|N is a 3-cocycle of N , and α is âalmostâ a lift of α|N to Gexcept that it is obstructed by Ï â H4[G/N,CÃ]. This means α may be constructed from aquadruple (Ï, γ, p, β)(with Ï = α|N ) satisfying d2Ï = 0, iÏ,Î³ = 0, and Ï = jÏ,Î³,pdÎ². Thereforethe following sequence is exact.GCrExt(Vec, N)O4âââ H4[G/N,CÃ] pâââ H4[G,CÃ] (6.79)A class Ï â H3[G,CÃ] is an extension of Ï|N to G. This means a Ï â H3[G,CÃ] givesa class Ï|N â H3[N,CÃ] together with a 3-group morphism G/N â BrPic(VecÏ|NN ). By trun-cating this 3-group morphism to a 2-group morphism we obtain an element ofGCrExt(Vec, N).This defines a group homomorphismR : H3[G,CÃ]â GCrExt(Vec, N). (6.80)This can be understood as an restriction map: a class α â H3[G,CÃ] defines a G-crossedbraided fusion category VecαG. Then by restricting to the normal subgroup N we obtain an el-ement of GCrExt(Vec, N). Another way of understanding it is to view a class α â H3[G,CÃ]as its restriction α|N plus extension data γ, p, β that specifies how α|N is extended to α. ThenR simply forgets the last piece β, and α|N , γ, p determine an element of GCrExt(Vec, N).An element in GCrExt(Vec, N) has trivial O4-obstruction exactly when it can be lifted tocertain VecαG. This means the following sequence is exact.H3[G,CÃ] Rââ GCrExt(Vec, N) O4âââ H4[G/N,CÃ]. (6.81)Take a class α â H3[G,CÃ], which we view as α|N plus extension data γ, p, β. Then αlies in the kernel of R if and only if α|N , γ, p are trivial. This means α comes from a class186β â H3[G/N,CÃ] in the LHS spectral sequence. In other words, the following sequence isexact.H3[G/N,CÃ] pâââ H3[G,CÃ] Rââ GCrExt(Vec, N) (6.82)Put together, we have the following.Theorem 6.8.2: There is an exact sequenceH3[G/N,CÃ] pâ3ââ H3[G,CÃ] Rââ GCrExt(Vec, N) O4âââ H4[G/N,CÃ] pâ4ââ H4[G,CÃ]. (6.83)This is the critical tool that we will use to compute the set of minimal condensablealgebras. As a set we haveGCrExt(Vec, N) = ker(O4)Ã im(O4) = im(R)Ã ker(pâ4) = H3[G,CÃ]/im(pâ3)Ã ker(pâ4)(6.84)Therefore the problem reduces to group cohomology calculation.6.8.4 Physical interpretationHere we provide a simple physical interpretation of the exact sequence Eq. (6.83). If weaccept that minimal condensable algebras are dual to 2+1-D G-gSPTs(which we will provein Sec. 6.9), then the exact sequence may be written in physical terms asSPTG/N3D â SPTG3D â gSPT(G,N)3D â SPTG/N4D â SPTG4D. (6.85)Here gSPT(G,N)3D stands for 2+1-D G-gSPTs with gapped symmetry N and the gaplesssector transforms faithfully under G/N . Consider the right half of the sequence. For a G-gSPT, the IR symmetry G/N may be anomalous, this anomaly is the image in SPTG/N4D .However this anomaly can not be arbitrary: it must be emergent, meaning that it trivializeswhen pulled back to G. Conversely, for every such emergent anomaly, one can construct aG-gSPT using the Wang-Wen-Witten construction [123]. This means the right half of thesequence is exact.Take a 2+1-D G-SPT, we can tune to a critical point where a quotient G/N becomescritical. This type of gSPT is called weak or non-intrinsic. This defines the map from SPTG3Dto gSPTG,N3D . Clearly a weak gSPT has no emergent anomaly: we can tune back to theG-SPT symmetrically without introducing topological order, which means the gapless sector187has non-anomalous G/N -action. Conversely if a G-gSPT has no emergent anomaly, then itsgapless sector can be gapped out without breaking symmetry or introducing topological order.This shows the sequence is exact at the middle term.Finally we look at the left half of the sequence. If a G-SPT comes from a G/N -SPT viapull-back, then as we tune to the G/N -critical point, the G/N -SPT information should allbe lost, therefore a weak gSPT coming from a G/N -SPT should be equivalent to a trivialgSPT(whatever a trivial gSPT means). The other direction is: if a G-SPT becomes a trivialgSPT when tuned to G/N -critical point, then it must come from a G/N -SPT. This is moresubtle as it requires a precise definition of a âtrivial G-gSPTâ. We can only provide anexplanation using the decorated domain wall picture, which is discussed in Sec. 6.9.From Eq. (6.84), we see that the classification of G-gSPT has two factors:gSPTG,N3D = H3[G,CÃ]/im(pâ3)ï¸¸ ︷︷ ︸weak gSPTÃ ker(pâ4)ï¸¸ ︷︷ ︸emergent anomaly(6.86)The second factor determines the image of the gSPT in H4[G/N,CÃ], i.e. the emergentanomaly of the gSPT. The first factor means that for a fixed emergent anomaly, differentG-gSPTs are related by stacking with G-SPTs. However stacking with a G-SPT does notchange the gSPT class if the G-SPT comes from a G/N -SPT. Therefore for a fixed emergentanomaly, the set of G-gSPTs is a torsor over H3[G,CÃ]/im(pâ3). The first factor can be calledthe weak gSPT factorâ it is the classification of weak gSPTs.6.9 Physics of 2+1-D G-gSPT: decorated domainwall and spectral sequenceIn the previous section, we defined a class of condensable E2-algebras in Z1[2VecG], namedminimal condensable algebras, and claimed they are dual to 2+1-D G-gSPTs. The mathemat-ical structure of minimal condensable algebras does seem to resemble some physical propertiesof G-gSPTs, as discussed above. However, to rigorously establish a correspondence, we needto develop a pure 2+1-D theory for 2+1-D G-gSPTs, which is the goal of this section. Inthis section we give a pure 2+1-D