Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Sheinbaum Frank, Daniel
Doctor of Philosophy - PhD
University of British Columbia
The present thesis consists of 2 parts. Chapter 1 is about applications of twisted equivariant K-theory to condensed matter. We consider non-interacting electrons on a half-crystal (a crystal with a boundary), with a gapped bulk condition, under quasi-adiabatic evolution. In A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 we found that Fermi surfaces for these systems under quasi-adiabatic evolution are classified by the Kâ»Â¹-group of the surface Brillouin zone Tdâ»Â¹. Systems with time-reversal and particle-hole symmetry were also considered and we obtained different KR-groups for the different cases. In Chapter 1 I rewrite A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 in a more function-analytic language and further solve technical issues to extend it to include crystallographic symmetries on the directions parallel to the boundary. In Chapter 2 I reproduce the relevant parts of my joint work with C. Okay (C. Okay and D. Sheinbaum arXiv:1905.07723). There we explored a connection between twisted equivariant K-theory to contextuality in quantum mechanics. We also reformulated the sheaf-theoretic framework of S. Abramsky and A. Brandenburger New Journal of Physics, 2011 for contextuality and connect it to another one employing a group cohomology approach of C. Okay, S. Roberts, S.D Bartlett, and R. Raussendorf Quantum Information and Computation, 2017. This leads to the construction of a classifying space for contextuality, from which Wigner functions are classes in its twisted K-theory.
Applications and Connections betweenTwisted Equivariant K-theory,Quantum Mechanics and CondensedMatterbyDaniel Sheinbaum FrankB.Sc., Universidad Nacional Auto´noma de Me´xico, 2013M.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Posdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2019c© Daniel Sheinbaum Frank 2019The following individuals certify that they have read, and recommendto the Faculty of Graduate and Postdoctoral Studies for acceptance, thedissertation entitled:Applications and Connections between Twisted Equivariant K-theory,Quantum Mechanics and Condensed Mattersubmitted by Daniel Sheinbaum Frank in partial fulfilment of the re-quirements for the degree of Doctor of Philosophy in Mathematics.Examining Committee:Alejandro Adem, MathematicsSupervisorCihan Okay, PhysicsCo-supervisorLior Silberman, MathematicsSupervisory Committee MemberGordon Walter Semenoff, PhysicsSupervisory Committee MemberSven Bachmann, MathematicsSupervisory Committee MemberKalle Karu, MathematicsUniversity ExaminerFei Zhou, PhysicsUniversity ExamineriiAbstractThe present thesis consists of 2 parts. Chapter 1 is about applicationsof twisted equivariant K -theory to condensed matter. We consider non-interacting electrons on a half-crystal (a crystal with a boundary), with agapped bulk condition, under quasi-adiabatic evolution. In A. Adem, O. An-tol´ın, G. Semenoff and D. Sheinbaum JHEP, 2016 we found that Fermi sur-faces for these systems under quasi-adiabatic evolution are classified by theKâ1-group of the surface Brillouin zone Tdâ1. Systems with time-reversaland particle-hole symmetry were also considered and we obtained differentKR-groups for the different cases. In Chapter 1 I rewrite A. Adem, O.Antol´ın, G. Semenoff and D. Sheinbaum JHEP, 2016 in a more function-analytic language and further solve technical issues to extend it to includecrystallographic symmetries on the directions parallel to the boundary. InChapter 2 I reproduce the relevant parts of my joint work with C. Okay (C.Okay and D. Sheinbaum arXiv:1905.07723). There we explored a connectionbetween twisted equivariant K -theory to contextuality in quantum mechan-ics. We also reformulated the sheaf-theoretic framework of S. Abramsky andA. Brandenburger New Journal of Physics, 2011 for contextuality and con-nect it to another one employing a group cohomology approach of C. Okay,S. Roberts, S.D Bartlett, and R. Raussendorf Quantum Information andComputation, 2017. This leads to the construction of a classifying spacefor contextuality, from which Wigner functions are classes in its twistedK -theory.iiiLay SummaryWhat makes a material conduct electricity? It depends on the existence of asurface (called Fermi surface) inside an abstract space, that comes out fromthe periodic arrangement of the atoms in a solid. First we study systemswhich have a boundary and do not have a periodic arrangement in thedirection perpendicular to the boundary. There are materials that can onlyconduct electricity on their surface or not conduct at all. We classify Fermisurfaces of such materials according to whether we can change one into theother as we change a physical property (like the strength of a magneticfield acting on the material or by including extra symmetries). What makesquantum mechanics weird? One of the main contenders is contextuality. Inour 2nd chapter we reconcile two different approaches to contextuality bycreating another abstract space whose topology (shape) yields many featuresof quantum mechanics.ivPrefaceChapter 1 is based on results published in A. Adem, O. Antol´ın and G.W.Semenoff and D. Sheinbaum JHEP, 2016. All authors worked equally onthe results presented in Sections 1.1 to 1.7. I solely extended the resultsappearing in Section 1.8. Chapter 2 has been published in very similarform in C. Okay and D. Sheinbaum arXiv:1905.07723. Both authors workedequally on the results presented there. I solely worked in the discussionpresented in Chapter 3.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 K-theory and Fermi Surfaces . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Systems on a half-solid . . . . . . . . . . . . . . . . . . . . . 31.2.1 The spectrum of the fiber . . . . . . . . . . . . . . . . 61.2.2 Surfaces, bulk states and spectral projections . . . . . 71.3 Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Gapped bulk condition and Fredholm operators . . . 101.4 Topology on the fibers and a red herring . . . . . . . . . . . 111.4.1 The Riesz metric for unbounded operators . . . . . . 121.5 Adiabatic and quasi-adiabatic evolution . . . . . . . . . . . . 131.6 Quasi-adiabatic equivalence relation of Fermi surfaces and K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6.1 Spectral flow . . . . . . . . . . . . . . . . . . . . . . . 171.6.2 Local stability of Fermi surfaces . . . . . . . . . . . . 191.6.3 K -theory or vector bundles? . . . . . . . . . . . . . . 201.7 Adding global symmetries . . . . . . . . . . . . . . . . . . . . 221.7.1 Real and quaternionic structure on a complex vectorspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 22vi1.7.2 Particle-hole symmetry . . . . . . . . . . . . . . . . . 231.7.3 Real vector bundles and KR-theory . . . . . . . . . . 241.7.4 A splash of equivariant homotopy theory . . . . . . . 261.7.5 Class D . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.6 Time-reversal symmetry . . . . . . . . . . . . . . . . 291.7.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 311.7.8 Higher dimensional cases and weak topological phases 331.8 Extending to crystallographic groups . . . . . . . . . . . . . 352 K-theory and Contextuality . . . . . . . . . . . . . . . . . . . 402.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2 Contextuality for Pauli observables . . . . . . . . . . . . . . 422.2.1 Contexts in quantum mechanics . . . . . . . . . . . . 422.2.2 Sheaf-theoretic description of contextuality . . . . . . 432.2.3 Pauli group . . . . . . . . . . . . . . . . . . . . . . . . 442.2.4 Extension class . . . . . . . . . . . . . . . . . . . . . 452.2.5 Isotropic subspaces . . . . . . . . . . . . . . . . . . . 462.2.6 Contextuality for Pauli observables . . . . . . . . . . 472.3 Wigner function as a global section . . . . . . . . . . . . . . 492.3.1 Twisted representations . . . . . . . . . . . . . . . . . 492.3.2 Computing the inverse limit . . . . . . . . . . . . . . 502.3.3 Wigner functions . . . . . . . . . . . . . . . . . . . . 532.3.4 Contextuality . . . . . . . . . . . . . . . . . . . . . . 552.4 Classifying space for contextuality . . . . . . . . . . . . . . . 562.4.1 Construction of the space . . . . . . . . . . . . . . . . 572.4.2 Homotopy theory . . . . . . . . . . . . . . . . . . . . 582.4.3 Coset poset . . . . . . . . . . . . . . . . . . . . . . . . 602.4.4 Cohomology revisited . . . . . . . . . . . . . . . . . . 612.5 Contextuality and twisted K -theory . . . . . . . . . . . . . . 622.5.1 Sketch of the computation . . . . . . . . . . . . . . . 632.5.2 Twisted equivariant K -theory . . . . . . . . . . . . . 642.5.3 Twisted K -theory . . . . . . . . . . . . . . . . . . . . 663 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.0.1 On K -theory and Fermi surfaces . . . . . . . . . . . . 683.0.2 On K -theory and contextuality . . . . . . . . . . . . 683.0.3 Connections between the two topics . . . . . . . . . . 69Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70viiList of Tables1.1 Classes of topologically stable Fermi surfaces for half-spacesystems with a gapped bulk in dimension d = 1, 2 and 3. Îis the time-reversal operator and Î is the particle-hole sym-metry operator. We use the names given in [1] (AZ). 0on the right-side denotes the trivial group. 0 on the left-side indicates absence of said symmetry. When Î = Î = 0the surface Brillouin zone Xs is a regular torus Tdâ1 and foreither Î2 = Â±I or Î2 = Â±I it is a torus with involutionT¯dâ1. Higher-dimensional classes relate to weak bulk topo-logical phases. Table taken from [2]. . . . . . . . . . . . . . . 34viiiList of Figures1.1 d = 2 Fermi points. the ~kâ-axis represents the surface Bril-louin zone from âpi to pi. Solid regions and dashed lines be-long to the continuous spectrum (Bulk states and resonancestates respectively) Solid lines belong to the discrete spectrum(Surface states). The Fermi points ΣF are those where thesolid lines cross the ~kâ-axis. In this example the spectral flowis equal to 2(+) crossings - 1(-) crossing = 1. Figure takenfrom [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Real part of the wave function of the distinct types of states ona half-space as a function of râ¥. All solutions decay exponen-tially outside the crystal. 1.2I is a bulk state which is periodicinside the crystal. 1.2II Surface state, decays very rapidlyaway from the boundary. 1.2III Resonance state, the decayeventually stops and the solution is again periodic. Dashedlines represent the envelope of the solutions. Figure takenfrom [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 d = 2 Topologically stable Fermi points. Solid regions anddashed lines belong to the continuous spectrum Ïac(H(~kâ))(bulk states and resonance states respectively). Solid lines toÏpp(H(~kâ)) (surface state). + (â) signs label the sign of theslope at crossings. 1.4I corresponds to class A (IQHE) witha net spectral flow = 1. Figure taken from [2] . . . . . . . . . 18ix1.4 d = 2 Topologically stable Fermi points. Solid regions anddashed lines belong to the absolutely continuous spectrumÏac(H(~kâ)) (bulk states and resonance states respectively).Solid lines to Ïpp(H(~kâ)) (surface state). + (â) signs labelthe sign of the slope at crossings. 1.4I corresponds to class A(IQHE) with a net spectral flow = 1. 1.4II corresponds to aspectral flow of 2, that may occur for both classes C and D atnon fixed-points in S 1,1. 1.4III represents two Majorana zeromodes in class D. 1.4IV represents a helical Dirac fermionedge mode (SQHE). Figure taken from [2]. . . . . . . . . . . . 33xAcknowledgementsTo my advisor, Professor Alejandro Adem, who took a gamble by accept-ing a Physics student, suggested a topic of research and encouraged me topursue my own sidelines on the subject which lead to our joint paper withO. Antol´ın and G. W. Semenoff. I thank him for being extremely patientwhen my background proved problematic for the different requirements ofthe department and encouraged me when I overcame these trials. I am alsograteful for his teachings on crystallographic groups and equivariant coho-mology. Finally, I thank him for having in mind my future goals and alwaysallowing me to explore research topics that interested me.To my co-advisor, Dr. Cihan Okay, for inviting me to collaborate on ourjoint work and in the process teaching me about simplicial homotopy the-ory, spectral sequences, p-adic completion and contextuality. I appreciatehis energy and willingness to discuss ideas even after 5 hour sessions.To my Committee members, Professors Sven Bachmann, Gordon W.Semenoff and Lior Silberman for their comments and suggestions. I amparticularly grateful to Professor Sven Bachmann for our long discussionsteaching me about the difference between quasi-adiabatic and adiabatic evo-lution.To Omar Antol´ın Camarena, who would often enter a room, see an ex-pression written on a board and would seem to be instantly transported tothe Platonic realm. If he found the expression there he would nod, withoutreally needing to see the rest to know it was right. My eternal gratitude forteaching me about any topic of mathematics I was interested in or neededto know, being KR-theory for our joint paper or Complex Analysis for myqualifying examinations. For taking seriously all of my ideas and finding aconcrete example why they didnât work. Finally for willing to learn expo-nentially faster than me any topic that seemed necessary in order to continuethe next day with our fantastic meetings.xiTo Bernardo Villareal Herrera, Javier Go´nzalez Anaya and Mateus ArantesFandino for our long discussions that often cleared my head and for lettingme use their brain power to have a better understanding and purpose.To Sara´ı Herna´ndez Torres, also known as Sara´ı Sensei, because withouther extensive lectures and problem solving sessions on Real Analysis I wouldhave been unqualified to be a Mathematician at UBC, and for becoming aclose friend in the process.To Karina, for listening day and night to my incoherent ramblings. Forreminding me at hard times to remain true to my principles and beliefs inphysics and for supporting my crazy plans with her time, intuition, intellectand phenomenal patience. Finally for making me enjoy life.To my parents, Mayra and Julio, to whom I owe everything. May thisand my future work begin to repay their tireless, relentless efforts, and theirfaith in what others thought of as their feeble, troubled child.To my Sister Tamara, whose perseverance made me finish this thesis ontime.To Sergio, Emilia and the newborn member of our family.To the memory of my grandparents Carlos and Martha.Finally, I would like to thank CONACyT, which provided a scholarshipthat payed both tuition and living expenses.xiiChapter 1K-theory and Fermi Surfaces1.1 IntroductionClassification of locally stable Fermi surfaces and topological phases of mat-ter using K -theory was first introduced in the pioneering work of HorËava[3] and Kitaev [4]. This chapter is a presentation and further continuationof [2]. To our knowledge, [2] was the first publication to directly obtaina K -theoretic classification for the global stability of Fermi surfaces with-out resorting to physically-unjustified mathematical procedures, as had beendone previously in the literature. A typical example of a Fermi surface ΣF ofinterest for the systems considered here (see section 1.3 and previous ones)is represented in figure 1.1. More generally a Fermi surface is usually a codi-mension 1 subspace (though there are degenerate cases) inside the surfaceBrillouin zone, a dâ1-dimensional torus, where the spectrum of an operator,the fiber single particle Hamiltonian which comes about in 1.2, is equal toa real number known as the Fermi energy εF . The Fermi surface of a givenmaterial determines its physical properties such as whether it is a conductoror an insulator [5], [6].Once we have a Fermi surface ΣF for a given Hamiltonian H, we can askwhat happens to it as we evolve our Hamiltonian by changing a parameterH(s). Can we always connect 2 different Fermi surfaces? Classifying Fermisurfaces up to this evolution is the whole point of this work.Physicist have used the notion of spectral flow [7] to describe a physicalphenomena known as a chiral anomaly in some condensed matter models[8]. This topological invariant comes out in our construction as well, beingthe number of crossings of the eigenvalues over an axis with positive slopeminus the number of crossings with negative slope. Figure 1.1 is an exampleof a spectral flow equal to 1 (see section 1.6.1).Remark 1.1.1. For readers more familiar with the physics literature andterminology we note that Ryu et al. [9],[10] and Witten [11] among others,have developed a formalism for studying gapped bulk systems by classifyingtheir gapless edge states and the bulk-boundary correspondence, where the1+  +"kkâ¡â¡Figure 1.1: d = 2 Fermi points. the ~kâ-axis represents the surface Brillouinzone