Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Wamer, Kyle Patrick
Doctor of Philosophy - PhD
University of British Columbia
In this thesis, we study the low energy properties of SU(n) chains in various representations. We are motivated by Haldane's conjecture about antiferromagnets, namely that integer spin chains exhibit a finite energy gap, while half-odd integer spin chains have gapless excitations. Haldane was led to this conclusion by deriving a sigma model description of the antiferromagnet, and this is what we generalize here to SU(n). We find that most representations of SU(n) admit a mapping to a sigma model with target space equal to the complete flag manifold of SU(n). These theories are not automatically relativistic, but we show that at low energies, their renormalization group flow leads to Lorentz invariance. We also show explicitly in SU(3) that the theory is asymptotically free, and contains a novel two-form operator that is relevant at low energies. For all n, these sigma models are equipped with n-1 topological angles which depend on the SU(n) representation at each site of the chain. For the rank-p symmetric representations, which generalize the spin representations of the antiferromagnet, these angles are all nontrivial only when gcd(n,p)=1. This observation, together with recent 't Hooft anomaly matching conditions, and various exact results known about SU(n) chains, allow us to formulate the following generalization of Haldane's conjecture to SU(n) chains in the rank-p symmetric representation: When p is coprime with n, a gapless phase occurs at weak coupling; for all other values of p, there is a finite energy gap with ground state degeneracy equal to n/gcd(n,p). We offer an intuitive explanation of this behaviour in terms of fractional topological excitations. We also predict a similar gapless phase for two-row representations with even n. The topological content of these chains is the same as the symmetric ones, with p now equal to the sum of row lengths of the representation. Finally, we show that the most generic representation of SU(n) will admit a sigma model with both linear and quadratic dispersion; such theories requires further understanding before their low energy spectra can be characterized.
Generalization of the Haldane conjecture toSU(n) chainsbyKyle Patrick WamerB.Sc., The University of Toronto, 2016M.Sc., The University of British Columbia, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2021c© Kyle Patrick Wamer 2021The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Generalization of the Haldane conjecture to SU(n) chainssubmitted by Kyle Patrick Wamer in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Physics.Examining Committee:Ian Affleck, Physics and Astronomy, UBCSupervisorSven Bachmann, Mathematics, UBCUniversity ExaminerMona Berciu, Physics and Astronomy, UBCUniversity ExaminerFei Zhou, Physics and Astronomy, UBCSupervisory Committee MemberPhilippe Lecheminant, Universite´ de Cergy-PontoiseExternal ExaminerAdditional Supervisory Committee Members:Marcel Franz, Physics and Astronomy, UBCSupervisory Committee MemberKirk Madison, Physics and Astronomy, UBCSupervisory Committee MemberiiAbstractIn this thesis, we study the low energy properties of SU(n) chains in various representations. We aremotivated by Haldane’s conjecture about antiferromagnets, namely that integer spin chains exhibita finite energy gap, while half-odd integer spin chains have gapless excitations. Haldane was ledto this conclusion by deriving a sigma model description of the antiferromagnet, and this is whatwe generalize here to SU(n). We find that most representations of SU(n) admit a mapping to asigma model with target space equal to the complete flag manifold of SU(n). These theories are notautomatically relativistic, but we show that at low energies, their renormalization group flow leadsto Lorentz invariance. We also show explicitly in SU(3) that the theory is asymptotically free, andcontains a novel two-form operator that is relevant at low energies.For all n, these sigma models are equipped with n − 1 topological angles which depend on theSU(n) representation at each site of the chain. For the rank-p symmetric representations, whichgeneralize the spin representations of the antiferromagnet, these angles are all nontrivial only whengcd(n, p) = 1. This observation, together with recent ’t Hooft anomaly matching conditions, andvarious exact results known about SU(n) chains, allow us to formulate the following generaliza-tion of Haldane’s conjecture to SU(n) chains in the rank-p symmetric representation: When p iscoprime with n, a gapless phase occurs at weak coupling; for all other values of p, there is a finiteenergy gap with ground state degeneracy equal to n/ gcd(n, p). We offer an intuitive explanation ofthis behaviour in terms of fractional topological excitations.We also predict a similar gapless phase for two-row representations with even n. The topolog-ical content of these chains is the same as the symmetric ones, with p now equal to the sum ofrow lengths of the representation. Finally, we show that the most generic representation of SU(n)will admit a sigma model with both linear and quadratic dispersion; such theories requires furtherunderstanding before their low energy spectra can be characterized.iiiLay SummaryThis thesis classifies new types of magnetic materials. Magnets are made up of atoms that, inaddition to having a mass and electric charge, also have a property called spin. The spin of amagnet can be thought of as a microscopic arrow: it has a fixed length, and points in a particulardirection. In familiar fridge magnets, all of these spins point along the same direction, and this iswhat causes them to be attracted to metallic surfaces. In this thesis, we consider magnetic materialsthat are different in two ways: 1) instead of a microscopic arrow, the spin is now a more complicatedshape, 2) instead of aligning in the same direction, the spins now point in opposite directions, anddo not stick to metals. For these new types of magnets, we classify for the first time how differentspin shapes lead to different phases of matter.ivPrefaceThe contents of this thesis were published in the following five articles:• [1] Miklo´s Lajko´, Kyle Wamer, Fre´de´ric Mila, and Ian Affleck. Generalization of the Haldaneconjecture to SU(3) chains. Nuclear Physics B, 924:508 – 577, 2017• [2] Kyle Wamer, Francisco H. Kim, Miklo´s Lajko´, Fre´de´ric Mila, and Ian Affleck. Self-conjugate representation SU(3) chains. Phys. Rev. B, 100:115114, 2019• [3] Kyle Wamer, Miklo´s Lajko´, Fre´de´ric Mila, and Ian Affleck. Generalization of the Haldaneconjecture to SU(n) chains. Nuclear Physics B, 952:114932, 2020• [4] Kyle Wamer and Ian Affleck. Mass generation by fractional instantons in SU(n) chains.Phys. Rev. B., 101:245143, 2020• [5] Kyle Wamer and Ian Affleck. Flag manifold sigma models from SU(n) chains. NuclearPhysics B, 959:115156, 2020The first three articles were cowritten with the Mila group at E´cole Polytechnique Fe´de´rale deLausanne, and all five articles were written with my supervisor, Ian Affleck. The first publication,[1], constitutes most of Chapter 4, and is the only article from the above list in which I was not theprimary writer. My main contribution to [1] was the renormalization group calculations appearingin Section 4.5. The flavour-wave calculations in Section 4.2 were carried out by my collaboratorsbefore I joined the project. And while the original sigma model derivation in [1] was worked out byAffleck, Lajko´ and Mila, the derivation appearing in Section 4.3 is my own, and takes the differentapproach of varying the matrix degrees of freedom at each site of the chain, instead of over the 3-site unit cell. The advantage of this method is that the topological terms do not receive contributionsfrom the uniform fluctuation fields, Lαβ, as in [1], which is important when one extends to moregeneral representations of SU(n). I also used this approach to present an original derivation of theCP1 sigma model from an antiferromagnet in Chapter 2The contents of [2] appear at the end of Chapter 4. Excluding the flavour-wave calculations,which were done by the Mila group, everything else presented in Section 4.7 was led and writtenby me.The third publication, [3], was written entirely by me, with the exception of the strong cou-pling analysis appearing in Section 5.5.1. Indeed, the SU(n) flavour-wave theory (Section 5.2), thederivation of the flag manifold sigma model (Section 5.3), and the renormalization group analysisvPreface(5.4) were all my calculations. The final two articles listed above, [4] and [5], constitute Chapter 3,Chapter 6 and Chapter 7. I wrote these two articles in their entirety, and carried out all of thecalculations.Finally, the content of these five publications can also be found in the review article [6], whichI cowrote with Ian Affleck and Dmitri Bykov. The general discussions appearing in Chapter 1 andChapter 8 are also included in this review.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A Review of the Haldane Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Spin-wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Sigma Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Coherent state path integral . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Properties of the Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Topological angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 The Lieb-Schultz-Mattis Theorem . . . . . . . . . . . . . . . . . . . . . . 162.4.2 The Affleck-Kennedy-Lieb-Tasaki model . . . . . . . . . . . . . . . . . . 162.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 SU(n) Chain Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 SU(n) Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Classical Ground States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22viiTable of Contents3.3.1 Pictorial representation of classical ground states . . . . . . . . . . . . . . 243.4 Classification of SU(n) Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.1 Representations with n − 1 rows . . . . . . . . . . . . . . . . . . . . . . . 273.4.2 Representations with row number dividing n . . . . . . . . . . . . . . . . 293.4.3 Remaining representations . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.4 Summary of classification . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Exact Results for SU(n) Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5.1 Integrable SU(n) chains and SU(n)k critical points . . . . . . . . . . . . . 333.5.2 The Lieb-Schultz-Mattis-Affleck theorem . . . . . . . . . . . . . . . . . . 353.5.3 Affleck-Kennedy-Lieb-Tasaki models . . . . . . . . . . . . . . . . . . . . 363.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 SU(3) Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Flavour-Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Derivation of the Flag Manifold Sigma Model . . . . . . . . . . . . . . . . . . . . 454.3.1 Coherent state path integral . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.2 Complete field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Properties of the Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.1 The flag manifold as coupled CP2 sigma models . . . . . . . . . . . . . . 524.4.2 Gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4.3 Z3 symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4.4 Other discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Renormalization Group Calculations . . . . . . . . . . . . . . . . . . . . . . . . 554.5.1 A primer on renormalization group theory . . . . . . . . . . . . . . . . . 554.5.2 Renormalization of the SU(3) chain . . . . . . . . . . . . . . . . . . . . . 574.5.3 Integration over a Wilson shell . . . . . . . . . . . . . . . . . . . . . . . 614.5.4 Results of the RG calculations . . . . . . . . . . . . . . . . . . . . . . . . 644.6 Phase Diagram of the Symmetric SU(3) Chain . . . . . . . . . . . . . . . . . . . 654.6.1 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.6.2 Strong coupling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.6.3 Wess-Zumino-Witten models and the explicit breaking of Z3 . . . . . . . 714.6.4 ’t Hooft anomaly matching . . . . . . . . . . . . . . . . . . . . . . . . . 724.7 Other SU(3) Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.7.1 Classical ground states and mixed order parameters . . . . . . . . . . . . 744.7.2 Flavour-wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7.3 Low energy field theory and the absence of Lorentz invariance . . . . . . . 764.7.4 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78viiiTable of Contents5 Symmetric SU(n) Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 SU(n) Chains and Flavour-Wave Theory . . . . . . . . . . . . . . . . . . . . . . . 805.2.1 Flavour-wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Derivation of the Flag Manifold Sigma Model . . . . . . . . . . . . . . . . . . . . 825.3.1 The complete field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.3 Embedding into complex projective spaces . . . . . . . . . . . . . . . . . 855.3.4 Translation invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Renormalization Group Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4.1 Goldstone mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . 875.4.2 Dimensional regularization . . . . . . . . . . . . . . . . . . . . . . . . . 905.4.3 Renormalization group equations . . . . . . . . . . . . . . . . . . . . . . 915.4.4 Beta functions of the Goldstone velocities . . . . . . . . . . . . . . . . . 935.4.5 Renormalization of velocity differences . . . . . . . . . . . . . . . . . . . 945.5 General Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5.1 Strong coupling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5.2 ’t Hooft anomaly matching . . . . . . . . . . . . . . . . . . . . . . . . . 985.5.3 A Haldane conjecture for SU(n) chains . . . . . . . . . . . . . . . . . . . 995.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006 Flag Manifold Sigma Models From Other Representations of SU(n) . . . . . . . . . 1026.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.2 Flag Manifold Sigma Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3.1 Generalizing the coherent state path integral construction . . . . . . . . . 1056.3.2 Representations with 1 row . . . . . . . . . . . . . . . . . . . . . . . . . 1076.3.3 Representations with n − 1 rows . . . . . . . . . . . . . . . . . . . . . . . 1076.3.4 Representations with row number dividing n. . . . . . . . . . . . . . . . . 1086.3.5 Remaining representations . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.3.6 Summary of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.4 Topological Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.4.1 Representations with one row . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.2 Representations with n − 1 rows . . . . . . . . . . . . . . . . . . . . . . . 1146.4.3 Representations with row number dividing n . . . . . . . . . . . . . . . . 1146.4.4 Remaining representations . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.5 A New Generalization of Haldane’s Conjecture . . . . . . . . . . . . . . . . . . . 1216.5.1 Representations that admit transitive Zn actions . . . . . . . . . . . . . . 1226.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123ixTable of Contents7 Fractional Instantons and the Haldane Gap in SU(n) Chains . . . . . . . . . . . . . 1257.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2 Mass Generation in the O(3) Nonlinear Sigma Model . . . . . . . . . . . . . . . . 1267.2.1 The Kosterlitz-Thouless transition . . . . . . . . . . . . . . . . . . . . . . 1267.2.2 Mass generation in the CP1 sigma model . . . . . . . . . . . . . . . . . . 1307.2.3 The breakdown of mass generation at θ = pi. . . . . . . . . . . . . . . . . 1337.3 Reducing the Global Continuous Symmetry to U(1) . . . . . . . . . . . . . . . . 1337.4 Mass Generation by Fractional Instantons . . . . . . . . . . . . . . . . . . . . . . 1357.4.1 Example: An explicit fractional instanton in SU(3) . . . . . . . . . . . . . 1367.4.2 Multiple copies of the sine-Gordon model . . . . . . . . . . . . . . . . . 1377.5 Destructive Interference in the Presence of Topological Angles . . . . . . . . . . . 1377.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145AppendicesA Proof of tr[U∂U†] = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156B Low Energy Expansion of the Symmetric SU(n) Hamiltonian . . . . . . . . . . . . . 157C Factorization of SU(n) Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159D Polyakov’s Renormalization of the O(n) nonlinear Sigma Model . . . . . . . . . . . 161D.1 Renormalization in the Real-Space Vector Language . . . . . . . . . . . . . . . . 161D.2 Renormalization in the Flag Manifold Language . . . . . . . . . . . . . . . . . . 163E Identities Involving fABC and TA in SU(3) . . . . . . . . . . . . . . . . . . . . . . . . 165E.1 Structure Factor Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165E.1.1 Identity 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166E.1.2 Identity 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166E.1.3 Identity 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167E.1.4 Identity 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167E.2 A Rewriting of the Sigma Model Lagrangian . . . . . . . . . . . . . . . . . . . . 170F Two Identities Involving SU(n) Structure Factors . . . . . . . . . . . . . . . . . . . 173F.1 Identity 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173F.2 Identity 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174xTable of ContentsG Dispersion Relations For SU(n) Chains . . . . . . . . . . . . . . . . . . . . . . . . . 176G.1 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176G.2 Topological Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178xiList of Tables3.1 Classification of ground states of SU(n) chains. . . . . . . . . . . . . . . . . . . . 324.1 A Haldane conjecture for SU(3) chains. . . . . . . . . . . . . . . . . . . . . . . . 786.1 Copy of Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Topological angles for representations with n − 1 rows. . . . . . . . . . . . . . . . 1146.3 Topological angles for representations with k = nλ rows . . . . . . . . . . . . . . . 1156.4 Topological angles for representations of SU(n), with n = kλ + 1, with k > 2, λ odd. 1176.5 Topological angles for representations of SU(n) with n = kλ + c, with c , 1, k − 1. . 1196.6 Topological angles for representations of SU(n) with n = kλ + (λ − 1). . . . . . . . 1206.7 SU(n) representations with a gapless phase. . . . . . . . . . . . . . . . . . . . . . 124xiiList of Figures2.1 Phase diagram of the antiferromagnetic spin chain. . . . . . . . . . . . . . . . . . 162.2 The AKLT ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 SU(n) Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Representations of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Colour dictionary for SU(n) ground states. . . . . . . . . . . . . . . . . . . . . . . 243.4 Energy costs of SU(n) ground states . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Ground state of an SU(7) chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 AKLT state for a symmetric SU(3) chain. . . . . . . . . . . . . . . . . . . . . . . 373.7 Degenerate AKLT states in an SU(4) chain. . . . . . . . . . . . . . . . . . . . . . 373.8 AKLT state for a self-conjugate SU(3) chain with p = 1. . . . . . . . . . . . . . . 383.9 AKLT state for a self-conjugate SU(3) chain with p = 2. . . . . . . . . . . . . . . 383.10 AKLT state of an SU(4) chain with p1 = 3, p2 = 1. . . . . . . . . . . . . . . . . . 394.1 Phase diagram of the symmetric SU(3) chain. . . . . . . . . . . . . . . . . . . . . 664.2 Monte Carlo simulations of the SU(3) chain. . . . . . . . . . . . . . . . . . . . . . 684.3 Monte Carlo simulations of the SU(2) chain. . . . . . . . . . . . . . . . . . . . . . 694.4 Free energy of the SU(3) chain at strong coupling. . . . . . . . . . . . . . . . . . . 705.1 Interactions in a symmetric SU(n) chain. . . . . . . . . . . . . . . . . . . . . . . . 805.2 Feynman diagram occurring in SU(n) velocity renormalization. . . . . . . . . . . . 935.3 Simplified phase diagram of the symmetric SU(n) chain. . . . . . . . . . . . . . . 99xiiiAcknowledgementsI would first like to thank my supervisor, Ian Affleck, whose tremendous expertise in the field ofquantum spin chains proved to be invaluable while carrying out the work of this thesis. Ian’s incom-parable work ethic and curiosity inspired me to produce meaningful research, and his dedication tomentorship and teaching had a huge positive impact on my time at UBC.I would like to acknowledge my collaborators Dmitri Bykov, Francisco Kim, Miklo´s Lajko,and especially Fre´de´ric Mila. My visit to EPFL as a Masters student was a very formative experi-ence, and over the years I have benefited greatly from conversations with Fre´de´ric regarding SU(n)physics.I would also like the thank the following list of individuals, all of whom with I had meaningfuldiscussions that led to my better understanding of theoretical physics: Sven Bachmann, OguzhanCan, Stepan Fomichev, Marcel Franz, Samuel Gozel, E´tienne Lantagne-Hurtubise, Chengshu Li,Pedro Lopes, Kirk Madison, Alberto Nocera, Nathan Seiberg, Shu-Heng Shao, Tin Sulejmanpasic,Alexei Tsvelick, Tarun Tummuru, Mithat U¨nsal, Juven Wang, Jordon Wilson-Gerow, Wang Yang,Yuan Yao, Xiao-Xiao Zhang and Fei Zhou.I would like to acknowledge the funding support that I received from both NSERC, and theStewart Blusson Quantum Matter Institute.Finally, I would like to thank my family who has supported me lovingly and unconditionallyover these past five years. To Gaby, Mom, Dad, and Kelsie, thank you so much.xiv“It was rejected by multiple journals, and was labelled a “conjecture” even though itwas, in my mind, a clear prediction. ... Of course, my predictions were later vindicatedboth by numerical studies and experiments.”F.D.M. Haldane2016 Nobel LaureatexvChapter 1IntroductionThe Haldane conjecture is a statement about quantum spin systems in one spatial dimension. Suchsystems are described by the Heisenberg model, and are either ferromagnetic or antiferromagnetic,depending on the sign of the interaction term between neighbouring spins on the chain [7]. Whilethe ferromagnet’s ground state is the same for both classical and quantum chains (it is the statewith all spins aligned along a common direction), this is not true for the antiferromagnet. Clas-sically, the ground state is the so-called Ne´el state, with spins alternating between being alignedand anti-aligned along a common direction, but quantum mechanically the Ne´el state is no longeran eigenstate of the Heisenberg Hamiltonian [7]. This fact can be understood from Coleman’stheorem, which forbids the spontaneous ordering of a continuous symmetry in one spatial dimen-sion [8].1The absence of an ordered ground state in the antiferromagnet has long been of interest tothe physics community. Indeed, shortly after Heisenberg introduced his model of a ferromagnet in1921 [11], Bethe discovered an exact solution of the antiferromagnetic chain with spin s = 12 at eachsite [12].2 However, despite this initial progress, spin chains with s > 12 were not amenable to suchtechniques, and fifty years would pass before their low energy properties could be characterized. In1981, Duncan Haldane proposed a radical classification of antiferromagnetic chains: Those withinteger spin have a finite energy gap above their quantum ground states, and exponentially decayingcorrelation functions. Meanwhile, those chains with half-odd integer spin have gapless excitations,with algebraically decaying correlation functions [13, 14].3Despite being consistent with Bethe’s 1931 solution, Haldane’s “conjecture” as it came to beknown, was met with widespread skepticism [16]. This was likely due to the fact that spin-wavetheory, a method that allows one to calculate the energy spectrum of antiferromagnets in higherdimensions, largely agreed with Bethe’s one-dimensional results. We now know this to be a coin-cidence, but at the time, this suggested to the community that spin-wave results might be reliablein one dimension for all values of s. This would imply that all antiferromagnets would exhibit gap-less excitations at low energies [7, 17]. Of course, this was in direct contradiction with Coleman’stheorem, that invalidated spin-wave theory in one dimension, but nonetheless, by the 1980s it was1Coleman’s theorem is often confounded with the Mermin-Wagner-Hohenberg theorem, which forbids an orderedground state in two spatial dimensions at finite temperature, and applies equally well to both ferromagnets and antifer-romagnets [9, 10]. In Section 2.1.1, we review this subject further, and explain why Coleman’s theorem breaks down forthe ferromagnet.2Unless otherwise specified, it is assumed that the spin value at each site of the chain is identical, so that an antifer-romagnetic spin chain is uniquely defined by a positive half integer s.3Haldane’s original preprint, which was rejected from publication in 1981, has since been made available online [15].1Chapter 1. Introductionwidely accepted that gapless excitations were universal among one dimensional antiferromagnets.In fact, Haldane’s conjecture was met with surprise in other research areas of physics as well.As will be reviewed in great detail later, Haldane’s conjecture hinges on a correspondence betweenantiferromagnets and the O(3) nonlinear sigma model, a quantum field theory that was being usedas a toy model for quantum chromodynamics at the time [18]. The role of the spin, s, manifests asa topological angle θ in the sigma model, so that integer s translates to θ = 0 and half-odd integers translates to θ = pi. Thus, Haldane’s claim about antiferromagnets was also a claim about