analysis for 2+1-D G-gSPTs, and confirm that they aredescribed by exactly the same structure as minimal condensable algebras. Our tool is thedecorated domain wall construction.1717The discussion here is inspired by a simialr analysis in [82, 81] for average SPTs(ASPT). In factASPT and gSPT are expected to be in 1-1 correspondence [82].188Let us start by reviewing the standard decorated domain wall construction for gapped(bosonic)SPTs. The discussion here follows [82, 126]. If the protecting symmetry G sits in agroup extension1â N â Gâ K â 1 (6.87)Then we may construct an SPT of G by decorating defects of K(domain walls, junctionsbetween domain walls, etc.) with SPTs of N . We start with a K-broken gapped trivial state,then then we put SPTs of N on defects of K, finally we proliferate the K-domain walls. Thisprocess yields a gapped SPT of G if and only if the conditions for the decoration pattern hold:â¢ The defects of K in each dimension have no tâ Hooft anomaly of N . This makes surecondensing the domain walls of K results in a gapped state with no ground state degen-eracy.â¢ The charge of N is preserved under a continuous deformation of K-defect configuration.This makes sure the resulting state preserves the symmetry G.â¢ There is no berry phase accumulated under a closed continuous deformation of defectconfiguration.Mathematically, the decorated domain wall construction is described by the Atiyah-Hirzebruch spectral sequence, whose E2-page isEp,q2 = Hp(K,hq(N)), (6.88)where hq(N) is the classification of invertible phases in q spacetime dimension with symmetryN , and p+q = d+1 is the spacetime dimension. If N is finite and unitary, then in low dimen-sions hq(N) â Hq[N,CÃ] and the spectral sequence reduces to the LHS spectral sequence.Physically the term Hp(K,hq(N)) describes decorating N -invertible states on codimension-pdefects of K. The condition that the K-defects are free of N -anomaly leads to the coyclecondition. Not all terms on the E2-page describe valid decoration pattern. For a decorationpattern Ïp,q â Ep,q2 to correspond to a valid G-SPT, there are certain obstructions that mustvanish. These obstructions are described by the differentialsdr : Ep,qr â Ep+r,qâr+1r . (6.89)For example, for a term Ï0,3 â H0(K,h3(N)) to give rise to a valid 2+1-D G-SPT, we musthave d2(Ï0,3) = 1 in E2,22 , d3(Ï0,3) = 1 in E3,13 and d4(Ï0,3) = 1 in E4,04 .These obstructions correspond to the conditions for the decoration pattern discussedabove:189â¢ For r â¤ dâ p, dr(Ï) = 0 means the defects of K carry no anomaly of N .â¢ For r = d â p, dr(Ï) = 0 means the charge of N is preserved under a continuousdeformation of K-defects.â¢ For r = d, dr(Ï) = 0 means there is Berry phase accumulated under closed continuousdeformation of K-defects.Similar to our discussion in Sec. 6.8.2, when these conditions are met, each conditiondrÏ = 0 means certain equation has a solution, and we need to choose a solution. Thissolution will then define a consistent decoration pattern therefore a G-SPT.Now we consider a 2+1-D gapless SPT with exact symmetry G and gapped symmetryN . Although the system is gapless, the symmetry N only acts on gapped degrees of freedom,and it is still legitimate to talk about the SPT/SET class of N . This class is a topologicalinvariant of the system, in the sense that it will not change as long as the N -charge gap is notclosed. If the N -charges form a nontrivial topological order, then it should be considered as agapless SET. Furthermore, we can talk about the gapped N -phases on defects of G. These arealso topological invariants of the system. These N -phases must be gapped and short-rangedentangled(otherwise we call it a gSET). Therefore for any gSPT we can identify a domain walldecoration pattern. This domain wall decoration pattern is exactly the topological skeletonof the gSPT. This decoration pattern must still satisfy the first two conditions for gappedSPTs:â¢ Since N is gapped and not broken, there should be no N -anomaly on defects of K.â¢ Since N is not broken, the charge of N should be preserved under a continuous defor-mation of K-defect configuration.However, we no longer require the last condition to hold. This is because the K-actionon the gapless state can be anomalous. This means upon a closed deformation of K-defects,the N -SPTs decorated on K-defects may contribute a Berry phase O, and the anomaly of theK-action on the gapless state contributes a phase â¦. Then the wavefunction of the gaplessstate is single valued as long as O = â¦â1. This is exactly the anomaly cancellation mechanismfor igSPTs: the anomaly from the gapped sector cancels the anomaly of the gapless sector.We summarize the above discussion as the following physical theorem.Theoremph 6.9.1: A 2+1-D G-gSPT is described by a term Ï â H3[N,CÃ] such thatd2Ï = 0, d3Ï = 0 together with a solution to these equations (γ, p). The emergent anomalyof the gSPT is jÏ,Î³,p, where jÏ,Î³,p is the O4-obstruction of lifting Ï to G.At this point, it should become clear that 2+1-D G-SPTs are exactly dual to minimalcondensable algebras in the SymTOZ1[2VecG]. Furthermore, our discussion on minimalcondensable algebras immediately leads to the following.190Corollary 6.9.1: A 2+1-D G-gSPT with gapped symmetry N is equivalently described bya G-crossed braided