from âpi to pi. Solid regions and dashed lines belong to the continuousspectrum (Bulk states and resonance states respectively) Solid lines belongto the discrete spectrum (Surface states). The Fermi points ΣF are thosewhere the solid lines cross the ~kâ-axis. In this example the spectral flow isequal to 2(+) crossings - 1(-) crossing = 1. Figure taken from [2].2edge states have an anomaly inflow interpretation [12],[13]. This paradigmis physically similar to that of [2], however these arguments rely upon takinga low-energy limit of a lattice Hamiltonian to obtain an effective relativisticfield theory [14] such as Chern-Simons for the IQHE, with a correspondencebetween topological field theories and symmetry protected topological phases[11],[15]. [2] did not rely on such approximations, but it does have otherstrong assumptions such as discrete translation-invariance.We continue the study of systems on a half-solid (see section 1.2). Asdiscrete translation-invariance is broken by the surface Rdâ1 Ã {0}, only areduced version of Blochâs theorem holds. Nevertheless this opens up thepossibility of having Surface states and so called Bulk states (1.2.2). This,together with a gapped bulk condition (1.3.1) will play the central role inthe derivation of the K -theoretic classification of quasi-adiabatically stableFermi surfaces (section 1.5) shown in theorem 1.6.1. We reproduce the state-ment of theorem 1.6.1Quasi-adiabatic equivalence classes of Fermi surfaces ΣF for non-interacting, gapped bulk Hamiltonians on a d-dimensional half-crystal arein one to one correspondence with the complex K -theory group Kâ1(Tdâ1).Figure 1.1 represents a class in the group Kâ1(S 1) â Z for d = 2 (seesection 1.6.1), specifically the class equal to 1. We include the cases withparticle-hole symmetry and time-reversal symmetry (and their different vari-ations) in theorems 1.7.2, 1.7.3, 1.7.4, 1.7.5. These results were originallypresented in [2]. Finally we extend [2] to half-solids with a surface crystal-lographic symmetry (section 1.8) in theorem 1.8.3.Remark 1.1.2. Issues of aperiodicity, disorder and unitary-antiunitaryambiguities in representations of symmetry operators are considered in amore general framework in [16],[17] for topological phases of non-interactingfermions and are not dealt with in this thesis. However, neither of [16],[17]attempt to classify Fermi surfaces.1.2 Systems on a half-solidLet H = L2(Rd;W ) be the Hilbert space of equivalence classes of squareintegrable functions on Rd valued in W , the standard 2-dimensional repre-sentation of the Lie algebra su(2).3H is an unbounded self-adjoint operator with a dense domainD(H) âHof the formH = ââ2 + V. (1.1)To represent the half-solid, one condition is that our potential V satisfiesV (râ¥, ~râ) ={W(râ¥, ~râ) râ¥ â¤ 00 otherwise.(1.2)where we have split Rd = Râ¥ â Rdâ1â , which will correspond to the direc-tions parallel and perpendicular to the boundary of the solid respectively.Another condition [18],[5] is periodicity of V (and hence H) with respect totranslations in the direction parallel to the boundary. Let Îâ â Zdâ1 be asublattice of Râ and letT : Îâ â U(H ) (1.3)be a unitary representation of Îâ. The periodicity condition is given by[H, TÎ³â ] = 0 âÎ³â â Îâ (1.4)Finally, as in [18] there is a technical condition on the characteristics ofthe potential V so that H has empty singular continuous spectrum (seesubsection 1.2.1). This condition is that V â L1loc(Rd), the space of locallyintegrable functions on Rd. With this condition we have that the densedomain on which H is defined is given byD(H) = D(ââ2) â©D(V ). (1.5)Here D(ââ2) is Câ0 (Rd), the subspace of infinitely differentiable functionswith compact support.At this point we should also mention that one can include a magneticfield ~B (to break time-reversal symmetry) through a vector potential ~A :Rd â Rd, ~B = âÃA, such that our Hamiltonian is of the formHs( ~A) = (iââ ~A)2 + ~Ï · ~B + V, (1.6)where ~Ï is the Pauli-Lubanski vector, made out of the generators of su(2)representing non-relativistic spin [19]. Here one has to be careful since thevector potential ~A may not have the periodicity of the lattice unless sometechnical conditions are imposed on the magnetic field.4Remark 1.2.1. There has been an extensive treatment including this tech-nicalities which are relevant when considering disorder, but in what followshere there is no strong loss of generality by assuming implicitly the magneticfield in the notation and we shall only write H.Because of the periodicity in the parallel direction our Hilbert space Hand our Hamiltonian H are unitary equivalent to a direct integral decompo-sition [5],[20]H 'â« âTdâ1H (~kâ)d~kâ , (1.7)H 'â« âTdâ1H(~kâ)d~kâ . (1.8)Let us dissect the different pieces appearing in (1.7) connecting analy-sis and topology. The integral sign should be thought of as a direct sumof vector spaces, one for each point on the measure space Tdâ1 but us-ing the measure d~kâ as Tdâ1 is uncountable. The measure space Tdâ1 isin fact the Pontryagin dual Hom(Îâ,S 1) of our lattice Îâ, equipped withits Haar measure d~kâ, which makes Tdâ1 a topological group homeomor-phic to a d â 1-dimensional torus. Therefore it is a well-posed problem toask questions about continuity (or differentiability, analyticity even) of thespectrum Ï(H(~kâ)) of the fiber or Bloch Hamiltonian H(~kâ), as one varies~kâ. H(~kâ) is an unbounded self-adjoint operator on the fiber Hilbert spaceH (~kâ) â¡ L2(Rd/Îâ;L{~kâ}ÃRdâ1/ÎââW ). Here L~kâÃRdâ1/Îâ denotes the fiberat ~kâ of the Poincare´ line bundle L â Rd/Îâ Ã Tdâ1 given by the quotientunder the action of Îâ [16], [21]Rd Ã Tdâ1 Ã CÃ Îâ â Rd Ã Tdâ1 Ã C ,(~r,~kâ, z, Î³â) 7â (~r + Î³â,~kâ, z â ei~kâ·Î³â) (1.9)Remark 1.2.2. To define the above unitary it is useful to first show (see[21] for the construction) L2(Rd;W ) is unitary equivalent toL2Tdâ1(Tdâ1ÃRdâ1;W ) = {Ï : Tdâ1ÃRdâ1|Ï(~kâ, ~r+Î³â) = ei~kâ·Î³âÏ(~kâ, ~r) âÎ³â â Îâ}(1.10)In the physics literature this is known as the Bloch transform[6] andTdâ1 is known as the Brillouin zone. We remark that equation (1.7) meansthat our Hilbert space is unitarily equivalent to L2-sections of the Hilbertbundle E, where5E =â~kââTdH (~kâ), (1.11)which is a Hilbert bundle over the surface Brillouin zone Tdâ1 [16], [21].However this bundle turns out to be trivializable, i.e.E = Tdâ1 ÃH â² (1.12)where Hâ²= L2(Rd/Îâ;C âW ). However, it is not equivariantly trivi-alizable as we shall see in section 1.8.1.2.1 The spectrum of the fiberLet us briefly remind our selves that the spectrum of an operator A, Ï(A)is the set of λ â C such that A â λI is not invertible [22]. For self-adjointoperators like H(~kâ) the spectral theorem implies that their spectrum isreal. One can show the spectrum is âclosedâ, it is also bounded for boundedoperators and conversely unbounded for unbounded ones. Continuity of thespectrum means viewing Ï as a functionÏ : Tdâ1 ââ C(R), (1.13)~kâ 7â Ï(H(~kâ)) (1.14)where C(R) is the metric space of closed subsets (the spectrum is always aclosed subset of R) under the Hausdorff distance.d(C1, C2) = maxxâC1minyâC2{d(x, y)} (1.15)and there are interesting variants of this notion explored in [23]. SinceH(~kâ) is self-adjoint, by the spectral theorem for unbounded self-adjointoperators, Ï(H(~kâ)) is real and the Hilbert space can be decomposed asa direct sum H (~kâ) = Hpp(~kâ) âHac(~kâ) âHsc(~kâ) where Hpp(~kâ) con-sists of the span of eigenvectors of H(~kâ) whose associated spectral measureon subsets of R is pure point or atomic, whereas the spectral measure as-sociated to elements of the absolutely continuous and singular continuoussubspaces are absolutely continuous and singular continuous with respectto the Lebesgue measure on R. Let us remark that though Hpp(~kâ) maybe finite, Hac(~kâ) and Hsc(~kâ) are always infinite dimensional [22]. Thus,the spectrum Ï(H(~kâ)) = Ïpp(H(~kâ)) âª Ïac(H(~kâ)) âª Ïsc(H(~kâ)). With theconditions for V stated in 1.2, Davies [18] and Simon show Ïsc(H(~kâ)) = â .61.2.2 Surfaces, bulk states and spectral projectionsNow that we have introduced more properties of the spectrum of the fiberHamiltonian, we can use them to partially characterize the solutions to theSchro¨edinger equation with our choice of V . Here we shall simply follow[18]:ConsiderPS =â« âTdâ1Ppp(H(~kâ))d~kâ . (1.16)Here Ppp(H(~kâ)) denotes the projection onto the subspace Hpp(H(~kâ)).It is shown in [18] that under certain restrictions on the potential V , η âRanPS, the rank of PS if and only iflimaââ suptâ«râ¥>a|eâitHÎ·(~r)|2d~r = 0 (1.17)This means that in a suitable sense η may not be found sufficiently faraway from the boundary (surface of the half-solid) for any choice of timet. Therefore as one can guess, the S stands for surface, and any element inRanPS â L2(Rd;W ) will be called a Surface state.Similarly, we may definePB =â« âTdâ1Pac(H(~kâ))d~kâ (1.18)= I â PS . (1.19)and elements Ï â RanPB are called Bulk states since for any t one hasa non-zero probability of finding the particle (the integral in the limit) in-finitely deep inside the crystal.We point out that the spectrum of the original Hamiltonian Ï(H) is abso-lutely continuous and comes in bands so both RanPS and RanPB belong inHac(H). A crucial point to make is that though Hpp(~kâ) and Hac(~kâ) areorthogonal subspaces, Ïpp(H(~kâ)) â© Ïac(H(~kâ)) 6= â . When this occurs wehave a Surface resonance depicted in 1.2III, together with a Surface 1.2IIand Bulk 1.2I state.Remark. The continuity explained in the previous section may be repre-sented by having functions εα : U â Tdâ1 â R, which are continuous. Inthe literature these are known as bands. Bands consist either of absolutelycontinuous spectrum (Bulk band) or pure point spectrum (Surface bands).The7(I)r?Re(âµ(~kk))crystal vacuum(II)r?Re(âµ(~kk))crystal vacuum(III)Re(âµ(~kk))r?crystal vacuumFigure 1.2: Real part of the wave function of the distinct types of states ona half-space as a function of râ¥. All solutions decay exponentially outsidethe crystal. 1.2I is a bulk state which is periodic inside the crystal. 1.2IISurface state, decays very rapidly away from the boundary. 1.2IIIResonance state, the decay eventually stops and the solution is againperiodic. Dashed lines represent the envelope of the solutions. Figuretaken from [2]index α can be used to denote either of these and we call it band index. Thereason why they are defined on an open subset is that Surface bands andBulk bands may intersect producing a surface resonance.1.3 Fermi surfaceWe have thus far only considered the single particle problem moving in thesemi-periodic potential of a half-crystal. We now wish to employ this picturetogether with the Pauli exclusion principle and the non-interacting electronapproximation to describe a system in the thermodynamic limit (number ofparticles N ââ). This conveniently allows us to avoid discussing the Fockspace construction Fa = ânâ¥0H â§n , where wedge denotes the antisymmet-ric tensor product and we can very conveniently ignore the so-called second8quantization formalism.The intuitive picture to define the Fermi energy εF is that each electron (2electrons if we take into account spin without a magnetic field) occupies aspectral subspace coming from a small ball around a point Îµ(~kâ) â Ï(H(~kâ))starting at min~kââTdâ1 Îµ(~kâ) â Ï(H(~kâ)). The reason why it has to be a smallball and not just a single point is due to Heisenbergâs uncertainty principle.The above mentioned process continues until one runs out of electrons (inthe thermodynamic limit the radius of the balls tends to zero as the volumeand number of particles goes to infinity). Surprisingly, it is not that easyto find a proper definition for εF in the mathematical literature. To ourknowledge the most rigorous one is the following:Consider the extended Hamiltonian for the full solidH = ââ2 + VË (râ¥, ~râ), (1.20)withVË (râ¥, ~râ) ={V (râ¥, ~râ) râ¥ â¤ 0V (ârâ¥, ~râ) otherwise.(1.21)We now have full periodicity Î â Zd, so we can perform a direct integraldecomposition over the Pontryagin dual Td. The difference now is that thefibers H(~k) are elliptic operators on a compact manifold so their spectrumÏ(H(~k)) is pure point and ordered [20]. It therefore allows us to define thedensity of states (density of levels) measure Ï as [5]Ï(ââ, E) = 2V ol(Td)ân{~k â Td | εn(~k) = E}(1.22)where the 2 is to take into account spin. We can then define the Fermienergy εF asεF ={inf E | Ï(ââ, E) = l} (1.23)where l is the number of atoms in the Wigner-Seitz cell of the lattice Î.When we truncate the system back into the half-solid this number remainsthe same (we could have attempted to modify the definition to employ itdirectly on the half crystal but then we would have to deal with the abso-lutely continuous spectrum of the fibers) and so, as defined, it has the samephysical interpretation.We can now define the Fermi surface ΣF for the half-solid asΣF ={~kâ â Tdâ1 |εF â Ï(H(~kâ))}(1.24)9At this point we mention in passing that an empty Fermi surface repre-sents a material which is an insulator (does not conduct electricity) whereas a non-empty one is a metal, with different possible variants of these basedon the size of the gap between the highest occupied state and the lowest oneunoccupied [6],[5].1.3.1 Gapped bulk condition and Fredholm operatorsWhen physicists study condensed matter systems (interacting or not) witha boundary, they often impose an extra constraint on their system called theGapped Bulk condition. To present this condition in full generality wouldderail our attention too much, however we will present a formulation of thisconstraint for the type of systems consider in this work (non-interactingelectrons in a half-solid). If one looks at the graph of the spectrum of thefiber Ï(H(~kâ)) vs ~kâ the d â 1-dimensional hyperplane corresponding toÏ(H(~kâ)) = εF may intersect the continuous spectrum or not. IfÏac(H(~kâ)) â© {Ï(H(~kâ)) = εF } = â â~kâ (1.25)then we say that our system satisfies the gapped bulk condition, which fmeans that εF is between Bulk bands. In physical terms it means that oursystem only conducts electricity through its surface! (if ΣF = â then it issimply an insulator). We shall assume from now on that our systems satisfythe gapped bulk condition and see that it is almost equivalent to our fibersbeing H(~kâ) in a certain subset of unbounded self-adjoint operators. Wemust further assume a slightly stronger conditionÏess(H(~kâ)) â© {Ï(H(~kâ)) = εF } = â â~kâ (1.26)This is to avoid the unphysical situation of having an infinite number ofsurface bands intersect at a given point in the surface Brillouin zone.Here we must start looking at a slightly different family of operators forthe following reason: By itself the kernel of H(~kâ) has no physical meaning.To add a physical meaning to the kernel we must shift the spectrum byan amount that has a physical characteristic of the system. Consider theoperator H â εF I and perform the same direct integral decomposition toobtain fibers H(~kâ)â εF I~kâ , where I~kâ is the identity operator in the fiberHilbert space H (~kâ). Let us enumerate what are the conditions which aresatisfied by the fibers H( ~kââ)â εF I~kâ :10â¢ dim KerH(~k)â εF I~kâ <â ,â¢ H(~k)â εF I~kâ is self-adjoint.Note that the 2nd condition is the slightly stronger version of the GappedBulk condition 1.26 (considering the essential spectrum). Operators satis-fying these conditions are known in the literature as unbounded self-adjointFredholm operators[24]. Let us denote the set of not necessarily bounded self-adjoint Fredholm operators on a Hilbert spaceHâ²by CFsa(H â²). CFsa(H â²)may be decomposed asCFsa(H â²) = CFsa+ (Hâ²) unionsq CFsaâ (Hâ²) unionsq CFsaâ (Hâ²). (1.27)CFsa+ (Hâ²) is the subset of CFsa(H â²) such that there exists a finite codi-mension subspace V âH â² , such thatãÏ,FÏã â¥ 0 âÏ â V . (1.28)and operators in CFsa+ (Hâ²) are called essentially positive. An analogous def-inition is employed for CFsaâ (Hâ²), which is called the set of essentially neg-ative not necessarily bounded self-adjoint Fredholm operators. CFsaâ (Hâ²)is the complement of these two, that is operators which are neither essen-tially positive nor essentially negative. Because our fibers have an infinitenumber of Bulk bands that means that there is absolutely continuous spec-trum above and below the kernel of H(~kâ)âÎµF I~kâ , therefore the spectrumsubspaces associated to that subset of the spectrum are both infinite dimen-sional by definition of the absolutely continuous spectrum [22]. We maytherefore conclude [2]:H(~kâ)â εF I~kâ â CFsaâ (H (~kâ)) â~kâ â Tdâ1. (1.29)This means that in the special case of a half-solid with no interactions,the slightly stronger version of the gapped bulk condition is equivalent tothe fibers being in CFsaâ , as stated previously.1.4 Topology on the fibers and a red herringSo far we have concluded that the fibers H(~kâ)âÎµF I~kâ are in CFsaâ (H (~kâ)).We have yet to discuss what is the appropriate topology for CFsaâ (H (~kâ))that represents the physical properties of the