massgaps in the O(3) nonlinear sigma model. At that time, it was widely believed that a finite gapwould exist for all values of θ. This was known exactly for θ = 0 [19], and suggested numericallyfor small values of θ [20, 21].Over the next few years, Haldane’s conjecture would defy these skeptics, thanks to verificationfrom multiple areas of research. Experimentally, neutron scattering on the organic nickel com-pound NENP, which is a quasi-one dimensional s = 1 chain, detected a finite energy gap above theground state [22, 23]. Numerically, studies using exact diagonalization, Monte Carlo, and densitymatrix renormalization group methods were able to detect a finite gap in s = 1, 2 and 3 [24–29].Very recently this has been extended to s = 4, where a tiny gap of 7.99(5)×10−4 was measured [30].Meanwhile, in the O(3) nonlinear sigma model picture, Monte Carlo methods were used to numer-ically verify the absence of a mass gap when θ = pi [31–36], and a related integrable model waseventually discovered by the Zamolodchikov brothers [37].In many cases, the studies carried out in order to verify Haldane’s claims were scientific break-throughs in their own right. Indeed, the fields of density matrix renormalization group [38–40], andmore generally tensor networks [41–43], as well as symmetry protected topological matter [44–47]all originated, in part, due to Haldane’s conjecture. It is thus not a leap to claim that any generaliza-tion of Haldane’s conjecture would be an impactful result to the physics community. And indeed,this is what led physicists, including Affleck, Read, Sachdev and others to extend Haldane’s workto SU(n) generalizations of spin chains in the late 1980s [48–51]. At the time, these were purelyhypothetical models with no experimental realization, but thanks to the correspondence betweenspin chains and sigma models, they were still interesting in their own right. Another motivationwas a proposed relation between nonlinear sigma models and the localization transition in the quan-tum Hall effect [48, 52, 53]. And while this unsolved problem remains a motivator to study suchmodels, recent developments from the cold atom community have revealed that SU(n) chains arenow experimentally realizable, offering a much more physical motivation [54–66]. These propos-als have led to a renewed theoretical interest in the field of SU(n) physics [61, 67–77]. Moreover,generalizations of the O(3) nonlinear sigma model, called flag manifold sigma models, have alsogarnered recent attention, as they realize so-called ’t Hooft anomalies, which are interesting to highenergy physicists [71, 72, 78, 79]. This calls for not only a generalization of Haldane’s predictionof gapped vs. gapless behaviour (driven by topological terms), but also for developing a more gen-eral understanding of the spin-chain/sigma model paradigm, so as to understand which families ofSU(n) chains correspond to a particular quantum field theory.In this thesis, we study SU(n) Heisenberg models on one-dimensional chains in various repre-sentations. Unlike the familiar spin chains with SU(2) symmetry, for n > 2, multiple non-negative2Chapter 1. Introductionintegers are required in order to completely specify the representation. Ultimately, our goal is toestablish a generalization of the Haldane conjecture for these models. That is, we seek to determinewhich families of SU(n) representations give rise to chains with gapless ground states, and whichfamilies give rise to chains with unique, gapped ground states. Essential to Haldane’s argument is alow energy equivalence between the antiferromagnet and the O(3) nonlinear sigma model, and so inextending Haldane’s conjecture, we also generalize this equivalence. That is, we determine whichSU(n) representations correspond to chains that admit mappings to flag manifold sigma models,which are the SU(n) generalization of the O(3) nonlinear sigma model.We begin in Chapter 2 of this thesis with a review of the various arguments that, collectively,provide a modern proof of Haldane’s conjecture. A large part of this chapter is dedicated to thederivation of the O(3) nonlinear sigma model description of the spin chain, in an equivalent CP1language, as this will serve as our starting point in later chapters. The notion of topological angle,and how it arises from the spin chain as a Berry curvature term is explained. We also introduce spin-wave theory and explain how, despite Coleman’s theorem, it is a useful tool in analyzing quantumspin chains in one spatial dimension. In Chapter 3, we promote the symmetry group of the antifer-romagnet from SU(2) to SU(n), and discuss these so-called SU(n) chains in full generality. Thisrequires a detailed discussion of the irreducible representations and Casimir operators of SU(n),which we provide. We also explain the issue of local degeneracies in SU(n) chains, and how thisrequires the addition of longer-range terms in the Heisenberg Hamiltonians. Experimental realiza-tion of SU(n) chains is also mentioned. Finally, we finish Chapter 3 by reviewing various exactresults that are known for SU(n) chain systems, including the LSMA theorem [80, 81], generalizedAKLT models [67, 82, 83], and integrable SU(n) chains [12, 84–89]. Next, in Chapter 4, we focusour attention on the simplest SU(n) chains: the SU(3) chains in the totally symmetric representa-tions. We extend much of the analysis from Chapter 2 to these chains: we introduce ‘flavour-wave’theory, which generalizes spin-wave theory, and derive the corresponding low energy quantum fieldtheory description of the SU(3) chain. In this case, the O(3) nonlinear sigma model is replaced withthe SU(3)/[U(1)]2 flag manifold sigma model, equipped with two topological angles. We performextensive renormalization group calculations in this field theory, and combine these with resultsfrom a Monte Carlo simulation and a strong coupling analysis to obtain a phase diagram for theSU(3) chain. We discuss the ’t Hooft anomalies present in this model, and present an SU(3) ver-sion of Haldane’s conjecture. We complete this chapter with a discussion of other representationsof SU(3), including the self-conjugate ones. Next, in Chapter 5, we consider the symmetric repre-sentation SU(n) chains, thus extending the calculations from Chapter 4 from 3 to n. The low energyphysics of these chains is captured by the SU(n)/[U(1)]n−1 flag manifold sigma model with n − 1topological angles. Here, new difficulties arise, perhaps most importantly the absence of Lorentzinvariance in the corresponding sigma model. We present extensive calculations that establish thatunder renormalization, Lorentz invariance is restored. We also extend the strong coupling analysisand ’t Hooft anomaly conditions from SU(3) to SU(n). This leads to a version of Haldane’s conjec-ture for SU(n) chains in symmetric representations. Having identified the SU(n) sigma models ofinterest, we turn in Chapter 6 to the task of reverse engineering the correspondence between SU(n)chains and sigma models. That is, given the SU(n)/[U(1)]n−1 flag manifold, we classify all possible3Chapter 1. Introductionrepresentations of SU(n) chains that admit a mapping from chain to sigma model with this targetspace. We identify the key property that ensures the presence of an ’t Hooft anomaly, namely atransitive Zn action on the field content, which is a manifestation of n-site translation invariancein the SU(n) chain. We also discuss the emergence of quadratic Goldstone bosons, and the subtlecorrespondence between non-Lorentz invariance, Goldstone’s theorem, and spontaneous symmetrybreaking in general SU(n) chains. This leads to detailed classification tables of dispersion relationsfor each SU(n) representation. Finally, in the last major chapter of this thesis, Chapter 7, we pro-vide a physical picture of the mass-generating mechanism in SU(n) chains. That is, we explain howtopological excitations with fractional charge produce a finite energy gap in the flag manifold sigmamodel, akin to the vortex-generated gap in the Kosterlitz-Thouless transition [90, 91].4 This is firstdone in the absence of topological angles; upon restoring the angles, we show how interference ef-fects inhibit this mass-generating mechanism, and lead to a gapless ground state, for certain SU(n)representations. Remarkably, these are precisely the representations that our generalized Haldaneconjecture predicts will have a gapless phase, and can be understood as a consequence of the tran-sitive Zn action, which ensures n − 1 distinct topological angles. In Chapter 8, we summarize ourresults and provide an outlook towards the rapidly developing field of SU(n) physics.4Throughout, we will refer to the Kosterlitz-Thouless transition, but perhaps a more apt name is the Berezinskii-Kosterlitz-Thouless transition.4Chapter 2A Review of the Haldane Conjecture2.1 IntroductionAntiferromagnetic chains are described by the Heisenberg model,H = J∑j~S ( j) · ~S ( j + 1). (2.1)Here, the sum runs overs all lattice sites j of the chain, and the interaction strength J is positive.The vectors ~S are 3-tuples of spin operators,~S =(S x S y S z)T(2.2)and satisfy the su(2) Lie algebra:5[S a, S b] = iabcS c. (2.3)These chains are defined by their spin s, which corresponds to the irreducible representation ofSU(2) under which the spin operators S a transform.6 The spin can be read off from the quadraticCasimir operator,~S · ~S = s(s + 1)I. (2.4)On the right hand side of this expression, I is the identity operator, whose dimension is equal to2s + 1, the dimension of the spin-s representation of SU(2).Before Haldane, most physicists based their understanding of antiferromagnetic chains on spin-wave theory, which corresponds to taking the limit s → ∞ [17]. This amounts to a semi-classicallimit, as can be seen by inserting (2.4) into the Heisenberg uncertainty relation:∆S a∆S b ≥ 12 |〈[S a, S b]〉| ∆S a :=√〈S 2a〉 − 〈S a〉2. (2.5)For large values of s, relative fluctuations of the spin operator, ∆S a/|S a| are order O(s−1). Thus inthe limit s→ ∞, the spin operators can be simply treated as three-dimensional vectors constrainedto have length√s(s + 1), and the Hamiltonian (2.1) resembles that of a classical antiferromagnet.Spin-wave theory proceeds from here, seeking to describe low energy fluctuations about the clas-sical ground state, which has its spins adjacently antiparallel (the so-called Ne´el state). As we will5Here and throughout, we set Planck’s constant ~ = 1.6A formal treatment of the representation theory of SU(n) will be provided in Section 3.2.1.52.1. Introductiondemonstrate shortly, this leads to the prediction of gapless excitations, regardless of the value of s,and largely agrees with Bethe’s s = 12 exact solution [12]. This agreement was sufficient evidencefor many, and as a consequence, the physics community largely accepted the spin-wave theoryprediction of gapless excitations as a universal property of antiferromagnetic chains [7, 16].However, what most physicists were forgetting, or at least choosing to ignore, was the fact thatspin-wave theory inherently assumes an ordered quantum ground state. This is perfectly reasonablein higher-dimensional systems, but is in direct contradiction with Coleman’s theorem for one-dimensional chains [8]. In order to demonstrate this fact, we now present a brief review of spin-wave theory, and discuss the Goldstone modes that lead to infrared divergences. Despite its generalinvalidity, it has been appreciated rather recently that spin-wave theory can still be a useful tool inone dimensional systems [92]. As such, this review will also set the stage for flavour-wave theory,a generalization of spin-wave theory to larger symmetry groups, that we will introduce in laterchapters.2.1.1 Spin-wave theoryTo proceed with spin-wave theory, we begin by rewriting the Hamiltonian (2.1) in terms of bosonoperators that satisfy canonical commutation relations. Since the classical ground state of theantiferromagnet is the Ne´el state, which has a two-site unit cell, we introduce two bosons b1 andb2, each via a Holstein-Primakoff transformation [93]. For example, on the even sites, we maywrite,S z = s − b†1b1 S x − iS y =√2sb†1√1 − b†1b12s, (2.6)which preserves the algebra (2.3). For the odd sites, we use a similar expansion, except nowthe presence of bosons should correspond to deviations of S z from −s: S z = −s + b†2b2. Now,under the assumption of large s, we may expand the square root in powers of s−1, and substitutethis expression into (2.1). After performing a Bogoliubov transformation, we obtain a diagonalquadratic Hamiltonian, with dispersion relation7E(k) = 2Js| sin(k)|. (2.7)In other words, spin-wave theory predicts gapless excitations that disperse linearly, with a velocityof v = 2Js. In this context, these linear modes are referred to as Goldstone modes, since they ariseas a consequence of the spontaneous breaking of the spin symmetry, a result known as Goldstone’stheorem [94].It is these Goldstone modes that are the foundation of Coleman’s argument for the absence ofspontaneous symmetry breaking in one spatial dimension. To see this, note that the evaluation ofany physical expectation value, such as the staggered magnetization, necessarily involves a sum7Throughout, we set the lattice spacing a = 1.62.2. Sigma Model Derivationover these modes. Indeed, inverting the Bogoliubov transformation that led to (2.7), we find〈S z(x)〉 = s − 〈b†1(x)b1(x)〉 → s −12∫dk2pi1 − | sin(k)|| sin(k)| (2.8)in the continuum limit (here x is the continuous coordinate along the chain). For small values ofk, the integrand behaves like 1k , indicating that 〈S z(x)〉 suffers from an infrared divergence. Suchdivergences are a clear sign of an ill-defined theory, and are ubiquitous in one-dimensional systemswhenever one assumes the spontaneous breaking of a continuous symmetry. As an exception, theferromagnet evades this problem, which is related to the fact that its Goldstone bosons dispersequadratically instead of linearly. The mechanism behind this quadratic dispersion has been com-pletely understood only recently [95–97], and we will have more to say on this subject in Chapter6.As we’ve highlighted above, the fact that spin-wave theory leads to infrared divergences, and isthus an inconsistent theory, did not deter researchers in the field from accepting its predictions. Andas it turns out, information can still be gleaned from this formalism, so long as the right calculationsare performed. By this, we mean that if one restricts to the calculation of quantities that preservethe symmetry that is being spontaneously broken, then infrared divergences will be absent. Thiswas recently demonstrated explicitly for spin chains for spins up to s = 52 , and is expected to be truefor all spins [92]. However, for spin-wave theory to be valid when doing numerical simulations, itis important that the system size is much smaller than the correlation length ξ ∼ s−1epis.And so, perhaps it is not surprising that the spin-wave prediction at s = 12 roughly reproducesBethe’s exact solution. However, the explanation for why spin-wave theory was fundamentallywrong for s = 1 could not have been anticipated at the time. This is because it had to do with atopological distinction between integer and half-odd integer spin chains, and such concepts wereforeign to condensed matter physicists in the early 1980s. As we will now detail, Haldane’s simpleapproach of maintaining the large-s limit, while weakening the assumption of a Ne´el ground state,leads directly to this exciting new field of theoretical physics.2.2 Sigma Model DerivationIn this section, we derive a low energy mapping between the spin-s antiferromagnetic chain anda particular sigma model. By sigma model, we mean a quantum field theory whose fields definea map from (Euclidean) space time to some curved manifold. Recall that in the large-s limit, wemay replace the 3-tuple of operators ~S with a classical vector in R3, constrained to have length√s(s + 1) by the quadratic Casimir. In this case, we are eventually led to a quantum field theorywhose field ~n is constrained to live on the sphere S 2, a so-called S 2-sigma model. This was theapproach originally taken by Haldane, although he followed the widespread convention of referringto the model as the O(3) nonlinear sigma model, even though O(3) is not isomorphic to S 2 as agroup. Instead, what holds isS 2 = O(3)/O(2), (2.9)72.2. Sigma Model Derivationwhere the quotient by O(2) corresponds to the gauge symmetry of choosing a particular directionfor S z [98].A slightly different approach is to make use of a true isomorphism between the groups S 2 andCP1, and instead work with a complex field ϕ ∈ C2 that satisfies |ϕ| = 1. This is the approach thatwe will follow. Our reasoning for this choice will become clear in Chapter 3, when we promotethe on-site symmetry of the chain from SU(2) to SU(n). In this more general case, it is much morenatural to work with complex vectors ϕ ∈ Cn than their real-vector equivalents.To this end, let us introduce the following matrices of spin operators,S =(S z S x − iS yS x + iS y −S z). (2.10)The su(2) Lie algebra (2.3) manifests itself in the following commutation relations:[S αβ, S µν] = δανS µβ − δµβS αν. (2.11)In terms of these matrices, the Heisenberg interaction can be rewritten according to~S (i) · ~S ( j) = 12tr[S (i)S ( j)]. (2.12)As before, in the limit of large s, quantum fluctuations tend to zero and we can treat these matri-ces as classical objects. The quadratic Casimir constraint (2.4) restricts the eigenvalues of thesematrices to be ±s.The main strategy behind Haldane’s derivation, which still carries over here to the complexformalism, is to begin as in spin-wave theory with a classical ground state ansatz, but then promotethe matrices S to objects that can vary in space and in time. Therefore, we must first establish aconvenient way of representing the Ne´el state in the complex formalism, before proceeding. Sincethe eigenvalues of S are constrained in the limit s→ ∞, we may writeS = V†(s 00 −s)V, (2.13)with V a 2 × 2 unitary matrix. We may shift S by the constant matrix sI, since this only alters theHamiltonian by a fixed amount. This allows for a convenient representation in terms of the firstrow of V , φ ∈ C2:S αβ = sφ∗αφβ, (2.14)Since V is unitary, we have that |φ|2 = 1. Inserting this into the Heisenberg interaction, we findtr[S ( j)S ( j + 1)] = s2|φ∗( j) · φ( j + 1)|2. (2.15)In words, this equation demonstrates that the Heisenberg interaction is minimized when neighbour-ing complex vectors φ are orthogonal. Thus, the classical Ne´el state in this formalism correspondsto a pattern of alternating, orthogonal vectors: At even sites we have φ = ϕ1, and at odd sites, we82.2. Sigma Model Derivationhave φ = ϕ2, which satisfy ϕ1,∗ · ϕ2 = 0. Pictorially, we may represent this ground state using twocoloured nodes := ϕ1 and := ϕ2:· · · · · · (2.16)This diagram can be compared with the typical Ne´el pattern · · · ↑↓↑↓ · · · , in which neighbouringobjects are antiparallel instead of orthogonal – this is a distinguishing feature between the twoformalisms.Now that the ground state ansatz has been established, we introduce fluctuations about the ϕα,for α = 1, 2. In terms of the original φα appearing in (2.13), we write8φ1 =1sL12ϕ2 +√1 − |L21|2s2ϕ1 φ2 =1sL21ϕ1 +√1 − |L12|2s2ϕ2. (2.17)This somewhat cumbersome notation has been chosen in order to be consistent with similarderivations in later chapters of this thesis. The coefficients L12 = L∗21 correspond to fluctuationswithin the two-site unit cell of the Ne´el state, and are functions of space and time, as are the basisvectors ϕα. The remaining terms in (2.17), proportional to the radical, correspond to uniformrotations of the entire two-site cell. We have chosen, by convention, to include a factor of 1s in frontof the L12 and L21 coefficients.In passing, we note that the expansion (2.17) can be related to the more familiar real-spacederivation of the spin chain/sigma model correspondence,~n =1s~l ±√1 − l2s2~m, (2.18)using the equivalences~m · ~σ∗ = U†σzU ~l · ~σ∗ = 2U†LU, (2.19)where we’ve defined a unitary matrix U whose rows corresponds to ϕα (i.e. Uαβ = ϕαβ ) and setL11 = L22 = 0. Here ~l, ~m and ~n are all vectors in R3, and the Pauli matrices ~σ are defined accordingtoσ1 =(0 11 0)σ2 =(0 −ii 0)σ3 =(1 00 −1). (2.20)In order to derive a sigma model, we now express the matrices S in terms of these new basisvectors, ϕα, according toS αβ(2 j) = sφ∗,1α φ1β S αβ(2 j + 1) = sφ∗,2α φ2β. (2.21)8It is not necessary to work with both φ1 and φ2, since they are linearly dependent. We do so for convenience only.92.2. Sigma Model DerivationWe expand this expression in powers of s−1, and insert it into the Heisenberg Hamiltonian in orderto obtain a low energy description of the antiferromagnetic chain. UsingS αβ(2 j) = 2sϕ∗,1ϕ1 + U†LU + 1s U†(−|L12|2 00 |L12|2)U + O(s−2), (2.22)S αβ(2 j + 1) = 2sϕ∗,2ϕ2 + U†LU + 1s U†(|L12|2 00 −|L12|2)U + O(s−2), (2.23)and Taylor expanding the terms that occur on different sites, the Heisenberg interactions becometr[S (2 j)S (2 j + 1)] = (2.24)4s2tr[U∂xU†Λ1∂xUU†Λ2] + 4|L21|2+4s(L21[∂xUU†]12 + L12[U∂xU†]21),tr[S (2 j + 1)S (2 j + 2)] = (2.25)4s2tr[U∂xU†Λ2∂xUU†Λ1] + 4|L12|2−4s(L12[∂xUU†]21 + L21[U∂xU†]12),where we have defined9Λ1 :=(1 00 0)Λ2 :=(0 00 1). (2.26)Above, we have suppressed terms that are at least O(s−2). At this point, we are tasked with av-eraging over the L12 fluctuations in order to obtain a theory written in terms of U only. Haldaneachieved this by turning from the Hamiltonian formalism to the Lagrangian formalism, and thiswill be our approach as well.2.2.1 Coherent state path integralIn order to derive the Lagrangian description of the antiferromagnetic chain, we introduce a com-plete set of coherent states. Starting with φ ∈ C2, we define|φ〉 := 1√(2s)!(φ · a)2s|0〉 (2.27)where a is a 2-tuple of boson creation operators (known as Schwinger bosons) [99, 100]. Withthese objects, we may construct a path integral by inserting them between thin time slices of widthδτ of the partition function:Z =∫D[φ]N−1∏k=0〈φ(τk)|e−Hδτ|φ(τk + δτ)〉. (2.28)9Throughout, we will denote by Λα the matrix with a single nonzero entry (equalling 1), at row and column α. Asan abuse of notation, we will not specify the dimensions of Λ, which will change from 2 to n as we switch from SU(2)to SU(n) in later chapters.102.2. Sigma Model DerivationHere |φ(τk)〉 is actually a direct product of coherent states, one for each site of the chain,|φ(τk)〉 = ⊗Lj=1|φ( j, τk)〉, (2.29)and the integration measure is defined to beD[φ] :=∫ ∏k∏jdΩφ( j,τk), (2.30)where dΩφ is the normalized Haar measure of the coherent state [98]. To linear order in δτ, wemay expand〈φ(τk)|e−Hδτ|φ(τk + δτ)〉 ≈ 〈φ(τk)|φ(τk + δτ)〉 − δτ〈φ(τk)|H|φ(τk)〉. (2.31)The second term will generate the Hamiltonian contribution to the Lagrangian. Meanwhile, theoverlap term can be approximated using〈φ(τk)|φ(τk + δτ)〉 ≈ (1 + φ∗ · ∂τφ)2s. (2.32)By taking the product over all time slices τk, we can then reexponentiate according to∏k〈φ(τk)|φ(τk + δτ)〉 = exp∑klog〈φ(τk)|φ(τk + δτ)〉 ≈ exp∑k2sφ∗ · ∂τφ. (2.33)This is the so-called “Berry phase contribution” to the path integral. It is a geometric quantity, andmathematically speaking, corresponds to the symplectic form on the coherent state manifold [71,72]. Adding to it the Hamiltonian term δτ〈φ|H|φ〉, and taking the number of time slices N → ∞,we obtain a path integral description of the partition function,Z, with Lagrangian densityL = LBerry − 〈φ(τ)|H|φ(τ)〉. (2.34)In the previous subsection, we were able to express H in terms of the matrices U and L. Using(2.17), we also do this for LBerry. First note thatLBerry = s1∑j=0φ( j)∗ · ∂τφ( j) (2.35)≈ s[φ1,∗ · ∂τφ1 + φ2,∗ · ∂τφ2 + ∂xφ2,∗ · ∂τφ2 + φ2,∗ · ∂x∂τφ2].To see this, note that the LBerry is a Lagrangian density, obtained by summing over both sites ofthe unit cell, and dividing by the unit cell size, which is 2. It is not important which unit cell wechoose, so we have taken sites 0 and 1 of the chain. We have also made use of the Ne´el ansatz (asreflected in (2.22), to identify φ( j = 0) = φ1 and φ( j = 1) = φ2. The approximation follows fromTaylor expanding all terms about site j = 1, to linear order. Using (2.17), we findLBerry = −sµνtr[∂µU†Λ2∂νU] − 2tr[L∂τUU†], (2.36)112.2. Sigma Model Derivationwhere we have made use of the identity tr[U†∂U] = 0. This identity will be used repeatedlythroughout this thesis, and is proven for n×n unitary matrices in Appendix A. Combining this resultwith (2.24) and (2.25), we arrive at the following Lagrangian description of the antiferromagneticchain:L = 4s2Jtr[U∂xU†Λ1∂xUU†Λ2] + 4J|L21|2 + sµνtr[∂µU†Λ2∂νU] − 2tr[L∂τUU†]. (2.37)Again, we have divided the Hamiltonian terms by 2, which is the size of the unit cell, to obtain theHamiltonian density. As a final step, we now integrate over the fluctuations L12, using the identitydzdz∗e−z∗ωz+uz+vz∗ =piωeuv/ω. (2.38)This results in a Lagrangian in terms of U only:L = 14Jtr[Λ1U∂τU†Λ2∂τUU†] + 4s2tr[Λ1U∂xU†Λ2∂xUU†] + sµνtr[∂µU†Λ2∂νU]. (2.39)There are two names for this quantum field theory. As it is written, it resembles a sigma model interms of matrices U ∈ U(2). However, U can actually be restricted to a smaller manifold, due tothe invariance of the theory under diagonal transformations:U →(eiϑ1 00 eiϑ2)U. (2.40)These transformations correspond to the group U(1) × U(1), and result in the restrictionU(2)→ U(2)U(1) × U(1) =SU(2)U(1), (2.41)which is an example of a flag manifold. Thus, the above field theory can be referred to as theSU(2)/U(1) flag manifold sigma model.However, this naming convention is somewhat recent, and is related to extensions of Haldane’swork to SU(n) chains, which is the subject of later chapters of this thesis. More commonly, physi-cists choose to write the theory in terms of the first row of U, that is, in terms of ϕ := ϕ1 ∈ C2 [7].In this formulation, (2.39) readsL = 4s2(|∂xϕ|2 − |ϕ∗ · ∂xϕ|2)+14J(|∂τϕ|2 − |ϕ∗ · ∂τϕ|2)− sµν∂µϕ∗ · ∂νϕ. (2.42)Since ϕ ∈ C2 and is