fusion structure on VecN .Corollary 6.9.2: The set of 2+1-DG-gSPTs with gapped symmetryN , denoted as gSPT(G,N)3D ,is given bygSPTG,N3D = H3[G,CÃ]/im(pâ3)ï¸¸ ︷︷ ︸weak gSPTÃ ker(pâ4)ï¸¸ ︷︷ ︸emergent anomaly(6.90)where pâ3, pâ4 are the maps on cohomology induced by p : Gâ G/N .6.10 Examples of minimal condensable algebras andG-gSPTsIn this section, we apply our general results in Sec. 6.8, Sec. 6.9 to study concrete examples.We compute the classification of 2+1-D gSPTs for several G andN , and discuss their emergentanomalies. Our main tool is the formulagSPTG,N3D = H3[G,CÃ]/im(pâ3)ï¸¸ ︷︷ ︸weak gSPTÃ ker(pâ4)ï¸¸ ︷︷ ︸emergent anomaly, (6.91)wherepâ3 : H3[G/N,CÃ]â H3[G,CÃ], (6.92)andpâ4 : H4[G/N,CÃ]â H4[G,CÃ], (6.93)are induced by p : Gâ G/N .When N = G, pâ3, pâ4 are zero maps and we havegSPTG,G3D = H3[G,CÃ] = SPTG3D. (6.94)When N = {e}, pâ3, pâ4 are identity maps and we havegSPTG,{e}3D = 0. (6.95)191Therefore in the following we only consider N to be a proper normal subgroup of G.6.10.1 G = Z4, N = Z2We need to understand the map on cohomology pâ3 : H3[Z2, U(1)]â H3[Z4, U(1)]. NoticeH3[Z2, U(1)] = H4[Z2,Z] = â¨x âª xâ©where x generates H2[Z2,Z] = Z2. Similarly H3[Z4, U(1)] = â¨a âª aâ©, with a generatingH2[Z4,Z] = Z4. The map pâ maps x to 2a, therefore xâª x to 4aâª a = 0. We see that pâ3 = 0.Since H4[Zn, U(1)] = 0, we also have pâ4 = 0. Therefore we havegSPTZ4,Z23D = Z4︸︷︷︸weak gSPTÃ 0︸︷︷︸emergent anomaly(6.96)H3[(Z4,Z2), U(1)] â H4[Z4, U(1)] = Z4. Physically, this means1. There is no igSPT for G = Z4, N = Z2 since H4[G/N,U(1)] = 0.2. All four SPTs of Z4 become four different weak gSPTs when G/N = Z2 is made critical.6.10.2 G = Zn, N = ZpLet n = pq, and N = Zp. Since H4[Zn, U(1)] = 0, there can not be an emergent anomaly andwe only need to understand the map pâ3 : H3[Zq, U(1)]â H3[Zn, U(1)].We notice thatH3[Zn, U(1)] â H4[Zn,Z] is generated by aâªa, with a generatingH2[Zn,Z].Similarly H3[Zq, U(1)] â H4[Zq,Z] is generated by xâªx with x generating H2[Zq,Z]. pâ mapsx to pa, therefore maps x âª x to p2a âª a. We havepâ3 : H3[Zq, U(1)] â Zq Ãp2âââ Zn â H3[Zn, U(1)]Therefore impâ3 = Z n(n,p2) = Z q(p,q) and H3[Zn, U(1)]/impâ3 â Zp(p,q). We see thatgSPTZn,Zp3D = Zp(p,q)ï¸¸ ︷︷ ︸weak gSPTÃ 0︸︷︷︸emergent anomaly(6.97)In total, the number of 2+1-D Zn-gSPTs isâp|n,p Ì¸=np(p, q) (6.98)192All of them are weak gSPTs. However, the restriction map R is not an isomorphism ingeneral: different SPTs can become identical gSPT when tuned to critical points.6.10.3 G = Z2 Ã Z4, N = Z2 = â¨0, 2â©The general case Zn1ÃZn2 will be treated in the next subsection. As a teaser, here we considerthe case where N is the Z2 subgroup of the Z4 factor, and G/N = Z2 Ã Z2.The relevant cohomology groups are H4[Z2ÃZ4, U(1)] â H4[Z2ÃZ2, U(1)] = Z22, H3[Z2ÃZ4, U(1)] = Z2 Ã Z4 Ã Z2 and H3[Z2 Ã Z2, U(1)] = Z32. The cohomology ring of Z2 Ã Z2 andZ2 Ã Z4 can be found in [91]. We haveHâ[Z2 Ã Z4,Z] â Z[a, b, c]/â¨2a, 4b, 2câ©with |a| = |b| = 2, |c| = 3. In degree â¤ 6, we also haveHââ¤6[Z22,Z] â Z[x, y, z]/â¨2x, 2y, 2zâ©.with |x| = |y| = 2, |z| = 3. The projection p : Z2ÃZ4 â Z2ÃZ2 induces a map on cohomologythat maps x 7â a y 7â 2b, z 7â c. Therefore we haveimpâ3 = â¨a2â© â Z2, H3[Z2 Ã Z4, U(1)]/impâ3 â Z2 Ã Z4,ker pâ4 = â¨y âª zâ© â Z2,from which we deducegSPTZ2ÃZ4,â¨(0,2)â©3D = Z2 Ã Z4ï¸¸ ︷︷ ︸weak gSPTÃ Z2︸︷︷︸emergent anomaly(6.99)The second factor ker pâ4 = Z2 means that there are igSPTs: the element y âª z â H4[Z2 ÃZ2, U(1)] is trivialized when pulled back to Z2 Ã Z4, thus can be a valid emergent anomaly.Fixing an emergent anomaly(either trivial or y âª z), different gSPTs are related by stackingwith Z2ÃZ4-SPTs. However not all different Z2ÃZ4-SPTs give different gSPTs by stacking.Specifically, stacking with the SPT protected purely by the Z2 factor does not affect the gSPTclass.In conclusion, there are 16 gSPTs with symmetry Z2 Ã Z4 and gapped symmetry N =Z2 = â¨(0, 2)â©, 8 of which are igSPTs.1936.10.4 G = Zn1 Ã Zn2, N = Zp = â¨(0, q)â©We may assume n1|n2 are the elementary divisors of the abelian group. Let n2 = pq andN = Zp = â¨(0, q)â©, G/N = Zn1 Ã Zq. The integer cohomology rings Hâ in degree â¤ 6 areHââ¤6[Zn1 Ã Zn2 ,Z] â Z[a, b, c]â¤6/â¨n1a, n2b, n1câ©Hââ¤6[Zn1 Ã Zq,Z] â Z[x, y, z]â¤6/â¨n1x, qy, (n1, q)zâ©.with |a| = |b| = |x| = |y| = 2, |c| = |z| = 3. pâ maps x 7â a, y 7â pb, z 7â n1(n1,q)c. Thus on H4we have x2 7â a2, y2 7â p2b2, xâªy 7â paâªb, on H5 we have xâªz 7â n1(n1,q)aâªc, yâªz 7âpn1(n1,q)bâªc.From this one can deduce thatH3[Zn1 Ã Zn2 , U(1)]/impâ3 â Zp(p,q) Ã Z(n1,p),andker pâ4 = Z(p,q,n1)).Therefore gSPTs with symmetry Zn1 Ã Zn2(with n1|n2) and N = Zp = â¨(0, q)â© haveclassificationgSPTZn1ÃZn2 ,Zp=â¨(0,q)â©3D = Zp(p,q) Ã Z(n1,p)ï¸¸ ︷︷ ︸weak gSPTÃ Z(p,q,n1)ï¸¸ ︷︷ ︸emergent anomaly. (6.100)We see that there exists 2+1-D igSPT with symmetry Zn1 Ã Zn2(with n1|n2) and N =Zp = â¨(0, q)â© if and only if (p, q, n1) Ì¸= 1.6.10.5 G = S3, N = Z3Let S3 = â¨r, s|r3 = s2 = 1, srs = r2â©. The only proper normal subgroup is N = â¨râ© = Z3,and we have G/N = Z2. Since H4[G/N,U(1)] = 0, there is no emergent anomaly thus noigSPT. We next classify weak gSPTs. We need to understand the map pâ3 : H3[Z2, U(1)] âH3[S3, U(1)]. Consider the LHS spectral sequence Hp[Z2,Zq[Z3, U(1)]] â Hp+q[S3, U(1)].For p + q = 3 we only have two nonzero terms on the E2-page: H3[Z2, U(1)] â Z2 andH0[Z2,H3[Z3, U(1)]] = Z3. We know that H3[S3, U(1)] = Z6. Therefore all terms inH3[Z2, U(1)] survive to H3[S3, U(1)]. Thus impâ3 = Z2 and H3[S3, U(1)]/impâ3 = Z3. Weconclude194gSPTS3,Z33D = Z3︸︷︷︸weak gSPTÃ 0︸︷︷︸emergent anomaly(6.101)In conclusion, there are 3 gSPTs with G = S3, N = Z3, and all of them are weak gSPTs.6.10.6 G = D4, N = Z(D4) = Z2Let D4 = â¨r, s|r4 = s2 = (rs)2 = 1â©. There are in total 4 nontrivial normal subgroups. Letus consider the case that can potentially have an emergent anomaly: N = Z2 = â¨r2â©, andG/N = Z22. We are interested in maps pâ3 : H3[Z22, U(1)] â Z32 â H3[D4, U(1)] â Z22ÃZ4, andpâ4 : H4[Z22, U(1)] â Z22 â H4[D4, U(1)] â Z22. We again use the LHS spectral sequence. LetHâ(Z22,Z) = Z[a, b, c]/â¨relationsâ© with |a| = |b| = 2, |c| = 3. It turns out a âª b â H4[Z22,Z] âH3[Z22, U(1)] is hit by a differential(d2) from H1[Z22,H1[Z2, U(1)]], and impâ3 = Z22. ThereforeH3[D4, U(1)]/impâ3 â Z4. For pâ4, it turns out all elements in H4[Z22, U(1)] are hit by adifferential from H2[Z22,H1[Z2, U(1)]], therefore ker pâ4 = Z22. We concludegSPTD4,Z(D4)3D = Z4︸︷︷︸weak gSPTÃ Z22︸︷︷︸emergent anomaly(6.102)There are three different types of nontrivial emergent anomalies, all of which are trivializedin D4 by a d2 differential.6.11 2+1-D Gapless SPT with generalized symme-tryIn this section we apply the SymTOclub-sandwich construction to construct 2+1-D gaplessSPTs with generalizes symmetries. For a symmetry category C, this amounts to choosing tonon-Lagrangian condensable E2-algebra that preserves the symmetry C. In this section wewill only consider âsimpleâ condensable algebras, meaning that the higher structures of thealgebra such as the associator, the braiding, are taken to be trivial, and we will only definecondensable algebras at the object level.1956.11.1 igSPT with 2-group symmetry 2Vec(Z[1]2 ÃZ[1]2 )âZ[0]2We consider a SymTOZ1[2VecD4 ]. We take D4 = â¨r, s|r4 = s2 = (rs)2 = 1â©. Some propertiesof this topological order are summarized below.â¢ The gauge charges form a symmetric fusion category Rep(D4). There are six simpleobjects.1. Four 1-dimensional representations, with r = Â±, s = Â±. The corresponding gaugecharges are denoted as BÂ±,±.2. One 2-dimensional representation, withs =(0 11 0), r =(w 00 w3)(6.103)where w = e2Ïi/4. The corresponding non-abelian gauge charge is denoted as Q.They have fusion ruleBÂ±,± âQ = QâBÂ±,± = Q, QâQ = B+,+ âB+,â âBâ,+ âBâ,â. (6.104)â¢ There are five flux sectors labeled by the five conjugacy classes: 1,M{s,sr2},Mr2 ,M{r,r3},M{sr,sr3}.On the symmetry boundary, we choose gapped minimal boundary C[D4,Z22, 0, 0], whereZ22 = â¨s, r2â©. By Example. 6.2.3, the symmetry category is the 2-group symmetry 2Vec(Z[1]2 ÃZ[1]2 )âZ[0]2 ,where the 0-form symmetry exchanges the two Z2-factors of the 1-form symmetry. We canderive this symmetry category by analyzing the corresponding Lagrangian condensation.The Lagrangian condensation on the symmetry boundary can be decomposed into twosteps. In the first step the condensable E2-algebra is Func[D4/(Z2ÃZ2),Vec] = Î£C[D4/(Z2ÃZ2)] = Î£(1âBâ+). This means the charge Bâ+ is condensed. This condensation reduces theD4-gauge theory to a Z22-gauge theory. In particular, we have:â¢ The charge B+â stays nontrivial and becomes a gauge charge of the Z22-gauge theory.The charge Q splits into two irreducible representations of Z2ÃZ2 = â¨s, r2â©: Qâ q1âq2.Together with the trivial charge they form the fusion category Rep(Z2 Ã Z2).â¢ The fluxes M{r,r3}, M{sr,sr3} braid nontrivially with B+â and get confined by the con-densation. The flux Mr2 and M{s,sr2} stay deconfined. Furthermore M{s,sr2} splits intotwo fluxes of Z22: M{s,sr2} âMs âMsr2 .In the second step, we condense the fluxes Mr2 ,Ms. This is described by the Lagrangianalgebra VecZ2ÃZ2 in Z1[2VecZ2ÃZ2 ]. In total, the charge Bâ+ and the fluxesM{s,sr2},Mr2 arecondensed.196We next construct an igSPT with this 2-group symmetry by choosing a non-Lagrangiancondensable algebra in Z1[2VecD4 ]. Consider the condensable E2-algebra Func[D4/Z4,Vec].This is a pure charge condensation that Higgses the gauge field to Z4 = â¨râ©. We haveFunc[D4/Z4,Vec] = ΣC[D4/Z4] = Σ(1 â B+â). I.e. the charge B+â is condensed. Thiscondensation reduces the D4-gauge theory to a Z4-gauge theory. In particular, we haveâ¢ The charges Bâ+ and Q stay nontrivial. Since r = â1 in Bâ+, Bâ+ becomes the doublecharge e2 of the Z4-gauge theory. The non-abelian charge Q splits into two irreduciblerepresentations of Z4: Qâ e+ e3.â¢ The fluxM{s,sr2} andM{sr,sr3} braid nontrivially with B+â and get confined. The fluxesMr2 ,Mr,r3 stay deconfined. Furthermore Mr,r3 splits: Mr,r3 â mr +mr3 . These are thefluxes of the Z4-gauge theory.MBphysâ¨B+ââ© â¨Bâ+,M{s,sr2},Mr2â©Z1[2VecZ4 ] Z1[2VecD4 ]e+ e3 â Qe2 â Bâ+m+m3 âMr,r3m2 âMr2Bsymâ Z1[2VecZ4 ]â¨e2,m2â©BIRsymBphysFigure 6.4: The club-sandwich construction of a (Z[1]2 Ã Z[1]2 )â Z[0]2 -igSPT.The structure of the club-sandwich is shown in Fig. 6.4. We note that this choice ofcondensable algebra preserves the 2-group symmetry 2Vec(Z[1]2 ÃZ[1]2 )âZ[0]2 : The domain wall Mand the symmetry boundary Bsym share no common charge or flux condensation. Thereforethere is no 0-form nor 1-form symmetry breaking, and the 2-group symmetry is preserved.Next we analyze the IR symmetry of this 