system. Different topologies11for CFsaâ (Hâ²) have been explored in [24]. Nonetheless one could arguethat because of the Stone-von Neumann theorem [22] on the isomorphismbetween differentiable one-parameter family of unitary operators and theirself-adjoint generators, the correct topology should be the strong operatortopology for whatever subset of the set of linear operators D(H â²) the Hamil-tonian operator H belongs to. Indeed this should be the correct topologyand it is the position taken in [16]. However, the properties of CFsaâ (Hâ²)have been hard to explore, mainly because of the unboundedness of the op-erators. Instead one could take the following perspective: What relevantinformation to the physical problem would we lose if we described CFsaâ asif they were bounded operators instead? One of the big differences withthe strong operator topology and the norm topology is semi-continuity ofthe spectrum but as we have referred to above, these families of operatorshave continuous spectrum in either topology. Furthermore as we shall seein the next section we are interested in the quasi-adiabatic character of thephysical system [25] (in physical terms the low energy behaviour) and we donot care about the asymptotics of the spectrum as it tends to infinity. Is itpossible to treat operators in CFsaâ (Hâ²) as if they were bounded?1.4.1 The Riesz metric for unbounded operatorsWe define the Riesz metric on CFsa(H â²) asL : CFsa â Fsa ,H 7â H(I +Hâ H)â 12 ,dR : CFsa Ã CFsa â Râ¥0 ,dR(H1,H2) 7â âL(H1)â L(H2)â, (1.30)where â  â denotes the standard operator norm and L is known as the Riesztransform [24]. The topology on CFsa induced by dR is known as the Riesztopology and the natural inclusion of bounded self-adjoint Fredholm opera-tors Fsa(H â²) âªâ CFsa with the norm topology is a homotopy equivalence.This fact will be very important to us in later sections. With this topologythe disjoint subsets of CFsa(H â²) have three path-components CFsaâ (Hâ²)and each component is homotopic to the corresponding bounded compo-nent of Fsa(H â²) [24]. One technical issue with this choice of topology isthat it is generally hard to determine whether a family of operators in CFsais continuous [24]. However, as is shown in [20],[5],[18] H(~kâ) with V (~r)bounded has a fixed domain for all ~kâ in the Brillouin zone and as explainedin [26], families of operators with a fixed domain are continuous in the Riesz12topology.An other issue that may arise is if we want to consider more symmetries.In that case we will view the direct integral decomposition of our Hilbertspace H as a Hilbert bundle (1.11) over the Brillouin zone Tdâ1 with anaction of a group G on the fibers, i.e.U(g) : H (~kâ)âH (g~kâ) âg â G. (1.31)For compact Lie groups this action is not continuous in the norm topologyin general. However since the groups of interest for these physical systemsare always discrete (they come from crystallographic groups or actions ofa single operator in the case of time-reversal symmetry) then the actionis indeed trivially continuous. We may also comment that the structuregroup on Projective Hilbert bundles is PU(H ) and one might worry thatits topological character is different with the norm topology or the strongoperator topology but due to Kuiperâs contractibility theorem [27] of U(H )for infinite dimensional separable H in both the strong and norm topologyPU(H )norm and PU(H )strong are homotopic.Since for any of the physical systems considered we have none of thetechnical issues mentioned above due to their characteristics (such as onlyhaving actions from discrete groups) it would seem that the issue of thechoice of topology for the space of our fibers is a Red Herring. It may wellbe that CFsaâ (Hâ²) in the Riesz topology or strong topology are homotopi-cally equivalent or if not, it may be due to the asymptotic character of thespectrum as it tends to infinity but it certainly does not seem relevant forthe problem at hand. Therefore we shall endow CFsaâ (Hâ²) with the Riesztopology from now on. Furthermore to simplify notation we shall set εF = 0and simply write H(~kâ).1.5 Adiabatic and quasi-adiabatic evolutionFrom previous sections we arrived at studying a half-crystal which may ormay not have a Fermi surface but must satisfy the gapped bulk condition.We further wish to study what happens to our system as we vary a param-eter such as the strength of the magnetic field ~B or change the potentialV (~r) through doping or other means. Since physically we mean changingany of those parameter with respect to time we can view each of them asparametrized by t , i.e. ~B(t), etc. In general this is represented by assuming13that we now have a family of single particle Hamiltonians H(s) that oftendepends differentiably on our parameter s = t/Ï , s being is rescaled timeand Ï represents the time scale of the physical problem. We will now pro-ceed to sketch briefly the notion of adiabatic evolution defined in [25] forany given quantum mechanical system.LetH(s) be a smooth family of self-adjoint operators. Consider the spec-tral projection P(s) onto a spectral subspace. We can define the adiabaticevolution generator asHad(s;P) = H(s) + iÏ [P(s),P â²(s)] (1.32)where P â²(s) is the derivative of P(s). This generator has been defined soas to decouple RanP (s) from its complement 1 â P(s) from the evolution.This decoupling has been forced, however what one is actually really inter-ested in is whether the actual physical evolution by H(s) approximates theevolution by Had(s;P) in a suitable sense. Let U(s) be the one-parameterfamily of unitary operators with generator H(s) and Uad(s;P) with gener-ator Had(s;P) respectively. Let P0 = P(0). Then we say the evolution byU(s) is adiabatic if there exists γ â¥ 0 such thatâ(U(s)â Uad(s))P0â â¤ O(ÏâÎ³). (1.33)Where the γ depends on different criteria of differentiability and O() asusual means terms up to said order in Ï . In physical jargon we are lookingat how close the ground state of the evolved Hamiltonian H(s) is to thestate corresponding to the evolved ground state of the initial HamiltonianH(0). For systems on a complete solid satisfying a gapped condition wherethe Fermi energy εF is not in any of the spectrums of the fibers Ï(H(~k)),so that we have an insulator, then for the choice of P0 the projection ontothe spectral subspace separated by the gap the evolution can be consideredadiabatic [25].However, the systems we are interested in are half-solids that only satisfya gapped bulk condition and there is no spectral gap formally separating thespectral subspaces. Therefore there is, to our knowledge, no known adia-batic theorem for this class of Hamiltonians and spectral subspaces (thosebelow the Fermi energy). There may not even be one. However in realexperiments crystals have a boundary and furthermore it would seem thatthe systems of interest often only satisfy the gapped bulk condition and notthe gapped condition (through the so-called bulk-boundary correspondence).14Adiabatic evolution is for our purposes a model of what is actually occur-ring in the laboratory, where experiments have shown that something likeadiabatic evolution is satisfied if we restrict the amount of energy that isinjected into the system because of the bulk gap. The concept of quasi-adiabatic evolution has been defined with the intent to model what occursexperimentally in systems with only a gapped bulk condition. To do thiswe first reinterpret adiabatic evolution as looking at paths on the space ofHamiltonians satisfying a gapped condition, that is to interpret adiabaticevolution H(s) as a homotopyH : [0, 1]â Gapped(L2(Rd;W )), (1.34)between two Hamiltonians H(0) and H(1), where for each s, H(s) sat-isfies the gapped condition. If the above is satisfied then we have that forsufficiently smooth paths (1.33) is satisfied. We can extend this to definequasi-adiabatic evolution to say that a quasi-adiabatic evolution is a homo-topy H(s) between H(0) and H(1), where each H(s) satisfies the gappedbulk condition.H : [0, 1]â GappedâBulk(L2(Rd;W )). (1.35)What is still missing (to our knowledge) in the mathematical physics lit-erature, specially on the one focused on the analytical side, is an analogueof (1.33). We leave this to future analysts and content ourselves with atopological interpretation of quasi-adiabatic evolution that seems to modelcorrectly what occurs in many experiments.1.6 Quasi-adiabatic equivalence relation of Fermisurfaces and K -theoryConsider a Fermi surface ΣF,0 associated to H0 and ΣF,1 associated to H1,both satisfying the gapped bulk condition. We say that these two are in thesame quasi-adiabatic equivalence relation if there exists a mapH : [0, 1]â GappedâBulk(L2(Rd;W ))satisfyingH(0) = H0 (1.36)H(1) = H1. (1.37)15We write ΣF,0 â¼qa ΣF,1 in this case. Therefore quasi-adiabatic equiva-lence classes of Fermi surfaces for bulk-gapped Hamiltonains is by defini-tion the set of path-components of gapped-bulk Hamiltonians pi0(GappedâBulk(L2(Rd;W )). We are at last finally ready to present the main result ofthis chapter, originally published in [2].Theorem 1.6.1. Quasi-adiabatic equivalence classes of Fermi surfaces ΣFfor non-interacting, gapped bulk Hamiltonians on a d-dimensional half-crystalare in one to one correspondence with the complex K -theory group Kâ1(Tdâ1).Proof. As shown before, the gapped-bulk condition onH is equivalent to say-ing the fibers H(~kâ) â Fsaâ (H (~kâ)). Using the direct integral decompositionand definition 1.11 we can view any d-dimensional half-crystal Hamiltoninanas a continuous section on the self-adjoint Fredholm bundle Fsaâ (E), i.e.H : Tdâ1 â Fsaâ (E), (1.38)Therefore pi0(Gapped-Bulk(L2(Rd;W )) â [Tdâ1,Fsaâ (E)], where [, ] meanshomotopy classes of sections, that is equivalence classes of sections H :Tdâ1 â Fsaâ (E) where two sections H0,H1 are equivalent if there existsa continuous section H : Tdâ1 Ã [0, 1] â Fsaâ (E) such that H(0) = H0 andH(1) = H1 [28]. Because the Hilbert bundle E (defined in 1.2) is trivial,that is, E = Tdâ1 Ã H â² , this is equivalent to homotopy classes of maps[Tdâ1,Fsaâ (Hâ²)] We now employ a theorem of Atiyah and Singer [29] thatshows Fsaâ (H â²) is a classifying space for the Kâ1-functor. Hencepi0(GappedâBulk(L2(Rd;W )) â [Tdâ1,Fsaâ (H â²)] ,â Kâ1(Tdâ1). (1.39)Remark. Systems considered here may have more symmetry than discretetranslation invariance in ~râ, but this symmetry may not be preserved un-der quasi-adiabatic evolution. In later sections we will include paths whichpreserve more symmetries.Here, Kâ1(X) â¡ KË 0(ΣX+), where KË 0(ΣX+) denotes the Grothendieckcompletion of the monoid of stably equivalent complex vector bundles overΣX+, the reduced suspension on the space X+, which is the topologicalspace X with a disjoint point attached [30].161.6.1 Spectral flowThe most famous example is the case for d = 2, called the spectral flow [7].Let us computeKâ1(S 1) â KË (Î£S1+)â KË 0(S 2 â¨ S1),â KË 0(S 2)â KË 0(S 1),â Zâ 0â Z (1.40)were â means isomorphism of groups. We have used our theorem 1.6.1and the above computation to deduce that there is an infinite number ofquasi-adiabatic equivalence classes of Fermi surfaces for d = 2 from purealgebra. Atiyah gave an interpretation of this invariant which is the numberof times Ïpp(H(~kâ)) crosses the ε = εF = 0 axis with a positive slope minusthe number of times it crosses with a negative slope. A non-trivial state isshown in subfigure 1.4IFor d â¥ 2 one can employ Jamesâs splitting [28]Σ(X Ã Y ) ' ΣX â¨ ΣY â¨ Σ(X â§ Y ) (1.41)to obtain a recursive formulaÎ£Tdâ1 ' S 2 â¨ Î£Tdâ2 â¨ Î£2Tdâ2. (1.42)Using this formula we further compute the case d = 3Kâ1(T2) â KË (Î£T2+)â KË 0(ΣT2 â¨ S 1),â KË 0(Î£T2)â KË 0(S 1),â KË 0(Î£T2)â KË 0(S 2 â¨ S 2 â¨ S 3)â Z2 (1.43)Note that for d = 1 (half-line solid) T1â1 = â, a point. ThereforeKâ1(â) â KË (ΣS 0)â KË 0(S 1),â 0 (1.44)17+  +"kkâ¡â¡Figure 1.3: d = 2 Topologically stable Fermi points. Solid regions anddashed lines belong to the continuous spectrum Ïac(H(~kâ)) (bulk statesand resonance states respectively). Solid lines to Ïpp(H(~kâ)) (surfacestate). + (â) signs label the sign of the slope at crossings. 1.4I correspondsto class A (IQHE) with a net spectral flow = 1. Figure taken from [2].181.6.2 Local stability of Fermi surfacesTo put 1.6.1 in context, we now present an adaptation of HorËavaâs work[3] on a classification of locally stable Fermi surfaces, that is, pieces of theFermi surfaces which are robust to small perturbations near a given ~k0 onthe Fermi surface. Consider a fully periodic system with Brillouin zone Tdwith a non-empty Fermi surface. Let us denote one of the path-componentsof the Fermi surfaces by â¦ and let dim â¦ = d â p â 1. We now look at thep+1-transverse directions to â¦ in Td around a chosen point ~k0 â â¦. HorËavaâsstability analysis is to consider the boundary of a p + 1- dimensional ballâBr0(~k0) â Td, which does not intersect â¦. The boundary is topologicallya sphere Sp of dimension p, whose radius r0 â¡ r0( ~k0) depends on the choiceof ~k0 â â¦. We can once again consider quasi-adiabatic evolution of thesystem H(~k, s). As long as we assume that the quasi-adiabatic evolutionof the system does not generate a new point of the Fermi surface on theboundary, i.e. with ~k â Sp, we have that H(~k, s) satisfies the gap conditionfor all ~k â Sp and for all s. Operators satisfying the gapped condition, whichwe denote T P(H â²) have been widely studied [16],[17],[31],[32] and they are(under the Riesz topology) homotopy equivalent to the colimit of infinitedimensional Grassmannians Grn(Câ). This can be easily seen by takingthe spectral subspace for each H(~k, s) under εF , which is finite dimensionalbecause of the full periodicity, and because of the global gapped conditionthe dimension of this subspace is constant in S p. If we consider all possibledimensions n for this spectral subspace we get our colimit. Therefore thedifferent quasi-adiabatic equivalence classes of this component Fermi surfaceâ¦ with âsufficiently smallâ perturbations are given by[S p, T P(H â²)] â VectC(S p). (1.45)VectC(Sp) is the monoid of isomorphism classes of complex vector bundleswith base space S p [30]. Note that to derive Equation (1.45) we simplyrepeated the analysis in [16], section 10. The arguments presented in [16]follow through, except that in our case, we must restrict the parameterspace for our families of Bloch Hamiltonians to âBr0(~k0) â S p instead of thewhole Brillouin zone Td. HorËava then performs a mathematical operation,which is not warranted by any physical property of the system and also losesinformation. HorËava stabilizes, i.e. sets two vector bundles to be in the sameequivalence class if adding a trivial vector bundle produces the same (to theoriginal) isomorphism class of vector bundles [30]. One hence obtains the19reduced K -theory groupKË (S p) ={Z p even0 otherwise.