constrained by |ϕ| = 1, this is known as the CP1 sigma model. It shouldbe emphasized that these two models are not just equivalent: they are identical. We are merelydistinguishing between two names for the same manifold, and thus the same sigma model. Thisshould be contrasted with the O(3) nonlinear sigma model originally obtained by Haldane. Thistheory is equivalent, but distinct from the quantum field theory derived here.122.3. Properties of the Sigma ModelThis completes the derivation of Haldane’s low energy mapping between antiferromagnet andsigma model. As mentioned previously, these calculations will serve as a template when we con-sider chain systems with larger symmetry groups in the remaining chapters of this thesis. Butbefore embarking on this generalization program, it is necessary to review additional argumentsthat led Haldane to his famed conjecture.2.3 Properties of the Sigma ModelWe begin with arguably the most distinguishing property of the quantum field theory derived above;namely, its topological angle.2.3.1 Topological anglePerhaps the most surprising aspect of Haldane’s conjecture is its discreteness; that is, how it distin-guishes between chains with spin 2s and 2s + 1. In many aspects, our above derivation resemblesspin-wave theory, which is a large-s expansion and parallels a typical semiclassical perturbativeexpansion with s−1 ∼ ~. But if this were the case, and our calculations were purely perturbative,then it would be impossible to detect a difference between even and odd s, in the limit s→ ∞. Thisparadox is resolved by the final term in (2.39),10Ltop = sµνtr[∂µU†Λ2∂νU]. (2.43)The subscript ‘top’ stands for ‘topological’, and refers to the fact that it is left unaffected by con-tinuous transformations of U. In other words, a topological quantity cannot be detected by contin-uously varying a small parameter, such as s−1. To see that this explicitly holds for (2.43), note thatit is a total derivative: ∫d2xLtop = s∫d2x∂µ(µνϕ∗ · ∂νϕ)∈ 2pisiZ. (2.44)Therefore,Ltop is affected only by the behaviour of ϕ on the boundary ofR2, and will be unchangedby transformations that alter ϕ in the bulk of R2 only. Since any field configuration that contributesto the overall partition functionZ must have finite action, we may assume that ϕ attains a constantvalue at some radius R away the origin. This allows us to identify R2 with the sphere S 2, so that ϕcan be viewed as the following mapping:ϕ : S 2 7→ SU(2)U(1). (2.45)The topological properties of such maps from S 2 to the flag manifold SU(2)/U(1) are classified bythe second homotopy group, pi2(CP1),pi2(CP1) = Z. (2.46)10We have used Λ2 = I − Λ1 and tr[U†∂U] = 0 to rewrite Ltop in terms of ϕ = ϕ1 instead.132.3. Properties of the Sigma ModelThis is the formal justification for the appearance of Z on the right hand side of (2.44). Actually,Ltop is usually identified with an element of H2(CP1,Z), the second cohomology group of CP1.However, for all of the theories that we study, the groups H2 and pi2 coincide, and so we will nothave to worry further about this distinction [71, 72]. And as is becoming a repeated phrase in thischapter, this machinery may seem unnecessarily complicated but will prove to be incredibly usefulwhen we consider more general symmetry groups later on.The prefactor occurring in (2.44) motivates the following definition:θ := 2pis. (2.47)This is the so-called topological angle of the sigma model. This nomenclature can be understoodby realizing that Ltop leads to the following prefactor in the path integral of the partition function:e−∫d2 xLtop = eiθk k ∈ Z. (2.48)In other words Ltop contributes a U(1) phase to particular field configurations, with phase angle θ.Much later, in Chapter 7 of this thesis, we will explain how the energy gap predicted by Haldaneis related to this U(1) phase. In short, a nontrivial θ leads to interference effects among fractionaltopological excitations, resulting in the closing of the energy gap.At this point, the distinction between half-odd integer s and integer s becomes clear: In theformer case, the corresponding sigma model has a topological angle of θ = pi, while in the lattercase, the topological angle is trivial. And while little was understood of the CP1 sigma model withnonzero θ, it was known that at θ = 0, the model exhibited a finite mass gap [19]. This result, whencompared with Bethe’s proof that the s = 12 chain is gapless [12], led Haldane to argue that thistopological angle, which is ultimately determined by the parity of 2s, drives a phase transition inthe CP1 sigma model [13, 14]. That is, when θ = pi, a gapless phase is possible, while when θ = 0,the model should exhibit a finite gap in all cases.11 And thus, we arrive at the famous Haldaneconjecture for spin chains: Integer spin chains are gapped, while half-odd integer spin chains aregapless. In the following subsection, we review additional studies that help elucidate the generalphase diagram of these chains.2.3.2 Renormalization group flowWe may rewrite the sigma model in (2.39) asL = 1gtr[U∂µU†Λ1∂µUU†Λ2] +12piθµνtr[∂µU†Λ2∂νU], (2.49)where we’ve definedg =1sv = 2Js, (2.50)11The condensed matter community tends to use the phrase “energy gap”, while the high energy community tends toinstead use “mass gap”. This reflects the fact that historically, the CP1 sigma model was used by particle physicists as atoy model for quantum chromodynamics.142.4. Exact Resultsand rescaled τ → vτ. With this notation, it is clear that the Lagrangian is invariant under the ex-change of spatial and temporal coordinates. In other words, Lorentz invariance has emerged in thelow energy theory of the antiferromagnet. This feature will prove to be essential when generalizingHaldane’s conjecture to larger symmetry groups, as we will demonstrate in great detail in laterchapters of this thesis. Note that if one repeats this field theory derivation for the ferromagnet,such Lorentz invariance is not achieved. This is deeply related to the fact that the correspondingGoldstone bosons of the ferromagnet have quadratic dispersion relations [95–97].The sigma model (2.49) at θ = 0 exhibits another interesting property, which is known asasymptotic freedom [18, 92]. This means that as we increase the energy scale of our theory, thecoupling constant g tends to zero. This is a statement about renormalization group flow, a subjectthat will be reviewed in detail in Section 4.5, and so we will not prove this result here. As a con-sequence of asymptotic freedom, perturbatively massless excitations will become massive, whichexplains the finite mass gap when θ = 0. Moreover, recent work suggests that asymptotic free-dom may render spin-wave theory applicable to chains after all, so long as the physical quantitiesconsidered preserve the spin symmetry [92]. This fact motivates our consideration of generalizedspin-wave theory in later chapters.Since Haldane’s original papers, there has been a large research effort on broadening our un-derstanding of the CP1 sigma model phase diagram. In [86], it is established that for bare couplingg less than some critical gc, and for θ = pi, the model flows to another sigma model which is confor-mally invariant: the SU(2)1 Wess-Zumino-Witten (WZW) model [101, 102].12 While it is naturalto expect a flow to some SU(2)-invariant conformal field theory at the critical point, this particularmodel is singled out due to the fact that all other theories (which are SU(2)k WZW models fork > 1) are unstabilized due to relevant operators [86]. This flow behaviour is shown in Figure 2.1.Note that for g > gc, the model goes into a gapped phase with a spontaneously broken Z2 symme-try, which is a manifestation of the underlying translation symmetry in the spin chain.13 In [86], itis explained how all regions of the phase diagram in Figure 2.1 can be reached: the bare couplingg can be increased by adding antiferromagnetic next-nearest-neighbour interactions, and θ can bechanged by adding alternating exchange interactions, which explicitly break translation symmetry.In both cases, the scaling of the mass gap has been calculated using the SU(2)1 WZW model. Fi-nally, the right hand edge of the diagram, corresponding to the strong coupling limit g → ∞, hasalso been considered [104, 105].2.4 Exact ResultsIn addition to the exact result of Bethe for the s = 12 chain [12], which predicts a gapless groundstate with no broken symmetries, there are two additional pieces of rigorous evidence for the an-tiferromagnet phase diagram, Figure 2.1. They are both in agreement with Haldane’s conjecture,12Another name for these models is WZNW, after Sergei Novikov [103].13If a translationally invariant chain with a d-site unit cell maps onto a sigma model, translation symmetry willmanifest as a Zd action.152.4. Exact ResultsSU(3)1SU(3)11×2×3×1×3×gaplessgaplessgappedgappedSU(2)11×2×1×gappedgapless02π4π/3π2π/30θgc g(b)π2π(a)ggcθFigure 2.1: Phase diagram of the antiferromagnetic chain.and are worth mentioning now, since their generalizations will appear later in the text.2.4.1 The Lieb-Schultz-Mattis TheoremThe Lieb-Schultz-Mattis (LSM) theorem is a rigorous statement about ground states in translation-ally invariant Hamiltonians [80]. Applied to the spin-s antiferromagnet, this theorem requires thatif s is not an integer, then either the ground state is unique with gapless excitations, or there is aground state degeneracy. This is consistent with the θ = pi line in Figure 2.1. In Section 3.5.2, weprove a more general result (an extension of the LSM-A theorem, where A stands for Affleck [81]),from which a proof of the LSM theorem follows automatically.2.4.2 The Affleck-Kennedy-Lieb-Tasaki modelOne of the most important theoretical results to follow Haldane’s conjecture and support its conclu-sion was the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction [82]. By forming singlet bondsbetween adjacent spin- 12 ’s of a spin-1 chain, the AKLT Hamiltonian was able to exhibit a unique,translationally invariant ground state with a finite excitation gap. This result compliments the LSMtheorem, and corresponds to the θ = 0 line in Figure 2.1. In Figure 2.2, we provide a pictorialrepresentation of this construction, which also serves to introduce notation that we will use later onto draw generalized AKLT states.162.5. ConclusionFigure 2.2: The AKLT ground state in an SU(2) antiferromagnetic chain. Each node correspondsto a spin- 12 representation. Two nodes arranged vertically in single column are symmetrized, andcorrespond to a spin-1 representation. A dashed edge denotes a singlet bond. Notice that this stateis translationally invariant.2.5 ConclusionIn this chapter, we reviewed the various arguments that support Haldane’s conjecture about antifer-romagnetic spin chains. Our primary focus was the derivation of the correspondence between spinchain and sigma model, which becomes exact in the limit s→ ∞. As we proceed to generalize theHaldane conjecture to larger symmetry groups in the following chapters, we will follow the samestructure of argument as was presented in this chapter. For clarity, we repeat the steps followedhere:1. Identify the classical ground states of the Hamiltonian, and the target space manifold of thesigma model.2. Perform a low energy expansion of the Hamiltonian, taking into account fluctuations that bothpreserve and destroy the classical ground state structure.3. Generate a Lagrangian from the path integral formalism, using coherent states.4. Identify the (set of) topological angle(s) of the sigma model.5. Determine the renormalization group flow of the sigma model, and make use of existing exactsolutions.146. Consult (and generalize) exact results for the original Hamiltonian, such as the LSMA theo-rem and the AKLT construction.7. Combine the results of Steps 5 and 6, and formulate a conjecture about the ground statebehaviour of the Hamiltonian.In later chapters, we will attempt to follow this agenda for a variety of quantum chain systems.In doing so, we will run into a variety of technical challenges, and in overcoming these challenges,we will be led to exciting new physical phenomena.14A large class of results pertaining to the sigma models that we encounter go by the name of ‘t Hooft anomalymatching conditions [106, 107]. Such results also exist for the CP1 sigma model, and offer a modern interpretation ofHaldane’s conjecture. We will discuss ‘t Hooft anomalies extensively in Section 4.6.4.17Chapter 3SU(n) Chain Hamiltonians3.1 IntroductionThe defining symmetry of an antiferromagnet is the Lie group SU(2), the set of special unitary2 × 2 matrices. The two complex dimensions of these matrices can be thought of as correspond-ing to two species of particle: a ‘spin-up’ particle and a ‘spin-down’ particle. The fact that thedimensions are complex follows from quantum mechanics, and is related to the invariance of thewave function under U(1) phase transformations. These two particles correspond to the s = 12representation of SU(2). Larger spin representations are constructed by taking tensor products ofthese 2-dimensional spaces. For example, a spin-1 particle corresponds to the (symmetric) tensorproduct of two 2-dimensional spaces, or 2 spin- 12 particles. Adding more spin-12 ’s will increase therepresentation (and thus the total spin), but the underlying symmetry group will remain SU(2).In order to modify the group SU(2) itself, we must change the number of “species” of particle.For example in quantum chromodynamics, the theory of strong interactions, the presence of threespecies of quark (red, blue, and green) leads to an internal SU(3) symmetry [108]. However, quarksare confined within hadrons, and do not form long chains like an SU(2) antiferromagnet. In fact, wedo not know of any naturally occurring chain systems with an on-site SU(n) symmetry, for n > 2.This statement holds true not just for exact symmetries, but also for approximate symmetries, likethe SU(3) ‘flavour’ symmetry that relates the up, down and strange quark [108].Nonetheless, shortly after Haldane’s conjecture was presented, work began on extending hiscalculations from SU(2) to SU(n) chains [48–51]. Initially, this was a purely theoretical exercise,with hopes of understanding the correspondence between sigma model and chain system on moregeneral grounds. It had been predicted that a certain family of sigma models could be used tocharacterize the localization transition in the quantum Hall effect, and so there was promise that ageneralized Haldane conjecture could shed light onto this problem [48, 52, 53].After this initial wave in the 1980s, interest in SU(n) chains diminished, likely due to theirdisconnect from physically realizable systems. However, this all changed in the early 2000s, whenadvances in cold atom experimentation suggested that chains with larger symmetry groups mightbe possible to construct using optical traps [54–66]. In the past decade, these ideas have come tofruition, and have resulted in a resurgence in theoretical interest in SU(n) chains.The idea behind engineering SU(n) chains using cold atoms is the following. Alkaline Earthatoms have vanishing electronic angular momenta. As a result, their nuclear spin Zeeman statesare all degenerate, and serve the roll of ‘species of particle’, in the sense mentioned above. If acollection of these atoms are trapped in a chain-like formation, the interatomic potential between183.2. SU(n) Hamiltonianssites will not distinguish between these Zeeman states, and an SU(2I + 1) symmetry will emerge,where I is the atom’s nuclear spin [57, 64]. So far, the largest symmetry group that has beenrealized in this way is SU(10), using Strontium, which has I = 92 [63].One might ask why is it that alkaline Earth atoms are necessary in order to construct such SU(n)chains. After all, doesn’t a spin-s SU(2) chain have 2s + 1 ‘species of particle’ at each site? Theissue is that while spin chains may have the correct number of states at each site, their Hamiltoniansare not invariant under larger symmetry groups except at fine-tuned points.15 This is because thevector ~S of spin operators does not transform nicely under the larger SU(2s + 1) symmetry group.On the other hand, when one writes down the Hamiltonian for a chain of alkaline Earth atoms, theinteractions are approximated by a pseudopotential that transforms trivially under the SU(2I + 1)group [64].Of course, the symmetry group is only half of the story. A more significant experimentalchallenge is how to realize different representations of the SU(n) chain, which requires multipleatoms trapped at the same site. There has been some progress is realizing 2-atom representations(the analogue of spin-1) by making use of an additional degeneracy between the so-called s and fstates [57]. However, realizing the most general representations of SU(n) remains an outstandingexperimental problem. Even so, theoretical physicists have forged ahead, attempting to characterizeSU(n) chains in a variety of different representations.In this chapter, we will introduce our systematic approach to studying SU(n) chain Hamilto-nians in the most general representation. Our work builds off the seminal contributions of Af-fleck [48–50] and Bykov [71, 72], and offers a modern perspective on how SU(n) representationsultimately determine the target space manifold of the sigma model describing the chain’s low en-ergy physics. In Section 3.2, we introduce the SU(n) Heisenberg Hamiltonian, and review elementsof the representation theory of SU(n), including Casimir operators and Young tableaux, that we willrequire later on. Then, in Section 3.3, we discuss the classical ground states of these Hamiltonians.We explain how longer-range interaction terms are necessary in order to stabilize these groundstates, and how the notion of flag manifolds arise from SU(n) representations. We also introducegraphical notation that will be used throughout this thesis when discussing the structure of SU(n)chains. Next, in Section 3.4, we classify all SU(n) chains that lead to an SU(n)/[U(1)]n−1 sigmamodel at low energies. Finally, in Section 3.5, we review various exact results that exist for SU(n)chains, including the Bethe ansatz models [12, 84, 85], the AKLT models [67, 82], and the Lieb-Schultz-Mattis-Affleck theorem [80, 81]. We will make frequent reference to this section later on,when we formulate our SU(n) versions of Haldane’s conjecture.3.2 SU(n) HamiltoniansIn the previous chapter, we introduced 2 × 2 matrices in (2.10) that contained the 3 generators ofsu(2). We now repeat these steps for general n. We define a traceless n × n matrix S αβ whose15For an example of this fine-tuning, it is well known that the spin-1 bilinear-biquadratic chain has two points whereSU(3) symmetry emerges [48].193.2. SU(n) Hamiltoniansentries correspond to the n2 − 1 su(n) generators. The entries of S αβ satisfy the same commutationrelations as in SU(2), (2.11), which we repeat here for convenience:[S αβ, S µν] = δανS µβ − δµβS αν. (3.1)Using these matrices, the Heisenberg interaction takes the same form as the right hand side of(2.12), which isJtr[S (i)S ( j)]. (3.2)As before, we take J > 0, which corresponds to an antiferromagnetic-like interaction. Thus, wearrive at the SU(n) Heisenberg model:H = J∑jtr[S ( j)S ( j + 1)]. (3.3)As it will turn out, this Hamiltonian will often be insufficient for our analysis. Instead, we will haveto consider a modified Hamiltonian which includes longer-range interaction terms. This is forcedon us by the representation theory of SU(n), which we now review.3.2.1 Representation theoryUnlike SU(2), a representation of SU(n) is not labeled by a single number (the ‘spin’). Instead, itis specified by n − 1 integers, which are related to so-called Casimir constraints:tr[S m] = CmI m = 2, 3, . . . , n. (3.4)In SU(2), the only constraint is the quadratic Casimir, tr[S 2] = s2 (s+1)I, which uniquely defines thespin, s. For n > 2, a representation’s defining integers are conveniently expressed in terms of rowlengths of a standard Young tableau. Indeed, the most general representation [p1, p2, . . . , pn−1] ofSU(n) corresponds to a unique diagram of boxes arranged in n− 1 rows, of lengths p1, p2, . . . , pn−1respectively. The row lengths must satisfy p1 ≥ p2 ≥ · · · ≥ pn−1 ≥ 0. See Fig. 3.1 for some exam-ples. This choice of labelling is related to the limit of large representation: in the limit p1 → ∞, thematrix of operators S is again replaced with a classical matrix, whose eigenvalues are completelydetermined by the row lengths pα [50]:λα = pα − p p := 1nn∑α=1pα. (3.5)Here we’ve defined pn := 0. Note that since the quadratic Casimir tr[S 2] → ∞ when p1 → ∞,it is sufficient to only consider large p1 to obtain the classical limit, as reflected in (2.5); it is notnecessary to also require pα → ∞ for α > 1.It now becomes apparent how different representations of SU(n) may lead to different types ofsigma models. Indeed, the matrix S is constrained to live on the manifold U(n)/H, whereH = U(m1) × U(m2) × · · · × U(mk) (3.6)203.2. SU(n) HamiltoniansFigure 3.1: Examples of Young tableaux in SU(n). A diagram with k nonzero rows corresponds toa representation in SU(n) with n ≥ k + 1.is a product of k unitary groups, one for each distinct value of λα, and mα is the degeneracy ofeach λα. These spaces are known as flag manifolds, and can be thought of as generalizations ofGrassmannian manifolds, which correspond to m1 = m and m2 = n − m (i.e. k = 2). Complexprojective space CPn−1 is the Grassmannian with m1 = 1. Flag manifolds are very interestingmathematical objects, and possess a variety of rich geometric structures, including symplectic andKa¨hler forms. For an in-depth account of this structure, we encourage the reader to consult therecent review [6].According to (3.6), it is possible to fix the target space manifold of the matrices S by choos-ing the appropriate representation of SU(n) on each site. This leads to a very important questionthat we should ask as we embark on a program to generalize Haldane’s conjecture: what is theappropriate generalization of the spin chain/sigma model correspondence beyond SU(2)? Recallthat in the previous chapter, we established two equivalent mappings for the antiferromagnet: onein terms of the SU(2)/U(1) sigma model (which equals CP1), and one in terms of the S 2 sigmamodel. Critically, when n > 2 these two descriptions are not equivalent. Moreover, the manifoldsSU(2)/U(1) and CP1 are no longer isomorphic when n > 2! This leads to at least three candidatefamilies of sigma model as we increase n.One may protest that our above inquiry about the ‘appropriate’ generalization of Haldane’sconjecture is misguided. After all, we have just established that the target space manifold of Sis uniquely determined by its representation, and so wherein lies our freedom to choose how togeneralize Haldane’s conjecture? In our defense, we stress a distinction that must be made betweenthe target space manifold of S , and the target space manifold of the corresponding sigma model.In SU(2), these two spaces coincided but this is not a requirement for n > 2. Indeed, as we willshow in the following section, it is possible to work with matrices S that lie in smaller manifolds,and then combine these degrees of freedom over consecutive sites of a unit cell to produce a largermanifold. For example, our focus in Chapter 4 is the SU(3)/[U(1)]2 sigma model. One mayarrive at this sigma model by considering representations of SU(3) that restrict S to SU(3)/[U(1)]2directly – these are so-called self-conjugate representations of SU(3). However, a second approachis to consider representations that restrict S to CP2, and combine these degrees of freedom over athree-site unit cell using orthogonality conditions to arrive at the same flag manifold. See Figure 3.2213.3. Classical Ground Statesfor examples of Young tableau in either case.p︷ ︸︸ ︷ p p︷ ︸︸ ︷Figure 3.2: Examples of Young tableaux that correspond to representations of SU(3) that lead to theSU(3)/[U(1)]3−1 flag manifold sigma model. These representations will be studied in Chapter 4.This second approach turns out to be a rich source of new physics, as was first emphasizedby Bykov [71, 72]. As such, we repeat this construction in Chapter 5 for general n: we chooserepresentations that restrict S to CPn−1, and then combine n such fields across an n-site unit cellto obtain the SU(n)/[U(1)]n−1 sigma model. The flag manifold SU(n)/[U(1)]n−1 corresponds tomα = 1, for α = 1, ..., n in (3.6), and is referred to as the ‘complete’ flag manifold associated withU(n). By construction, these sigma models will also have a Zn translation symmetry, which is amanifestation of the n-site unit cell. These combined symmetries will lead to a rich classificationbased on ‘t Hooft anomaly matching [106, 107], among other exact results. However, varioustechnical hurdles must be cleared along the way.In passing, it is worth mentioning that the initial studies of SU(n) chains did not considersuch multi-site constructions. Instead, those authors focused on sigma models whose target spacescoincided with the target space of the S matrices themselves [49]. In particular, this approach wasused to obtainCPn−1 sigma models and their Grassmannian generalizations, with symmetry groupsU(n)/[U(m) × U(n − m)] [48, 50, 51]. In these papers, the desired sigma model was generated byidentifying the representationR that directly restricts S to the sigma model’s full manifold, and thenplacing R and its conjugate R on even and odd sites of the chain, respectively. For representationssatisfying R = R, this construction reduces to the on-site method mentioned previously for SU(3)self-conjugate chains.3.3 Classical Ground StatesIn the limit of large representation, we replace S with V†diag(λ1, . . . , λn)V , where V is a unitarymatrix, and the eigenvalues λα are given in (3.5). Again, this structure of S follows from the n − 1Casimir constraints. The Heisenberg interaction then readstr[S (i)S ( j)]→∑α,β,γ,δλαλβ|φα,∗(i) · φβ( j)|2, (3.7)where