2Vec(Z[1]2 ÃZ[1]2 )âZ[0]2 -gSPT. As explained in Sec. 6.5,this is obtained by fusing the domain wallM withBsym to obtain a new symmetry boundary ofthe condensed phase Z1[2VecZ4 ]. To determine the nature of this new symmetry boundary, weonly need to determine the charge and flux condensation on it. See Fig. 6.4. The charge e, e3become the non-abelian charge Q, while the charge e2 becomes the abelian charge Bâ+. Sinceon Bsym only the charge Bâ+ is condensed, we see that the effective symmetry boundary ofthe Z4-gauge theory condenses the charge e2. The flux m,m3 of the Z4-gauge theory becomes197the flux M{r,r3} when moved into the D4-theory, and the flux m2 becomes the flux Mr2 .Since on Bsym the flux Mr2 is condensed, we see that the effective symmetry boundary ofthe Z4-gauge theory condenses the flux m2. Therefore the effective gapped boundary for theZ4-gauge theory is defined by the property that the charge e2 and the flux mr2 condense.This means the IR symmetry of this gSPT is C[Z4,Z2, 0, 0]. According to Example. 6.4.1, thissymmetry is the anomalous 2-group symmetry2VecÏZ[0]2 ÃZ[1]2 . (6.105)Therefore the above construction yields an igSPT with full symmetry 2Vec(Z[1]2 ÃZ[1]2 )âZ[0]2and anomalous IR symmetry 2VecÏZ[0]2 ÃZ[1]2.6.11.2 igSPT with symmetry 2VecZ4 Ã 2Rep(D4)The SymTOis Z1[2VecZ4ÃD4 ] = Z1[2VecZ4 ]â Z1[2VecD4 ]. The symmetry boundary is definedby the Lagrangian algebra Func[Z4,Vec] â VecD4 . Physically, all charges of the Z4-gaugefield are condensed, and all fluxes of the D4-gauge field are condensed. To construct a gSPTwith this symmetry, we construct a non-Lagrangian condensable algebra in two steps. In thefirst step, we condense all abelian charges of D4: BÂ±,±. This will reduce the gauge group toZ4 Ã Z2, with the second Z2-factor generated by the central element r2 â D4. Specifically,â¢ The non-abelian charge Q remains deconfined and splits into eZ2 â eZ2 , with eZ2 theZ2-charge.â¢ The flux Mr2 is the only deconfiend flux in the D4-sector and becomes the Z2-fluxMr2 â mZ2 .In the second step, we condense the flux m2Z4mZ2 , where mZ4 is the elementary flux of theZ4-gauge theory. This reduces the gauge group to Z4. Specifically,â¢ The charge eZ4eZ2 stays deconfined, where eZ4 is the elementary charge in the Z4-gaugetheory. It generates the charges of Z1[2VecZ4 ].â¢ The flux mZ4 stays deconfined and generates the fluxes of Z1[2VecZ4 ].Combine the above two steps, we obtain a non-Lagrangian condensable algebra in Z1[2VecZ4 ]â Z1[2VecD4 ]. We notice that this defines a 2+1-D gapless phase that preserves the symme-try 2VecZ4 â 2Rep(D4): The domain wall M condenses {BÂ±,±,Mr2m2Z4}, which has trivialintersection with the condensation on the symmetry boundary.To deduce the IR symmetry of this gSPT, we again fuse the domain wall M with Bsymto obtain a new gapped boundary of Z1[2VecZ4 ]. The nature of this gapped boundary is198M1 M2Bphys Bsymâ¨m2Z4mZ2â© â¨BÂ±,±â© â¨eZ4 ,MD4â©Z1[2VecZ4 ] Z1[2VecZ4ÃZ2 ] Z1[2VecZ4ÃD4 ]eâ eZ4eZ2mâ mZ4 eZ2 + eZ2 â QmZ2 âMr2Figure 6.5: The domain wall M is obtained via two steps of condensation.determined by the condensation on it. Observe that e2 = (eZ4eZ2)2 = e2Z4 can be moved intothe Z1[2VecZ4 ]â Z1[2VecD4 ] region and is condensed onBsym. The fluxm2 â m2Z4 âMr2 canbe moved into the Z1[2VecZ4 ]â Z1[2VecD4 ] region and is also condensed on Bsym. Thereforethe effective gapped boundary of the Z4-gauge theory condenses e2Z4 and m2Z4 . We see thatthe IR symmetry is again C[Z4,Z2, 0, 0], an anomalous 2-group symmetry. In conclusion, theabove construction yields an igSPT with exact non-invertible symmetry 2VecZ4 â 2Rep(D4)and an emergent anomalous 2-group symmetry 2VecÏZ[0]2 ÃZ[1]2in the IR.199Chapter SevenOutlookWe have outlined a general framework for understanding gapless quantum phases. The frame-work is based on the notion of quantum order and symmetry-enriched quantum order, whichgeneralize naturally the notion of topological order and symmetry-enriched topological order.We have also given a classification of symmetry-enriched quantum orders, using the notion oftopological skeleton. We found that a symmetry-enriched quantum order is given by a functorfrom the symmetry category to the category of topological skeleton.For a class of simple symmetry-enriched quantum orders called gSPTs, we have provideda complete algebraic classification, a detailed lattice model construction and discussed theirrole in the topological holography paradigm in 1+1D and 2+1D.A lot more needs to be done in order to establish a complete theory of all quantum phaseswith or without symmetry. Here we point out some future directions, some of which arerelatively straightfoward, some of which are more challenging.7.1 Lattice model of symmetry-enriched quantumordersPhysical realization of the symmetry-enriched quantum orders we predicted would be veryinteresting. In Chapter 4, we discussed lattice models of gapless SPTs. Gapless SPTs arethe simplest symmetry-enriched quantum orders. According to the framework developed inChap 3, there are infinite many SEQs for a given quantum order. It would be very interestingto realize new SEQs beyond gapless SPTs in lattice models.It is almost impossible to enforce gaplessness in a lattice model without finetuning. In200practice what one often enforces is a symmetry