(1.46)Here again KË (S p) denotes the reduced Grothendieck completion of VectC(Sp),the monoid of isomorphism classes of complex vector bundles with base spaceS p [30].Limitations of local stabilityTo perform HorËavaâs analysis we had to impose the strong condition ofgapped fibers at the boundary of a ball âBr0(~k0). There is no universal wayto tell whether during a more general quasi-adiabatic evolution the fiber willproduce a new Fermi point on the boundary or not. Thus, we would have tolook at the particularities of each chosen point ~k0 â â¦ to define an adequateS p and relative to that what would be a âsmallâ perturbation. In particularthis construction is not well suited for modeling globally stable gapless edgemodes observed at the boundaries of topological phases of matter. Thatbeing said, because of the role of T P(H â²) in HorËavaâs analysis, under theright codimension dâpâ1, the local analysis imitates several aspects of theglobal analysis of gapped (not bulk gapped) topological phases.1.6.3 K -theory or vector bundles?HorËavaâs analysis allowed us to introduce indirectly what has been done inthe literature for the classification of so-called fermionic gapped topologicalphases for non-interacting systems. These constructions often arrive at anequivalence between a phase and an isomorphism classes of n-dimensionalvector bundles over the d-dimensional Brillouin zone Td for fully periodicsystems, VectnC(Td). In general, passing from VectnC(Td) to K -theory in-volves a series of steps. First we must allow addition of isomorphism classes[E1] â [E2] = [E1 â E2] and adding 1-dimensional vector bundles (i.e. sin-gle bands), which forces us to consider isomorphism classes of vector bun-dles of any dimension over Td, arriving then at the semigroup VectC(Td).Then, we have 2 options available to us to produce K -theory, either imposethe stabilization equivalence relation [30], as we did in subsection 1.6.2 toobtain the reduced KË (Td) group or we can define another equivalence rela-tion, defined on pairs ([E1], [E2]), ([Eâ²1], [Eâ²2]) of isomorphism classes, turningVectC(Td) into the abelian group K (Td) [30]. The latter process is called20a Grothendieck completion or Grothendieck construction and is usually ig-nored in the physics literature, where most works compute the unreducedK 0(Td) group directly to classify fermionic gapped phases, without justi-fying the transition to K -theory. As a matter of fact they often prefer tolook at so-called strong phases in which they replace the torus Td by a d-dimensional sphere [9] without any rigorous justification to our knowledge(although it does make computing the K -theory a little bit simpler). Notethat though related by the inclusion of a point to the space X being consid-ered, the computations of K 0 and KË 0 yield different results. As adequatelyput by Thiang [17], this has produced a conflation between the full K 0(Td)groups and their reduced versions KË 0(Td).The mathematical literature on the other hand addresses the issue indifferent ways. As examples we can consider the work of De Nittis andGomi [31],[32], which uses n-dimensional vector bundles and their charac-teristic classes to classify fermionic phases, without appealing to K -theory.In a different vein Freed and Moore [16] employ a process equivalent to aGrothendieck completion but emphasize that they are not aware of an apriori good physical motivation for its use. There are also examples in theCâ-algebraic approach of Prodan and Schulz-Baldes [33] together with ref-erences therein. Prodan and Schulz-Baldes motivate the use of K -theoryfor lattice Hamiltonians in the so called tight-binding approximation [6],through its applications in the bulk-boundary correspondence when com-bined with cyclic cohomology. Though a classification using K -theory is nota direct logical consequence of their formalism, its use in their approach iscertainly well motivated and it employs K -theory of Câ-algebras for bothsurface states and bulk states, with similar results as ours for the surfacestates in symmetry class A, employing the functor Kâ1 but on a real spacealgebra of surface observables.After publishing [2] we became aware of further work by Thiang [34]where a physical interpretation to elements in K 0(Td) is given to relativephases (as a difference between 2 topological phases where there is a bound-ary) although his analysis is more general and includes disorder in the Câ-algebraic framework. From the physical point of view we believe this inter-pretation to be unsatisfactory. When systems have a boundary the surfaceBrillouin zone is a torus of one lower dimension, thus the moment we putan interface we lose the periodicity and the Câ-algebra involved would haveto include said interface in its construction. Therefore we hold that theconstruction for systems on a half-crystal [2] was the first one for whichK -theory arises naturally from the simple mathematical representations of21quasi-adiabatic evolution and a gapped bulk spectrum for surface BlochHamiltonians on half-crystal. We also do not restrict to Hamiltonians inthe tight-binding approximation nor lattice systems. In the opposite direc-tion note that our construction has the rather strong assumption of discretetranslation-invariance on directions parallel to the boundary. It also hasthe more common assumption of a gapped bulk spectrum. Both of theserestrictions are better handled in [33],[17].1.7 Adding global symmetriesThus far, we have only considered half-crystals whose only symmetry isdiscrete translation invariance of the surface lattice Îâ. Thus our systemsdonât even need to respect time-reversal symmetry, since they may includea magnetic field, which in fact is what allows for the spectral flow to benon-zero. We now wish to extend the above analysis to include two dif-ferent possible symmetries, which are independent of the crystallographicstructure. Therefore, we will only study systems on half-crystals for whichthe boundary conditions preserve these symmetries. Thus, both the abso-lutely continuous and pure point spectrum of our Hamiltonians Ïpp(H(~kâ)),Ïac(H(~kâ)) respectively, inherit the same structure when a symmetry is im-plemented.1.7.1 Real and quaternionic structure on a complex vectorspaceWe must first digress and introduce the following mathematical definitionstaken from [35]. A real structure on a complex vector space V is an anti-linear operator K, such that K2 = I. In this case V â W âR C, whereW is vector space over R. A quaternionic structure on a complex vectorspace V is an anti-linear operator K, such that K2 = âI. The operatorsI,K, iI and J = iK satisfy the quaternion relations. As will be shown belowour symmetry operators defined below can be considered as either real orquaternionic structures on our Hilbert space. This will modify the complexK -theory that appears in our classification to a different (but related) ex-traordinary cohomology theory [28]. We will use this modified theory tocompute the distinct classes of globally topologically stable Fermi surfacesin a given symmetry class. Note that adding symmetries to the above anal-ysis is one of the main extensions done in [2], lacking in [36] among otherimportant differences.221.7.2 Particle-hole symmetryThe first symmetry we consider is the so called particle-hole symmetry. Thissymmetry as normally implemented, cannot be realized by non-relativisticcondensed matter systems, however we could slightly modify its implementa-tion to make it correct and we shall describe this modification. Systems withonly particle-hole symmetry and discrete translation-invariance are denotedas symmetry classes C and D in the physicistâs Altland-Zirnbauer classifica-tion [1]. The operator representing said symmetry is traditionally denotedby Î. There are different choices of implementations of Î one can make,such as whether it is a unitary or antiunitary operator [16],[17] and eachone of them represents systems in condensed matter with different physicalproperties. Our choice of implementation is the one most commonly usedthroughout the physics literature [4],[37] [38], where our symmetry operatorÎ acts on the fiber Hamiltonians in the following way [16]ÎH(~kâ) = âH(â~kâ)Î , (1.47)Îi = âiÎ . (1.48)This implies that for each band index α of H(~kâ), there exists a band indexβ of H(â~kâ), such thatÎÏÎ±(~kâ) = ÏÎ²(â~kâ) , (1.49)ÎµÎ±(~kâ) = âÎµÎ²(â~kâ) , (1.50)where ÏÎ±(~kâ) lives in the spectral subspace associated to the band indexα (which may be finite dimensional if ÎµÎ±(~kâ) â Ïpp(H(~kâ)) or infinite di-mensional if ÎµÎ±(~kâ) â Ïac(H(~kâ))). Our surface Brillouin zone always hada canonical involution Ï : Tdâ1 â Tdâ1, Ï(~kâ) = â~kâ, making Tdâ1 a realspace in the sense of [39]. However, up until now, because our fibers didnot need to satisfy a symmetry condition, this real structure was irrelevantfor our classification. This however will drastically change once we includesymmetries. We shall denote (Tdâ1, Ï) by Xs to remind ourselves we arealways taking into account its real structure. In general β 6= α to respectthe symmetry, however if ÎµÎ±(~kâ) â Ïpp(H(~kâ)) passes through an involutionfixed-point ~kâ = â~kâ, then β may equal α. This is a subtle but non-trivialpoint as it permits the existence of some topologically non-trivial Fermi sur-faces for all symmetry classes considered here (see subfigures 1.4II, 1.4III).At this point, looking at 1.50, the keen reader will note an inconsistency23with previous established facts about Ï(H(~kâ)). We had mentioned ear-lier that Ï(H(~kâ)) is bounded from below and unbounded from above [20].This is clearly inconsistent with 1.50. This is an other instance where theunboundedness of the Hamiltonian is more of a technical difficulty than atruely qualitative constraint. We could surmount this difficulty by insteadsettingÎH(~kâ)PE(~kâ) = âH(~kâ)PE(â~kâ)Î (1.51)Where PE(~kâ) would be the projection onto the spectral subspace corre-sponding to Ïâ¤E(H(~kâ)) of elements of the spectrum of the fiber which aresmaller than a cutoff E. Note that E should be independent of ~kâ. Froma physical point of view, as with quasi-adiabatic evolution and our choiceof the Riesz topology in section 1.4, we are not interested in modelling thehigh energy asymptotics of the system. Therefore we will simply continueto work with 1.47 and treat H(~kâ) as if it were bounded above as well.1.7.3 Real vector bundles and KR-theoryWe must make another digression in order to introduce some concepts whichmay not be that well known to topologists nor physicists. Given a space Xwith an involution Ï , one can define a Real vector bundle over (X, Ï) as acomplex vector bundle E that is also a real space and satisfies the followingconditions [39]â¢ The projection P : E â X commutes with the involutionâ¢ The map Ex â EÏ(x) is anti-linearCÃ Ex ExCÃ EÏ(x) EÏ(x)where the horizontal maps (z, v) 7â zv and the vertical maps are in-volutions, where the involution on C is the standard complex conju-gation.This anti-linearity property is the main difference with normal Z/2Z-equivariantvector bundles (see 1.7.4), where the structure maps must be unitary.One can define the Grothendieck ring of Real bundles over X, KR(X).Naturally there is a topological suspension with an involution Î£Ï together24with the usual suspension Σ on Real spaces to get a bi-graded cohomologytheory KRp,q(X) (see [39] for more details). The number p counts the num-ber of involution suspensions one performs on X and q the number of normalsuspensions. However Atiyah proves in [39] that there is an isomorphismKRp,q(X) â KR0,qâp(X) â¡ KRqâp(X). (1.52)He further showed there is an isomorphismKR(X) â KRâ8(X), (1.53)which is another instance of Bott periodicity. Dupont [40] was the first togeneralize this construction to symplectic vector bundles E, which have thesame definition as above except their involution is of order 4 (the involutionon the base X still being order 2), that is when Ex â EÏ(x) , Ï2 = âI.Analogously, one defines KH (X). Gukov shows in [41] the isomorphismKHân(X) â KRânâ4(X). (1.54)To compute examples we will use the fact that KRj is a functor fromthe category (see [42] for the basics of category theory) of spaces with aninvolution to abelian groups, the inclusion of the fixed point i : {x0} â Xis a morphism (Ï -equivariant maps) in that category and furthermore, thereis another one, r : X â {x0}, such that the composition r â¦ i is the identity.Hence we haveKRn(x0) KRn(X) KRn(x0)râIiâ (1.55)Meaning that the long exact sequence of the pair (X, x0)KRnâ1(X) KRnâ1(x0) KRn(X, x0) ââKRn(X) KRn(x0) KRn+1(X, x0)iâ 0râ 0(1.56)splits and we have an isomorphismKRn(X) â KRn(X, x0)âKRn(x0). (1.57)25We can generalize this to the pair (XÃY, {x0} ÃY)KRnâ1(XÃY) KRnâ1({x0} ÃY) KRn(XÃY/{x0} ÃY) ââKRn(XÃY) KRn(Y ) KRn+1(XÃY/{x0} ÃY)iâ 0râ 0(1.58)to obtainKRn(XÃY) â KËRn(XÃY/{x0} ÃY)âKRn(Y). (1.59)Equation 1.59 will be a fundamental tool in computing examples. We mustmake a last comment before we move to equivariant homotopy theory. If oneconsiders vector bundles, where the fibers are vector spaces over R insteadof being vector spaces over C, these are called real vector bundles. One canrepeat the above process and define the group KO(X) of real vector bun-dles. There is an equivalence between Real subbundles over the fixed pointsunder the involution Ï and real bundles over these fixed points. There-fore KRn(x0) â KOn(x0) and this is useful because we can use real Bottperiodicity [30].1.7.4 A splash of equivariant homotopy theoryAn action of a group G on a space X is a continuous mapÏ : GÃX â X (1.60)such thatÏ(g1, Ï(g2, x)) = Ï(g1g2, x) (1.61)The action is usually left implicit implicit in the notation by writing gxinstead of rho(x, g). Given spaces X,Y with a G-action, one can definemaps that are equivariant, i.e.f : X â Y (1.62)f(gx) = gf(x) (1.63)And so one can construct a homotopy theory of equivariant maps [43] andstudy homotopy classes of equivariant maps [ , ]G. The subject is not just atrivial extension of ordinary homotopy theory and it gets quite complicated.For example, up until this point the results presented only cared aboutthe homotopy type of our spaces but did not put any other restrictions26(i.e. compact,connected, Hausdorff nor CW-structure). In contrast, resultsin equivariant homotopy theory often require our G-spaces to be G-CWcomplexes (see [43] for a definition). Here we present a theorem which is anequivariant version of Whiteheadâs theorem of weak homotopy equivalencesbetween two G-CW complexes.Theorem 1.7.1. (Whitehead) An equivariant map f : X â Y of G-CWcomplexes is a G-homotopy equivalence if and only if fH : XH â Y H isa weak homotopy equivalence for all H-fixed points XH , Y H and all closedsubgroups H â G.We will make extensive use of this theorem in the next sections.1.7.5 Class DSystems in class D are defined by the conditionÎ2 = I . (1.64)Thus, Î represents a real structure on our complex fiber Hilbert space H â²,as defined above, which we denote as H â²R. We extend our previous analysison quasi-adiabatic stability of Fermi surfaces by employing equivariant ho-motopy theory, where the above implementation of the symmetry naturallyinduces the action H(~kâ) 7â âÎH(â~kâ)Î. We wish to use a slightly differentaction, defined in [44] (so that we can use the results in there). To do so wepass to the skew-adjoint operator iH(~kâ), with the actioniH(~kâ) 7â ÎiH(â~kâ)Î. (1.65)This leads to the following result:Theorem 1.7.2. ([2]-Class D) Systems on a half-crystal with gapped bulkand particle-hole symmetry in class D are given by KRâ1(Xs).Proof. Quasi-adiabatic equivalence classes of Fermi surfaces in symmetryclass D are defined as [Xs, FË(H â²R)]C2 . We can employ a result of Matumoto[44][Xs,F(H â²R)]C2 â KR(Xs) . (1.66)where F(H â²R) are Fredholm operators that anti-commute with the involutionon H â²R and C2 denotes the cyclic group of order 2 and also denotes thechoice of the action defined above (1.65). Using Whiteheadâs equivalence27(theorem 1.7.1) it suffices to have plain homotopy equivalence on the fixed-points of the subgroups of C2 to have the full equivariant equivalence. Theoriginal result of Atiyah and Singer [45] gives the equivalence when H = eis the trivial subgroup. For the full group C2, (1.66) gives an equivarianthomotopy equivalence between F(H â²R) and the classifying space associatedto the functor KR. Thus, Matumotoâs result trivially implies (â¦F(H â²R))C2is equivalent to the fixed points of the loop space of the classifying space ofKR (ÏF(H â²R)C2 ' (ÏKR)C2). Since the Atiyah-Singer map is equivariantthis means that there is an equivalence FË(H â²R)C2 ' (â¦F(H â²R))C2 , where FËdenotes skew-adjoint Fredholm operators [45]. Using the triviality of ourHilbert bundle E we conclude[Xs, FË(H â²R)]C2 â KRâ1(Xs) . (1.67)Class CNow we move on to the case whereÎ2 = âI (1.68)also known as symmetry class C. For this case, the particle-hole symmetryoperator induces a quaternionic structure on H , iÎ being the third genera-tor. Let us denote this quaternionic structure by H â²Q. Again we employ thesame strategy as above, passing to skew-adjoint operators. Following [44]we denote by D2 the cyclic group of order 2 but with a different action onF(H â²Q)iH(~kâ) 7â âÎiH(â~kâ)Î . (1.69)Matumoto also proved the equivalence[Xs,F(H â²Q)]D2 â KH (Xs) . (1.70)We have the following analogous result:Theorem 1.7.3. ([2]-Class C) Systems on a half-crystal with gapped bulkand particle-hole symmetry in class C are given by KHâ1(Xs).Proof. Repeating the fixed-point argument as above we have that stableisomorphism classes of Fermi surfaces in class C are given by[Xs, FË(H â²Q)]D2 â KHâ1(Xs) . (1.71)281.7.6 Time-reversal symmetryThe other symmetry we will study is far more ubiquitous in quantum me-chanics and is that of time-reversal symmetry through an operator Î [4],[38]ÎH(~kâ) = H(â~kâ)Î , (1.72)Îi = âiÎ . (1.73)Similarly to particle-hole symmetry, this implies that for each band index αof H(~kâ), there exists a spectrum index β of H(â~kâ), such thatÎÏÎ±(~kâ) = ÏÎ²(â~kâ) , (1.74)ÎµÎ±(~kâ) = ÎµÎ²(â~kâ) . (1.75)Unlike particle-hole symmetry, time-reversal symmetry is always imple-mented by an anti-unitary operator and the issues of upper unboundednessof Ï(H(~kâ)) with (1.50) do not arise with (1.75). There are two classes inthe Altland-Zirnbauer classification that correspond to systems with onlydiscrete translation-invariance and time-reversal symmetry, namely classesAI and AII, whose properties we describe below.Class AISystems in class AI must obeyÎ2 = I . (1.76)We now have the appropriate action directly on self-adjoint operators sinceH(~kâ) 7â ÎH(â~kâ)Î . (1.77)Theorem 1.7.4. ([2]-Class AI) Systems on a half-crystal with gapped bulkand time-reversal symmetry in class AI are given by KRâ7(Xs).Proof. These are [Xs,Fsaâ (HR)]C2 . The main difference with the above the-orems is that the fixed points are homotopic to the fixed points of the seventhloop space. Looking again at the fixed-points of the subgroups of C2 usingBott periodicityFsaâ (HR)C2 ' â¦7F(Hâ²â²) (1.78)proven in [45], which holds for the fixed-points of all subgroups of C2 andwhere Hâ²â²is a â-representation of the Real C6 Clifford