we’ve defined Vαβ(i) = φαβ (i). Since the φα are rows of a unitary matrix, they must bemutually orthonormal on the same site. Implicit in this expression is our assumption that thesame representation occurs at each site of the chain, which we will always take to be true. Sinceλα = pα − p for the representation with Young tableau row lengths pα, we opt to shift S by pIto simplify our calculations (this merely shifts the interaction term (3.7) by an overall constant).223.3. Classical Ground StatesHaving done this, the simplest SU(n) chain Hamiltonian, namely the nearest-neighbour model,becomesH = J∑jn−1∑α,β=1pαpβ|φα,∗( j) · φβ( j + 1)|2, J > 0. (3.8)Note that the sums over α and β stop at n − 1, since pn = 0 by definition. This nearest-neighbourmodel is the logical starting point for any SU(n) generalization of the antiferromagnetic spin chain.However, in most cases, we will be required to consider Hamiltonians with longer-range interac-tions. For instance, if one considers an SU(n) chain in the totally symmetric representation, withpα = 0 for α > 1, then S is restricted to the manifold SU(n)/U(n − 1) = CPn−1, meaning it isentirely specified by a single complex unit vector φ ∈ Cn. Now, in the nearest-neighbour model(3.8), a Ne´el-like state that alternates between two orthogonal directions in Cn, such as φ(2 j) = ϕ1and φ(2 j + 1) = ϕ2, would be a classical ground state. For n = 2 this is the end of the story, andled us to an appropriate flag manifold description in Chapter 2. But for n > 2, there is a nontrivialsubspace, namely Cn−2, that is spanned by vectors orthogonal to the Ne´el vectors ϕ1 and ϕ2. Asa consequence, any field theory derivation will be plagued with local zero modes – zero energyexcitations above the ground state - that correspond to transformations into this unused subspace.In order to resolve this obstruction in deriving a field theory, we are forced to modify theHamiltonian (3.8) by adding longer-range interactions. This will couple more lattice sites together,and result in a classical ground state that is made up of vectors that span all of Cn. For the totallysymmetric representations, this will require up to n−1-neighbour interactions, and will result in ann-site unit cell structure for those chains. This is consistent with the exact Bethe ansatz solutions(reviewed below in Section 3.5) that exhibit such n-site structure.16 Moreover, it has been shown inSU(3) that zero point fluctuations are minimal in the 3-site state when p→ ∞, and this is expectedto hold for all n [109]. Accordingly, we believe that the first effect of quantum fluctuations in thesymmetric chains will be to select the n-sublattice state by an ‘order-by-disorder’ mechanism thatgenerates effective additional couplings of order 1/p that lift the classical degeneracy. It shouldbe kept in mind however that all the properties discussed for these longer-range Hamiltonians areexpected to apply to nearest-neighbour Hamiltonians, and that the longer-range couplings havebeen introduced as a first effect of quantum fluctuations in those models. This sentiment appliesnot only to the symmetric chains, but to all of the SU(n) chains that we consider.In the following section, we explain how this construction generalizes as we increase the num-ber of nonzero pα. Loosely speaking, the number of nonzero rows k in the representation willcorrespond to the number of fields ϕ ∈ Cn that exist at each site of the chain. We will then takeλ ∼ nk consecutive sites together to produce a mapping from the chain to the flag manifold sigmamodel, SU(n−1)/[U(1)]n−1. As we will show, any attempt to derive a sigma model with a differenttarget space manifold will ultimately fail; again, this will be due to spurious zero modes. Beforewe embark on this classification task, it will be useful to introduce some graphical notation that16By n-site structure, we are referring to the presence of an order-n discrete symmetry Zn in the low energy theory.It is not the case that the ground state has n-site order, which would correspond to the spontaneous breaking of thissymmetry.233.3. Classical Ground Stateswill aid us in our calculations.3.3.1 Pictorial representation of classical ground statesAccording to (3.7), to each site of the chain we should assign a set of orthonormalized vectors ϕα.We may use the same basis on each site of the chain, since any local change of basis transformationleaves the Hamiltonian invariant (and fortunately, no superpositions of states arise). Further, wewill use coloured circles to represent the first few elements of this basis, in an effort to visually aidthe reader. This generalizes what we originally did for the Ne´el state in the previous chapter (see(2.16). Our colour dictionary, for the first eight basis elements, can be found in Figure 3.3.ϕ1 = ϕ2 = ϕ3 = ϕ4 =ϕ5 = ϕ6 = ϕ7 = ϕ8 =Figure 3.3: Colour dictionary for the first eight basis elements in Cn. These coloured circles willbe used to pictorially represent ground states throughout.When drawing a classical ground state, we will arrange the same-site vectors into a singlecolumn, and use a white space to separate neighbouring chain sites. For example, the Ne´el state ofthe SU(2) antiferromagnet is· · · · · · , (3.9)while a classical ground state of the adjoint SU(3) chain is· · · · · · (3.10)· · · · · · . (3.11)That this is the ground state of the SU(3) adjoint chain will be demonstrated below.The benefit of these ground state pictures is that it makes it easy to read off the energy cost ofa term tr[S (i)S ( j)]. Indeed, in a classical ground state, each colour corresponds to a standard unitvector ϕα, and we have according to (3.8),tr[S (i)S ( j)] =∑α,βpαpβ|ϕα,∗(i) · ϕβ( j)|2. (3.12)The right hand side of this expression vanishes unless one of the complex unit vectors (i.e. one ofthe colours) at site i equals one of the complex unit vectors at site j. In this case, the right hand243.4. Classification of SU(n) Chainsside equals pα0 pβ0 , where α0 and β0 are the respective positions of the unit vector/colour in columni and column j. To visualize this, it is useful to imagine bonds between all of the circles of the twocolumns, as in Figure 3.4. These bonds are inactive (meaning zero energy cost), unless two nodesare the same colour. For example, the Ne´el state in (3.9) has an energy cost of zero per site (recallwe have shifted the S αβ matrices by a constant), while the classical ground state of the adjointchain, (3.10), has an energy cost of p22 per site.p1p3p3p2Figure 3.4: Left: Imaginary bonds between two sets of vectors ϕα and ϕβ at sites i and j of thechain. Right: The energy cost of each bond is pα0 pβ0 whenever two nodes have the same colour;α0 and β0 are the column positions of these nodes.3.4 Classification of SU(n) ChainsAs a warm-up to the more complicated representations below, we begin with the totally symmetricrepresentations that have Young tableaux with a single row of length p1. According to (3.12),any configuration that has no energy cost per bond will be a classical ground state of the nearest-neighbour SU(n) Hamiltonian. Since only one node is present at each site, the Ne´el state is such anexample. However, as mentioned above, the basis at each site is larger than 2 for n > 2 (i.e. thereare more than 2 colours available), and this leads to an infinite number of ground states. Indeed,the ground state· · · · · · , (3.13)which exists for n > 2, is related by a zero-energy transformation to the Ne´el state. This lo-cal degree of freedom is an example of a zero-energy mode, and destabilizes any candidate groundstate above which we would like to derive a quantum field theory. As a result, the nearest-neighbourHamiltonian must be modified if we would like to proceed. Since it is believed that longer-range in-teractions may be dynamically generated from the nearest-neighbour model [109], we add further-neighbour interactions to realize a stable ground state. Since there are n possible colours, werequire up to (n−1)-neighbour interactions, all of which are taken to be antiferromagnetic, in order253.4. Classification of SU(n) Chainsto remove the zero modes.17 For example, in SU(5), with interactions up to 4th-neighbour, onesuch ground state is· · · · · · . (3.14)While this large number of interaction terms may seem contrived, there is a second reason whyone should consider adding them. Arguably, it is the simplest way to restrict to classical groundstates that have a Zn symmetry, which is to be expected for the symmetric chains, since this is afeature of the integrable SU(n) chains, that correspond to p = 1 (see Section 3.5). Moreover, inthe following two chapters, we will explicitly demonstrate how the SU(n)/[U(1)]n−1 sigma modelarises as the low energy description of this longer-range Hamiltonian. In short, while the on-sitematrix S lines in CPn−1, by coupling n neighbouring sites, our underlying degrees of freedom areactually n orthogonally coupled CPn−1 fields, which is equivalent to SU(n)/[U(1)]n−1.In the more general representations with more than one Young tableau row, we will see a simi-lar pattern: collections of CPn−1 fields from neighbouring sites will become orthogonally coupled,ultimately leading to the flag manifold SU(n)/[U(1)]n−1. Actually, the fields at each site will cor-respond to a Grassmannian manifold in this more general case. If the following discussion seemstoo abstract, we encourage the reader to first turn to Chapter 4, where the symmetric representationSU(3) chains are considered in great detail.Loosely speaking, the number of nonzero rows k in the representation will correspond to thenumber of fields ϕ that exist at each site of the chain. We can think of these k fields as living in theproduct space Pk,n := [CPn−1]k of k copies of CPn−1. We will then take λ := nk consecutive sitestogether to produce a mapping to the complete flag manifold SU(n)/[U(1)]n−1. In other words, weconstruct an embedding,SU(n)/[U(1)]n−1 ↪→ Pk,n × · · · × Pk,n, (3.15)into the λ-fold product of Pk,ns. However, this isn’t the whole story, since the k fields on eachsite (those within Pk,n) can still be locally rotated amongst one another, leading to additionalzero modes. This is resolved by adding a λ-neighbour interaction that freezes out these addi-tional degrees of freedom, which is essentially mimicking what happens when a representation Ris coupled to its conjugate representation R. It corresponds to the embedding Gk,n ↪→ Pk,n, whereGk,n := U(n)/[U(k) ×U(n − k)] is a Grassmannian manifold. In other words, by adding a sufficientnumber of interaction terms to our Hamiltonian, we end up with the following embedding of a flagmanifold into a λ-fold product of Grassmannians:SU(n)/[U(1)]n−1 ↪→ Gk,n × · · · ×Gk,n. (3.16)The argument that we have just sketched assumes that it is always possible to derive a completeflag manifold sigma model description of an SU(n) chain. However, this is not entirely true: thereis a large family of representations that do not admit such a mapping, namely those representationsthat satisfy pα = pβ , 0 for at least two rows of the Young tableau. For such representations, this17This is consistent with [109], which found that the energetically favoured state should have n colours.263.4. Classification of SU(n) Chainsproduces a factor of U(2) in the quotient group H in (3.6). While this also occurs when we havetwo rows of zero length, as in the symmetric representations of SU(4), the result is fundamentallydifferent. In both cases, there are spurious local degrees of freedom on each site (correspondingto rotating the ϕ fields into each other); however, the trick of adding a λ-range interaction doesnot freeze this additional symmetry in the case of pα = pβ , 0. This should become apparentbelow. As a result, for the most general representation of SU(n), we do not expect that a well-defined mapping to the SU(n)/[U(1)]n−1 sigma model exists, and so henceforth we restrict to therepresentations that satisfy pα , pβ for all nonzero row lengths. Our justification for this restrictionshould be clear in Chapter 5, and it has to do with the classification of ’t Hooft anomalies in thesetheories. As we will explain later, any generalization of Haldane’s conjecture is greatly bolsteredby the presence of an ’t Hooft anomaly, and at this time, only the SU(n)/[U(1)]n−1 sigma modelshave been show to have this property. A second justification follows from Chapter 7, where weinterpret the Haldane gap as being generated by fractional instantons.3.4.1 Representations with n − 1 rowsWe now consider representations of SU(n) whose Young tableau have n − 1 nonzero rows. Aspreviously stated, we will assume throughout that no two rows have the same length. Arguably,these representations are simpler than the symmetric ones considered above. Since the on-siterepresentation of the S matrix already corresponds to the manifold SU(n)/[U(1)]n−1, a nearest-neighbour Heisenberg interaction is sufficient to derive the associated sigma model. Let us firstdemonstrate this in SU(3). The interaction term for a classical ground state,tr[S ( j)S ( j + 1)] =2∑α,β=1pαpβ|ϕα,∗( j) · ϕβ( j + 1)|2, (3.17)is never zero for two adjacent sites, which requires choosing the colour for four nodes. Sincep1 > p2, the minimum is p22, which is achieved when the two same-colour nodes are in the secondposition of the column. Thus, a typical ground state in SU(3) looks like· · · · · ·· · · · · · ,which is precisely what we drew above for the adjoint SU(3) chain (which corresponds to the casep1 = 2, p2 = 1). Note that no local transformations exist that cost zero energy: all of the p2 nodesmust stay the same colour in order to minimize the tr[S ( j)S ( j + 1)] term, and the remaining twocolours behave just like the Ne´el state of SU(2). We will return to these chains near the end ofChapter 4.In SU(4), the term tr[S ( j)S ( j + 1)] requires the introduction of six coloured nodes. Using the273.4. Classification of SU(n) Chainsinequality p22 + p23 ≥ 2p2 p3, we see that the ground states have the following form:· · · · · ·· · · · · ·· · · · · · .This pattern extends to general n: the first row of nodes establishes a Ne´el-like state, while theremaining n−2 rows have a “reverse-ordered” pattern: the colour ordering along a column switchesdirection between even and odd sites. Let us explain how this works. For general n, our task is tominimize (3.12). Using orthonormality, this reduces totr[S ( j)S ( j + 1)] =n∑α,β=1pαpβ|ϕβα( j + 1)|, (3.18)where we’ve defined pn := 0. Since ϕβ( j + 1) = ϕα′for some α′ (and all of the α′ are distinct), wemay introduce a permutation operator on the set of n elements, σ : {1, 2, . . . , n} → {1, 2, . . . , n} thatobeysϕβ(i + 1) = ϕσ(β)and rewrite (3.18) astr[S ( j)S ( j + 1)] =n∑α=1pαpσ(α). (3.19)Thus, our task amounts to finding the permutation σ that minimizes (3.19). By defining a vector~x := (p1, p2, . . . , pn), we can think of σ as specifying a second vector ~y; (3.19) is then their dotproduct. Since the entries of both ~x and ~y are nonnegative and nondegenerate, it is clear how tochoose σ so that ~y is as orthogonal to ~x as possible:• Since p1 is the largest component of ~x, we assign to σ(1) the smallest possible component of~y, which is pn.• Next, assign to σ(2) the second smallest possible component of ~y, which is pn−1.• Repeating this procedure, we see that indeed the classical Hamiltonian is minimized by thereverse-ordered ground state, corresponding to a permutation operatorσ : i 7→ k + 1 − i.The basis states at site j + 1 are ϕβ( j + 1) = ϕσ(β).The above procedure will work for all n. For another example, here is a ground state in SU(5):· · · · · ·· · · · · ·· · · · · ·· · · · · · .For these representations, the unit cell is always 2 sites in length, which leads to a Z2 translationsymmetry in the sigma model.283.4. Classification of SU(n) Chains3.4.2 Representations with row number dividing nLet k be the number of nonzero rows in the Young tableau, and write n = kλ. In this case, thematrix S at each site of the chain lies neither in CPn−1 nor SU(n)/[U(1)]n−1. While it would bestraightforward to derive other families of flag manifold sigma models from these representations,we are only interested in those with target space SU(n)/[U(1)]n−1, as these models have been shownto exhibit an ’t Hooft anomaly in certain cases. Thus, some care must be taken in order to realizethe appropriate degrees of freedom. As before, we begin with an example, this time with k = 2 inSU(4). This requires choosing four colours for four nodes in order to minimize the tr[S ( j)S ( j + 1)]term, which is easily done. For example:· · · · · ·· · · · · · .However, such a configuration does not lead to the manifold SU(n)/[U(1)]n−1, because the fourcolours do not behave like four orthogonal CP3 fields. Indeed, at each site, we may additionallyrotate the two colours into each other at no energy cost, which corresponds to another type of zeromode. In order to achieve the correct flag manifold, we “freeze out” these additional degrees offreedom by adding a weaker second-neighbour interaction, tr[S ( j)S ( j + 2)]. The effect of thisterm is to invoke a “reverse-ordering” pattern between sites and their second-neighbours. The newground state is· · · · · ·· · · · · · .The fact that this ground state minimizes the combined J1tr[S ( j)S ( j+1)]+J2tr[S ( j+1)S ( j+2)]term (for antiferromagnetic couplings J1 J2) follows from the identity p21 + p22 ≥ 2p1 p2. In asense, this second-neighbour interaction generates the same behaviour that we saw in the previouscase of k = n − 1: The nearest-neighbour term partitions the colours into subsets, and the second-neighbour term reverse-orders these subsets, effectively breaking the additional on-site rotationsymmetry between colours. In the k = n − 1 case, both of these steps are achieved by the sameinteraction term: first the colours are partitioned into 3 sets: e.g. { }, { }, { , , }, and then eachset is reverse-ordered compared to the previous time it occurred. It will turn out that this reverse-ordering is a generic feature of all the representations that we consider.As a next step, we extend from 4 to general even n, and consider k = n2 . A nearest-neighbourinteraction will again serve to partition the colours into two sets, leaving a local rotation symmetryamong the k colours on each site. In order to freeze out these degrees of freedom, we again add asecond-neighbour interaction, which reverse-orders each set. For example, in SU(6) we have· · · · · ·· · · · · ·· · · · · · .293.4. Classification of SU(n) ChainsClearly, the ground state will always have a 4-site unit cell for k = n2 .Now, when k < n2 , the full set of colours is no longer used up when the nodes on two neigh-bouring sites are filled. As a result, additional zero modes are present that cannot be removed byreverse-ordering the colours within a set. To resolve this, we first add up to (λ − 1)-neighbour in-teractions (always with antiferromagnetic couplings), to properly partition the full set of n coloursinto λ sets of k elements (λ := nk ). Then, we add a weaker λ-neighbour interaction which serves toreverse-order within each set of the partition. For example, in SU(6) with k = 2, the Hamiltonianwe should consider isH =∑j(J1tr[S ( j)S ( j + 1)] + J2tr[S ( j)S ( j + 2)] + J3tr[S ( j)S ( j + 3)]), (3.20)with J1 > J2 J3 > 0, which has, for example, the following ground state:· · · · · ·· · · · · · .The J1 and J2 terms serve to partition the colours into three sets: { , }, { , }, { , }, and the J3terms serve to reverse-order within each of these three sets. Based off of this example, we can seethat the unit cell has size 2λ for these representations.3.4.3 Remaining representationsFinally, we consider all remaining values of k (the number of nonzero rows in the Young tableau).Let c = n mod k, so that n = λk + c for some λ ∈ Z. As in the previous case of n = kλ, localzero modes will be present unless sufficiently long-range interactions are included to use up all ofthe available colours. We add up to λ-neighbour terms, which partitions the colours into λ sets ofk elements, and one set of c elements. Briefly, we return to the notation ϕα for the basis vectorsinstead of coloured nodes. Then, a possible partitioning of the colours is:{ϕ1, . . . ,ϕk}, {ϕk+1, . . . ,ϕ2k}, . . . , {ϕ(λ−1)k+1, . . . ,ϕλk}, {ϕλk+1, . . . ,ϕλk+c}. (3.21)Now, in order to minimize the interaction term tr[S ( j)S ( j+λ)], the remaining nodes on the (λ+1)thsite will be the reverse-ordered set {ϕk,ϕk−1, . . . ,ϕc+1}. For example, in SU(7) with k = 3, threeconsecutive sites may look like· · · · · ·· · · · · ·· · · · · · .303.4. Classification of SU(n) ChainsThe nodes of the next site (which is a (λ + 1)th neighbour), will then begin to be filled with theremaining {ϕ1, . . . ,ϕc} colours from the first site. In our present SU(7) example, this looks like:· · · · · · (3.22)· · · · · ·· · · · · · .Since c = 1 in this example, the drawn ground state is stable. However for c > 1, there will stillbe zero modes associated with rotating among the set {ϕ1, . . . ,ϕk}. Thus, an additional (λ + 1)-neighbour interaction must also be added! The following ground state for SU(5) with k = 3demonstrates this:· · · · · · (3.23)· · · · · ·· · · · · · .Thus, we are led to the following conclusion for this class of representations: If c = 1, ourHamiltonians should contain up to λ-neighbour interactions, and if c > 1, we should also add anadditional (λ + 1)-neighbour interaction term.Using the emerging patterns in the previous examples as a guide, we may now determine theunit cell size for the most general representation. This quantity is very important, as it determinesthe translation group symmetry that is present in the flag manifold sigma model. Note that in both(3.22) and (3.23), there are two competing types of order among the coloured nodes. The first crows exhibit one type of order, which has periodicity λ + 1 when c = 1, and 2(λ + 1) otherwise.Meanwhile, the remaining k − c rows have a periodicity 2λ for all c except c = k − 1, in whichcase the periodicity is λ. In order to determine the overall unit cell length, we must find the leastcommon multiple of these two periodicities. For example, in our SU(7) example, we see that theunit cell will have length 12, leading to a Z12 symmetry in the field theory. This can be seen inFigure 3.5.3.4.4 Summary of classificationIn Table 3.1, we summarize our results from the previous subsections. In the first column, wespecify the number of nonzero rows in the Young tableau, k, and the integer c := n mod k. In thesecond column, we write down the longest-range interaction that must be included in the Hamilto-nian in order to eliminate any local zero modes. As always, it is understood that each interactionterm is Jrtr[S ( j)S ( j + r)] for some coupling Jr > 0, and that Jr > Js for r < s. Finally, in thethird column, we specify the order d of the translation group Zd that acts on the corresponding flagmanifold sigma model. This order equals the size of the unit cell in the classical ground states ofthe Hamiltonian. In the final column of the table, the following identities are useful:lcm[2λ, (λ + 1)] =λ(λ + 1) λ is odd2λ(λ + 1) λ is even (3.24)313.5. Exact Results for SU(n) Chainsperiod 3︷ ︸︸ ︷· · · · · ·· · · · · ·· · · · · ·︸ ︷︷ ︸period 4Figure 3.5: Potential ground state of an SU(7) chain. Since the first row has 3-site periodicity, andthe remaining rows have 4-site periodicity, the unit cell is 12 sites in length.lcm[λ, 2(λ + 1)] =2λ(λ + 1) λ is oddλ(λ + 1) λ is even (3.25)Representation Longest Interaction Translation Group Orderk = 1 Jn−1tr[S ( j)S ( j + n − 1)] nk = n − 1 J1tr[S ( j)S ( j + 1)] 2k = nλ , λ < n Jλtr[S ( j)S ( j + λ)] 2λn = 2λ + 1, k = 2 Jλtr[S ( j)S ( j + λ)] λ(λ + 1)n = kλ + 1, λ > 1, k > 2 Jλtr[S ( j)S ( j + λ)] lcm[2λ, (λ + 1)]n = kλ + c, c , 1, k − 1 Jλ+1tr[S ( j)S ( j + λ + 1)] 2λ(λ + 1)n = kλ + (k − 1) Jλ+1tr[S ( j)S ( j + λ + 1)] lcm[λ, 2(λ + 1)]Table 3.1: Classification results of all SU(n) representations satisfying pα , pβ for all nonzero pα.We use the notation lcm[a, b] to denote the least common multiple of a and b.3.5 Exact Results for SU(n) ChainsIn this final section, we now review various exact results that are known for the SU(n) Hamiltoni-ans considered above. These exact results will prove to be especially useful when we attempt togeneralize the Haldane conjecture to certain SU(n) chains in later chapters.323.5. Exact Results for SU(n) Chains3.5.1 Integrable SU(n) chains and SU(n)k critical pointsLet us begin our discussion with the simplest representation of SU(n): the fundamental represen-tation whose Young tableau consists of a single box. While this is a special case of the symmetricmodels discussed above, we do not need to consider longer-range interactions, since in this case,the model is exactly solvable. This was first realized by Sutherland [84], who used the Betheansatz to find the exact spectrum of the p = 1 SU(n) chain, and show that it was gapless. Later,it was discovered that the low energy degrees of freedom of this chain correspond to the SU(n)1WZW model [50, 85, 110]. This generalizes the SU(2) result of Bethe that was mentioned in Chap-ter 2: The low energy spectrum is gapless, with n − 1 relativistic bosons propagating with velocityv = 2piJ/n [83, 84].In order to understand this result, we may rewrite the nearest-neighbour SU(n) Hamiltonian interms of fermion operators. To this end, we introduce n flavours of fermion ψα and rewrite thematrices S of SU(n) generators according toS αβ( j) = ψ†β( j)ψα( j). (3.26)There is only one independent Casimir constraint for the p = 1 model, tr[S 2], which in the fermionlanguage, corresponds to the condition of 1 particle on each site:ψ†α( j)ψα( j) = 1. (3.27)The nearest-neighbour Hamiltonian becomes a simple exchange term:H = J∑jψ†β( j)ψα( j)ψ†α( j + 1)ψβ( j + 1). (3.28)As in SU(2), we may obtain this model from an SU(n) Hubbard model,H =∑j(−t[ψ†α( j)ψα( j + 1) + h.c.] + U[ψ†α( j)ψα( j) − 1]2)(3.29), in the limit U/t 1. Starting