constraint. From our perspective, this isenforcing a symmetry category S, and hope that the model realizes a nontrivial functor F :Sâ C, where C is the topological skeleton of a quantum order. For instance, the anyonic chainis a model that can enforce a general fusion categorical symmetry in 1+1D. In general it doesnot guarantee to realize a CFT with the input fusion category as topological skeleton. Thiscan be understood as the fact that the functor can be non-identify and further does not evenneed to be nontrivial. In many examples enforcing a fusion categorical symmetry leads to afull symmetry-breaking phases. Nevertheless enforcing a symmetry category can be a strongconstraint, failing to produce a nontrivial functor F can be viewed as our inability to identifythe correct parameters, since the space of gapless phases has zero measure in the parameterspace. In 1+1D, there is no known way of realizing an arbitrary given functor F. Noticethat whether all fusion categories can be realized as topological skeleton of RCFTs is still anopen problem, and this is clearly a sub-problem of constructing general symmetry-enrichedRCFT. However we can ask if for a given RCFT can we construct an arbitrary functor withthe topological skeleton of this RCFT as target. One should be able to achieve this by firstattaching this RCFT to the boundary of the Levin-Wen model with the topological skeletonof the RCFT as input, and then apply the SymTFT construction to construct symmetry-enriched version of the RCFT. We hope to explore this in details in the future.In 2+1D the situation is in fact more manageable. The first realization is that any 2+1D(rational) quantum order is essentially gauged version of symmetry-breaking transitions, upto stacking with topological orders. This is due to the classification of fusion 2-categories [31].Therefore one can realize a general quantum order as a sandwich of a 3+1D finite gauge theory,with the physical boundary being a simple G-SSB transition boundary. Then by extendingthe sandwich to a club-sandwich we obtain a general symmetry-enriched quantum order.7.2 Gauging of quantum order and symmetry-enrichedquantum orderWe have not discussed gauging for quantum orders. There should be a notion of gauginga quantum order and a notion of gauging a symmetry-enriched quantum order. Gauging aquantum order can be thought of as gauging the topological skeleton of the quantum order, in1+1D RCFTs this corresponds to orbifolding. We can also gauge the protecting symmetry ofa symmetry-enriched quantum order, which would change both the protecting symmetry andthe topological skeleton of the underlying quantum order. When both changes are recorded,gauging should be an invertible process. Gauging the topological skeleton could also change201the local operator algebra, but there should be a notion of Morita equivalence of local operatoralgebras, such that gauging leaves local operator algebra invariant up to Morita equivalence.Apart from the effect on local operator algebra, the rest of the theory of gauging in quantumorders and symmetry-enriched quantum orders should be relatively simple and described bystandard abstract nonsense of higher category theory. Of course, implementing (generalized)gauging in lattice models would be very interesting as well.7.3 The theory of local operator algebra.So far we have not treated local operator algebra in quantum orders rigorously. Developingthe theory of LOA is a major challenge to our framework. The theory of LOA in quantumfield theory has some significant development in recent years, and a rigorous mathematicaltheory seems to be emerging [21]. However for condensed matter purposes, a lattice version ofLOA would be needed. There have also been plenty of developments in this direction [48, 4].Ideally one should be able to construct a braided fusion higher category from the represen-tations of LOA, and there should be a notion of Morita equivalence of LOA that leaves thecorresponding braided fusion higher category invariant. The Kitaev gapless spin liquid wouldbe an interesting case to consider, since the LOA should simply be the gauged version of thatof a fermi surface.7.4 Irrational quantum ordersSo far we have been mainly focusing on quantum orders with finite many topological defects.However in a gapless phase there can be infinite many topological defects. To describe suchcases, a generalization of fusion higher category is needed, where the number of simple ob-jects is allowed to be infinite. A prototypical example of such categories is the category ofrepresentations of a continuous Lie group. E.g. Rep(SU(2)) has simple objects labeled byj â 12Z, this category satisfies almost all axioms of fusion category except for the finitenesscondition. Almost all known examples of infinite topological skeleton come from continuousgauge theory, but there could be infinite topological skeleton beyond gauge theory. One cancompare with the development of topological order: finite gauge theories are typical examplesof topological order, but once the abstract theory of topological order is established, it is clearthat topological orders can be more than gauge theories. Similarly, perhaps an abstract theoryof infinite topological skeleton can lead to new non-rational orders beyond gauge