algebra. This equiv-alence is equivariant with respect to (1.77), hence, together with the trivialfact thatFsaâ (H ) ' â¦7F(H ) (1.79)29due to Bott periodicity, the same argument with (1.66) runs through and[Xs,Fsaâ (HR)]C2 â KRâ7(Xs) , (1.80)Class AIIFinally, for stable Fermi surfaces in symmetry class AII, we must assumeÎ2 = âI . (1.81)By repeating the same process for self-adjoint operators as we did for skew-adjoint operators in class C, sinceH(~kâ) 7â âÎH(â~kâ)Î (1.82)and we obtainTheorem 1.7.5. ([2]-Class AII) Systems on a half-crystal with gapped bulkand time-reversal symmetry in class AII are given by KHâ7(Xs).Proof. Repeating same arguments as in theorem 1.7.4 with 1.82 we canconclude[Xs,Fsaâ (HQ)]D2 â KHâ7(Xs) . (1.83)We remark that both Î and Î are either real or quaternionic structureson the fiber Hilbert space and that the difference between them lies in theset of fixed-points under (1.65), (1.69), (1.77), (1.82) in the space of Hamil-tonian operators Fsaâ (H â²R), Fsaâ (H â²Q) or their action-induced skew-adjointcounterparts. We also note that this does not contain all of the Altland-Zirnbauer classification appearing in other classifications [16],[17] as we aremissing the cases with chiral symmetry (classes BDI, DIII, CI, CII). Forthese classes there is instead a unitary operator which is assumed in generalto be S = ÎÎ (see [16] for the general case) and they will not be consideredin this thesis.301.7.7 ExamplesWe should now compute a few examples. Let d = 2, then Xs = S1,1, theunit circle in the complex plane, where the involution is given by complexconjugation [39], [46]. In general S p,q is the (p + q â 1)-dimensional sphereembedded in Rp,q = iRp â Rq.KRj(S 1,1) â KRj(S 1,1, {x0})âKO j (1.84)where KO j denotes the real (trivial involution) j-th K -group of a point.Now by definitionKRj(S 1,1, {x0}) â¡ KËRj(S 1,1),â KËRj+1(S 0,1),â KËO j+1(S 0),â KO j+1. (1.85)We hence obtainKRj(S 1,1) â KO j âKO j+1. (1.86)For class D, j = â1 [46]KRâ1(S 1,1) â KOâ1 âKO0 â Zâ Z/2. (1.87)The Z-factor again corresponds to a spectral flow of surface states similarto class A, but now â ÎµÎ±(~kâ) â Ï(H(~kâ)), there exists ÎµÎ²(â~kâ) â Ï(H(â~kâ))such that ÎµÎ²(â~kâ) = âÎµÎ±(~kâ). Thus for each zero mode at ~kâ, there mustalso be one at â~kâ, as shown in subfigure 1.4II. Hence the parity of thespectral flow in class D depends on the existence of an odd number of zeromodes at one of the involution fixed-points.We first choose one of the involution fixed-points. If there is an even/oddnumber of zero modes at the chosen point, we can determine if there is aneven/odd number of zero modes at the other fixed-point by computing theparity of the spectral flow. In subfigure 1.4III we represent a state withno net spectral flow that has two zero modes one at each fixed-point. Thisis the non-trivial 0 â 1 state in Z â Z/2. Let us remark that there is nocanonical way of defining this invariant and a choice of a fixed-point mustbe taken. We also comment that the stability of this class depends on 1)31respecting particle-hole symmetry and 2) the existence of 2 fixed-points onXs.Examples of systems with a non-trivial class on the Z-factor are spinlessp-wave superconductors [38], where Bogoliubov surface zero modes at oneof the involution fixed-points {â1, 1}, satisfy an analogous reality conditionto that of Majorana fermions, hence they are called Majorana zero modes.Bogoliubov quasi-particles are effectively charge-less, thus, our zero modesonly contribute to the thermal current. Such a current could be induced by athermal gradient on our system, similar to the electric field for class A [47].This spectral flow has an analogy with an anomaly inflow interpretation,where the anomaly would be a gravitational one (thermal energy) [47].For symmetry class C, using eqs. (1.54), (1.86) we haveKHâ1(S 1,1) â KOâ5 âKOâ4â Z . (1.88)Since now Î2 = âI, Fermi points at the involution fixed-points must bedoubly degenerate and the mod 2 invariant must always vanish. Thus thespectral flow must come in pairs because of the symmetry. There is the samegravitational anomaly inflow physical interpretation as in symmetry class D(subfigure 1.4II) for the Bogoliubov zero modes. Condensed matter systemswith Î2 = âI in d = 2 correspond to so-called d-wave superconductors [38].For symmetry classes AI and AII the spectral flow must vanish becauseof the symmetry of the spectrum ÎµÎ²(â~kâ) = ÎµÎ±(~kâ) for some α, β. Bycomputing class AI, d = 2KRâ7(S 1,1) â KOâ7 âKOâ6= 0 . (1.89)We see there is only the trivial class of Fermi points, i.e. all Fermi points arequasi-adiabatically gappable in class AI, d = 2. For symmetry class AII,d = 2 we obtainKHâ7(S 1,1) â KOâ3 âKOâ2â Z/2 . (1.90)Due to Kramers theorem, if Fermi points arise at the involution fixed-points{1,â1} of Xs, they must be doubly degenerate, and thus the invariant isthe number of double-crossed Fermi points mod 2 [38]. Such double-crossedFermi points have a U(1)-spin anomaly inflow interpretation [10] and are32(I)+  +"kkâ¡â¡(II)+ +"kkâ¡â¡(III)+ "kkâ¡â¡(IV)"kkâ¡â¡+Figure 1.4: d = 2 Topologically stable Fermi points. Solid regions anddashed lines belong to the absolutely continuous spectrum Ïac(H(~kâ))(bulk states and resonance states respectively). Solid lines to Ïpp(H(~kâ))(surface state). + (â) signs label the sign of the slope at crossings. 1.4Icorresponds to class A (IQHE) with a net spectral flow = 1. 1.4IIcorresponds to a spectral flow of 2, that may occur for both classes C andD at non fixed-points in S 1,1. 1.4III represents two Majorana zero modesin class D. 1.4IV represents a helical Dirac fermion edge mode (SQHE).Figure taken from [2].often called helical Dirac zero modes because of the analogous role to the he-licity of relativistic Dirac fermions played by spin-orbit coupling. A physicalexample is the quantum spin Hall effect (QSHE) [38].1.7.8 Higher dimensional cases and weak topological phasesFor d = 3 Xs = S1,1 Ã S 1,1, we will use 1.59, taking X = Y = S 1,1 andobtainKRj(S 1,1 Ã S 1,1) â KËRj(S 1,1 Ã S 1,1/{x0} Ã S 1,1)âKRj(S 1,1). (1.91)The latter term we have already computed for any j. To compute ËKRj(S 1,1ÃS 1,1/{x0} Ã S 1,1) we note that X Ã Y/({x0} Ã Y ) ' X â§ Y+ and X â§ Y+ â¡X Ã Y+/X â¨ Y+ since we are collapsing X Ã {+} to a point. This is Ï -equivariant as the involution is the product of the involutions and the spaceswe are collapsing are Ï -invariant.33ThusËKRj(S 1,1 Ã S 1,1/{x0} Ã S 1,1), â KËRj(S 1,1 â§ S 1,1+ ),â KËRj(Î£ÏS 1,1+ ),â KËRj+1(S 1,1+ ),â KRj+1(S 1,1). (1.92)Hence we concludeKRj(S 1,1 Ã S 1,1) â KO j âKO j+1 âKO j+1 âKO j+2 (1.93)We summarize the results for all symmetry classes and for d = 1, 2, 3 onTable 1.1.Symmetry dAZ Î Î 1 2 3A 0 0 0 Z Z2AI I 0 0 0 0AII âI 0 0 Z/2 (Z/2)3D 0 I Z/2 Zâ Z/2 Z2 â Z/2C 0 âI 0 Z Z2Table 1.1: Classes of topologically stable Fermi surfaces for half-spacesystems with a gapped bulk in dimension d = 1, 2 and 3. Î is thetime-reversal operator and Î is the particle-hole symmetry operator. Weuse the names given in [1] (AZ). 0 on the right-side denotes the trivialgroup. 0 on the left-side indicates absence of said symmetry. WhenÎ = Î = 0 the surface Brillouin zone Xs is a regular torus Tdâ1 and foreither Î2 = Â±I or Î2 = Â±I it is a torus with involution T¯dâ1.Higher-dimensional classes relate to weak bulk topological phases. Tabletaken from [2].Unfortunately we do not have as of yet a picture of these topologicalinvariants in terms of the spectrum of the fiber Hamiltonians, though onecan speculate that it will be analogous and one must look at codim onesubspaces such as intervals and circles but we leave this for future work.341.8 Extending to crystallographic groupsWe shall now extend the work presented in [2] to include spatial symmetriesfor our Hamiltonian. These symmetries correspond to crystals of one lowerdimension as it is only on the direction parallel to the boundary that theseare preserved. We shall follow [16] and [21] up to a point. For their con-struction, equivariant vector bundles over the Brillouin zone trivially yieldequivariant K -theory after Grothendieck completion. In our case we haveto see whether sections of our Fredholm bundle over our particular Hilbertbundle E, with an action of the group is equivalent to equivariant K -theory.A d-dimensional crystallographic group G is defined via the group ex-tension0 Îâ G P 0i pi (1.94)where P â O(d â 1) is known as the point group, the group of extrasymmetries besides discrete translations, the latter are given by the subgroupÎâ â Zdâ1. There are two types of crystallographic groups, those whichare simply a semi-direct product Îâ o P are known as Symmorphic andthose which arenât are known as Non-symmorphic. The twistings that willappear later are those arising when the symmetry group of our Hamiltonianis non-symmorphic [21]. Since the Brillouin zone is the Pontryagin dualHom(Îâ, U(1)), there is an action of P on Hom(Îâ, U(1)) given byp~kâ(m1, · · · ,mdâ1) = ~kâ(pi(pËâ1(mË1, · · · , mËdâ1))), (1.95)where (mË1, · · · , mËdâ1) is the inclusion of (m1, · · · ,mdâ1) â Îâ in G andpË â piâ1(p) where pi is the projection of G onto P in (1.94). Note that thechoice of pË is not unique but the outcome is independent of it.While we have described the action of the point group P on the base,we need to also describe the map induced by P on sections of our Hilbertbundle E. Any wave function Ï(~kâ, ~r) in L2Tdâ1(Tdâ1 ÃRd;W ) (see (1.2.2))can be viewed as section Ï â L2(Tdâ1;E) in the following wayÏ(~kâ) : Rd/Îâ â E|~kâ ,[~r] 7â [~kâ, ~r, Ï(~kâ, ~r)]. (1.96)This facilitates the understanding of the induced action of P on E as the35action of P on L2Tdâ1(Tdâ1 Ã Rd;W ) is given by(pÏË)(~kâ, ~r) = U(p)Ï(pâ1~kâ, pâ1~r + apâ1) ,= U(p)ei~kâ·apâ1Ï(pâ1~kâ, pâ1~r) (1.97)where U(p) : W â W and (p, ap) is a representation of P â G â Rdâ1 oO(d â 1). Here it is very important to remark that the phase ei~kâ·apâ1 willbe the source of the âtwistâ in the non-symmorphic case.Thus we havep : E|~kâ â E|p~kâ ,{[~r] 7â [~kâ, ~r, Ï(~kâ, ~r)]} 7â {[~r] 7â [p~kâ, ~r, U(p)ei~kâ·apâ1Ï(~kâ, pâ1~r)]} (1.98)And finally we get a Hilbert bundle map that covers the action on Tdâ1:E ETdâ1 Tdâ1ppand therefore an associated Fredholm bundle map:Fsaâ (E) Fsaâ (E)Tdâ1 Tdâ1ppNotice that this map is not in general an action since from (1.97) we seethat(p2(p1Ï)(~kâ, ~r) = U(p2)U(p1)Ï(pâ12 pâ11~kâ, pâ12 pâ11 ~r+ pâ12 apâ11+ ap2) (1.99)But(p2p1Ï)(~kâ, ~r) = U(p2p1)Ï((p2p1)â1~kâ, (p2p1)â1~r + ap2p1). (1.100)and these are equal if and only ifpâ12 apâ11 + apâ12 â a(p2p1)â1 = 0 (1.101)36which is true for symmorphic crystallographic groups and not for non-symmorphic. Since the action of P leaves Îâ â Rd we can in fact viewthe above equation as a group 2-cocycleν : P Ã P â Îâ , (1.102)(p1, p2) 7â ap1 + p1ap2 â ap1p2 (1.103)so that fiberwise we have the twisted actionp1(p2Ï(~kâ)) = Ï(p1, p2,~kâ)p1p2Ï(~kâ) (1.104)withÏ(p1, p2,~kâ) = ei~kâ·Î½(pâ12 ,pâ11 )). (1.105)All twists coming from projective Hilbert bundles with P -equivariant fibersover Tdâ1 are elements of the equivariant cohomology group H3P (Tdâ1;Z)since BPU(H ) is the Eilenberg-Maclane space K(3,Z) [16],[48]. Gomi[21] clarified using the arguments presented above that the twist Ï be-longs to a subgroup and in fact Ï â H2(P ;C0(Hom(Îâ, U(1)))), whereH2(P ;C0(Hom(Îâ, U(1)))) denotes the group cohomology [48] of P withcoefficients in C0(Hom(Îâ, U(1))). Note that ν(p1, p2) = ap1 +p1ap2âap1p2 ,ν â H2(P ; Îâ) that precisely classifies extensions of the form (1.94).We will now use that E|~kâ â¡ H (~kâ) = L2(Rd/Îâ;L{~kâ}ÃRd/Îâ â W ).But first we must make a few restrictions on the action: Let P be a discretegroup acting on a a topological measure space (X,µ) such that L2(X; dµ)is well defined and µ(pU)) = µ(U) âp â P . Furthermore we ask that X/Pis also a measure space with measure µP , the one inherited from (X,µ).DefineXfreeP = {x â X|px 6= x âp 6= e â P} (1.106)That is, XfreeP are the points which are neither fixed nor periodic. Then wehave the following LemmaLemma 1.8.1. If the P -action on X satisfies µ(X âXfreeP ) = 0 then thereis an equivalence between L2(X; dµ) and L2(P ; dµcount)â L2(X/P ; dµP ) asP -representations.Proof. There exists a non-canonical injective map qP : XfreeP /P Ã P â Xthat sends ([x], e) to any one of the representatives of [x], and then defineqP ([x], p) = pqP ([x], e) and this is injective because qP ([x], e) â XfreeP .37Let U : L2(X/P Ã P ; dµP Ã dµcount) â L2(X; dµ) that sends f in anequivalence class in L2(X/P Ã P ; dµP Ã dµcount) (µcount being the countingmeasure of a set) toU(f)(x) ={f(qâ1P (x)) if x â XfreeP0 if x not in XfreeP(1.107)and let Uâ : L2(X; dµ) â L2(XfreeP /P Ã P ; dµP Ã dµcount) that sends f inan equivalence class in L2(X; dµ) toU â (f)([x], p) = f(qP ([x], p)) (1.108)Since µ(X âXfreeP ) = 0, U(Uâ(f)) is in the same class as f and the samefor Uâ(U(f)), thus U,Uâ is a unitary equivalence that commutes with theaction of P on L2(X; dµ) and on L2(XfreeP /P ÃP ; dµP Ãdµcount), therefore,as representations of P , they are equivalent.The above lemma has probably been proven before (or something stronger)but we were unable to find such a statement in the literature.Remark 1.8.2. In general, crystallographic groups where P is non-trivialcombinations of rotations, reflections and translations satisfy 1.8.1. How-ever there are exceptions like G = Z Ã Z4, with the action of the generatorof Z4 being ~r 7â â~r. Hence at this point we have to check each case individ-ually.Let us now employ this result. Whenever the action of P satisfies 1.8.1,we can rewrite the fiber of our Hilbert bundle E|~kâ = L2(Rd/Îâ;L{~kâ}ÃRd/ÎââW ), as the equivalent G-representationE|~kâ â L2(P ;L2((Rd/Îâ)freeP /P ; L{~kâ}ÃRd/Îâ âW )) (1.109)Then we can apply a result in the appendix A.4 of Freed, Hopkins andTeleman [49] to proveTheorem 1.8.3. Quasi-adiabatic equivalence classes of Fermi surfaces ona half-crystal with point group P and total symmetry group G, satisfying0â Îâ â Gâ P â 0 and the P -action on Rd/Îâ as in 1.8.1 are in one toone correspondence with the twisted P -equivariant K -group K Ïâ1P (Tdâ1), Ïas in (1.105).38Proof. Since the fiber E|~kâ is of the form L2(P ; l2), then E is a locally uni-versal Hilbert bundle (see [49] for the properties of universal Hilbert bun-dles), and furthermore, using 1.7.1 and [45] on the fixed points proves thatFsaâ (E) is a classifying space for K Ïâ1P . Quasi-adiabatic equivalence classesof Fermi surfaces with symmetry group G are given by [Tdâ1,Fsaâ (E)]P âK Ïâ1P (Tdâ1)Remark 1.8.4. In [50] a similar result was stated without proof.It is straightforward to combine 1.8.3 to include time-reversal symmetryor particle-hole symmetry when these do not interact with the point groupP since we also have equivariant Bott peridocity. We obtain the groupsKRÏâ1P (Xs) , Î2 = I , (1.110)KRÏâ5P (Xs) , Î2 = âI , (1.111)KRÏâ7P (Xs) , Î2 = I , (1.112)KRÏâ3P (Xs) , Î2 = âI . (1.113)When there is a non-trivial relation between point group and time-reversal symmetry one has the magnetic point group satisfying0â P â PË â Z/2â 0 (1.114)and this has been formalized in [16],[21] and [51] as a new type of K -theorythat combines KR-theory and equivariant K -theory through twists but weshall leave that for future work.39Chapter 2K-theory and Contextuality2.1 IntroductionIn quantum mechanics only mutually commuting observables can be simul-taneously measured to reveal their joint outcome. Such a set of commut-ing observables is called a context. It is a fundamental property of quan-tum mechanics that its predictions cannot be reproduced by assuming pre-determined context independent outcomesâa phenomenon called contextu-ality [52, 53]. This special feature of quantum systems is expected to play animportant role in any computation scheme relying on quantum principals.Indeed, in quantum computation contextuality is established as a resourceto achieve universal quantum computation and quantum speed-up can onlybe attained if contextuality is present [54â58]. To better characterize con-textuality from the quantum computation perspective different frameworkshave been developed such as [59â63]. Here we extend the topological ap-proach of [63] and makes a connection to the sheaf-theoretic approach of[62].Classifying spaces are fundamental objects in algebraic topology thatplay a prominent role in the classification of principal bundles [64]. Thesegeometric objects appear in gauge theories in the standard model [65]. Mo-tivated by questions arising from gauge theory [66, 67], Adem et al. [68, 69]introduced the notion of a classifying space for commutativity in Lie groups.Here we introduce the classifying space for contextuality, which is a variantof this construction tailored for applications to contextuality. Not only is it aunifying object for contextual interpretations, it is an interesting mathemat-ical object in its own right because of its rich homotopy-theoretic structure.The classifying space for commutativity of extraspecial groups studied in[70â72] demonstrates such an interesting behavior. There is a