at small U, we may take the continuum limit, giving n flavours ofrelativistic Dirac fermion. Next, by using non-abelian bosonization, we obtain a charge boson plusthe SU(n)1 WZW model. The Hubbard interactions can be seen to gap the charge boson withoutaffecting the low energy behaviour in the spin sector, yielding the SU(n)1 WZW model as the lowenergy effective theory [86].It is worth mentioning that without any additional symmetries, the SU(n)1 model is actualunstable, due to the relevant operator tr[g], for WZW field g ∈ U(n). However, for the SU(n)chains presently discussed, at low energies there is an additional discrete symmetry, Zn, whoseaction on g isg 7→ e 2piin g, (3.30)and forbids the term tr[g]. In our longer-range models, a similar symmetry also emerges, as wewill see in the following chapters. There, it will be a manifestation of the underlying translation333.5. Exact Results for SU(n) Chainssymmetry on the chain. Thus, the results in Table 3.1 should be quite suggestive of how Haldane’sconjecture should be extended to SU(n).For larger p, there exist integrable SU(n) models that can also be solved exactly [87–89]. Thesemodels have more complicated nearest-neighbour interactions, and have been shown to correspondto the SU(n)p WZW models at low energies [50, 68, 111]. Again, we can understand this from non-abelian bosonization. In this case we must introduce fermions with p colours as well as n flavours,and write a generalized Hubbard model. Non-abelian bosonization now gives the SU(n)p WZWmodel in the flavour sector, together with a charge boson and an SU(p)n WZW model for the colourdegrees of freedom. However it is now seen that the Hubbard interactions will generally gap theflavour sector as well as the charge and colour sector, unless the interactions are fine-tuned. Thiscan be understood from the fact that the SU(n)p WZW models contain relevant operators allowedby symmetry for all p > 1, which are expected to appear in the Hamiltonian and destabilize thecritical theory. We understand the fine-tuned nature of the Bethe ansatz integrable models as,remarkably, corresponding to fine-tuning of the field theory to eliminate all relevant operators.In Section 2.3.2, we explained how the gapless phase predicted by Haldane in half-odd integerspin chains is characterized by the SU(n)1 WZW model. We can now better explain why this isthe case. For arbitrary s, the spin chain may be mapped to the SU(2)k WZW model, with levelk = 2s [112, 113]. As mentioned above, these models will generically contain relevant operators(unless they are fine-tuned, corresponding to the integrable models). Thus, under renormalization,we expect the SU(2) chain to be described by some other, stable critical theory. With the added Z2symmetry arising from a 2-site unit cell, the only candidate theory is SU(2)1. Thus, if a gaplessphase is to occur in the spin chain, we predict it will be characterized by this ‘attractor’ theory. Thisis consistent with the Zamolodchikov c-theorem, which requires that k be a nonincreasing functionof energy scale [114].Now, the obvious question is how to generalize this mechanism to SU(n). As hinted above,when the Zn symmetry (3.30) is present, the SU(n)1 WZW model is stable. In fact, it is the onlystable WZW model with SU(n) symmetry [78, 79]. For instance, the primary operator transformingunder the adjoint representation of su(n) is a relevant operator in all levels k > 1. Therefore, onemight reasonably conclude that whenever an SU(n) chain can be mapped to an SU(n)k criticaltheory, it will flow to SU(n)1 at low energies. Unfortunately, the flow from k 7→ k′ < k is actuallyquite subtle for general n, and is related to the concept of ’t Hooft anomaly matching [115, 116].We will return to this issue in the following two chapters.Let us conclude this subsection by making a distinction between the low energy sigma modelsthat we seek in this thesis, and the Bethe ansatz integrable models mentioned above. Like Haldane,our motivation is to predict gapless phases only at weak coupling, which lies outside the strong-coupling regime of the integrable models. Thus, it is not appropriate to view our models as aperturbation of the integrable ones, if we want to make any weak coupling predictions. However,this is not to say that we aren’t interested in what happens when we add relevant operators to SU(n)kfor k > 1. For example, the operator |tr[g]|2 is a relevant perturbation for k > 1 and may lead toa symmetry broken phase. However, we believe that this mechanism relies crucially on the signof the coupling constant in front of this term, and the gapped broken symmetry phase only occurs343.5. Exact Results for SU(n) Chainswhen this constant is negative. When it is positive, we instead expect a massless RG flow to SU(n)1(so long as the other constraints on the flow of k do not apply). Note that both of these scenarios areconsistent with the LSMA theorem (reviewed below), and are present in the SU(2) phase diagramFigure 2.1. This issue is discussed in detail by Affleck and Haldane in [86].3.5.2 The Lieb-Schultz-Mattis-Affleck theoremThe Lieb-Schultz-Mattis-Affleck theorem (LSMA) is a rigorous statement about ground states intranslationally invariant SU(n) Hamiltonians [80, 81]. It claims that if the number of boxes p in aYoung tableau is not a multiple of n, then either the ground state is unique with gapless excitations,or there is a ground state degeneracy. Recently, it was claimed in [67] that the LSMA theorem is notapplicable to models with longer-range interactions than nearest-neighbour. Here, we dispute thisclaim by extending the original proof in [81] to models with further-range interactions. Explicitly,we consider the following Hamiltonian on a ring of L sites:H =R∑r=1Hr Hr :=L∑j=1Jrtr[S ( j)S ( j + r)] (3.31)where S is defined as above. We assume that |ψ〉 is the unique ground state of H, and is transla-tionally invariant: T |ψ〉 = |ψ〉. We then define a twist operatorU = eA A :=2piinLL∑j=1jQ( j) (3.32)withQ =n−1∑α=1S αα − (n − 1)S nn = trS − nS nn = p − nS nn. (3.33)Note that A is a Hermitian operator. Here p =∑α pα is the total number of boxes occurring in theYoung tableau. Using the commutation relations (3.1), it is easy to verify that[tr[S ( j)S ( j + r)],Q( j) + Q( j + r)]= 0, (3.34)which then impliesU†tr[S ( j)S ( j + r)]U = e−rpiinL (Q( j+r)−Q( j))tr[S ( j)S ( j + r)]erpiinL (Q( j+r)−Q( j)). (3.35)Using this, one can show thatU†HU − U = [H, A] − H + O(L−1) (3.36)so that U |ψ〉 has energy O(L−1). Now, using the translational invariance of |ψ〉, we find〈ψ|U |ψ〉 = 〈ψ|T−tUT |ψ〉 = 〈ψ|Ue 2piin Q(1)e− 2piinL∑Lj=1 Q( j)|ψ〉. (3.37)353.5. Exact Results for SU(n) ChainsSince |ψ〉 is a ground state of H, it is an SU(n) singlet, and so must be left unchanged by the globalSU(n) transformation e−2piinL∑Lj=1 Q( j). Moreover, using (3.33), we have〈ψ|U |ψ〉 = e 2piipn 〈ψ|Ue2piiS nn |ψ〉. (3.38)In Chapter 2, we saw that the matrices S could be written in terms of Schwinger bosons. Thisis true for all n, as we will see in Chapters 4 and 5 when we discuss flavour-wave theory. Thediagonal elements S nn always correspond to number operators for these bosons. Thus, S nn actingon |ψ〉 will always return an integer, and e2piiS nn can be dropped. Thus, we find that so long as p isnot a multiple of n,〈ψ|U |ψ〉 = 0, (3.39)implying that U |ψ〉 is a distinct, low-lying state above |ψ〉. This completes the proof. Finally, wemay also comment on the ground state degeneracy in the event that a gap exists above the groundstate. Through the repeated application of (3.38), we have〈ψ|Uk|ψ〉 = e 2piipkn 〈ψ|U |ψ〉. (3.40)So long as k < r := n/ gcd(n, p), the family {Uk|ψ〉} is an orthogonal set of low-lying states. Ifan energy gap is present, this suggests that the ground state is at least r-fold degenerate. In thefollowing subsection, we introduce valence bond solids that exhibit translation invariance when pis a multiple of n. It is easy to see that when 1 < gcd(n, p) < n, these constructions lead to explicitr-fold degeneracies, consistent with our findings here. For example, the symmetric representationof SU(4) with p = 2 leads to 2 distinct valence bond solids that are both invariant under translationby 2 sites.3.5.3 Affleck-Kennedy-Lieb-Tasaki modelsIn Chapter 2, we introduced the AKLT model as an important theoretical result that served tobolster Haldane’s conjecture in its infancy. This is because it exhibits a unique, translationallyinvariant ground state with a finite excitation gap for a spin-1 chain. Note that this does not vi-olate the LSMA theorem since a spin-1 chain has a corresponding Young tableau with 2 = nboxes. Recently, the AKLT construction has been generalized by various research groups to SU(n)chains [61, 67, 69, 74–76, 83, 117]. In particular, symmetric representation AKLT Hamiltonianshave been constructed whenever p is a multiple of n [67]. Again, this is precisely when the LSMAtheorem does not apply. These Hamiltonians exhibit a unique, translationally invariant groundstate, and are constructed by forming SU(n) singlets from n fundamental representations on adja-cent sites. Since p is a multiple of n, this may be done in a translationally invariant manner, byshifting the left-most site of the singlet along the chain as one cycles through the p (symmetrized)fundamentals at each site of the chain. See Figure 3.6 for the case of n = p = 3. This picture canbe understood as follows. If we denote by αaj a fundamental representation of SU(3) at site j, thenat each site we have|α1j , α2j , α3j〉, (3.41)363.5. Exact Results for SU(n) Chainswhich is symmetric under permutation of its entries (by definition). By considering three adjacentsites, we may construct an SU(3) singlet out of three of these entries:α1jα2j+1α3j+2|α1j , αaj , αbj〉|α2j+1, αcj+1, αdj+1〉|α3j+2, αej+2, α fj+2〉. (3.42)Here ab is the antisymmetric unit tensor with 01 = −10 = 1. The remaining free representationsαaj , . . . , αej+2 are similarly combined with adjacent sites, eventually creating an SU(3) singlet out ofthe entire chain.Figure 3.6: AKLT state for an SU(3) chain in the symmetric representation with p = 3. Singletsare formed from three consecutive sites.Note that when n is not prime, similar valence bond solids may be constructed that are con-sistent with a nontrivial result from the LSMA theorem. That is, for p not a multiple of n, withr := n/ gcd(n, p), Hamiltonians may be constructed with r-fold degenerate ground states that areinvariant under translations by r sites. In Figure 3.7, we provide an example of two-fold degenerateground states in SU(4). All of these models have short range correlations, and are expected to havegapped ground states, based on arguments of spinon confinement [67].Figure 3.7: Two-fold degenerate AKLT states for an SU(4) chain in the symmetric representationwith p = 2.Another generalization of AKLT models is possible for self-conjugate representations of SU(n).We will make use of this fact explicitly in Section 4.7, when we discuss self-conjugate SU(3)chains. Recall that such representations are labeled by a single integer p > 0, and have a Young373.5. Exact Results for SU(n) Chainstableau of the form2p︷ ︸︸ ︷. (3.43)For all p, we may write down a translationally invariant valence bond state. This is done byforming SU(3) singlets at each site, by pairing each fundamental representation with an adjacentantifundamental representation. For example, in the adjoint representation of SU(3) with p = 1, ateach site we have|α1j , α2j ;α3j〉 (3.44)which is antisymmetric across the ‘;’ symbol. Then a singlet bond between sites j and j + 1 isconstructed as follows:α1j ,α2j+1,α3j+1|α1j , αaj ;αbj〉|αcj+1, α2j+1;α3j+1〉. (3.45)This is shown in Figure 3.8. Here and throughout, objects appearing within the same columnare symmetrized, and two nodes that appear close together in the same row are antisymmetrized.Dashed lines denote singlet bonds, as before.Figure 3.8: AKLT state for an SU(3) chain in the self-conjugate representation with p = 1. Thisstate spontanesouly breaks parity symmetry.However, only when p is even is this ground state also invariant under a Z2 parity symmetry,which corresponds to a mirror reflection about the center of a bond between two sites of the chain.This is demonstrated in Figure 3.9 for the case of p = 2. Similar constructions exist for all n.Figure 3.9: AKLT state for an SU(3) chain in the self-conjugate representation with p = 2. Thisstate is invariant under parity symmetry.Finally, there is one more version of AKLT model that we will discuss in this thesis. In Chap-ter 6, we will be led to SU(n) representations whose Young tableaux have two rows, with lengths383.5. Exact Results for SU(n) Chainsp1, p2 satisfying p1 , p2 , 0. We note here that when p1 + p2 = n, it is straightforward to writedown a translationally invariant AKLT state. Indeed, we may construct a singlet over p1 consec-utive sites using (p1 − p2) fundamentals and p2 antisymmetric doublets. Since each site has p1representations (either fundamentals or doublets), we may shift the singlet by one site as we movedown the rows of the valence bond solid. Note that the self-conjugate SU(3) AKLT model is aspecial case of this more general construction.As a new example, let us explain this construction in greater detail for the case of SU(4), withp1 = 3 and p2 = 1. We denote by αaj a fundamental representation of SU(4) at site j. Then on eachsite of the chain, we have the representation|α1j , α2j , α3j ;α4j〉, (3.46)which is symmetric under permutations of the first three entries, and antisymmetric under ex-changes with the fourth entry. For instance,|α1j , α2j , α3j ;α4j〉 = |α2j , α1j , α3j ;α4j〉 = |α2j , α3j , α1j ;α4j〉,|α1j , α2j , α4j ;α3j〉 = −|α1j , α2j , α3j ;α4j〉 = −|α1j , α4j , α3j ;α2j〉.Using two fundamental representations αaj , and one antisymmetric doublet representationFigure 3.10: AKLT state for an SU(4) chain with p1 = 3 and p2 = 1. Singlets are formed fromthree consecutive sites, using two fundamentals and one antisymmetric doublet.|αaj ;αbj〉 = − |αbj ;αaj〉, we may contract indices to form a singlet across three sites according toα1jα2j+1α3j+1α4j+2|αaj , α1j , αbj ;αcj〉|αdj+1, αej+1, α2j+1;α3j+1〉|α4j+2, α fj+2, αgj+2;αhj+2〉. (3.47)The remaining free representations αaj , . . . , αhj+2 are then contracted into different singlet bonds,over different sets of three sites. By using the pattern shown in Figure 3.10, a translationallyinvariant valence bond solid can be constructed, that is also parity symmetric. In fact, for generaleven n, we may always choose a p1-site singlet bond that is symmetric under parity, leading to aparity-symmetric AKLT state. See Figure 3.11a and Figure 3.11b for two additional examples inSU(6).393.6. Conclusion(a) AKLT state for an SU(6) chain with p1 = 5 and p2 = 1. Singlets are formed from five consecutive sites,using four fundamentals and one antisymmetric doublet.(b) AKLT state for an SU(6) chain with p1 = 4 and p2 = 2. Singlets are formed from four consecutivesites, using two fundamentals (single circles), and two antisymmetric doublet (double circle).3.6 ConclusionWe began this chapter by motivating the study of SU(n) chains as a natural extension to the familiarHeisenberg model. Thanks to recent advances in cold atom experimentation, what started out asa purely theoretical study has since developed into an exciting cross-disciplinary effort. Indeed,SU(n) chains now garner interest from condensed matter physicists, cold atom physicists, and eventheoretical high energy physicists.We first introduced the SU(n) Hamiltonian, and the n × n matrices of operators, S . Then, byreviewing the representation theory of SU(n), we were able to relate Young tableaux to the eigen-values of S , and ultimately to a classical Hamiltonian. Then, in exhaustive detail, we consideredall of the possible ways of realizing a particular quantum field theory, the SU(n)/[U(1)]n−1 flagmanifold sigma model, by adding sufficiently long-range interactions to the Hamiltonian in orderto stabilize zero mode fluctuations. The results of this classification are presented in Table 3.1.Then, in Section 3.5, we proceeded to review various exact results that exist for SU(n) Hamilto-nians, namely the LSMA theorem, the AKLT construction, and the Bethe-ansatz integrable SU(n)models. We also hinted at the renormalization flow of these SU(n) theories; this subject will treatedmore thoroughly later on.With this groundwork in place, we would now like to focus in more detail on particular repre-sentations of SU(n). Admittedly, this chapter has been quite abstract, and quite removed from our403.6. Conclusionultimate goal of obtaining an SU(n)-version of the Haldane conjecture. In the following chapter,we turn to SU(3) chains, and connect back to this overarching goal. The results of this chapterwill prove to be useful here, so that the appropriate Hamiltonian can be used, depending on whichSU(3) chain we consider. However, we will also draw on ideas from Chapter 2; in particular, wewill attempt to follow the prescription outlined in Section 2.5. This will lead to unexpected chal-lenges, including the absence of Lorentz invariance, and also the presence of pure-imaginary, yetnon-topological terms in the sigma model.41Chapter 4SU(3) Chains4.1 IntroductionIn the previous chapter, we introduced SU(n) chains in full generality, and discussed how theirclassical ground states, and consequently their underlying symmetries, depended on the chosenSU(n) representation at each site. An important realization was that longer-range interaction termsare necessary to include in the SU(n) Heisenberg Hamiltonian in order to stabilize local zero modesthat are present due to the larger symmetry available.At this point, we are now prepared to derive our first mapping between SU(n) chain and quan-tum field theory, and ultimately arrive at a generalization of Haldane’s conjecture. As will becomeclear quite shortly, this is by no means a trivial task, and will require us to develop substantialrenormalization group results in order to arrive at any meaningful conclusion. As such, we restrictin this chapter to the simplest extension possible; namely, we consider SU(3) chains in the totallysymmetric representations. According to the classification in Table 3.1, these chains require bothfirst and second-neighbour interactions in their Hamiltonian. Therefore, the model that we considerin this chapter isH =∑j(J1tr[S ( j)S ( j + 1)] + J2tr[S ( j)S ( j + 1)]). (4.1)As always, we take J1, J2 > 0. Each matrix S transforms under the symmetric representation,labeled by p, which corresponds to the length of the single row in a Young tableau:p boxes︷ ︸︸ ︷(4.2)Later on, we will comment on the remaining representations of SU(3), and explain how our resultscan be extended to the self-conjugate representations of SU(3), with two-row Young tableaux.For the most part, this chapter will follow the outline presented in Section 2.5. We will beginwith extending the familiar theory of spin-waves to SU(3), and derive the predicted velocities ofsuch low-lying excitations. As mentioned in Chapter 2, despite the lack of spontaneous symmetrybreaking in one spatial dimension, this perturbative treatment still serves its purpose; in particu-lar, we will use this so-called flavour-wave theory to verify the absence of zero-mode excitationswhen second-neighbour interactions are added to the SU(3) Heisenberg Hamiltonian. Next, in Sec-tion 4.3, we derive the SU(3)/[U(1)]2 sigma model description of the symmetric SU(3) chain ingreat detail. Our approach will be generalized in Chapter 5 to all n > 3, and so we make sure to be424.2. Flavour-Wave Theoryvery explicit in our calculations. Following this, we turn to our renormalization group (RG) calcu-lations in Section 4.5. Since this is the first appearance of RG technology in this thesis, we offer abrief introduction to the subject, before turning to the flag manifold sigma model in particular. InSection 4.6, we discuss the implications of these RG results, and present a phase diagram of theSU(3) chain, which generalizes the SU(2) story in Figure 2.1. This phase diagram is supported bythe various exact results that we mentioned in the previous chapter, and we recall them here. Thephase diagram is also supported by a strong coupling analysis, and some Monte Carlo simulations,which we review. At this stage, we are able to formulate our SU(3) version of Haldane’s conjecturefor symmetric chains. Finally, we finish this chapter by turning to the remaining representations ofSU(3) in Section 4.7.The above work on symmetric SU(3) chains was published in 2017 in [1]. Section 4.7 waspublished in [2]. Preceding our papers, Greiter and Rachel considered symmetric SU(n) chainsin [67], which was published in 2007. By constructing valence bond solids, a` la AKLT, they pre-dicted a finite energy gap for those representations whose row length p is a multiple of 3. Theresult that we obtain will be consistent with this prediction. Similar results followed [61, 69, 74].Furthermore, in 2012, Bykov established in two separate papers the first correspondence betweenthe symmetric SU(3) chains and the SU(3)/[U(1)]2 flag manifold sigma model [71, 72]. We werenot aware of Bykov’s contribution at the time of our research, and as such there is a large overlapbetween his calculations and our derivation in Section 4.3. However, the remaining sections ofthis chapter, including the extensive renormalization group calculations in Section 4.5, are entirelynovel. Finally, it is worth noting that various numerical studies have been carried out on the sym-metric SU(3) chains, with contradicting predictions [70, 118–120]. However, these results wererecently surveyed in [77], and these contradictions are now better understood. This publicationalso presented the most convincing results to date, using the density matrix renormalization groupto study the symmetric SU(3) chain with p = 3. Their findings are in complete accord with ourgeneralized Haldane conjecture, namely that a finite energy gap exists in this representation.4.2 Flavour-Wave TheoryAccording to Coleman’s theorem [8], we do not expect spontaneous symmetry breaking of theSU(3) symmetry in the exact ground state of our Hamiltonian. Nonetheless, we may still expandabout the classical (symmetry broken) ground state to predict the Goldstone mode velocities. Ifthe theory is asymptotically free (as in SU(2)), then at sufficiently high energies the excitationsmay propagate with these velocities. In the familiar antiferromagnet, this procedure is known asspin-wave theory; in SU(n) it is called flavour-wave theory [121, 122].To begin, we rewrite the Hamiltonian (4.1) in terms of Schwinger bosons with three flavours.The generators at site j can be written asS αβ( j) = b†β( j)bα( j) (4.3)and the local Hilbert spaces are defined by putting p bosons at each site, i.e. by the Casimir434.2. Flavour-Wave Theoryconstraint ∑αb†α( j)bα( j) = p. (4.4)The resulting Hamiltonian is quartic in bosonic operators:H =∑j(J1b†β( j)bα( j)b†α( j + 1)bβ( j + 1) + J2b†β( j)bα( j)b†α( j + 2)bβ( j + 2)). (4.5)As in SU(2), we proceed by identifying a classical ground state which corresponds to a true groundstate of (4.1) in the limit p → ∞. It is a product wave function in which, at each site, all bosonsare in the same state. Note that if J2 were zero, there would be an infinite degeneracy of potentialstates, and it would not be clear around which state the quantum fluctuations should be introduced.Moreover, this degeneracy would lead to local zero modes with vanishing velocity.The introduction of J2 resolves these issues and imposes a 3-site structure for the referencestate.18 As mentioned previously, this 3-site structure is consistent with the p = 1 Bethe ansatzsolution [12, 84, 85], and has been shown to be the state with minimal zero point fluctuations forlarge p [109]. We view J2 as originating from quantum fluctuations in the nearest-neighbour model,and accordingly, we will apply our conclusions about (4.1) to the nearest-neighbour model withouthesitation. Unfortunately, the exact mechanism by which J2 is generated is not yet understood,and we cannot express J2 as a function of J1; however, on general grounds we at least know that itshould be suppressed by a factor of p−1.Since we have 3 bosons for each of the 3 sites, we have actually introduced 9 bosonic operators.To emphasize this, we introduce the following notation:jα := 3 j + (α − 1). (4.6)The classical ground state involves only ‘diagonal’ bosons of the type bγ( jγ) and b†γ( jγ). The ‘off-diagonal’ bosons are Holstein-Primakoff bosons, like those defined in (2.6). Flavour-wave theoryallows for a small number of Holstein-Primakoff bosons at each site, captured byν( jγ) =∑α,γb†α( jγ)bα( jγ), (4.7)and writes the Hamiltonian in terms of these 6 bosons. In the large-p limit, we have p ν( jγ),and we can expandS γγ = p − ν( jγ),S αγ( jγ) ≈ √pb†α( jγ),S γα( jγ) ≈ √pbα( jγ),18Another coupling that works is a ferromagnetic third-neighbour interaction. This was considered in [1].444.3. Derivation of the Flag Manifold Sigma Modelto findtr[S ( jγ)S ( jη)] = p[b†γ( jη)bγ( jη) + b†γ( jη)bη( jγ) + b†η( jγ)b†γ( jη) + bη( jγ)bγ( jη)]+ O(p0). (4.8)In terms of these degrees of freedom, the Hamiltonian (4.1) decomposes into a sumH = H12 + H13 + H23, (4.9)where Hγη is a Hamiltonian involving only the two boson flavours bγ( jη) and bη( jγ). In momen-tum space, this gives 3 different 2 × 2 matrices, each of which can be diagonalized by the sameBogoliubov transformation:Hγ,γ+1 = const. +∑kω(k)2∑j=1(d†j (k)d j(k) +12)(4.10)whereωt(k) = 2p√J1J2∣∣∣∣∣sin 3ka2∣∣∣∣∣ . (4.11)Therefore, the corresponding flavour-wave velocity isv = 3p√J1J2, (4.12)which is equal for each of the six Goldstone bosons. Later, when we consider SU(n) chains withn > 3, we will learn that different branches of boson will generically propagate with differentvelocities. This will inhibit the automatic emergence of Lorentz invariance in the correspondingflag manifold sigma models.4.3 Derivation of the Flag Manifold Sigma ModelHaving verified that a unique, nonzero Goldstone velocity exists for the J1 − J2 Heisenberg Hamil-tonian in (4.1), we now derive a low energy quantum field theory description of the chain. Sincethe classical ground state has 3-site sublattice order, with unit vectors ϕα ∈ Cn defined on each site(as per Chapter 3), we may define a unitary matrix U, byUαβ = ϕαβ . (4.13)Throughout, a superscript index labels the vector, and a subscript index labels the component ofthe vector. Applying the pictorial notation from Chapter 3, we may represent the classical groundstate at hand by· · · · · · . (4.14)Since the Hamiltonian is invariant under changing the overall phase of each of the three vectorsϕα, (see (3.12)), we see that the ground state manifold is isomorphic to SU(3)/[U(1)]2. Two phases454.3. Derivation of the Flag Manifold Sigma Modelcan be changed independently, later referred to as gauge symmetries, while the third phase is fixedby setting the determinant of U to 1. This manifold has 6 generators, namely the six off-diagonalGell-Mann matrices of SU(3), which corresponds to the same amount of Goldstone bosons foundin the previous flavour-wave calculations.To describe fluctuations about the ϕα, and thus about the ground state manifold, we write:φα =∑β1pLαβϕβ +√1 − µ(α)ϕα. (4.15)Here Lαα = 0 (no sum) andp2µ(α) :=∑β|Lαβ|2. (4.16)These complex coefficients Lαβ describe general fluctuations about the ϕα. The vectors φα arethose that appear in the formula for the SU(n) matrices S in the limit of large p:S αβ( jγ) = φ∗,γα ( jγ)φγβ( jγ). (4.17)This is the generalization to SU(3) of (2.21). By redefining the unitary matrix U, we may take L tobe Hermitian.19 Now, by letting U and L vary uniformly from site to site, we writeS αβ( jγ) = pφ∗,γα ( jγ)φγβ( jγ) = p∑δ,σL˜δγ( jγ)U†αδ( jγ)L˜γσ( jγ)Uσβ( jγ), (4.18)where we’ve defined L˜αβ = 1p Lαβ when α , β, and L˜αα =√1 − µ(α). Explicitly, the L˜ matrix isL˜ =1pp√1 − 1p2 (|L12|2 − |L13|2) L12 L13L∗12 p√1 − 1p2 (|L12|2 + |L23|2) L23L∗13 L∗23 p√1 − 1p2 (|L13|2 + |L23|2) .