theories, andalso give a unified and pure algebraic formulation of gauge theories with finite or continuous202gauge groups.7.5 Continuous symmetryThis is also a major challenge to our framework. The main problem is that a categoricaldescription of continuous symmetry is unknown. This is related to the previous problem bygauging. In special cases, this problem is not a problem. For example, consider a 2+1D quan-tum order with topological skeleton ΣC where C is a potentially-degenerate braided fusioncategory. Then a G-enriched ΣC can still be defined, since the notion of G-crossed braidedfusion category can be defined for continuous G. Another example is gSPTs with continuoussymmetry, we have formulated gSPTs as functors dVecG â dVecÏK , but this can be formu-lated in elementary terms as well, namely such a functor is the same as a group homomorphismG â K and a trivialization of Ï upon pull-back. This elementary formulation makes sensefor continuous symmetry groups as well. However it is not clear if generalizing in this waycovers all possible symmetry-enriched quantum orders. When we derive the classification ofsymmetry-enriched quantum order, one important assumption is that the symmetry defectsmap to topological defects of the underlying quantum order. For a continuous symmetry thisdoes not need to be the case.7.6 Spacetime symmetrySo far we have only discussed quantum phases with internal symmetries. A spacetime sym-metry can be anti-unitary, such as time-reversal or parity inversion. 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Advances in Theoretical andMathematical Physics, 27(2):583â622, 2023.213AppendicesA.1 Some useful higher linear algebraHere we review some useful results about (braided)fusion higher categories and their modulecategories. Most results are due to [28, 29, 33, 68, 69]. They can all be viewed as special casesof the higher condensation theory [63].A.1.1 Morita theory of fusion 2-categoriesCondensable E1-algebras in fusion 2-categories âDefinition A.1.1: Let C be a fusion 2-category, an algebra in C is an object A togetherwith a unit i : I â A, a product µ : AâA â A together certain coherence 2-isomorphismsensuring the algebra is unital and associative.The class of algebras relevant for physics is more restrictive.Definition A.1.2: An algebra A in C is called rigid if the product µ admits a right adjointµâ : A â AâA as an AâA bimodule 1-morphism.Definition A.1.3: A separable algebra in C is a rigid algebra A together with a section ofthe counit µ â¦ µâ â idA as an AâA bimodule 2-morphism.Definition A.1.4: An algebra A in C is called connected if the unit i : I â A is a simple1-morphism.Separable algebras are used to define Morita equivalence of fusion 2-categories.Theorem A.1.1: Let A be a separable algebra in C, then BimodAâA(C), the 2-category ofAâA bimodules in C, is a multi-fusion 2-category. If A is also connected, then BimodAâA(C)is a fusion 2-category.This leads to the following physical definition.214Definition A.1.5: A condensable E1-algebra in a fusion 2-category C is a connected separablealgebra. We callBimodAâA(C) the dual fusion 2-category obtained by condensing the algebraA.Module 2-categories over fusion 2-categories âDefinition A.1.6: Let C be a fusion 2-category, a left module 2-category is a 2-category Mtogether with a monoidal 2-functor Câ End(M), i.e. a coherent left action of C on M.Lemma A.1.1: Let M be a finite semi-simple module 2-category over C, then there exists arigid algebra A in C, such that M âModA(C), the 2-category of right A modules in C.Definition A.1.7: We call a finite semi-simple module 2-categoryM separable if the algebraA is separable. This does not depend on the choice of A.Denote by CâM := EndC(M) the monoidal 2-category of left C-module 2-functors from Mto M.Theorem A.1.2: Let M âModA(C) be a separable left module 2-category over C. Then,CâM â BimodAâA(C)op. (7.1)as multi-fusion 2-categories.We will say CâM is the dual (multi-)fusion 2-category with respect to M.Definition A.1.8: We say two fusion 2-categories C,D are Morita equivalent if one is thedual fusion 2-category of the other in the above sense.A.1.2 Relative tensor product of module categoriesThe concept of relative tensor product is the mathematical foundation of the SymTOsandwichconstruction.Definition A.1.9: Let M,N be separable right/left module 2-categories over C. Their rela-tive tensor product isMâ C N = FuncC(Mop,N). (7.2)This is not the usual definition of relative tensor product. Usually one defines relativetensor product as an object with a balanced map from M â N that is universal among suchobjects. Then one can prove that if the relative tensor product exists, it is given by Eq.7.2.For our purpose Eq.7.2 suffices.215Lemma A.1.2: LetM =ModA(C), N =ModB(C) be separable left C-module 2-categories,thenMop â C N = BimodAâB(C). (7.3)A.1.3 CentersDefinition A.1.10: Let M be a finite semi-simple n-category, its E0-center isZ0(M) := End(M), (7.4)which is a multi-fusion n-category.Definition A.1.11: Let C be a multi-fusion n-category, its E1-center isZ1(C) := EndCâC(C), (7.5)which is a braided multi-fusion n-category.The E1-center is Morita invariant:Theorem A.1.3: If C,D are two Morita equivalent fusion n-categories, thenZ1(C) â Z1(D) (7.6)as braied fusion n-categories.Lemma A.1.3: Let C be a fusion n-category, then,ΣZ1(C) â Z0(ΣC). (7.7)A.1.4 E1-fusion module 2-category over braided fusion 2-categoryDefinition A.1.12: Let B be a braided fusion 2-category, a E1-fusion left module 2-categoryis a fusion 2-category D together with a braided 2-functor F : Bâ Z1(D).Remark. Equivalently, we have a monoidal 3-functor ΣF : Î£Bâ ΣZ1(D) â Z0(ΣD). There-fore a E1-fusion module 2-category D over B is the same as a usual module 3-category ΣDover ΣB.Definition A.1.13: Let D,E be right/left E1-fusion module 2-category over B, their relative216tensor product isDâ B E := â¦(Î£Dâ ΣB ΣE), (7.8)which is a multi-fusion 2-category.Theorem A.1.4: Let C be a fusion 2-category, and M,N be separable right/left module2-categories over C, then we have Z1(CâM) â Z1(C) â Z1(CâN), which means the dual fusion2-categories CâM, CâN are right/left E1-fusion module 2-categories over Z1(C). Then we haveCâM â Z1(C) CâN â Z0(Mâ C N). (7.9)217A.2 Condensable E2-algebras in braided fusion 2-categoriesHere we review the definition of condensable E2-algebras and their (local)modules.Definition A.2.1: Let C be a braided fusion 2-category, an E2-algebra in C is an object Atogether with a 1-morphism called the product: µ : A â A â A, a 1-morphism called theunit: i : 1â A, together with invertible 2-morphisms in the following diagram:â¢ The left and right unitors l, r:AâAA A1i1l,AâAA A11irâ¢ An associator a:(AâA)âA Aâ (AâA)AâA AâAAÂµ1 1µµµaâ¢ A braiding Î²:AâA AâAAcA,AµµÎ²These 2-morphisms themselves need to satisfy a set of coherence conditions, which can befound in [32].An E2-algebra will play the role of the new vacuum in the condensed phase, therefore weneed impose a stability condition of the vacuum as explained in [63].Definition A.2.2: A condensable E2-algebra in C is a connected separable E2-algebra. Thecollection of condensable E2-algebras in C will be denoted as AlgcE2(C).Here connected means the unit i is a simple 1-morphism, and separable means the productµ admits a section(a co-product) as a map between A â A bimodules. These details can befound in [29].Definition A.2.3: Let A â AlgcE2(C). A right module over A is an object M â C togetherwith a 1-morphism ν :MâAâM, and two invertible 2-morphisms218â¢ A unitorMâAM M11iâ¢ An associator(MâA)âA Mâ (AâA)MâA MâAMÎ½1 1µννThese 2-morphisms need to satisfy certain coherence conditions, which can be found in [28].By the results in [32], the collection of all separable right modules over A,denoted asModA(C), has a natural fusion 2-category structure.Definition A.2.4: Let A â AlgcE2(C). A local module over A is a right module M togetherwith an invertible 2-morphism in the following diagram called a holonomyMâA MâAMcA,Mâ¦cM,AÎ½Î½âThe holonomy needs to satisfy certain conditions, see [32].The collection of all local modules over A, denoted as Mod0A(C), has a natural braidedfusion 2-category structure.219A.3 Extension theory of (braided)fusion categoriesHere we review the theory of extending (braided)fusion categories by finite groups developedin [36].Definition A.3.1: The Brauer-Picard 3-group BrPic(C) of a fusion category C is a 3-category with one object C and whoseâ¢ 1-morphisms are invertible Câ C bimodule categories,â¢ 2-morphisms are equivalences of such bimodule categories,â¢ 3-morphisms are isomorphisms of such equivalences.The homotopy groups of BBrPic(C) areÏ1 = BrPic(C), Ï2 = Inv(Z1(C)), Ï3 = CÃ, Ïi = 0 for i â¥ 4 (7.10)where Inv(Z1(C)) stands for the group of invertible objects in Z1(C).An extension of a fusion category C by a finite group G is a G-graded fusion categorywhose trivial component is C. There is a natural notion of equivalence of G-extensions.Theorem A.3.1: (a) Equivalence classes of G-extensions of C are given by 3-group homo-morphisms from G to BrPic(C).(b) Such homomorphisms (and hence, G-extensions) are parameterized by triples (c,M, α),where c is a group homomorphism c : Gâ BrPic(C),M belongs to a certain H2(G, Ï2BrPic(C))-torsor T 2c , and α belongs to a certain H3(G, Ï1BrPic(C))-torsor T 3c,M .(c) Certain obstruction classes O3(c) â H3(G, Ï2BrPic(C)) and O4(c,M) â H4(G, Ï3BrPic(C))must vanish for (c,M, α) to determine an extension.Definition A.3.2: Let B be a braided fusion category, Autbr(B) is the 2-group of braidedauto-equivalences of B and natural isomorphisms.Proposition A.3.1: There is a canonical isomorphism of 2-groupsBrPic(C) â Autbr(Z1(C)) (7.11)Definition A.3.3: Let B be a braided fusion category, then every left module category ofB is automatically a B-bimodule category. The Picard 3-group Pic(B) of a braided fusioncategory B is the 3-category with one object and whose 1-morphisms are invertible left B-module categories, 2-morphisms are equivalences of module categories, and 3-morphisms areisomorphisms of such equivalences.220The homotopy groups of Pic(B) areÏ1Pic(B) = Pic(B), Ï2Pic(B) = Inv(B), Ï3Pic(B) = CÃ, ÏiPic(B) = 0 for i â¥ 4. (7.12)A G-crossed extension of B is a G-crossed braided fusion whose trivial component is B.Theorem A.3.2: Let B be a braided fusion category. Equivalence classes of G-crossed ex-tensions of B are in bijection with morphisms of 3-groups Gâ Pic(B).221
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On gapless topological phases and their holographic duals
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