close connec-tion to quantum computation since the basic observables called the Pauliobservables constitute a finite group which is an extraspecial group. Ourgoal here is to make this connection precise by showing how traditional quan-tities relevant to contextuality (such as Wigner functions) can be viewed aselements in algebro-topological groups coming from our classifying space.40A context specifies a set of observables that can be measured simultane-ously. In more mathematical terms for us a context in quantum computationrefers to an abelian subgroup of the Pauli group. Given a collection I ofcontexts we construct a topological space Bcx(I), called the classifying spacefor contextuality, and identify two classes:â¢ a cohomology class in the second cohomology group with mod-p coef-ficients[β] â H2(BcxI)â¢ a K -theory class, that depends on a quantum state Ï, in the β-twistedK -group[WÏ] â RâK β(BcxI).The cohomology class [β] is introduced in [63] using chain complexes ob-tained from a cover of contexts. The chain complex of BcxI specializes tothe construction given there for small dimensions. But the space itself is infi-nite dimensional analogous to the case of classifying spaces of finite groups.This class captures contextuality in the strong sense measuring only thefailure of assigning pre-determined measurement outcomes. The full formu-lation of contextuality involves probability distributions on outcomes overcontexts and can be formalized using sheaf theory [62]. Given a quantumstate Ï the empirical model eÏ provides a description as a probability distri-bution on measurement outcomes over each context. If the empirical modelcan not be described using a deterministic hidden-variable model then itis called contextual. When this approach is transferred to our topologicallanguage the empirical model eÏ can equivalently be described as the class[WÏ] which lives in the twisted K -group of BcxI. This class is essentiallythe discrete Wigner function [73] of the quantum state.We have the following structure for this chapter. In §2.2 we start withthe sheaf-theoretic definition of contextuality, and restrict this frameworkto Pauli observables [63]. This is summarized in Corollary 2.2.8. Section2.3 uses twisted representations to describe the space of empirical models.Theorem 2.3.9 explains the relationship between the empirical model eÏand the class [WÏ] corresponding to the Wigner function. The classifyingspace for contextuality is introduced in §2.4. We study its cohomology andexplain the distinction between the even and odd prime cases. We refer tothe quantum computation literature to point out the consequences of theseobservations. Representation theoretic approach of §2.3 and the topologicalapproach of §2.4 come together in §2.5. In this section we compute the β-41twisted K -theory of BcxI in Theorem 2.5.4 whose proof mostly follows thecomputation in [70] for the untwisted version.2.2 Contextuality for Pauli observablesIn this section we introduce the basic ingredients of quantum computation.We define contextuality in the sheaf-theoretic language of [62]. Then wespecialize to the Pauli group, the basic observables of quantum computation.Our description makes a connection to the topological framework of [63]. Forbasics of quantum computation we refer to [74].2.2.1 Contexts in quantum mechanicsIn quantum mechanics the state of a system is specified by a density matrixÏ acting on a Hilbert space H. A density matrix is a Hermitian matrixthat is positive and has trace Tr(Ï) = 1. We will denote the space ofdensity matrices by Den(H). An observable is a Hermitian matrix actingon the Hilbert space. In quantum computation the Hilbert space is finitedimensional. Here we will only consider the case when the dimension is apower of a prime p.Remark 2.2.1. When p = 2 the observables are Hermitian matrices asusual, but for p > 2 the type of observables we will consider are unitarymatrices that are not necessarily Hermitian. In fact, the results here can beapplied to the more general observables introduced in [63]. But we restrict toPauli observables since for this case the topology of the classifying space forcontextuality is well-understood, and independently studied (see Theorem2.4.6).An observable A can be measured on a system at state Ï. The mea-surement result is specified by an eigenvalue λA of the observable. Bornâsrule says that the probability of a measurement result λA occurs is givenby the trace Tr(ÏPÎ»A) where PλA is the projector onto the eigenspace corre-sponding to the eigenvalue λA. This can be generalized to a set of pairwisecommuting observables, also known as a context. Let C be a context and λbe a function that assigns an eigenvalue to each observable in the context.The probability of obtaining λA as a measurement result for each observ-able A â C is given by Tr(ÏPÎ») where Pλ projects onto the simultaneouseigenspace of the observables in C.422.2.2 Sheaf-theoretic description of contextualityAbramsky and Brandenburger introduced a sheaf theoretic approach in [62]to study probability distributions for measurements over various contexts.The general framework consists ofâ¢ a set X of measurements,â¢ a set O of outcomes,â¢ a collection M of contexts such thatX =âCâMC.The collection M is sometimes referred to as a cover of contexts. We willassume that M is closed under intersections and regard it as a partiallyordered set ordered under inclusion. Given this data one can construct asheaf of eventsE :Mop â Setwhich assigns the set OC of all functions C â O to a given context C. For aset U let D(U) denote the set of probability distributions over U with finitesupport. This gives a functorD : Setâ Setwhere for a function f : U â U â² the corresponding function D(f) : D(U)âD(U â²) is defined byD(f)(d) : U â² â Râ¥0, uâ² 7ââuâfâ1(uâ²)d(u).We are interested in the inverse limit of the composite functor DE :Mop âSet. An element d of the inverse limit is a collection of distributions{ d | C | C âM}, where d|C means restriction to C, such that further re-striction to pairwise intersections coincide: (d|C)|Câ©Câ² = (d|Câ²)|Câ²â©C . Thusthe inverse limit is a subset of the productlimâ DE ââCâMDE(C).In physics literature an element of the inverse limit is called an empiricalmodel.43An important example comes from quantum mechanics. Given a stateÏ we can define an element eÏ of the inverse limit. Let us denote thisassignment by a mape : Den(H)â limâ DE , Ï 7â eÏwhere eÏ|C is defined on λ : C â O by the quantum mechanical probabilityeÏ|C(Î») = Tr(ÏPÎ») â Râ¥0.Let S(Ï) denote the set of functions s â E(X) â¡ OX such that the restrictionof the empirical model satisfies eÏ|C(s|C) 6= 0 for all C âM.Definition 2.2.2. A state Ï is called non-contextual if there exists a dis-tribution d â DE(X) such that d|C = eÏ|C for all C â M, otherwise, it iscalled contextual (with respect toM). A state Ï is called strongly contextualif the set S(Ï) is empty.In sheaf theory language we can summarize this definition by the state-ment that a state Ï is contextual if eÏ does not comes from a global section,also known as a deterministic hidden-variable model.2.2.3 Pauli groupIn quantum computation there is a distinguished set of observables calledthe Pauli observables acting on a Hilbert space H of dimension d. Moreprecisely, H is the complex group ring C[Z/d] of the additive group Z/d ofintegers modulo d. Pauli observables acting on H can be defined by theunitary representationZ ãa| = Ïa ãa| X ãa| = ãa+ 1|where a â Z/d and Ï is the d-th root of unity e2pii/d. We are using thequantum mechanical notation ãa| to denote the basis elements of the Hilbertspace. This representation is usually called a 1-qudit, or more popularly a1-qubit when d = 2. Here we will only consider the case where d is a primep.Qudits can be composed to obtain larger systems. An n-qudit has theHilbert spaceHn = Hâ · · · â Hï¸¸ ︷︷ ︸n44which can be identified with the complex group ring C[(Z/p)n]. Let us setV = (Z/p)n Ã (Z/p)n. For an element (a, b) â V the corresponding Paulioperators are defined as the tensor powersZ(a) = Za1 â · · · â Zan X(b) = Xb1 â · · · âXbn .When it comes to define the Pauli group, or the group of Pauli observables,there is a distinction between the cases p = 2 and p > 2. Let us setµ ={ âÏ if p = 2Ï if p > 2.The Pauli group of an n-qudit system is defined byPn = ãµI, Z(a), X(b)| (a, b) â V ã (2.1)as a subgroup of the unitary group U(pn).Example 2.2.3. The Hilbert space of a 1-qubit system is given by H =C[Z/2]. Pauli observables constitute a subgroup of the unitary group U(2)generated by the Pauli matricesX =(0 11 0)Z =(1 00 â1)together with iI.2.2.4 Extension classAs an abstract group Pn is an extraspecial p-group.A p-group is group Gsuch that to each element g, there is a power n(g) so that gpn(g)= 1. Afinite group is a p-group if its order is a power of p. A finite p-group G isextraspecial if its center, Z(G) is cyclic of order p and G/Z(G) is abelian oforder p. When p = 2 it is of complex type, and for p > 2 it has exponent p.The subgroup ãµIã â Pn is isomorphic to the additive groupZµ ={Z/4 p = 2Z/p p > 2under the canonical homomorphism I 7â µ. Under this identification thePauli group can be written as a central extension0â Zµ â Pn piââ V â 0 (2.2)45since up to an element of ãµIã a Pauli operator is specified by an element ofV . Given a function γ : V â Zµ we can write down a set theoretic sectionηγ(v) = µγ(v)Z(vz)X(vx)of the projection homomorphism and the extension cocycle can be computedfrom the formula β(v, vâ²) = Î·(v)Î·(vâ²)Î·(v + vâ²)â1. The cohomology class[β] â H2(V,Zµ) of the extension is independent of the choice of the section.Let η0 denote the section corresponding to the zero function γ = 0, thenthe corresponding cocycle can be computed asβ0(v, vâ²) ={vx · vâ²z p > 22 vx · vâ²z p = 2 (2.3)where the notation v Â·w stands for the standard inner product. For any otherchoice of γ we have that βγ = β0+dγ where dγ(v, vâ²) = Î³(v)âÎ³(v+vâ²)+Î³(vâ²).2.2.5 Isotropic subspacesCommutation properties of Pauli operators are determined by a symplecticform b : V Ã V â Z/p defined byb(v, vâ²) = vx · vâ²z â vâ²x · vz. (2.4)This fact about Pauli operators can be seen from the commutator [η(v), Î·(vâ²)]and the expression b(v, vâ²) = β(v, vâ²) â Î²(vâ², v) (for p = 2, µ = âÏ and wecan identify 2(Z/4) with Z/2) that relates the symplectic form to the ex-tension cocycle. We can choose a symplectic basis { z1, · · · , zn, x1, · · · , xn }for V where xi and zi are the symplectic pairs satisfying b(zi, xi) = 1. Thesuitable choice compatible with the representation of Pn is to take zi thecanonical basis for the first factor of V = (Z/p)n Ã (Z/p)n and xi the basisfor the second factor.Definition 2.2.4. A subspace I â V is called isotropic if b|I = 0 i.e.b(v, vâ²) = 0 for all v, vâ² â I. We write I(V ) for the collection of all isotropicsubspaces in V .We will derive a property of β restricted onto an isotropic subspace. Forthis purpose we make a canonical choice for the extension cocycle:β = β0 + dq/2 (2.5)46where q : V â Zµ denotes the functionq(v) ={2vx · vz p = 2vx · vz p > 2.The motivation for this choice is provided by the following result.Proposition 2.2.5. For any isotropic subspace I the restricted cocycle sat-isfies 2β|I = 0. Moreover, for p > 2 we have that 2β = b.Proof. For p = 2 as a consequence of 2.3 we have that 2β = dq = 2b since2q can be thought of a quadratic form on 2(Z/4) â¼= Z/2. Thus 2β restrictedto I vanishes. In the case of p > 2 one can calculate to obtain β = b/2.2.2.6 Contextuality for Pauli observablesThe observables that are used in quantum computation have eigenvaluesgiven by the set { 1, Ï, · · · , Ïpâ1 }. These observables are precisely the ele-ments of order p in the Pauli group.In the sheaf theoretic description of contextuality for Pauli observableswe consider the triple (Σ, I,Z/p) whereâ¢ the collection I of contexts is given by a subcollection of I(V ) satis-fying the property that if I, I â² â I then I â© I â² â Iâ¢ the set of measurements is { η(a) | a â Σ(I) } whereΣ(I) =âIâIIâ¢ the set of outcomes is Z/p.Remark 2.2.6. As a consequence of Proposition 2.2.5 the cocycle β|I is thezero function when p > 2, and is divisible by 2 when p = 2. For the p = 2case we identify 2(Z/4) with Z/2 and regard β|I as a function I Ã I â Z/2.Next we introduce a modified version of the event sheaf. In the Paulicase we will use the set of functionsEβ(I) = { s : I â Z/p | β|I = ds }which as an assignment gives a functorEβ : Iop â Set.Note that this is a subfunctor of the sheaf of events: Eβ â E (as a subcategoryof the image). The motivation for this definition is the following observationin quantum mechanics.47Proposition 2.2.7. If a function s : I â Z/p does not belong to Eβ(I) theneÏ|I(s) = 0.Proof. If eÏ|I(s) 6= 0 then there is a common eigenstate ãs| with eigenvalueÏs(a) for the observable η(a). Thus we can writeÏs(a)+s(b) ãs| = η(a)η(b) ãs| = ÏÎ²(a,b)Î·(a+ b) ãs| = ÏÎ²(a,b)+s(a+b) ãs|which implies that s â Eβ(I).This means that the distribution eÏ|I â DE(I) can be regarded as anelement of DEβ(I) for each I â I. As a consequence the empirical model eÏassociated to a state Ï gives a functione : Den(H)â limâ DEβ.Proposition 2.2.7 has another implication. To demonstrate this we in-troduce the setEβ(I) = limâ Eβas the inverse limit of the functor Eβ : Iop â Set. More explicitly, thisset consists of functions that are obtained byâgluingâ functions Eβ(I) overcontexts. By naturality of the inverse limit we have a commutative diagramDen(H) limâ DEβ DEÎ²(I)limâ DE DE(Σ)e θ(2.6)where the union of all contexts is denoted by Σ = Σ(I).Corollary 2.2.8. Let Ï be a state and I be a cover of contexts.1. Ï is strongly contextual if Eβ(I) = â .2. Ï is contextual if and only if eÏ /â Im(θ).Proof. Recall from Definition 2.2.2 that Ï is strongly contextual if S(Ï) isempty. By definition s â E(Σ) belongs to this set if eÏ|I(s|I) 6= 0 for allI â I. By Proposition 2.2.7 the restriction eÏ|I belongs to Eβ(I) for each I.48This implies that s|I is in Eβ(I). But this suffices to conclude that s belongsto Eβ(I), since it is obtained by patching the local functions.For the second statement observe that if eÏ can be written as Î¸(dâ²) forsome dâ² â DEβ(I) then regarding dâ² as an element of DE(Σ) in the obviousway implies that Ï is non-contextual. Conversely, it suffices to check thatif eÏ = θ(d) for some d â DE(Σ) then d belongs to DS(Ï), which is asubset of DEβ(I) by the previous paragraph. Observe that if d(s) 6= 0 theneÏ|I(s|I) = θ(d)|I(s|I) 6= 0 for all I, and thus s belongs to S(Ï).2.3 Wigner function as a global sectionThe purpose of this section is to interpret empirical models described inthe previous section in representation theoretic terms. The approach takenhere is to extend probability distributions over Râ¥0 to the whole field of realnumbers. After extending the coefficients the empirical model map becomese : Den(H)â limâ Râ Eβ.Our main result is proposition (2.3.5) that gives an identification of theinverse limit as the representation ring RâR(V ). Under this identificationthe empirical model eÏ corresponds to the Wigner function WÏ of the stateÏ. In general WÏ is not a probability distribution and may assume negativevalues. An important application to contextuality, for p > 2, says that WÏis a probability distribution if and only if Ï is non-contextual. This resultdisplays a distinct feature of the odd prime case as first demonstrated in[54, 55].2.3.1 Twisted representationsLet ι : Z/p â U(1) denote the embedding ι(1) = Ï where Ï is the p-th root of unity e2pii/p. Under this identification the cohomology class β|Icorresponds to a class in H2(I, U(1)), also denoted by the same symbol.Such a class can be used to define a twisted representation [75].In general, for a finite group G and a cohomology class α â H2(G,U(1))an α-twisted representation can be thought of as a linear representation ofthe extension GË corresponding to α. Therefore most of the properties oflinear representations can be transferred to twisted representations. Thereis a Grothendieck group of twisted representations [76], denoted by Rα(G).49Proposition 2.3.1. The embedding ι : Z/p â U(1) induces a natural iso-morphism of R-vector spacesÎ¹â : REÎ²(I)â RâRÎ²(I).Remark 2.3.2. Note that when p > 2 the twisted representation groupcoincides with the ordinary representation group R(I) as a consequence ofProposition 2.2.5.The benefit of passing to representations is the character map [77]:ch : RÎ±(G)â Clα(G) (2.7)where Clα(G) denotes the C-vector space of α-class functions. A functionf : Gâ C is an α-class function if it satisfies the formulaf(hghâ1) =α(h, hâ1)Î±(h, ghâ1)Î±(g, hâ1)f(g) (2.8)for all g, h â G.Lemma 2.3.3. Let I â V be an isotropic subspace. Then Clβ(I) coincideswith the C-vector space CI of complex valued functions on I.Proof. The p > 2 case follows from Remark 2.3.2. For p = 2 we check thatÏÎ²(w,âw)ÏÎ²(w,vâw)ÏÎ²(v,âw)=Î·(w)Î·(w)Î·(0)Î·(w)Î·(v + w)η(v) η(v)η(w)η(v + w)= 1for all v, w â I. Recall that each η(v) squares to the identity matrix, η(0) =I, and [η(v), η(w)] = I since v, w belong to an isotropic subspace.2.3.2 Computing the inverse limitLet I â I(V ) be a cover of contexts. We are interested in the inverse limitof the functor REβ : Iop â Set. We will compute it using the charactermap 2.7. The first step is to extend the scalars to complex numbers.Lemma 2.3.4. There is a natural isomorphism of C-vector spacesCâ (limâ REÎ²)â¼= CΣ.50Proof. Up to isomorphism we can consider the functor R â Rβ instead(Proposition 2.3.1). Since the character map 2.7 is natural it gives a natu-ral transformation ch : R â Rβ â Clβ, where Clβ is regarded as a functoron Iop. Tensoring R â Rβ with C the character map becomes an isomor-phism of C-vector spaces. We denote the resulting natural isomorphism bych : CâRÎ² â Clβ. Since taking inverse limit commutes with tensoring withC we haveCâ (limâ RâRÎ²)â¼= limâ CâRÎ²â¼= limâ Clβ.Using Lemma 2.3.3 we can compute the inverse limit of Clβ. The functorClβ is isomorphic to the functor Câ which sends I to the set CI of complexvalued functions. Inverse limit of Câ is simply the set of functions from theunion Σ = Σ(I) to complex numbers.Next step is to identify the inverse limit over R. We take I = I(V ) andΣ = V since smaller sets of measurements can be dealt with by naturality,i.e. restricting from I(V ). The analysis depends on the prime p. Let usdefine a mapÏp : RâR(V )â limâ RâRÎ²as follows:â¢ p = 2: A β-twisted representation is a linear representation (compati-ble with the embedding ι) of the extension0â Z/2â IË â I â 0corresponding to the cocycle β. By definition of β (2.5) every elementof the group IË has order at most 2. Therefore the class function of arepresentation takes values in R. This means that the character map2.7 factors through the isomorphismchR : RâRÎ²(I)â RI .A similar observation is true for R â R(V ), i.e. the character mapinduces an isomorphism to RV . Using these identifications we candefineÏ2 : RâR(V )â limâ RâRÎ²by sending a representation to the restriction of the associated charac-ter to each isotropic subspace. More explicitly, we have a commutative51diagramRâR(V ) limâ RâRÎ²RV limâ RâÏ2chRchRâ¼=where Râ is the functor I 7â RI . The lower horizontal map is anisomorphism by an analogous argument given in the proof of Lemma2.3.4. The point is that for p = 2 extending coefficients to real numbersis sufficient.