(4.19)The expression (4.18) can be rewritten asS αβ( jγ) = pL˜γγ( jγ)∑δ,γ(L˜δγ( jγ)U†αδ( jγ)Uγβ( jγ) + U†αγ( jγ)L˜γδ( jγ)Uδβ( jγ))(4.20)+p[L˜γγ( jγ)]2U†αγ( jγ)Uγβ( jγ) + p∑δ,σ,γL˜δγ( jγ)U†αδ( jγ)L˜γσ( jγ)Uσβ( jγ).UsingL˜γγL˜δγ = L˜δγ + O(p−2) (4.21)19The skew components of L generate unitary transformations, and can be recombined with the matrix U.464.3. Derivation of the Flag Manifold Sigma Modelwe haveS αβ( jγ) =∑δ,γ(Lδγ( jγ)U†αδ( jγ)Uγβ( jγ) + U†αγ( jγ)Lγδ( jγ)Uδβ( jγ))(4.22)+p(1 − µ(γ))U†αγ( jγ)Uγβ( jγ) + p−1∑δ,σ,γLδγ( jγ)U†αδ( jγ)Lγσ( jγ)Uσβ( jγ).In matrix form, this isS ( jγ) = pU†ΛγU + U†{L,Λγ}U + p−1U†LγU, (4.23)where Λγ is zero except at entry (γ, γ) where it equals 1:Λ1 = 1 0 00 0 00 0 0 , Λ2 = 0 0 00 1 00 0 0 , Λ3 = 0 0 00 0 00 0 1 . (4.24)We’ve also definedLαβ( jγ) = Lαγ( jγ)Lγβ( jγ) − p2µ(γ)[Λγ]αβ. (4.25)Let us pause for a moment and reflect on our calculations. Note that at no stage did we explicitlyuse the fact that U, L and Λγ are 3 × 3 matrices. Indeed, the above calculations carry over directlyto SU(n) when we promote the objects ϕα to lie in Cn instead of C3. This observation will allowus in Chapter 5 to immediately use the result (4.23) without further proof.Using (4.23), we now evaluate the trace terms appearing in (4.1). Again, this derivation willhold for general n×n matrices, and we will refer to these calculations in Chapter 5. Since the matri-ces U and L are evaluated at different sites, we Taylor expand which introduces spatial derivatives.For exampleU( jγ) = U(n j + (γ − 1)) = U( jη) + (η − γ)∂xU( jη) + 12(η − γ)2∂2xU( jη) + · · · , (4.26)where we’ve assumed the derivative is uniform: ∂xU( jη) = ∂xU( j′λ). Expanding in powers of Land p−1, we find after some tedious algebra,tr[S ( jγ)S ( jη)] = p2(η − γ)2tr[U∂xU†Λγ∂xUU†Λη] (4.27)+2(η − γ)p(Lηγ[∂xUU†]γη + Lγη[U∂xU†]ηγ)+ 4|Lηγ|2 + const.For the complete derivation of this result, see Appendix B. This completes the second item on therecipe presented in Section 2.5.474.3. Derivation of the Flag Manifold Sigma Model4.3.1 Coherent state path integralHaving rewritten the Hamiltonian in terms of U and L, we now derive the Lagrangian by using acoherent state path integral approach [99, 100, 123]. Our approach mimics the procedure outlinedin Section 2.2.1. As a complete set of states, we introduce|φ〉 = (φ · a)p|0〉, (4.28)with φ ∈ Cn. As in SU(2), the a are bosonic operators, known as Schwinger bosons. Again,we promote all of the fields to n-complex dimensional objects, so that we may make use of thisderivation again in Chapter 5. These states correspond to an element of the rank-p symmetricrepresentation of SU(n), [φ]p, acting on a highest-weight state in the Hilbert space:|φ〉 = [φ]p|highest weight〉. (4.29)The resolution of the identity is then the integration over all of SU(n) of the projection |φ〉〈φ|:1 =∫Dφ|φ〉〈φ|. (4.30)Inserting this between each time slice τi of the partition function, we obtain terms of the form〈φ(τi)|e−Hδτ|φ(τi+1)〉 = 〈φ(τ)|φτ+δτ〉e−Hδτ. (4.31)Exponentiating these terms, we find the following contribution to the action:∏i〈φ(τi)|φ(τi+1)〉 ∝ (1+φ∗(τi) ·∂τφ(τi))p = exp p log∑i(1+φ∗(τi) ·∂τφ(τi)) ≈ exp p∫dτφ∗ ·∂τφ,(4.32)where we’ve used 〈φ|φ′〉 = (φ∗ · φ′)p. Now, using (4.15), we have∂τφαβ =∑γ∂τL˜αγUγβ + L˜αγ∂τUγβ, (4.33)where L˜ is defined below (4.18). We neglect time derivatives of L˜, which are already small fluctu-ations. Then we haveφ∗,α · ∂τφα =∑δ,γ,βL˜δαU†βδL˜αγ∂τUγβ (4.34)=∑δ,α∑β[LδαU†βδ∂τUαβ + U†βαL˜αδ∂τUδβ + (1 − µ(α))U†βα∂τUαβ]+ O(p−2)= tr[Λα∂τUU†] + p−1tr[{Λα, L}∂τUU†] + O(p−2). (4.35)Therefore, we see that the Lagrangian receives the following Berry contribution for each of then sites of the unit cell:484.3. Derivation of the Flag Manifold Sigma ModelLB = −1n(ptr[Λα∂τUU†] + tr[{Λα, L}∂τUU†])+ O(p−1). (4.36)We have divided by n, the size of the unit cell, since LB is technically a Lagrangian density thatwill be integrated against the two-dimensional measure dxdτ. In SU(3), the explicit contributionfrom a sum over 3 sites is, to O(p−1),LB = − p3 tr[Λ1∂τU(x)U†(x) + Λ2∂τU(x + 1)U†(x + 1) + Λ3∂τU(x + 2)U†(x + 2)](4.37)−13tr[{Λ1, L(x)}∂τU(x)U†(x)+{Λ2, L(x+1)}∂τU(x+1)U†(x+1)+{Λ3, L(x+1)}∂τU(x+2)U†(x+2)].In SU(2), such an expression led directly to the topological term (2.44) upon Taylor expansionand integrating out the L matrices. Here, we will find that such a procedure will lead to additionalterms that are purely imaginary in imaginary time (like a topological term), yet are not quantized.These so-called ‘λ-terms’, which are absent in SU(2), can be interpreted as torsion terms in the flagmanifold metric tensor. In Section 4.5, we will devote considerable effort to analyzing the λ-termin SU(3) using the renormalization group.4.3.2 Complete field theoryWe now are in a position to derive the complete field theory of the SU(3) chain in the symmetricrepresentation. First, we Taylor expand the result (4.37). Note that since the L matrices are alreadysubleading, this Taylor expansion is only nontrivial for the first line in (4.37). We findLB = − p3 µνtr[(Λ2 + 2Λ3)∂µU∂νU†] − 23tr[L∂τUU†], (4.38)where we again made use of the identity tr[∂UU†] = 0, which is proven in Appendix A. Now,let us determine the Lagrangian contribution from the Hamiltonian. Using the formula (2.34), wehaveL − LB = − J13(tr[S (x)S (x + 1)] + tr[S (x + 1)S (x + 2)] + tr[S (x + 2)S (x)])(4.39)− J23(tr[S (x)S (x + 2)] + tr[S (x + 2)S (x)]).Note that we have remembered to divide by 3, the size of the unit cell, when adding (4.27) intothis expression. We have also made use of the 3-site order to make the identification x ∼ x + 3.Inserting (4.27), we then haveL − LB = − J13[ 3∑α=1p2tr[U∂xU†Λα∂xUU†Λα+1] + 2ptr[{L,Λα}∂xUU†Λα+1] + 4|Lα,α+1|2](4.40)494.3. Derivation of the Flag Manifold Sigma Model− J23[4p23∑α=1tr[U∂xU†Λα∂xUU†Λα+2] + 4ptr[{L,Λα}∂xUU†Λα+2] + 4|Lα,α+2|2],where all indices are defined modulo 3. Now, the L matrices can be integrated out using the sameGaussian identity we used in Chapter 2, (2.38), as well as the result (4.38). For example, integratingout L12 and L∗12 leads to∫dL12dL∗12 exp[−43(J1 + J2)|L12|2 − 2L12([∂τUU†]21 + 2(2J2 − J1)[∂xUU†]21)](4.41)× exp[−2L21([∂τUU†]12 − 2((2J2 − J1)[∂xUU†]12)]= exp[− 13(J1 + J2)tr[Λ1U∂τU†Λ2∂τUU†] + p2(2J2 − J1)23(J1 + J2)tr[Λ1U∂xU†Λ2∂xUU†]]× exp[p2J2 − J13(J1 + J2)µνtr[∂µUU†Λ2∂τUU†Λ1]].Similar integrals can be done for the L23 and L13 terms as well. In the end, we arrive at thefollowing action:S [U] =∫dxdτ( 3∑α=11g[vtr[Λα−1U∂xU†Λα∂xUU†] +1vtr[Λα−1U∂τU†Λα∂τUU†]](4.42)+iθ2piiµνtr[(Λ1 − Λ3)∂µU∂νU†]+ iλ2piiµν3∑α=1tr[Λα−1U∂µU†Λα∂νUU†]).Let us define the various parameters introduced above. The coupling constants g and λ obey1g= p√J1J2J1 + J2λ =2pip32J2 − J1J1 + J2, (4.43)the velocity satisfiesv = 3p√J1J2, (4.44)and the topological angle is defined byθ =2pip3. (4.45)This action defines a sigma model quantum field theory. That is, a theory whose fields, (in thiscase the matrices U), are restricted to lie in a particular smooth manifold. For us, this space is theflag manifold SU(3)/[U(1)]2: the space of special unitary 3x3 matrices that are invariant under twoU(1) gauge transformations. The terms proportional to 1g correspond to a symmetric metric tensoron this manifold, and the λ-term corresponds to an antisymmetric metric torsion. For more detailson the mathematical structure of both sigma models, as well as flag manifolds in general, we refer504.4. Properties of the Field Theorythe reader to the papers of Bykov [71, 72], as well as the recent review [6].20 In the followingsection, where we list the various properties of this model, these gauge transformations will beverified explicitly.But before moving on, we make an additional useful observation. Since our quantum fieldtheory was derived to capture small fluctuations about a classical ground state, we may treat U asa matrix close to the identity operator, and writeU = ei∑A ωATA , (4.46)for small real fields ωA, and su(3) Lie algebra generators TA. Indeed, this is a defining property ofany Lie group, including SU(3) in which U belongs. Now, in Appendix C, we prove a factorizationproperty of SU(3) matrices. Namely, any matrix U can actually be written asU = ei(ω3T3+ω8T8)ei∑A,3,8 ωATA , (4.47)where T3 and T8 are diagonal matrices, and the remaining generators are off-diagonal. Insertingthis factorization into (4.42), the gauge invariance cited above will ensure that the fields ω3 and ω8do not appear in the action. In other words, the flag manifold sigma model action is described byonly 6 fields, which agrees with the real dimension of SU(3)/[U(1)]2. This number of degrees offreedom is consistent with the fact that we found 6 Goldstone bosons in our flavour-wave theorycalculations. Moreover the ωA fields propagate with precisely the same velocity, v. This is tobe expected, since it was flavour-wave theory that provided our reference state for deriving thequantum field theory in the first place!4.4 Properties of the Field TheoryThe first symmetry to notice in (4.42) is the invariance under exchanging the spatial and temporalcoordinates, x ↔ τ; i.e. the theory is Lorentz invariant just as in SU(2). This can be made moreobvious by rescaling τ 7→ vτ.Next, we comment on the coupling constants. The first, g, is a direct analogue of the CP1coupling (2.50), and as in that theory, it tends to zero in the limit of large representation, p → ∞.We will make use of this fact in the following section when we perform perturbation theory todetermine various renormalization group flows in the sigma model. Meanwhile, the coupling con-stant λ is novel, and arises due to a mismatch between the first and second-neighbour interactionsin the Heisenberg Hamiltonian. Only for the fine-tuned choice of J1 = 2J2 does this term vanish.Unfortunately, we are unable to identify a physical symmetry of the SU(3) chain that would ruleout such a term, and so we are forced to analyze its consequences. Interestingly, this λ-term isa pure-imaginary term in imaginary time, and contributes only a phase to the underlying SU(3)partition function, much like the topological term analyzed in Section 2.3.1. However, unlike atopological term, it is not related to an invariant, and so is not quantized in any way.20The ‘flag’ structure is more obvious when we recall the isomorphism U(3) ≡ SU(3) × U(1), so that the manifold isU(3)/[U(1)]3. In [6], additional equivalent definitions of flag manifolds can be found.514.4. Properties of the Field TheoryFinally, let us direct our attention to the terms involving θ. As our notation suggests, these aretopological invariants, and correspond to the second cohomology group of the manifold SU(3)/[U(1)]2:H2(SU(3)/[U(1)]2) = Z × Z. (4.48)Recall that this group is equivalent to the second homotopy group for flag manifolds, which clas-sifies topologically distinct maps from S 2 to the target space. As we explained in Chapter 2, sinceany finite-action field configuration must attain a constant value at some radius R away from theorigin in spacetime, we may identify R2 with the sphere S 2, and view any field ϕ ∈ SU(3)/[U(1)]2as a map from S 2 to the flag manifold. Explicitly, the topological invariant isQα :=12piiµν∫dxdτtr[Λα∂µU∂νU] ∈ Z, (4.49)so that topological action arising from (4.42) readsS top = i3∑α=1θαQα, (4.50)with θ1 = −θ3 = θ = 2pip3 , and θ2 = 0. Actually, we have some freedom in our definition of the θα,since it is easily verified using tr[U∂U†] = 0 that the topological charges Qα satisfyQ1 + Q2 + Q3 = 0. (4.51)In fact, not only do the topological charges sum to zero, but so do their densities, qα := tr[Λα∂µU∂νU†].4.4.1 The flag manifold as coupled CP2 sigma modelsIn Section 3.3, we explained how longer-range interactions in the Heisenberg Hamiltonian served tocouple degrees of freedom together across sites of the chain. According to (4.18), in the symmetricrepresentations we presently consider, to each site of the chain we may assign a field ϕα, whichcorresponds to a CP2 degree of freedom. In a nearest-neighbour model, the unit cell would involveonly two such CP2 fields, and local zero modes would persist. But as we have seen, by adding aJ2 term in (4.1), we couple three sites together, and obtain a field theory involving three separateCP2 fields. We now rewrite our sigma model in a manner that makes this threefold CP2 structureexplicit. Using the identity3∑α=1|ϕ∗,α−1 · ∂µϕα|2 = 123∑α=1(|∂µϕα|2 − |ϕ∗,α · ∂µϕα|2), (4.52)we see that the flag manifold sigma model resembles that of three CP2 fields, coupled via orthog-onality and the λ-term. Indeed, after some algebra we findS =12g3∑α=1(|∂µϕα|2 − |ϕ∗,α · ∂µϕα|2)+µν2pii3∑α=1[iλ(ϕ∗,α+1 · ∂µϕα)(ϕα+1 · ∂νϕ∗,α) + iθα∂µϕα · ∂νϕ∗,α],(4.53)524.4. Properties of the Field Theorywhere we have rescaled τ by setting v = 1. The first term occurring above is precisely the actionfor a CP2 field. This expression makes explicit the embedding of the flag manifold SU(3)/[U(1)]2into three copies of CP2:U(3)U(1) × U(1) × U(1) ↪→ CP2 × CP2 × CP2. (4.54)Since CP2 is the Grassmannian G1,n, this embedding is a special case of (3.16).4.4.2 Gauge symmetryWe now turn to the U(1) × U(1) gauge structure of the sigma model. In (4.42), this manifests asinvariance of the action under the transformation U′(x, τ) = D(x, τ)U(x, τ), whereD(x, τ) =eiϑ1(x,τ) 0 00 eiϑ2(x,τ) 00 0 eiϑ3(x,τ) (4.55)with ϑ3 = −ϑ1 − ϑ2. It is actually easier to verify the gauge symmetry in terms of the ϕα. In termsof these fields, this transformation corresponds to ϕ′α = eiϑαϕα. Let us verify each term in (4.53)separately. Since|ϕ′∗,α±1 · ∂µϕ′α|2 = |ϕ∗,α±1e−iϑα±1 · (∂µϕαeiϑα + ϕα∂µeiϑα)|2 = |ϕ∗,α±1 · ∂µϕα|2 (4.56)because the ϕα are mutually orthogonal, we can use (4.52) to verify that the real terms in (4.53) aregauge invariant. The λ-term can be similarly shown to be invariant since the phase factors cancelout. Finally, the topological charges transform asµν(∂µϕ′α · ∂νϕ′∗,α) = µν(∂µϕα · ∂νϕ∗,α) + µν(∂µ(iϑα)∂ν(−iϑα)) (4.57)+µν(∂µϕα · ϕ∗,α∂ν(−iϑα)) + µν(∂µ(iϑα)ϕα · ∂νϕ∗,α).The second term on the right hand side vanishes by symmetry: µν is antisymmetric while (∂µϑα∂νϑα)is symmetric. Meanwhile, the third and fourth terms cancel due to the identity∂(ϕα · ϕ∗,α) = 0. (4.58)Thus, the topological term is invariant, and (4.42) possesses the gauge symmetry (4.55) as claimed.It is perhaps useful to relate these gauge symmetries to the familiar U(1) gauge symmetry ofthe CP2 sigma model. To this end, we introduce three fields Aαµ ∈ R2, which satisfyA1µ(x) + A2µ(x) + A3µ(x) = 0 (4.59)534.4. Properties of the Field Theoryfor all x and µ. This constraint is equivalent to the condition that ϑ3 = −ϑ1 − ϑ2 in (4.55). In termsof these Aαµ , there exists yet another rewriting of (4.42), which isS =∫dxdτ 3∑α=1[12g∣∣∣(∂µ + iAαµ)ϕα∣∣∣2 + iθα2piiµνAαν + µν2piiλ(ϕ∗,α+1 · ∂µϕα)(ϕα+1 · ∂νϕ∗,α)] . (4.60)To prove this, we first expand the real term in (4.60):∣∣∣(∂µ + iAαµ)ϕα∣∣∣2 = ∣∣∣∂µϕα∣∣∣2 + (Aαµ)2 − 2iAαµϕ∗,α · ∂µϕα (4.61)=∣∣∣∂µϕα∣∣∣2 − ∣∣∣ϕ∗,α · ∂µϕα∣∣∣2 + (Aαµ − iϕ∗,α · ∂µϕα)2.This form suggests we shift Aαµ by iϕ∗,α · ∂µϕα. In doing so, this changes the topological term in(4.60) according toµν∂µAαν 7→ µν∂µAαν + iµν∂µϕ∗,α · ∂νϕα. (4.62)Since the Aαµ are decoupled from the ϕα, we may now integrate out the gauge fields completely, andobtain the desired theory (4.53). Note that shifting the gauge fields does not disturb the constraint(4.59), sinceA1µ(x) + A2µ(x) + A3µ(x) 7→ A1µ(x) + A2µ(x) + A3µ(x) +3∑α=1ϕ∗,α · ∂µϕα, (4.63)and the second term vanishes since it is of the form tr[U∂U†].4.4.3 Z3 symmetryTranslation symmetry in the underlying SU(3) chain is realized in the sigma model as a global Z3symmetry: U 7→ U′ = RU, withR =0 1 00 0 11 0 0 . (4.64)This cyclically permutes the three ϕα fields, which can be represented pictorially as7→R 7→R 7→R . (4.65)Note that R3 = 1, as it must. Using the form of the action (4.53), the only nontrivial calculation wemust do to verify symmetry under R has to do with the topological term. Note that2pi3(Q1 − Q3) 7→ 2pi3 (Q2 − Q1) =2pi3(Q1 − Q3) + 2pi3 (Q1 + Q2 + Q3) − 32pi3Q1. (4.66)Since Q1 + Q2 + Q3 = 0, the second term vanishes. Now recall that the topological term enters thepartition function as a pure phase. Since Q1 ∈ Z, we havee2piip3 (Q1−Q3) 7→ e 2piip3 (Q1−Q3)e2piipQ1 = e 2piip3 (Q1−Q3), (4.67)which proves the symmetry. We will have more to say on this symmetry later on, when we discussthe concept of ’t Hooft anomaly matching.544.5. Renormalization Group Calculations4.4.4 Other discrete symmetriesAnother discrete global symmetry of the sigma model is parity symmetry, which corresponds toa mirror symmetry in the SU(3) chain. In fact, there are three non-equivalent mirror symmetries,which lead to three non-equivalent parity symmetries. For example, a reflection about the lineseparating and sites in the chain corresponds toϕ1(x, τ) 7→ ϕ3(−x, τ) ϕ2(x, τ) 7→ ϕ2(−x, τ) ϕ3(x, τ) 7→ ϕ1(−x, τ). (4.68)It is easily verified that such a transformation leaves (4.53) invariant. Note for instance that thetopological term is unchanged: Q1 and Q3 change sign, but so does the partial derivative, ∂x, andthese two changes cancel out. This parity symmetry will not be related to any ‘t Hooft anomalies,but will become important in Section 4.7 when we discuss other SU(3) representations.Finally, the sigma model is also unchanged under the symmetry ϕα(x, τ) → ϕα(x,−τ), as wellas i 7→ −i. This corresponds to time-reversal symmetry in the SU(3) chain.4.5 Renormalization Group CalculationsIn Chapter 2, once we derived the corresponding sigma model of the antiferromagnet, we wereimmediately able to draw conclusions about its phase diagram. This was because a great deal oftheoretical investigation had already been carried out for the CP1 sigma model. In particular, it waswell established that the theory is asymptotically free, meaning that the coupling constant g flowsto small values at high enough energies [18].Having derived the flag manifold sigma model description of the SU(3) chain in this chapter, wewould now like to proceed in a similar fashion. Ultimately, we hope to establish a phase diagramfor the chain that depends on the representation p, and uncover an SU(3) version of Haldane’sconjecture. Unfortunately, we have more work to do first. Since these flag manifolds are relativelynovel theories, they have not been studied to the same extend as in SU(2). Therefore, in this section,we are forced to do the necessary renormalization group (RG) calculations ourselves, in order toestablish asymptotic freedom in the SU(3) chain. We will also investigate the relevance of theλ-term, as it is an operator that is not often considered in standard quantum field theories.4.5.1 A primer on renormalization group theorySince the notion of renormalization group flow will be central to the results of this thesis, we offerhere a brief reminder of its theoretical foundation.21 The fundamental idea behind the renormal-ization group is that coupling constants change as we change the length scale of our theory. Inthe context of spin chains, this notion of different length scales is very natural: the atomic spacingof spins along the chain is a much smaller distance than the observable lengths defined by macro-scopic phenomena. This motivates the search for so-called relevant operators – those operators21The content of this subsection is partially transcribed from the author’s MSc thesis [124].554.5. Renormalization Group Calculationswhose coupling constants increase as we increase length scales. In contrast, irrelevant operators,those whose couplings decrease as we increase length scales, are less interesting to us as physicistswho seek to model and identify macroscopic phenomena.22 When we apply this thinking to ourflag manifold sigma model, what we hope to understand is which of the couplings g and λ areassociated with relevant operators.In order to achieve this understanding, we must determine the functions governing the evolutionof g and λ as a function of length scale. These are known as beta functions, and can be definedas follows. Given a family of length scales bΛ parametrized by b, the beta function of couplingconstant X is defined according toβX :=dXd log b. (4.69)Here Λ can be thought of as an underlying ultraviolet (i.e. high energy) cutoff in the theory, de-termined in part by the inverse lattice spacing, a−1.23 Since increasing b corresponds to increasinglength, relevant operators have positive beta functions. Now, there are various ways of calculatingβX; we choose to follow Ken Wilson’s approach, who is credited for much of our modern under-standing of the renormalization group [125]. According to Wilson, one reduces the length scale ofthe quantum field theory by integrating out those fields whose momentum modes are restricted tolie in a thin shell in momentum space. This shell consists of all momenta with magnitude lyingwithin (b−1Λ,Λ).24Let us explain how this integration is carried out for the case of a single scalar field, φ. Thenwe will turn to the flag manifold sigma model in the following subsection. The Lagrangian densityis taken to beLϕ = 12(∂ϕ)2 +∑iXiOi[ϕ], (4.70)where Oi[ϕ] is a generic operator involving the field ϕ and its derivatives. We begin by separatingthe field into slow and fast components,ϕ = ϕs + ϕ f , (4.71)where ϕs contains the momentum modes of ϕ with magnitude less than b−1Λ, and ϕ f contains themodes that lie within the shell. In terms of these new variables, the Lagrangian density can bereorganized as follows:Lϕ = Ls +L0f +Ls f . (4.72)The first term, Ls, equals the original Lagrangian density, but with ϕ replaced with ϕs. The sec-ond term, L0f , equals the free fast Lagrangian density, 12 (∂ϕ f )2. The remaining term contains all22For completeness, we note that operators that are neither relevant nor irrelevant are referred to as marginal.23In practice, Λ is a reduced cutoff, which is less than the bare cutoff Λ0 := a−1. This is because we typically workwith a Lagrangian that already corresponds to some lower energy description of the system than the original underlyingHamiltonian.24Note that since b is a length scale, b−1Λ corresponds to an energy scale less than Λ.564.5. Renormalization Group Calculationsoperators that mix slow and fast components. The partition function can then be rewritten asZ =∫Dϕsϕ f e−∫dd x(Ls+L0f +Ls f ) = Z0, f∫Dϕse−∫dd x(Ls+δL)), (4.73)whereZ0, f :=∫Dϕ f e−∫dd xL f 〈· · · 〉 f := Z−10, f∫Dϕ f · · · e−∫dd xL f (4.74)ande−∫dd xδL := 〈e−∫dd xLs f 〉 f . (4.75)In other words, integrating out the fast modes has generated new terms in the Lagrangian. Sincethe operators appearing in (4.70) were generic, we can writeδL = 12δZϕ(∂ϕs)2 +∑iδZiOi[ϕs] (4.76)in terms of renormalization constants δZϕ and δZi. To compare Ls + δL to the original theory, werescale coordinatesx→ b−1x (4.77)so that the new UV cutoff is once again Λ, and rescale the fieldϕs → ϕs√(1 + δZϕ)bd−2 (4.78)so that the new kinetic term is once again 12 (∂ϕ)2. Defining di and ni to be the mass dimensionand number of factors of ϕ, respectively, of Oi, we find that the coupling constants of the reducedtheory, Xi(b), satisfyXi(b) = Xi(1 +δZiXi)(1 + δZϕ)−nibdi . (4.79)This expression can now be differentiated with respect to log b, yielding the beta functions ofthe theory, βXi . Often, we use flow diagrams that capture the information provided by these betafunctions.4.5.2 Renormalization of the SU(3) chainAccording to the steps outlined above, we begin by deriving a perturbative expansion of the sigmamodel, (4.42). This is achieved by rewriting the matrices U in terms of the 8 generators TA ofSU(3), according toU = ei∑8A=1 ωATA . (4.80)These matrices are defined explicitly in (E.1), and can be thought of as generalized Pauli matrices.They satisfy the su(3) algebra[TA,TB] = 2i fABCTC , (4.81)574.5. Renormalization Group Calculationswhere the fABC structure constants are fully antisymmetric. Due to the factorization property ofSU(3) matrices (4.47), and the gauge symmetry (4.55), we can replace (4.80) with the simplerrelationU = eiωaTa . (4.82)Here and throughout, lowercase