â¢ p > 2: We have seen that a β-twisted representation is the same asan ordinary representation. Therefore for each isotropic subspace wehave a restriction map R(V ) â R(I). These maps induce a map tothe inverse limitÏp : RâR(V )â limâ RâRÎ²and again this map turns out to be an isomorphism as in the p = 2 case.This can be seen by extending the scalars to complex numbers andusing Lemma 2.3.4. Indeed, tensoring Ïp with C gives an isomorphismsince we obtain a commutative diagramCâR(V ) Câ (limâ RâRÎ²)CV limâ CâÏpchchâ¼=Here left-vertical map is an isomorphism by definition of class functionsand right-vertical arrow is an isomorphism by Lemma 2.3.4.Both diagrams imply that Ïp is an isomorphism, and under the identifi-cation of Proposition 2.3.1 we obtain:Proposition 2.3.5. There is a natural isomorphismÏp : RâR(V )â limâI(V )REβ for any prime p.52Remark 2.3.6. Note that the inverse limit of Rβ is a subgroup of theproductâIâI(V )Rβ(I). In particular, it is a torsion-free abelian group.Therefore Proposition 2.3.5 shows that the inverse limit of Rβ is isomorphicto Z|V |âthe free abelian group of rank |V | = p2n.2.3.3 Wigner functionsDensity matrices map to the inverse limit of REβ by the assignment Ï 7â eÏ.As a consequence of the isomorphism Ïp of Proposition 2.3.5 there must bea map W that fits in the commutative diagramDen(H) RâR(V )limâ REÎ²WeÏp (2.9)We write Ï 7â WÏ for this assignment. It turns out that WÏ, an elementof the representation group, is in fact the (discrete) Wigner function of Ï.Wigner functions are introduced to the quantum computation literature inthe work of Gross [73].Let us introduce the Wigner function of a density matrix. First we definethe point operatorsAv = |V |â1/2âuâVbu(v)Î·(u)where bu is the representation V â U(1) defined by bu(v) = Ïb(v,u) (b is thesymplectic form in 2.4).Definition 2.3.7. Wigner function of Ï is the function WÏ : V â R definedby the equationWÏ(v) = |V |â1/2Tr(ÏAv).We construct an element of the representation groupWÏ =âvâVWÏ(v)bv â RâR(V )by regarding the values WÏ(v) as coefficients of the representations bv. Theset {Av | v â V } is an orthonormal basis for the pn Ã pn matrices over Cwith respect to the inner product (A,B) = |V |â1/2Tr(Aâ B), where (â)â stands for the conjugate transpose. Therefore we can writeÏ =âvâVWÏ(v)Av.53Note that the Pauli observables { η(v) | v â V } is also an orthonormal basiswith respect to the same inner product.The operator inner product introduced above is related to the innerproduct of characters. The set of irreducible α-twisted representations of afinite group G constitutes an orthonormal basis of Clα(G) with respect tothe inner product(Ï, Ï) =1|G|âgâGÏ(g)ÏÂ¯(g).Let Ïs â Rβ(I) denote the twisted representation corresponding to s â Eβ(I)under the isomorphism of Proposition 2.3.1. The projector onto the commoneigenspace of { η(u) | u â I } corresponding to the eigenvalues specified bys can be written asPs =1|I|âuâIÏÂ¯s(u)Î·(u).Using this we can show the following.Lemma 2.3.8. We haveeÏ|I(s) = Tr(ÏPs) =âtâV(bt|I , Ïs)WÏ(t).Proof. This follows from the formula for Ps and the calculationTr(Ï η(u)) =âtâVWÏ(t)bt(u).The main result of this section is the identification of the empirical modeleÏ as an element of the representation group. We will write WÏ|I for therestriction of Ïp(WÏ) to an isotropic subspace I.Theorem 2.3.9. (OS 2019) Sending a density matrix Ï to the element WÏin RâR(V ) defines a mapW : Den(H)â RâR(V ) (2.10)that makes the diagram 2.9 commute.Proof. The interpretation of WÏ|I depends on the prime p since the map Ïpdoes as well.54â¢ p > 2: In this case Ïp is induced by the restriction maps R(V )â R(I).Lemma 2.3.8 implies thateÏ|I(s) = (WÏ|I , Ïs)which is what we want to prove.â¢ p = 2: The interpretation in this case is through the character map.The restriction is induced by the restriction maps RV â RI . Therepresentations bt are regarded as functions RV . Again Lemma 2.3.8finishes the proof.Remark 2.3.10. As an immediate corollary of this theorem and the factthat a state is uniquely determined by its Wigner function we observe thatthe map W is injective. Extending the coefficients to R in the domain of 2.10the map W becomes an isomorphism of R-vector spaces. Commutativity ofdiagram 2.9 implies that the map e is injective. A subtle point is that WÏ isnot necessarily a probability distribution as it may assume negative values.In quantum mechanics the negativity of the Wigner function indicates adivergence from the classical behavior and this is related to contextuality aswe will see next.2.3.4 ContextualityAs we emphasized in Remark 2.3.10 the coefficients WÏ(v) can take negativevalues. We write WÏ â¥ 0 when WÏ(v) â¥ 0 for all v â V , and say that theWigner function is non-negative. Recall that a state Ï is contextual if eÏ isnot in the image of the mapθ : DEÎ²(I)â limâ DEβ.We will look at the odd prime and p = 2 cases separately.â¢ p > 2: The odd prime case is nicer since β-twisted representationscoincide with ordinary representations. We can consider DEβ(V ) asa subset of R â R(V ), and θ is the restriction of Ïp to this subset.The Wigner function, regarded as an element of R â R(V ), can beused as a distribution when WÏ â¥ 0. Thus it lies in DEβ(V ) andfurthermore satisfies Î¸(WÏ) = eÏ by Theorem 2.3.9. Conversely, if Ï is55non-contextual, i.e. there exits d with θ(d) = eÏ, that means WÏ â¥ 0since d = WÏ because Ïp is an isomorphism. Thus we obtain a proofof the following result, which is first proved in [54] for n = 1 andgeneralized to all n in [55].Corollary 2.3.11. Assume that p > 2. Then WÏ â¥ 0 if and only if Ïis non-contextual.â¢ p = 2: This case is much trickier and the Wigner function does notdetermine contextuality as in p > 2. There are partial results in thephysics literature, for example see [56, 78]. For some I the set Eβ(I)turns out to be empty, thus for such a cover of contexts any state Ïis strongly contextual. For example, this happens if I contains one ofthe Mermin square I or Mermin star I? cover of contexts.Remark 2.3.12. The notion of non-contextuality we use relies on the ex-istence of deterministic value assignments for measurement outcomes, alsoknown as deterministic hidden-variable models. There is a different ap-proach that uses ontological models and a more flexible notion of contextu-ality, see [61]. For p = 2 although the characters bt do not correspond topossible outcome assignments (Proposition 2.2.7) a non-negative WÏ can beregarded as a distribution over the âontic spaceâ { bt | t â V }.2.4 Classifying space for contextualityIn [63] a topological approach is introduced to study contextuality in quan-tum mechanics. The essential ideal is that contexts can be regarded asgeometric simplices, and glued to each other to form a topological space.Strong contextuality is shown to be detected by the cohomology class [β]that lives in the second cohomology group of a certain chain complex. Herewe can introduce a space, called the classifying space for contextuality, whichrealizes this chain complex. The name âclassifying spaceâ stems from thefact that the resulting space classifies principal bundles whose transitionfunctions, whenever simultaneously defined, are given by contexts.In this section we give the construction of the classifying space BcxI for agiven cover I of contexts and prove its basic properties. Homotopy theoreticproperties of this space are independently studied in [72]. We recall theseproperties and show how to use them in contextuality. A crucial feature ofBcxI is that the higher homotopy groups are non-trivial in contrast to theusual classifying space of a finite group. In the next section we will interpret56the results of the previous section (§2.3) on Wigner functions in terms ofthe (twisted) topological K -theory of BcxI.2.4.1 Construction of the spaceLet I â I(V ) be a cover of contexts. We will construct a space out of thiscover. For this the language of simplicial sets, a generalization of simplicialcomplexes, is most suitable. A simplicial set consists of a sequence of setsX0, X1, · · · indexed by natural numbers together with face and degeneracymaps, see [79]. Each set Xn is referred to as the set of n-simplices. Facemaps reduce the dimension di : Xn â Xnâ1, whereas the degeneracy mapsincrease the dimension sj : Xn â Xn+1. This structure is enough to obtaina topological space, and the procedure is called the geometric realization|Xâ¢| =ânâ¥0Xn Ãân/ â¼where the identification â¼ is generated by the face and the degeneracy maps.In effect, each combinatorial n-simplex is replaced by the topological simplexân = { (t0, · · · , tn) â Rn+1 |nâi=0ti = 1, ti â¥ 0 }and glued to each other using the face and the degeneracy maps.The standard example of a simplicial set is the nerve of a category. A(small) category comes with a set of objects and a set of morphisms. Let Cbe a category. The nerve of C is the simplicial set whose set of n-simplicesconsists of composable arrows of the formA0 â · · · â An.We denote the nerve by N(C). Face maps correspond to composing twoadjacent morphisms, and degeneracy maps are given by inserting an identitymorphism.Definition 2.4.1. The classifying space for contextuality, denoted byBcx(I),is the geometric realization of the simplicial set Bcx(I)â¢ whose set of n-simplices is given byBcx(I)n = { (v1, v2, · · · , vn) | vi â I, 1 â¤ i â¤ n, for some I â I }57with the simplicial structure mapsdi(v1, · · · , vn) =ï£±ï£²ï£³(v2, · · · , vn) i = 0((v1, · · · , vi + vi+1, · · · , vn)) 0 < i < n(v1, · · · , vnâ1) i = nand si(v1, · · · , vn) = (v1, · · · , viâ1, 0, vi, · · · , vn) for 0 â¤ i â¤ n.Remark 2.4.2. This construction is analogous to a construction introducedin [68]. For a group G one can construct a classifying space Bcom(G) forcommutativity. When G is abelian the construction recovers the usual clas-sifying space BG. A similar definition works for an arbitrary collection F ofsubgroups of G which is closed under taking intersections. We can define aspace from the set of n-tuple of elements which belong to a subgroup in F .Taking F to be the collection of all abelian subgroups of G gives Bcom(G).Homotopy theoretic properties of these spaces are studied in [70â72].Note that the construction is a variation of the standard construction of aclassifying space for a group. The set of n-simplices of Bcx(I) are containedin the set of n-simplices, which is given by the n-fold direct product V n, ofthe classifying space BV . The simplicial structure of BV is defined similarly.We denote the inclusion map byι : Bcx(I)â BV.Remark 2.4.3. The name âclassifying spaceâ refers to the fact that BGclassifies principal G-bundles [80]. More precisely, given a CW-complex Xthe set [X,BG] of homotopy classes of maps is in one-to-one correspondencewith the set of isomorphism classes of principal G-bundles over X. SimilarlyBcomG is a classifying space for certain types of principal bundles. Thesebundles have the property that the transition functions commute wheneversimultaneously defined [69]. A similar conclusion can be made for BcxI.This space classifies principal V -bundles whose transition functions, when-ever simultaneously defined, specifies a context. We denote the equivalenceclasses of such bundles by PrinVcx(X).2.4.2 Homotopy theoryThe space BcxI is studied in [72]. Formally, there is a structural similarityto the space BcomG (see 2.4.2), which has been studied in [70, 71], and mostof the tools developed there also applies to BcxI. We sketch the relevanthomotopical properties whose proofs can be found in these papers.58By construction Bcx(I) can be expressed as the union of classifyingspacesBcx(I) =âIâIBI.This can be seen from the description of n-simplices. We can also thinkof the union as a colimit in the category of spaces. To study homotopytheoretic properties the union is usually replaced by a homotopy colimit[81]âa gluing process that has nice homotopical properties. We can definea functor B : I â Top that sends I to the classifying space BI. There is anatural maphocolimIB â Bcx(I)which turns out to be a homotopy equivalence, see [70]. This approach alsohelps us to compute cohomology of the space by using a spectral sequence.Cohomology (or even generalized cohomology) of a homotopy colimit canbe computed using a spectral sequence.Another benefit that we gain by switching to homotopy colimits is aformula for the fundamental grouppi1(BcxI) = ãev, v â Σ| evevâ² = ev+vâ² if { v, vâ² } â I for some I â Iã.