Latin letters index the off-diagonal Gell-Mann matrices, lowercaseGreek letters index the diagonal ones, uppercase Latin letters index all eight Gell-Mann matrices,and repeated indices are summed over. Now following Wilson, we rewrite the fields U in terms of‘slow’ (Us) and ‘fast’ (U f ) fields, asU = U f Us (4.83)and make use of (4.82) for each factor. These fast fields have momentum modes restricted to aWilson shell [b−1Λ,Λ), where Λ is an energy cutoff (less than the inverse lattice spacing), and b &1. The RG step of integrating over this shell is then equivalent to integrating out the fields U f fromthe theory. This factorization of U is motivated by Polyakov’s work on the O(3) nonlinear sigmamodel, which found the model’s RG equations by a quadratic expansion of the ‘fast fields’ [126]. InAppendix D, we review this calculation, and also demonstrate the equivalence between Polyakov’sdecomposition and (4.83) for the SU(2) case.We are now ready to insert (4.83) into our Lagrangian. But since we will be forced to make useof many identities involving the structure factors fABC , we choose to rewrite the Lagrangian in anequivalent form, which is derived in Appendix E.2. Neglecting the topological term, as it does notcontribute to perturbative RG equations, this equivalent form readsL = 18gtr[∂µ(U†TγU)∂µ(U†TγU)] + λ√32µνtr[∂µUU†T8∂νUU†T3]. (4.84)For convenience, we define the composite objectsMA = U†s TAUs Nµ = ∂µUsU†s (4.85)which will simplify our notation greatly. We now derive a perturbative Lagrangian by focusing oneach term in (4.84) separately.First term of (4.84)We start by inserting U = U f Us into the first term of (4.84), and expanding U f = I + iωaTa −12ωaωbTaTb + O(ω3). Then to leading orderU†f TγU f = Tγ − iωa[Ta,Tγ] + ωaωbTaTγTb −12ωaωb{TaTb,Tγ}. (4.86)The quadratic term can be simplified by noting that since ωaωb is symmetric in a and b,ωaωb(TaTγTb − 12 {TaTb,Tγ})=14ωaωb([Ta, [Tγ,Tb]] + [Tb, [Tγ,Ta]]). (4.87)584.5. Renormalization Group CalculationsNow using the su(3) algebra, and defininghabγD =(facD fγbc + fbcD fγac), (4.88)we obtainU†TγU = Mγ + 2 faγbωaMb − ωaωbhabγDMD. (4.89)Derivatives and traces are now taken, resulting intr[(∂µ(U†TγU))2] = (4.90)tr[(∂µMγ)2] + 4 faγc fbγd∂µωa∂µωbtr[McMd] + 4 faγc fbγdωaωbtr[∂µMc∂µMd]+ 4 faγb∂µωatr[∂µMγMb] + 4 faγbωatr[∂µMγ∂µMb]+ 8 faγc fbγd∂µωaωbtr[Mc∂µMd] − 4∂µωaωbhabγDtr[∂µMγMD]− 2ωaωbhabγDtr[∂µMγ∂µMD] + O(ω3).We can simplify some terms. Using (E.4), we havetr[McMd] = tr[U†s TcUsU†s TdUs] = tr[TcTd] = 2δcd. (4.91)Additionally,tr[MA∂µMB] = tr[Us∂µU†s TBTA] + tr[∂µUsU†s TATB] (4.92)= tr[∂µUsU†s [TA,TB]] (4.93)= 2i fABCtr[∂µUsU†s TC] (4.94)andhabγD fDγC = − fDaE fbγE fDγC − fDbE faγE fDγC = 0 (4.95)imply that the term proportional to ∂µωaωbhabγD vanishes. The result istr[(∂µ(U†TγU))2] = tr[(∂µMγ)2] + 8(∂µωa)2 + 4 faγc fbγdωaωbtr[∂µMc∂µMd]+ 8i fabe∂µωaωbtr[∂µUsU†s Te] + 16i fabρ∂µωaωbtr[∂µUsU†s Tρ]− 2ωaωbhabγDtr[∂µMγ∂µMD]plus linear terms in ωa and higher-order corrections. For now, we neglect the linear terms: this willbe justified shortly. Then, to quadratic order in ωa, the first term of (4.84) can be written as18gtr[(∂µ(U†TγU))2] = (4.96)18gtr(∂µMγ)2 +18g[8(∂µωa)2 + 4 faγC fbγDωaωbtr[∂µMC∂µMD]+ 8i fabe∂µωaωbtr[∂µUsU†s Te] + 16i fabρ∂µωaωbtr[∂µUsU†s Tρ]− 2ωaωbhabγDtr[∂µMγ∂µMD]].594.5. Renormalization Group CalculationsSecond term of (4.84)We now perform the same expansion of U f to simplify the second term of (4.84). DefineLq := µνtr[∂µUU†T8∂νUU†T3], Nµ := ∂µUsU†s . (4.97)Then we haveLq = µνtr[∂µU f U†f T8∂νU f U†f T3] + µνtr[∂µU f U†f T8U f NνU†f T3] (4.98)+ µνtr[NµU†f T8∂νU f U†f T3U f ] + µνtr[NµU†f T8U f NνU†f T3U f ].Using (4.89), this isLq = + µνtr[∂µU f U†f T8∂νU f U†f T3] (4.99)+ µνtr[∂µU f (T8 + 2 fa8bωaTb − ωaωbhab8DTD) NνU†f T3]+ µνtr[NµU†f T8∂νU f (T3 + 2 fa3bωaTb − ωaωbhab3DTD)]+ µνtr[Nµ (T8 + 2 fa8bωaTb − ωaωbhab8DTD) Nν (T3 + 2 fa3bωaTb − ωaωbhab3DTD)].Expanding the remaining U f , and dropping a term proportional to ∂µωa∂νωbµν which vanishesafter Fourier transforming, we findλLq = µνλ√32[tr[NµT8NνT3] + ∂µωaωbtr[NνTb (T3TaT8 − T8TaT3)] (4.100)+12∂µ(ωaωb)tr[Nν(T8TaTbT3 − T3TaTbT8])]+ ωaωb(4 fb3d fa8ctr[NµTcNνTd] + tr[NνTDNµ (hab8DT3 − hab3DT8)])+ 2i∂µωaωbtr[TaTcNν ( fb8cT3 − fb3cT8)],]where again we have omitted the terms linear in ωa.Discussion of linear termsIn both parts of (4.84), we have refrained from writing the terms linear in ωa explicitly. We nowjustify this choice. After integrating by parts, all such terms are of the formωaFa(~x) (4.101)where Fa(~x) is a function of the slow matrices Us, and involves two derivatives. Naively, wemay argue that∫d2xωaFa(~x) = 0, since ωa only contains fast modes, while Fa is made of slowfunctions. However, since Fa contains products of slow modes, it will generically have some fast604.5. Renormalization Group Calculationsmodes; thus a different argument is required to justify neglecting these terms. We present two sucharguments:First, upon integrating out the fast fields, the contribution to the effective Lagrangian isδL = 12∫d2~xd2~yFa(~x)Gab(~x − ~y)Fb(~y), (4.102)where Gab(~x) is the Green’s function of the ω fields. In ~k-space, this isδL(~k) = 12∫b−1Λ<|~k|<Λd2~k(2pi)21k2Fa(~k)Fa(−~k) (4.103)because the ω fields only have momentum modes in the Wilson shell. Since F(~k) involves no morethan four slow fields, k is restricted to |k| < 4Λ˜, where Λ˜ := b−1Λ. Following [126], we will takethe limit Λ˜ → 0, and argue that |δL(~k)| → 0. Indeed, since Fa(~k) contains two derivatives of slowmodes,|F(~k)| < Λ˜2|F˜(~k)| (4.104)for some operator F˜(~k) involving up to four slow fields, without derivatives. Therefore,|δL| < 32Λ˜3 maxΛ˜<|k|<4Λ˜|F˜(~k)||F˜(−~k)|. (4.105)Since F˜(~k) just involves products of slow fields, this maximum should be bounded by some Λ˜ inde-pendent constant as Λ˜ → 0, say max0<|k|<Λ|F(~k)||F˜(−~k)|. Therefore |δL| vanishes as Λ˜3. This is tobe compared with the marginal kinetic term, (∂µωa)2, which only vanishes as Λ˜2. This agrees withour naive intuition, that since Fa(~x) involves two derivatives, δL should consist of irrelevant oper-ators; however, we are more careful here, since the momenta of Gab(~x − ~y) in (4.102) is restrictedto the Wilson shell.An alternative argument for neglecting linear terms is as follows. In the perturbative La-grangian, the leading term that arises from ωa(x)Fa(x) will correspond to a Feynman diagramwith a single internal line. Since the leading g-dependent interaction is a four-point vertex, thesimplest O(λ) diagram arising from ωa(x)Fa(x) will serve to renormalize a five-point or six-pointinteraction. Such diagrams can be excluded from a first-order perturbative calculation of β(g) andβ(λ), which consider the four-point and three-point interactions, respectively.4.5.3 Integration over a Wilson shellHaving justified the dropping of linear terms, we now Fourier transform the sum of (4.96) and(4.100) and perform Gaussian integrals over the ωa. We use the general formula∫D[φ]e−∫d2 x(L−L0) = Ne−12 tr[log∫ O] (4.106)614.5. Renormalization Group Calculationswhere O is a formal expression involving the Green’s functions of ωa. Following Polyakov’s ap-proach (reviewed in Appendix D), we expand the tr[log] to second order, since higher-order termswill involve more than two derivatives acting on the Us, and correspond to irrelevant operators. Theslow functions appearing in the linear term of this expansion can be replaced by their averages, asin (D.7). Meanwhile, the quadratic terms of the tr[log] can be approximated using (D.11). Noticethat terms proportional to ∂µωaωb do not contribute to linear order, according to (D.8). The resultistr[logO] = (4.107)∫d2x( ∫d2k(2pi)2( 12k2faγC faγDtr[∂µMC∂µMD] − 14k2 haaγDtr[∂µMγ∂µMD])− 12∫d2k(2pi)2kµkνk4(fabEtr[NµTE] + fabρtr[NµTρ])2+√3∫d2k(2pi)2( 2gk2λ fa3d fa8cµνtr[NµTcNνTd] +g2k2λµνtr[NνTDNµ(haa8dT3 − haa3DT8)])+√3λg∫d2k(2pi)2kµkρρνk4(fabEtr[NµTE] + fabγtr[NµTγ])×(+i2(tr[TbT8NνTaT3] − tr[NνTaT8TbT3]) − tr[Nν ( fa8cT3TbTc − fa3cT8TbTc)]+i4tr[Nν (T8{Ta,Tb}T3 − T3{Ta,Tb}T8)]))+ higher-order corrections.Here the integration over k is restricted to the Wilson shell. The first line can be simplified usinghaaγD = 6δγD, and (E.7). The second line can be simplified using(fabEtr[NµTE] + fabρtr[NµTρ])2= −12(tr[NµTγ])2 − (tr[NµTa])2. (4.108)The third line can be simplified too. Since fa8c fa3d vanishes unless c = d, the first term is propor-tional to µνtr[NµTcNνTc] = 0. Using haaγD = 6δγD, the whole third line is6√3gλµν∫d2k(2pi)21k2tr[NνT8NµT3]. (4.109)The last line can be dropped, since it is symmetric in a, b and multiplies terms proportional to fabγ.Finally, we use the momentum-shell integrals∫b−1Λ<k<Λd2k(2pi)21k2=12pilog b∫d2kkµkνk4g(k2) =∫d2k12k2g(k2) (4.110)624.5. Renormalization Group Calculationsfor a general function g(k2). At last, we arrive attr[logO] ≈∫d2x(log b4pi(tr[(∂µMc)2] − 3tr[(∂µMγ)2])+log b8pi(12(tr[NµTA])2 − 11(tr[NµTa])2)(4.111)+ 3√3(log b)gλµνtr[NνT8NµT3] + i√3µνλg log b8pi(fabEtr[NµTE] + fabγtr[NµTγ])×((tr[NνTaT3TbT8] − tr[NνTaT8TbT3]) + 2itr[Nν ( fa8cT3TbTc − fa3cT8TbTc)])).Now, we make use of various identities proven in Appendix E. The identities (E.16) and (E.21)let us rewrite the first line as−5 log b16pitr[(∂µMγ)2]. (4.112)Now consider the last term:A := i√3µνλg log b8pi(fabEtr[NµTE] + fabγtr[NµTγ])(4.113)×((tr[NνTaT3TbT8] − tr[NνTaT8TbT3]) + 2itr[Nν ( fa8cT3TbTc − fa3cT8TbTc)]).The first term of the second line is simplified using (E.26). Then antisymmetry in a and b implyA = − √3µνλg log b8pi(fabEtr[NµTE] + fabγtr[NµTγ])(4.114)×(f3bctr[Nν(TaTcT8 − T8TaTc)] + f8bctr[Nν(T3TaTc − TaTcT3)]+ f3actr[Nν(T8TbTc − TbTcT8)] + f8actr[Nν(TbTcT3 − T3TbTc)]).Now we use (E.28) to simplify this to:A = −2√3iµνλg log b8pi(fabEtr[NµTE]+ fabγtr[NµTγ])tr[NνTdTc](f3bc fa8d+ f8bc f3ad+ f3ac f8bd+ f8ac fb3d).(4.115)Recognizing the antisymmetry, this isA = −√34iµνλg log b4pitr[NνTdTc](fabEtr[NµTE] + fabγtr[NµTγ])(f8bc f3ad + f3ac f8bd). (4.116)Now we use (E.33). The result isA =√3iµν√3λg log b2pi(tr[NµT1trNνT2] − tr[NµT4trNνT5] + tr[NµT6trNνT7]). (4.117)634.5. Renormalization Group CalculationsNow we need to prove that the operator appear here is proportional to Lq0. This is (E.41):µν(tr[NµT1trNνT2] − tr[NµT4trNνT5] + tr[NµT6trNνT7])= −i√3tr[NµT8NνT3]. (4.118)Therefore,A = λ√323g log b4piµνtr[NµT8NνT3]. (4.119)Now we return to the Lagrangian, which requires dividing (4.111) by 2. Using (4.119) and(4.112), this isL = 18g(1 − g5 log b16pi)tr[∂µMγ∂µMγ]+√32λµνtr[NµT8NνT3](1−23g2pilog b+2g log b4pi3). (4.120)This allows for the identification ofλeff = λ(1 − 3g2pilog b)geff = g(1 − 5g log b4pi)−1(4.121)from which we can read off the β functions:βg(λ, g) =5g24piβλ(λ, g) = −3gλ2pi . (4.122)4.5.4 Results of the RG calculationsLet us consider each beta function in turn. First off, we conclude that g is a relevant parameter:Since it has a positive beta function, it flows to large values at large length scales. Such behaviouris referred to as asymptotic freedom, since the opposite flow towards short length scales, or equiv-alently high energies, renders the theory ‘free’ (i.e. g → 0). This is precisely the behaviour thatoccurs in the CP1 sigma model considered in Chapter 2. In fact, asymptotic freedom is known tooccur in CPn−1 for all n > 2, and so our result should not be that surprising, since we have demon-strated in (4.53) that the flag manifold can be viewed as three coupled CP2 fields.25 A fact that wewill use repeatedly throughout this thesis is that in the absence of topological angles, asymptoticfreedom indicates the presence of a finite energy scale in the theory. In other words, when θ is triv-ial (corresponding to p being a multiple of 3), we expect our sigma model, and thus our underlyingSU(3) chain, to exhibit a finite energy gap. This is our first result that will lead us to an SU(3)generalization of Haldane’s conjecture.Now let us turn to the beta function for λ. At first glance, it appears that the λ-term is irrelevant:at large length scales λ will flow to zero and we may drop it from our analysis. Unfortunately, thisis not the complete story. Recall that we are doing RG calculations for a perturbative Lagrangian,25However, a critical difference between the theories CP1 and CPn−1 for n > 2 is that the latter is known to exhibit afinite mass gap even in the presence of topological angle. This is where the intuition of a flag manifold as being coupledCPn−1 sigma models dramatically breaks down.644.6. Phase Diagram of the Symmetric SU(3) Chainobtained by replacing U with eiωaTa in (4.42). The kinetic term that results upon this replacementisL0 = 1g∂µωa∂µωa. (4.123)In order to obtain the more familiar kinetic term of free bosons, we must rescale the fields accordingto ωa →√g2ωa. As a result, the perturbative Lagrangian we obtain has the following structureL = 12∂µωa∂µωa + iλg32 µνRabcωa∂µωb∂νωc + gPabcdωaωb∂µωc∂µωd + O(λg2) + O(g2) (4.124)where Rabc and Pabcd are tensors of real coefficients that can be expressed in terms of the SU(3)structure factors. At this order in perturbation theory, the imaginary term does not have couplingconstant λ. Instead, its coupling constant is λ˜ = λg3/2. Therefore, in order to accurately assess therelevance of the imaginary term, we must calculate the beta function of λ˜, and determine its sign.Sinceβλ˜(λ, g) = g32 βλ +32g12λβg =3g8piλ˜ > 0 (4.125)we conclude that the imaginary term in (4.124) is also relevant. Therefore, any comprehensivephase diagram of the symmetric SU(3) must include axes for both λ and g. Despite this fact,a novel interpretation of RG flows based on ’t Hooft anomalies suggests that it may be safe toneglect the λ-term in many cases. This will be explained shortly. As a consequence, in some of thearguments that follow, we will make the simplifying assumption that λ = 0.4.6 Phase Diagram of the Symmetric SU(3) ChainAt this stage, let us reflect on our findings. Given an SU(3) chain in the rank-p symmetric rep-resentation, we have obtained a mapping to an SU(3)/[U(1)]2 flag manifold sigma model, withtopological angles θ = ±2pip3 . In the absence of these angles, i.e. when p = 3k for some k ∈ Z+, weexpect this sigma model, and thus the underlying chain, to exhibit a finite mass gap.Meanwhile, we have learned from the exact results of Chapter 3 that when the angles arenontrivial (p , 3k), the LSMA theorem forbids such a finite mass gap in the SU(3) chain, unlesstranslation symmetry is spontaneously broken (see Section 3.5.2). This is consistent with the factthat the spectrum of the chain with p = 1 is known exactly, and it does not exhibit a finite energygap (see Section 3.5.1). When p = 3k, the LSMA theorem breaks down, which is consistentwith our sigma model conclusions, and also the SU(3) AKLT model, which explicitly constructs atranslationally invariant SU(3) chain with a finite energy gap, as shown in Figure 3.6.Together, these facts suggest that there should exist a gapless phase in the SU(3) chain driven bytopological effects. In other words, there should exist a critical coupling gc, below which the modelexhibits gapless excitations. Above gc, translation symmetry will be spontaneously broken with 3-fold ground state degeneracy. Indeed, this is consistent with the findings of Corboz et. al. [109],who found a trimerized phase in the J1 − J2 SU(3) Heisenberg model. According to our expression654.6. Phase Diagram of the Symmetric SU(3) Chainfor the coupling constant (4.43), g is an increasing function of J2, and so we expect that for J2sufficiently large, the system will correspond to g > gc, and will be gapped with a spontaneousbreakdown of the Z3 symmetry.See Figure 4.1 for our proposed phase diagram. So far, we haven’t discussed the model at θ = pi,which corresponds to a model that explicitly breaks the Z3 symmetry, and thus breaks translationsymmetry in the chain model. This aspect of the phase diagram, as well as the other ground statedegeneracy predictions, will be justified in the following subsections.SU(3)1SU(3)11×2×3×1×3×gaplessgaplessgappedgappedSU(2)11×2×1×gappedgapless02π4π/3π2π/30θgc g(b)π2π(a)ggcθFigure 4.1: Phase diagram of the SU(3) chain in the rank-p symmetric representation. The angle,θ =2pip3 , vanishes when p is a multiple of 3.Let us compare our phase diagram to what occurs in the CP1 sigma model of the SU(2) chain(see Figure 2.1). Both include gapless phases that are characterized by SU(n)1 Wess-Zumino-Witten CFTs, and occur when θ = 2pin . Indeed, this will be a general trait of symmetric SU(n)chains – a fact that we will prove in following chapter, after a great deal of work. Both also includegapped phases at θ = 0; it is here that the AKLT-like models reside. What is unique to SU(3) is the664.6. Phase Diagram of the Symmetric SU(3) Chain2-fold degeneracy at θ = pi, which can only be realized by explicitly breaking translation symmetry,and will be shown below.As it stands, our arguments are very similar to those initially put forward by Haldane, and soperhaps one is also tempted to caveat our prediction as a ‘conjecture’. However, just as moderndevelopments have bolstered Haldane’s original arguments, there exist additional results in SU(3)which support our findings. We now review each of these in turn.4.6.1 Monte Carlo simulationsIn order to verify the effect of topological angle on mass gap in the SU(3) chain, classical MonteCarlo simulations were carried out by Lajko´ in [1]. Typically, such a calculation is not possibledue to the sign-problem related to the pure-imaginary nature of the topological term. However, thisissue was resolved using a clever method of Alle´s and Papa, which first carries out the simulationsfor imaginary topological angles ϑ = −iθ, and then extrapolates to real angles [33, 36]. This proce-dure works only if we exclude the λ-term from our analysis (which is also a pure-imaginary term,and contributes to the sign problem). Explicitly, the topological pat of the action was discretizedby splitting each square plaquette (in 2d Euclidean space) into two triangles [127], and defining alocal topological charge density qα according toei2piqα(∆i jk) =(ϕ∗,α(~ri) · ϕα(~r j))(ϕ∗,α(~r j) · ϕα(~rk))(ϕ∗,α(~rk) · ϕα(~ri))|ϕ∗,α(~ri) · ϕα(~r j)||ϕ∗,α(~r j) · ϕα(~rk)||ϕ∗,α(~rk) · ϕα(~ri)| . (4.126)Here ∆i jk is the triangle with nodes i, j and k defined in clockwise order. Combining these topolog-ical charge densities with the discretized real action,− 12g2∑α=1∑µ=x,τ∣∣∣ϕ∗,α(~r j) · ϕα(~r j + ~δµ)∣∣∣2, (4.127)the algorithm proceeds by sampling the time averaged correlation functions,C(x) =1L∑τ〈(ϕ1(0, 0) · ϕ∗,1(x, τ))(ϕ∗,1(0, 0) · ϕ1(x, τ))〉. (4.128)Note that the λ-term is neglected in these simulations. The correlation length can then be extractedby fitting C(x) with an exponential; its inverse defines the mass gap above the ground state thatwe seek. The mass gap is extracted from imaginary to real angles using the fitting function (c1 +c2θ2)/(1 + c3θ2) [33, 36].In Figure 4.2, the results of these Monte Carlo simulations for the SU(3) chain can be found.For completeness, we also include similar results for the SU(2) chain in Figure 4.3. As expected,when θ = 0 a finite mass gap exists for all values of g, in both SU(3) and SU(2). However,for sufficiently small g, this mass gap closes for particular values of θ2: in SU(3), this occurs atθ = ± 2pi3 , and in SU(2), this occurs at θ = ±pi. These are precisely the topological angles that occur674.6. Phase Diagram of the Symmetric SU(3) Chain 0 0.2 0.4 0.6 0.8 1 1.2 1.4-10 -5 0 5 10m=1/ξθ2m=1/ξm=1/ξm=1/ξm=1/ξm=1/ξm=1/ξm=1/ξm=1/ξm=1/ξ0.2170.2270.2330.2440.2560.2700.2860.303g=0.323 0 0.2 0.4 0.2 0.25 0.3m(θ=2 pi/3)gFigure 4.2: Monte Carlo simulations in the SU(3)/[U(1)]2 sigma model, with λ = 0. By extrapo-lating from imaginary topological angles, we see that below a critical coupling, the mass gap closesat θ = ±2pi/3.in our flag manifold sigma models when p is not a multiple of n, for n = 2, 3. We believe the finitemass gap that persists beyond some critical coupling gc, which is approximately 2.55 in SU(3),corresponds to a phase transition from a gapless phase into a gapped phase with spontaneouslybroken translation symmetry, in accordance with the LSMA theorem. Therefore, these MonteCarlo simulations present strong evidence in favour of our proposed phase diagram in Figure 4.1.4.6.2 Strong coupling analysisA second calculation, also led by Lajko´ in [1], was to consider the g→ ∞ limit of the flag manifoldsigma model, with λ set to zero. In this limit, the action consists only of a topological term, andcan be used to extract information about ground state degeneracies. Indeed, a similar calculationwas originally carried out by Seiberg in [104] and Plefka and Samuel in [105] for CPn−1 sigmamodels, and their findings were in agreement with the far-right side of the SU(2) phase diagramin Figure 2.1. In particular, the presence of a finite gap for all values of θ follows from the fact684.6. Phase Diagram of the Symmetric SU(3) Chain 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4-10 -5 0 5 10m = 1/ξθL2m = 1/ξm = 1/ξm = 1/ξm = 1/ξm = 1/ξm = 1/ξm = 1/ξm = 1/ξm = 1/ξ0.5880.6250.6670.7140.7690.8330.9091.111g=1.429 0 0.2 0.4 0.8 1.2 1.6m(θ= pi)gFigure 4.3: Monte Carlo simulations in the CP1 sigma model (corresponding to the SU(2) chain).Below a critical coupling, we see that the mass gap closes at θ = ±pi.that when the action is purely topological, the field ϕα become correlation free. Thus, we may alsoconclude that in SU(3), a finite gap will be universal in the limit g→ ∞.In short, the strong coupling analysis of the sigma model involves obtaining the free energydensity, by first developing a lattice version of the gauge-field formulation of the action, (4.60). Itis found that in the limit g→ ∞, this density is given byf (θ1, θ3) = − log(maxm,nz(θ1 + 2pim, θ3 + 2pi)), (4.129)where12z(θ1, θ3) =(θ1 − θ3) cos(θ1−θ32)− θ1 cos(θ12)+ θ3 cos(θ32)θ1θ3(θ1 − θ3) . (4.130)Since (4.64) ensures that θ1 = −θ3 in our theories, we plot f (θ1, θ3) along this line in Figure 4.4.The cusp in the free-energy at 2pi/3 (the blue line) is indicative of a phase transition, and willbe what we consider in this section. The behaviour at θ = pi (the red line) corresponds to a model694.6. Phase Diagram of the Symmetric SU(3) Chain 0.3 0.4 0.5 0.6 0.7 0.8 0.9-4π/3 -2π/3 0 2π/3 4π/3(a)free energyθf(θ, -θ)f(θ, 0)-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8-π2 -4π2/9 0 4π2/9 π2(b)free enrgyθ2f(θ, -θ) f(θ, 0) Figure 4.4: Free energy of the SU(3)/[U(1)]2 sigma model at strong coupling. The blue line,corresponding to our model with two topological angles, exhibits a cusp at θ = ±2pi/3. The redline, corresponding to a single topological angle, exhibits a cusp at θ = ±pi. This corresponds tothe self-conjugate SU(3) chains, discussed in Section 4.7.with explicitly broken Z3 symmetry, but possessing a Z2 parity symmetry. We will consider sucha model in Section 4.7.Since a finite mass gap is present for all values of θ in the strong coupling limit, it should be thatthe Z3 symmetry is spontaneously broken. We verify this explicitly by calculating the expectationvalue of the topological charge densities,〈q1〉 = −i∂ f (θ1, θ3, g→ ∞)∂θ1〈q3〉 = −i∂ f (θ1, θ3, g→ ∞)∂θ3(4.131)depending on from which sector Rm,n we approach the point θ1 = −θ3 = 2pi/3. By sector, we mean704.6. Phase Diagram of the Symmetric SU(3) Chainwhich values of m, n that maximize the free energy in (4.129). We findθ1 → 2pi/3 〈q1〉 〈q2〉 〈q3〉θ3 → −2pi/3from R0,0 −i 32pi + i√38 0 i32pi − i√38from R−1,0 i 32pi − i√38 −i 32pi + i√38 0from R0,−1 0 i 32pi − i√38 −i 32pi + i√38 .(4.132)where 〈q2〉 was calculated using q2 = −(q1 + q3). These cases are connected by the Z3 transforma-tion, clearly showing the spontaneous breaking of the symmetry.4.6.3 Wess-Zumino-Witten models and the explicit breaking of Z3The final two arguments that we mention here are related to the renormalization group flow of theSU(3) chain. As discussed in Section 3.5, we expect that any gapless critical point that occurs in ourSU(3)/[U(1)]2 flag manifold sigma model will be described by a particular CFT, the SU(3)1 Wess-Zumino-Witten model. This relies crucially on the fact that the model possesses an additional Z3symmetry; otherwise the relevant operator tr[g] will destabilize the theory, where g is a fundamentalWZW field.26 We comment in passing that SU(3)1 has a marginal operator whose coupling constantwill change sign at the phase transitions identified in Figure 4.1. That is to say, at strong enoughcoupling g in the sigma model, a marginal operator becomes marginally relevant, and drives a phasetransition to a gapped phase with 3 ground states that spontaneously break translation symmetry.This is discussed further in Chapter 5.Now, we consider what happens when the Z3 is explicitly broken, in order to predict the phasestructure of the sigma model at topological angles other than θ = 0,± 2pi3 .Let us break the Z3 explicitly, but preserve a Z2 symmetry that exchanges g and g†. Then thefollowing relevant operator is permitted:V = ρ(tr[g] + tr[g†]). (4.133)Here ρ is an undetermined coupling constant. This is what occurs, for example, when we pre-serve the parity symmetry (4.68) that interchanges ϕ1 and ϕ3. Such an explicit breaking could beachieved by altering the nearest-neighbour bonds between the 1 and 3 sites from J1 to J1(1 + δ).As a consequence of this, the topological angle θ is altered accordingly:θ → 2pip3(1 +J2J1 + J2− J2J1(1 + δ) + J2). (4.134)We now consider two possible scenarios, distinguished by the sign of δ:26Despite the confusing notation, we insist on using the standard variable g for our WZW fields, despite the fact thatit has nothing to do with the sigma model’s coupling constant, g.714.6. Phase Diagram of the Symmetric SU(3) Chain• Case 1: δ > 0 In this case, the bond is weakened, so that the underlying chain exhibits thefollowing pattern· · · W S S W S S W S S W S S · · · (4.135)where ‘W’ refers to a weaker bond, and ‘S’ to a stronger bond. Such a chain will form SU(3)singlets from the three sites adjacent to two ‘S’ bonds, resulting in a unique ground state.Since θ is shifted below 2pi3 , we predict a unique ground state for all values of θ until thecritical value of θ = 0.