(2.11)We will denote the fundamental group by pi(I), or just by pi if the contextis understood. Note that each isotropic subspace I â I can be regarded asa subgroup of pi by sending an element u to the corresponding generator eu.Indeed, there is a set map Σ â pi. Moreover, this map induces a map ofspacesÏ : BcxI â Bpithat is defined by sending an n-simplex (v1, · · · , vn) to the tuple (ev1 , · · · , evn).It is an interesting fact that Ï is not a homotopy equivalence in general. Inother words, the higher homotopy groups pii(BcxI), i â¥ 2, can be non-trivialin contrast to Bpi whose homotopy groups are zero except pi1(Bpi) = pi.Example 2.4.4. Let us explain the simplest case of n = 1. Let X â¨ Ydenote the wedge sum of the spaces X and Y at a specified basepoint. Thehomotopy colimit decomposition for n = 1 is simply given byBcxI(V ) 'rpâ¨BZ/p.where rp = (p2n â 1)/(pâ 1) is the number of maximal isotropic subspaces.59From the homotopical point of view all the information about the higherhomotopy groups is hidden in the homotopy fiber of Ï. It turns out that thehomotopy fiber has a very concrete description in terms of a combinatorialobject called the coset poset.2.4.3 Coset posetWe introduce this space for an arbitrary group G.Definition 2.4.5. For a collection of subgroups F of G we define the posetCG(F) = { gA | A â F , g â G }ordered by the inclusion relation.We usually identify the poset with the associated space given by thegeometric realization of its nerve. Our convention for the morphisms ofthe category associated to a poset P is that we write A â B if A â¤ B.The associated space has n-simplices given by composable morphisms A1 â· · · â An.Given this construction and the fundamental group pi(I) of Bcx(I) wecan describe the homotopy fiber of Ï. Each subspace in the context I canbe regarded as subgroup of pi thus we can construct the coset poset Cpi(I).Theorem 2.4.6. [72] (OS 2019) Consider I = I(V ) where V = (Z/p)2nand n â¥ 2. The fundamental group pi1(BcxI) is given by the central extensionassociated to the cocycle bpi â¼= V Ãb Z/pand the higher homotopy groups are given bypii(BcxI) â¼= pii(d(p,n)â¨Sn)for all i â¥ 2, where the number of spheres is given by the formulad(p, n) = (â1)r+1 + p2r+1+r2 +râj=1(â1)jp2r+1âj+(râj)2(jâ1ât=0p2rât â ptpj â pt)Remark 2.4.7. We will see some applications of this result. However, wedo not know of an interpretation for the numbers d(p, n). This number60gives information about the global structure of the classifying space forcontextuality. Interpreted in terms of principal bundles we havePrinVcx(Sn) = Zd(p,n)where we used the notation of Remark 2.4.3. On the other hand, The setPrinV (Sn) of equivalence classes of ordinary principal bundles over the n-sphere, n â¥ 2, has a single element given by the trivial bundle.Remark 2.4.8. A better way to rephrase Theorem 2.4.6 is to say that forn â¥ 2 there is a fibration sequenced(p,n)â¨Sn â Bcx(I) Ïââ Bpi. (2.12)To see that the fiber is a wedge of spheres we refer the readers to the generalmethodology of [70],[72]. Moreover, the group pi turns out to beV Ãb Z/p â¼={V Ã Z/2 p = 2Pn p > 2as a consequence of the presentation given in 2.11.2.4.4 Cohomology revisitedHaving the homotopical description ofBcx(I) we can improve our statementsabout cohomology. The first application is to determine the extra factor inH1(BcxI) when p = 2.Corollary 2.4.9. Assume n â¥ 2. There is an isomorphismH1(BcxI) â¼={(V Ã Z/2)â p = 2V â p > 2where (â)â stands for the dual vector space.Proof. This follows form the description of the fundamental group pi, andthe standard fact that for a connected space X the first homology group isthe abelianization of the fundamental group [82].The Serre spectral sequence [82] of the fibration 2.12 gives us an isomor-phismÏâ : H i(Bpi)â H i(BcxI)61in degrees i â¤ n â 1 and an injection in degree i = n. When p = 2 wehave pi = V Ã Z/2 as observed in Remark 2.4.8, so its cohomology is easyto describe. Fixing the basis {xi, zi | 1 â¤ i â¤ n } âª {x0 } for V Ã Z/2 thecohomology ring is given by the polynomial ringHâ(Bpi) = F2[xâ0, xâ1, · · · , xân, zâ1 , · · · , zân].Let us define a homomorphismq : pi â V Ã Z/2 (v, t) 7â (v, t+ q(v))which turns out to be an isomorphism of groups. Now consider the compositemap of spacesqË : BcxI Ïââ Bpi Bqââ B(V Ã Z/2)As a consequence Corollary (2.4.9) we have:Corollary 2.4.10. The cohomology class [β] â H2(BcxI) is the image ofQ = xâ0 +nâi=1xâi âª zâi â H2(B(V Ã Z/2))under the map Hâ(B(V Ã Z/2))â Hâ(BcxI) induced by qË.2.5 Contextuality and twisted K -theoryThe goal of this section is to compute the β-twisted K -theory of BcxI. Theidea follows the computation of the topological K -theory of BcomG givenin [70]. We refer to this paper for the details of the constructions involved.Another reference that is useful for the homotopical properties of this spaceis [72]. The computation will bridge a connection to the representationtheoretic description of the Wigner function given in §2.3. This will allowus to interpret the Wigner function as a class [WÏ] in the β-twisted K -theoryof BcxI(V ) after extending the coefficients to R.Remark 2.5.1. Another form of K -theory, namely operator K -theory, hasbeen used in the study of contextuality by De Silva and Barbosa [83].Therein the authors relate operator K -theory to topological K -theory viathe Gelfand spectrum. However, there seems to be no direct connection tothe way K -theory appears in the present work.622.5.1 Sketch of the computationBefore embarking in the details, let us first summarize the idea of the proof.As an element of the inverse limit of RâRÎ² the Wigner function is a sumWÏ =âvâVWÏ(v) bvof 1-dimensional irreducible representations of the group V . Such a sum isusually referred to as a âvirtualâ sum when the coefficients assume nega-tive values. In this section we will explain a chain of ideas, due to Atiyahand Segal, that will allow us to pass from twisted representations to twistedK -theory. This is given by the âcompletionâ of the representation group inthe sense that Cauchy sequences are convergent in the resulting completedmodule. In the space level the completion process corresponds to a construc-tion known as the Borel construction. Let G be a group and let EGâ BGdenote the universal principal G-bundle [48, Chapter II]. For a space X withan action of G the Borel construction is defined to be the spaceEGÃG X = (EGÃX)/Gobtained as the quotient under the diagonal action. The AtiyahâSegal com-pletion theorem [84] says that the topological K -theory group K(EGÃGX)is the completion of the equivariant K -theory group KG(X). The equivari-ant K -theory group is a generalization of the representation group. WhenX is a point with the trivial G-action then KG(X) = R(G). All of this isalso true in the twisted setting [85]. We are only interested in twisting bythe cocycle β.In our case we take X to be the coset poset CpiI with the action of pi givenby left multiplication on the cosets. The Borel construction gives us BcxI,as shown in [72]. Therefore by the AtiyahâSegal completion theorem twistedK -theory of BcxI is the completion of the twisted equivariant K -theory ofthe coset poset. There are two steps of the computationâ¢ After extending the coefficients to R we show that K βpi (CpiI) is equalto the inverse limit of R â Rβ. Thus WÏ belongs to the β-twistedequivariant K -theory group.â¢ Completing K βpi (CpiI) we obtain K β(BcxI) and after extending coeffi-cients to R we conclude that the Wigner function can be regarded asa class [WÏ] in the β-twisted K -theory group.632.5.2 Twisted equivariant K -theoryWe can twist the equivariant K -theory groups by the cohomology class [β] âH2(BcxI(V )). Recall that [β] = 0 when p is odd. Thus in this case thetwisting disappears. For p = 2 we observed in Corollary 2.4.10 that [β]comes from a class in H2(Bpi,Z/2). We will treat the two cases in a uniformway. Therefore we think of [β] as living in H2(Bpi,Z/p) given by the classQ when p = 2, and understood to be zero when p > 2. We assume n â¥ 2 sothat pi is a finite group (Remark 2.4.8). The case n = 1 can be computeddirectly, and we defer this case to the proof of Theorem 2.5.4.As observed in [76] twisted equivariant K -group can be obtained fromthe untwisted version. To see this let us consider the group extension0â Z/pâ piβ â pi â 0corresponding to [β]. Note that the coset poset CpiI can be regarded as apiβ-space via the quotient homomorphism piβ â pi. Irreducible representionsof Z/p consist of the homomorphisms Ïj : Z/pâ U(1) sending the additivegenerator 1 to the element Ïj where 0 â¤ j â¤ pâ 1. Then we haveK âpiÎ² (CpiI) =pâ1âj=0K âpiÎ² (CpiI)(Ïj) (2.13)where the factors correspond to equivariant bundles on which Z/p acts bythe specified representation. We can identify the twisted equivariant groupsK Î²+âpi (CpiI) with the factor corresponding to the j = 1 representation:K Î²+âpi (CpiI) = K âpiÎ² (CpiI)(Ï).Note that this representation corresponds to the usual embedding Z/p âU(1).Equivariant K -theory of the coset poset can be computed using the meth-ods of [70]. The main tool is a spectral sequence that can be applied tohomotopy colimit decompositions. As shown in [72] the coset poset is thehomotopy colimit of the functor pi/â : I â Top which sends I to thecoset pi/I regarded as a discrete space. Furthermore, as a consequence of2.13 the spectral sequence decomposes and we can use it to compute thetwisted equivariant K -groups. Therefore the results for the twisted versionare essentially obtained in the same way as the untwisted case.Before discussing the spectral sequence we need a construction that ex-tends the construction of the inverse limit of a functor. Let F : Iop â Ab64be a functor that takes values in the category of abelian groups. One canconstruct a cochain complex Câ(I, F ) which is indexed over the n-simplicesof the nerve of the poset I:Cn(I, F ) =âA1â·Â·Â·âAnF (A1)and the boundary map comes from the simplicial structure of the nerve of I.The cohomology group H i(I, F ) gives the i-th derived functor of the inverselimit of F . Note that H0(I, F ) is canonically isomorphic to the inverse limitof F . We can also think of the groups H i(I, F ) as the CËech cohomologygroups of the presheaf F with respect to the cover I.For homotopy colimits we can use the (equivariant) Bousfield-Kan spec-tral sequenceâthe spectral sequence used in [70] for the untwisted case. Asa consequence of the direct sum decomposition in 2.13 the spectral sequencedecomposes:Ei,j2 = Hi(I,K β+jpi (pi/â)) =â K Î²+âpi (CpiI).The functor K β+jpi (pi/â) can be identified with the β-twisted representationgroup functor Rβ : Iop â Ab for j even and it is zero otherwise. This isthe key connection to §2.3.Remark 2.5.2. We would like to remind that [β] = 0 for the odd primecase (Remark 2.3.2). Therefore the twisting disappears in this case, and weare dealing with untwisted K -groups.We are interested in understanding the spectral sequence after tensoringwith R. The following serves to this purpose.Lemma 2.5.3. The derived limit functor H i(I, Rβ) is torsion for i > 0.Proof. This follows from [70, Theorem 4.7] once we show Rβ is a pre-Mackeyfunctor. Restriction and induction of twisted characters provide the neces-sary structure [76, Theorem 4.2].As a consequence of this result our spectral sequence collapses aftertensoring with R. Here we are also using the fact that the spectral sequenceis bounded in the i-directionâall the terms vanish for sufficiently large i.For i > 0 the terms RâEi,j2 are zero since Ei,j2 are torsion as proved above.When i = 0 and j even we have R â E0,j2 = H0(I,R â Rβ), and for j oddRâ E0,j2 = 0. Recall that H0(I,RâRÎ²) can be identified with the inverse65limit of the functor R â Rβ. For I = I(V ) we have computed this inverselimit in Proposition 2.3.5. Thus we obtainRâK β+ipi (CpiI) â¼={RâR(V ) i = 00 i = 1.(2.14)Therefore we can take the target of the function W in 2.10, that sends astate Ï to the Wigner function WÏ, to be the twisted equivariant K -theorygroup:W : Den(H)â RâK βpi (CpiI(V )). (2.15)We can express this by saying that a state Ï can be interpreted as a prob-abilistic combination of pi-equivariant β-twisted vector bundles over CpiI.Our next goal is to relate this to the twisted K -theory of the classifyingspace for contextuality.2.5.3 Twisted K -theoryThe twisted version [85] of Atiyah-Segal completion theorem gives an iso-morphismK Î²+â(BcxI) â¼= K Î²+âpi (CpiI)â§where (â)â§ means completion with respect to the augmentation ideal I(pi).The augmentation ideal is defined to be the kernel of the homomorphismR(pi)â Z which sends a representation to its dimension.Let us explain the completion process. First of all K Î²+âpi (CpiI) is a mod-ule over the representation ring R(pi). This is the case since K âpiÎ² (CpiI) isa module over the ring R(piβ) via the map CpiI â â which contracts thecoset poset to a point, and using the ring map R(pi) â R(piβ) induced bythe quotient homomorphism we see that the β-twisted K -group is a moduleover R(pi). Then the completion can be computed by tensoring with thecompletion of R(pi):K Î²+âpi (CpiI)â§ â¼= K Î²+âpi (CpiI)âR(pi) R(pi)â§.Therefore it suffices to understand the completion of R(pi). Recall fromRemark 2.4.8 that pi is a p-group. It is well-known that for p-groups com-pletion of the representation ring turns out to be essentially tensoring withthe p-adic numbers Zp:R(pi)â§ â¼= Zâ (Zp â I(pi)).66Theorem 2.5.4. (OS 2019) There is an isomorphismRâK β+i(BcxI(V )) â¼={Râ(Râ Z|V |â1p)i = 00 i = 1.Proof. The proof follows the proof of the untwisted version in [70, Theorem5.2]. For n = 1 the result is a direct consequence of the decomposition inExample 2.4.4. Assume n â¥ 2. The order of completion and tensoring withR matters. We first complete then tensor with R. Lemma 2.5.3 says thatfor i > 0 the groups Ei,j2 are torsion. Under the completion process, whichis essentially tensoring with Zp, these groups do not change. After tensoringwith R they disappear. Therefore we focus on i = 0 part. Completion ofH0(I, Rβ) gives us ZâZ|V |â1p (see Remark 2.3.6). Finally tensoring with Rgives the desired result.As a consequence of this result we can think of the W map as landinginside twisted K -theoryW : Den(H)â RâK β(BcxI(V )).Note that the completion mapK βpi (CpiI(V ))â K β(BcxI(V ))is injective since tensoring with Zp is. This means that we do not loseany information during the completion process and can identify the Wignerfunction WÏ as a class in the β-twisted K -group.67Chapter 3Conclusions3.0.1 On K -theory and Fermi surfacesFor chapter 1 we have rewritten [2] and clarified points, particularly of anal-ysis, such as what was meant by Bulk and Surface states in terms of pro-jections and employed the correct terminology of quasi-adiabatic evolutionequivalence relation. We have also argued that the choice of topology for un-bounded self-adjoint Fredholm operators which was discussed in [16] to be ared harring. Furthermore we have extended our results to the twisted equiv-ariant case for most crystallographic groups. What remains to be done infuture work on the crystallographic side is the computation of these groupsK Ïâ1P (Tdâ1) using an equivariant version of the Atiyah-Hirzebruch spectralsequence and multiplying in there by the class of the twist to take it intoaccount. We also need to extend to the case of non-trivial magnetic pointgroup. On the global symmetry side we must include all the classes in theAltland-Zirnbauer classification that we are missing, which are those withchiral symmetry (classes BDI, CI, CII, DIII). On the analytical side a properanalytical formulation of quasi-adiabaticity, when the systems in questionposses a gapped bulk would be an interesting task for analysts.3.0.2 On K -theory and contextualityOn chapter 2 we have presented the essential parts of [86] to show thatadapting the formulation of Abramsky et al [62], which is designed for gen-eral resource theories, to the formalism of [63], that is specific to finitedimensional quantum mechanics and observables for quantum computation,leads to reinterpreting any density matrix as a class in the twisted K -groupKÎ²(BcxI)âR. Motivated by this connection to physics we computed it forthe case I = I(V ). Though there is a natural interpretation that relates thealgebraic topology of BcxI to state-independent contextuality and Wignerfunctions, we do not as of yet have employed the K -theory interpretationto address a specific physical problem such as state-dependent contextual-ity. It is vital to continue the search for such an application to consolidatethe mathematical relevance of BcxI as a unifying object in contextuality for68quantum mechanics. Nonetheless from a mathematical point of view, thesespaces are interesting as they posses non-trivial higher homotopy, unlikethe classifying space which is an Eilenberg-Maclane space. Furthermore onemight use this mathematical richness as a motivation to look for new phys-ical phenomena related to these groups. From a mathematical perspectiveit would be interesting to compute more examples when I 6= I(V ).3.0.3 Connections between the two topicsBoth chapters employ twisted equivariant K -theory to reinterpret aspects inquantum mechanics (Fermi surfaces on a half crystal and density matrices onfinite dimensional Hilbert spaces). However it seems so far that these arisein a very different manner. In the former the context is infinite dimensionalHilbert spaces and it is through the dynamics (the Hamiltonian) which isan unbounded operator and its spectral decomposition, that one arrives atK-theory, without any reference to contexts. On the latter it is throughfinite dimensional representation theory and the higher dimensional torsionof a spectral sequence that one can reinterpret density matrices as classes ina K-theory group, and no connection to analysis was made nor any referenceto dynamics. However a generalization to non-interacting Fermionic topo-logical phases are so called SPT phases [15], [87], [88] these are supposedto be generalizations of topological phases that include short range entan-glement. Though a topological theory for them has been presented [87],[88] in terms of invertible topological quantum field theories, these rely on arelativistic field interpretation, which from the physical point of view seemsunwarranted given that it requires including extra symmetries (the Poincare´group), though there are a few exceptions[14]. Furthermore this topologicalconstruction has not been derived from an analytical point of view as in[16], [17], [33] and the present work. Non the less in the physics literaturethere are some examples of systems considered to be Bosonic SPT phaseswhich have been shown to have non-trivial quantum computational power[89]. Due to the relation between quantum computation and contextuality,this would suggest that a form of contextuality is hidden in Bosonic SPTphases since it is believed to be the necessary ingredient for non-trivial quan-tum computation. 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Applications and connections between twisted equivariant K-theory, quantum mechanics and condensed matter
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