• Case 2: δ < 0 In this case, the bond is strengthened, and the chain’s pattern becomes· · · S W W S W W S W W S W W S · · · (4.136)Instead of forming singlets, now the chain will form 3 representations out of the two sitesadjacent to a single ‘S’ bond.27 Such ‘33 chains’ are known to spontaneously dimerize andexhibit a finite energy gap [128, 129]. Since θ is shifted above 2pi3 , we predict a 2-fold degen-erate ground state for angles in the interval ( 2pi3 ,4pi3 ), including the angle θ = pi.The previous discussion justifies the right edge of Figure 4.1. Now let us return to the WZWoperator V , defined in (4.133). We claim that ρ should have the following dependance on θ:ρ ∝(θ − 2pi3). (4.137)Then, if we write the diagonal elements of the WZW fundamental g as eiϑ, V becomesV ∝(θ − 2pi3)∑αcosϑα. (4.138)For values of θ less than 2pi3 , this term has a unique minimum with ϑα = 0, and corresponds to aunique ground state. On the other hand, for θ > 2pi3 , V is minimized by the two values ϑα = ± 2pi3 .Thus, with the form (4.137), we obtain the desired degeneracies.4.6.4 ’t Hooft anomaly matchingFinally, we discuss the notion of ’t Hooft anomaly matching [106, 107]. In short, the presence of an’t Hooft anomaly signifies nontrivial low energy physics. In one spatial dimension this necessitatesa gapless phase so long as the symmetries of the model are not spontaneously broken. Often, thisconsequence is described as a field-theoretic version of the LSMA theorem. In this subsection, wefirst review the theory of ’t Hooft anomalies, and then cite results that apply to our SU(3) chains.In order to understand ’t Hooft anomalies, we must first define anomalies themselves. Considera quantum theory which has a symmetry group G that leaves the classical action invariant. We say27The 3 representation of SU(3) has a Young tableau with p1 = p2 = 1, that is, .724.6. Phase Diagram of the Symmetric SU(3) Chainthat G is anomalous if G is violated in the full quantum theory. Thus anomalous symmetries aresymmetries of classical theories which do not survive the transition to quantum mechanics. Forinstance, if G is a continuous global symmetry, then it has a conserved current jAµ with A labelingthe generators of G. If this symmetry is anomalous, then there are quantum corrections whichmakes the divergence of jAµ nonzero:∂µ jAµ = AA. (4.139)The quantity Aa on the right hand side of this equation is what is referred to as the anomaly. Thechiral anomaly in quantum electrodynamics is the prototypical example of this behaviour [130].So far, what we have said applies to global symmetries. If instead G is a gauge symmetry,then anomalies indicate a fundamental inconsistency of the theory and must vanish. Recall thatgauge symmetries are not conventional symmetries that act on the configuration space and leadto identical physics. Rather, they are redundancies in our description of the physics when wework in the space of gauge fields rather than its quotient by gauge transformations. While thesegauge anomalies tell us our theory is sick, they can be used as a field theoretical tool, as was firstshown by ’t Hooft [106]. Given a theory with an exact global symmetry G (no anomaly), wecan imagine ‘gauge-ing’ the symmetry by introducing gauge bosons BAµ , minimally coupled to thetheory according to the prescription∂µ 7→ ∂µ − igBAµT A (4.140)for generators T A of G. If this G-gauge theory is anomalous, we say that the original theory has an‘t Hoot anomaly. Thus, ‘t Hooft anomalies do not invalidate the global symmetry, but they lead topowerful constraints on the phases of the theory.The utility of ’t Hooft anomalies stems from their nonrenormalization property: the anomalyis independent of energy scale. This implies that it can be calculated exactly using the low energydegrees of freedom, which is precisely what we access when we derive field theory descriptionsof condensed matter systems. In one spatial dimension, if a symmetry with an ’t Hooft anomalyis not spontaneously broken, we therefore may conclude that there exist low-lying excitations inthe theory.28 Otherwise, there would be no degrees of freedom to couple to the gauge fields andgenerate the anomaly.Let us now turn to the CP1 sigma model of the spin chain, which will prepare us for our dis-cussion of ’t Hooft anomalies in SU(3) chains below. While there are no pure ’t Hooft anomaliesas we’ve defined them above, there are so-called mixed t’ Hooft anomalies in CP1. By this wemean that if we gauge a continuous global symmetry, a second, independent symmetry becomesanomalous. ForCP1, this mixed anomaly is between the continuous group PSU(2), and the discretegroup of translations Z2. Here it is essential that we properly identify the continuous global sym-metry of the model as PSU(2) = SU(2)/Z2, and not SU(2), since otherwise no anomaly would bedetectable [78, 79, 132].29 This is because SU(2) gauge fields have integer quantized topological28In higher spatial dimensions there is another possibility: the system is gapped with some topological quantum fieldtheory [131].29This quotient by Z2 follows from the fact that the center of SU(2) acts trivially on the spin chain.734.7. Other SU(3) Chainscharge, while PSU(2) gauge fields can correspond to nontrivial sections of a PSU(2) bundle withhalf-integer quantized charge. Now, it is found that if one tries to gauge PSU(2), the Z2 group oftranslations is violated whenever θ = pi: the corresponding counterterm added to the Lagrangianto account for half-quantized PSU(2) gauge fields is not invariant under Z2. This is the ’t Hooftanomaly, and implies that half-odd integer spin chains either have spontaneously broken transla-tion symmetry, or gapless excitations. These possibilities reaffirm Haldane’s conjecture, and areconsistent with the phase diagram in Figure 2.1.In the SU(3)/[U(1)]2 sigma model, a similar ’t Hooft anomaly result holds [78, 79]. Whenθ = 2pi3 , there is a mixed anomaly between the continuous group PSU(3) = SU(3)/Z3 and Z3, thelatter of which corresponds to translations in the SU(3) chain. Thus, when p is not a multiple of3, we may conclude that either a gapless phase exists, or translation symmetry is spontaneouslybroken. This is entirely consistent with the phase diagram presented in Figure 4.1.In fact, the nonrenormalizaiton of ’t Hooft anomalies provides further evidence that the criticalpoint characterizing these gapless phases indeed corresponds to the SU(3)1 universality class. In[78, 79], the anomalies for both our flag manifold sigma model and the SU(3)1 WZW modelwere calculated, and were shown to agree. Thus, a renormalization group flow from one to theother is completely plausible. We will have more to say on how ’t Hooft anomalies constrainrenormalization group flow of these flag manifold sigma models in the following chapter.4.7 Other SU(3) ChainsBefore concluding this chapter, we turn to the remaining representations of SU(3): those whichhave Young tableaux with row lengths p1 and p2 , 0. According to the discussion in Section 3.4,so long as p1 , p2, it suffices to consider a nearest-neighbour Heisenberg Hamiltonian (i.e. J2 = 0).The edge case p1 = p2 corresponds to a representation that is conjugate to a symmetric model withp = p1, and leads to the same sigma model as the symmetric chains presented above. In thefollowing, we attempt to generalize our symmetric SU(3) calculations to this more general settingof p1 , p2 , 0.4.7.1 Classical ground states and mixed order parametersIn Section 3.4, we already established the classical ground states of these more general SU(3)chains. Using the pictorial notation from that chapter, a typical ground state is· · · · · · (4.141)· · · · · · . (4.142)Unlike the symmetric ones, these representations correspond to states with 2-site order. Anothernovel property is the existence of both ferromagnetic and antiferromagnetic order parameters. In-deed, we have that〈S αβ(2 j) ± S αβ(2 j + 1)〉 , 0 (4.143)744.7. Other SU(3) Chainsfor both the choices of ‘+’ and ‘-’, which follows from the equation〈S αβ(·)〉 =p1φ∗,1α φ1β + p2φ∗,3α φ3β · = 2 jp1φ∗,3α φ2β + p2φ∗,3α φ3β · = 2 j + 1 (4.144)The expectation values taken in (4.144) are with respect to SU(3) coherent states, which are a directgeneralization of those in (4.28). We assume a lowest-order expansion, so that each φα is evaluatedat the same site. Explicitly, the positive sign corresponds to a ferromagnetic order parameter, F,Fαβ =12〈S αβ(2 j) + S αβ(2 j + 1)〉 = 2p2 − p12 φ∗,3α φ3β, (4.145)and the negative sign corresponds to an antiferromagnetic order parameter, A,Aαβ =12〈S αβ(2 j) − S αβ(2 j + 1)〉 = p12 (φ∗,1α φ1β − φ∗,2α φ2β). (4.146)If we recall our discussion of Coleman’s theorem and spontaneous symmetry breaking in one spa-tial dimension, the presence of both F and A is rather interesting. On the one hand, we havean antiferromagnetic order parameter, which leads to linearly dispersing Goldstone bosons whichare the source of infrared divergences when the symmetry is spontaneously broken. On the otherhand, we have a ferromagnetic order parameter, which leads to quadratically dispersing Goldstonebosons.30 Such bosons are not ruled out by infrared divergence considerations. Thus, in this gen-eral setting, it may be possible for the SU(3) symmetry to be partially broken spontaneously, so thatquadratic Goldstone bosons persist, while the remaining degrees of freedom correspond to a truesymmetry of the ground state [95–97]. In Chapter 6, we will further discuss these possibilities, andspeculate on an entire flag manifold sigma model hierarchy that may arise in generic SU(n) chains.But for now, we turn to flavour-wave theory, and explicitly check that both linear and quadraticmodes are indeed present in the spectrum.4.7.2 Flavour-wave theoryFollowing Mathur and Sen [123], we write the matrix of generators, S , in terms of two 3-componentcommuting boson operators b1α and b2α:S αβ = (p1 − p2)b†,1α b1β − p2b†,2β b2α. (4.147)30That quadratically dispersing modes arise from a ferromagnetic order parameter can be understood as follows.When one derives an effective Lagrangian for Goldstone bosons corresponding to broken symmetry generators Qα,there is a term linear in ∂τ that is proportional to 〈[Qα,Qβ]〉 [95]. In the case of spin chains, these broken symmetrygenerators are Sˆ x and Sˆ y, so that the linear time-derivative term depends on 〈Sˆ z〉. For ferromagnets, this is nonzero andquadratic dispersion results; for antiferromagnets it vanishes so that the leading time-derivative term in the Lagrangianis proportional to ∂2τ, and results in linear dispersion. These results extend in a straightforward way to the SU(n) orderparameters discussed in the present section.754.7. Other SU(3) ChainsThe classical ground state corresponds to (p1 − p2) bosons of type b1, and p2 bosons of type b2.Now, following the same steps as laid out in Section 4.2, including Holstein-Primakoff and Bogoli-ubov transformations, we obtain a diagonalized Hamiltonian involving 6 Goldstone bosons.31 Thedispersion relations of the bosons at small energies are as follows:ω1,2(k) = 2J1 p1|k| ω3,4(k) = ω5,6(k) = 2J1(p1−p2)p2 |k|2p1−2p2 p1 , 2p2Jp1|k| p1 = 2p2(4.148)As expected, we find both linear and quadratic dispersion relations. However, for the special caseof p1 = 2p2, only linear modes exist. This is consistent with the fact that the ferromagnetic orderparameter (4.145) vanishes in this case. These representations are the so-called self-conjugaterepresentations of SU(3), with Young tableaux2p︷ ︸︸ ︷. (4.149)In the mathematical language of Bykov, we say that the self-conjugate representations correspondto a Lagrangian embedding, in which the total Berry phase of the unit cell vanishes [6, 71, 72].Ultimately, we are interested in low energy theories that only possess linearly dispersing modes.This is so that the emergence of Lorentz invariance can be facilitated, which would lead to a flagmanifold sigma model like the ones considered earlier in this chapter. Accordingly, we now restrictto the self-conjugate SU(3) chains, and derive a low energy description.4.7.3 Low energy field theory and the absence of Lorentz invarianceThe field theory derivation for self-conjugate SU(3) chains (i.e. those with p1 = 2p2) is very similarto that of Section 4.3 for the symmetric chains. Indeed, the only modification has to do with thecoherent states that we use. Defining p according to p = p2 = p1/2, these coherent states are|φ1, φ2〉 = (φ1 · a†,1)p(φ2 · a†,2)p|0〉, (4.150)with φ∗,1·φ2 = 0 and |φα|2 = 1. The aα are Schwinger bosons. After a series of lengthy calculations,we obtain the following actionS =∫dxdτ(p2J1[4|ϕ∗,2 · ∂xϕ1|2 + |ϕ∗,3 · ∂xϕ2|2 + |ϕ∗,1 · ∂xϕ3|2](4.151)+14J[4|ϕ∗,2 · ∂τϕ1|2 + |ϕ∗,2 · ∂τϕ2|2 + |ϕ∗,1 · ∂τϕ3|2] )+ S top.31Actually, flavour-wave theory also produces two spurious ‘flat modes’, with momentum-independent dispersionrelations ω7,8(k) = 4(p1 − p2). However, these modes do not lie entirely in the representation, and should be removedfrom the analysis. Indeed, they do not appear in the field theory.764.7. Other SU(3) ChainsHereS top = ipip(Q1 − Q2), (4.152)where Qα is defined in (4.49), with Qα ∈ Z. Note that the topological angle is now θ = pip, insteadof the value ± 2pip3 found previously. Therefore, in the strong coupling limit the free energy of thismodel coincides with the red line in Figure 4.4.Moreover, we observe that while this theory involves three complex fields ϕα, it does not pos-sess the same symmetries of the symmetric chain’s sigma model derived above. Most notably,there are now two distinct velocities, which prevents the theory from being invariant under Lorentztransformations. In a perturbative expansion, in which we replace the matrix Uαβ = ϕαβ with eiωaTa ,we find that four of the ωa propagate with one velocity v, while the remaining two fields propa-gate with v/2. These velocities, as well as the counting of these modes, are in agreement with theflavour-wave findings of the previous subsection.A further point of interest is that while this theory lacks Lorentz invariance, it does possess asymmetry that rules out the possibility of a λ-term. Indeed, the Z2 translation symmetry, whichacts asϕ1 7→ ϕ2 ϕ3 → −ϕ3, (4.153)or↔ → ,changes the sign of each term proportional to λ in (4.42). This symmetry is also responsible forrestricting the number of velocities to 2, down from the most general possibility of 3. It also leadsto the 2-fold degeneracy at strong coupling that we see at θ = pi in the phase diagram, Figure 4.1.4.7.4 Phase diagramLet us consider how the phase diagram of these self-conjugate SU(3) chains should differ from thatof the symmetric chains, Figure 4.1. Unfortunately, the LSMA theorem does not apply to thesemodels, as can be seen by examining the proof in Section 3.5.2. The so-called twist operator thatserved to generate a distinct low-lying state acts trivially on the self-conjugate representations ofSU(3), so that no conclusions may be drawn.In fact, none of the arguments in favour of a gapless phase carry over to these SU(3) chains.There is no self-conjugate analogue of the Bethe-ansatz solvable model discussed in Section 3.5.1.Moreover, Monte Carlo analysis indicate a finite gap for all values of p [2]. And finally, the ’tHooft anomaly was related to the Z3 manifestation of translation invariance, and does not occurfor the Z2 symmetry of a chain with a 2-site unit cell. These facts lead us to predict that there is nogapless phase in the self-conjugate SU(3) chain: there is a finite energy gap for every value of p.That being said, there is still a way to distinguish between the cases of even and odd p. When pis even, we may write down a translationally invariant valence bond state, which is a self-conjugategeneralization of the SU(3) AKLT state. This state is also invariant under the Z2 parity symmetry,(4.153). When p is odd, any translationally invariant AKLT-like state we attempt to write downnecessarily breaks this parity symmetry. These states are shown in Figures 3.8 and 3.9. This leads774.8. Conclusionus to conclude that self-conjugate SU(3) chains with p odd will exhibit spontaneously broken paritysymmetry. This is also supported by a strong coupling analysis, which finds broken parity whenθ = pi, but not when θ = 0.4.8 ConclusionIn this chapter, we have generalized Haldane’s conjecture to SU(3) chains in various represen-tations. First, we considered the rank-p symmetric SU(3) chains, whose Young tableaux have asingle row of p boxes. Such chains admit a low energy mapping to the SU(3)/[U(1)]2 flag mani-fold sigma model, with topological angles θ = ±2pip/3. By combining extensive renormalizationgroup calculations with a collection of supporting arguments, including Monte Carlo simulations, astrong coupling analysis, ’t Hooft anomaly matching conditions and known exact results for SU(3)chains, we proposed the phase diagram in Figure 4.1. In particular, we argued that when p is amultiple of 3, the SU(3) chain will exhibit a unique, gapped ground state, in a fashion similar to theinteger-spin SU(2) chains. For all remaining values of p, we argued that gapless excitations wouldexist for small enough interaction strengths, with a transition to a phase with spontaneously brokentranslation symmetry at larger strengths.Next, we considered SU(3) chains with 2-row Young tableaux. We demonstrated that in thismore general setting, the spectrum is no longer purely linearly dispersing. Instead, there are bothlinear and quadratic low energy modes, which can be understood as arising from the mixed ferro-and antiferromagnetic order parameters that exist simultaneously in these models. This led us tofocus on the self-conjugate representations, with p1 = 2p2, since these are the only other SU(3) rep-resentations with linear modes only. Again, a flag manifold sigma model description was obtained,but this time it lacks Lorentz invariance due to the presence of two distinct velocities. Conse-quently, we conclude that a gapless phase does not occur in the self-conjugate chains for any valueof p1 = 2p2. When p2 is odd, we further predict that parity symmetry is always spontaneouslybroken.Our findings are summarized below in Table 4.1 for all SU(3) representations that lead to flagmanifold sigma model descriptions.Representation (p1, p2) Spectrum(3k, 0) gapped; unique(3k ± 1,0) gapless(4k, 2k) gapped; unique(4k+2, 2k+1) gapped; 2-foldTable 4.1: A generalization of the Haldane conjecture to SU(3) chains. We label representationsby the lengths of their Young tableau’s rows, p1 and p2. Here k is a positive integer.78Chapter 5Symmetric SU(n) Chains5.1 IntroductionIn the previous chapter, we established a low energy mapping between the symmetric SU(3) chainand the SU(3)/[U(1)]2 flag manifold sigma model. This model can be visualized as three copiesof CP2 that are coupled via orthogonality. And since each copy of CP2 has the same velocity andcoupling constant, the resulting flag manifold exhibits Lorentz invariance. This fact is not alwaysguaranteed, as we saw when we studied the self-conjugate SU(3) chains, and were led to a sigmamodel with two distinct velocities.In this chapter, we again focus on symmetric representations, but promote the symmetry groupfrom SU(3) to SU(n) in general. We will attempt to derive a flag manifold sigma model descriptionof these chains in a similar fashion; namely, by starting with flavour-wave theory and then derivinga path integral using SU(n) coherent states. As expected, the structure of an SU(n)/[U(1)]n−1manifold will emerge, equipped with n−1 topological angles that depend on the representation size,p. However, the unique velocity property in SU(3) that ensured Lorentz invariance will not proveto be a general property of SU(n). Instead, we will find that there exist b n2c unique velocities, whichat first appears to be a major impediment to any SU(n) generalization of Haldane’s conjecture.Indeed, any gapless phase should correspond to some Lorentz invariant conformal field theory, andso, perhaps these distinct velocities prevent such a phase from occurring.The majority of the calculations in this chapter are devoted to resolving this concern. In partic-ular, we make use of the renormalization group to study the flow of these distinct velocities as wechange the energy scale of our theory. Ultimately, we will find that these velocities flow to a singleunique value, so that Lorentz invariance emerges after all. At this point, we are then able to extendour SU(3) phase diagram to SU(n), and predict various gapless phases driven by topological terms.Again, we will rely on the presence of ’t Hooft anomalies, as well as more general results about therenormalization group flow of conformally invariant field theories with SU(n) symmetry, namelythe SU(n)k WZW models. Our main result is the prediction of gapless excitations above the groundstate when p and n have no common divisor greater than 1.In Section 5.2, we begin by writing down the SU(n) Heisenberg Hamiltonian with interactionsup to (n − 1)-th neighbour. As explained in Chapter 3, such long-range interactions are necessaryin order to stabilize local zero modes. We also present the findings of SU(n) flavour-wave theoryin this section. Next, in Section 5.3, we derive the low energy quantum field theory description ofthe chain, and obtain the same flavour-wave velocities in a perturbative expansion. In Section 5.4,we use the renormalization group to argue that at low energies, these (distinct) velocities flow795.2. SU(n) Chains and Flavour-Wave Theoryto a common value, so that Lorentz invariance emerges and the field theory becomes a Lorentzinvariant SU(n)/[U(1)]n−1 flag manifold sigma model. The first derivation of the SU(n)/[U(1)]n−1flag manifold sigma model from SU(n) chains was done by Bykov in [71, 72], who elected tofine-tune the various interaction strengths in the Heisenberg Hamiltonian to guarantee Lorentzinvariance from the outset. These sigma models were also studied systematically in [78] and [79].In Section 5.5, we present our proposed phase diagram for the SU(n) chain, incorporating the ’tHooft anomaly matching conditions of [78, 79] and [116], This phase diagram is the source of ourSU(n) version of the Haldane conjecture. We also perform a strong coupling analysis. Finally, ourconclusions follow in Section 5.6.5.2 SU(n) Chains and Flavour-Wave TheoryOur starting point is the following SU(n) Heisenberg Hamiltonian, with interactions J1, J2, . . . , Jn−1,all positive, that couple each site of the chain with up to its (n − 1)-th neighbours:H =∑jn−1∑r=1Jrtr[S ( j)S ( j + r)]. (5.1)In Figure 5.1, we draw the first few interactions in a typical SU(n) chain.Figure 5.1: Interactions in a typical symmetric SU(n) chain. Explicitly, we have drawn the nearest-neighbour interactions in blue, second-neighbour interactions in red, and third-neighbour interac-tions in green. For n > 4, further interactions must also be included.As always, S ( j) is an n × n matrix whose entries contain the n2 − 1 generators of SU(n). Weconsider only symmetric representations in this chapter, meaning that the chain is characterizedby a single integer p, according to tr[S ] = p. In the large-p limit, we are able to replace S witha matrix of classical numbers. To this order in p, the Casimir constraints of SU(n) completelydetermine the eigenvalues of S . We haveS αβ = pφ∗,αφβ (5.2)for φ ∈ Cn with |φ|2 = 1. The interaction terms appearing in (5.1) reduce totr[S ( j)S ( j + r)] = p2|φ( j)∗ · φ( j + r)|2. (5.3)805.2. SU(n) Chains and Flavour-Wave TheorySince φ lives in Cn, a classical ground state will possess local zero modes unless the Hamiltoniangives rise to n − 1 constraints. This is the justification for our study of the longer-range Hamil-tonian above, which removes any local zero modes by including longer-range interactions. Theseinteractions result in an n-site ordered classical ground state which gives rise to a Zn symmetry inany low energy field theory description. For example, in SU(5), one such ground state is· · · · · · . (5.4)This Zn symmetry is also present for the p = 1 Bethe ansatz-solvable models, as mentioned inSection 3.5.1 [12, 84, 85]. In fact, it is expected that quantum fluctuations may produce an n-sitestructure through an order-by-disorder mechanism that generates effective additional couplings oforder p−1 that lift the local zero modes [109].32 However, the explicit mechanism for generatingthese couplings has not yet been worked out, so that we cannot predict the numerical values ofthese longer-range interactions from a nearest-neighbour model.Since the classical ground state minimizing (5.1) has n-site order, it is characterized by n nor-malized vectors that mutually minimize (5.3). That is, the classical ground state gives rise to anorthonormal basis of Cn. Due to this n-fold structure, we rewrite the Hamiltonian as a sum overunit cells (indexed by j):H =∑jn∑α=1n−1∑r=1Jrtr[S ( jα)S ( jα + r)] jα := n j + (α − 1). (5.5)5.2.1 Flavour-wave theoryAs we’ve mentioned repeatedly above, we do not expect spontaneous symmetry breaking of theSU(n) symmetry in the exact ground state of our Hamiltonian, due to Coleman’s theorem [8].Nonetheless, we may still expand about the classical (symmetry broken) ground state to predictthe Goldstone mode velocities. If the theory is asymptotically free, as in SU(2) and SU(3), then atsufficiently high energies (set by the inverse correlation length) the excitations may propagate withthese velocities [92].To this end, we introduce n2 bosons in each unit cell to reproduce the commutation relations ofthe S matrices, (3.1):S αβ( jγ) = b†α( jγ)bβ( jγ). (5.6)The counting is n flavours of boson for each of the n sites of a unit cell. The condition tr[S ] = pimplies there are p bosons at each site. The classical ground state involves only ‘diagonal’ bosonsof the type bγ( jγ) and b†γ( jγ). The ‘off-diagonal’ bosons are Holstein-Primakoff bosons. Flavour-wave theory allows for a small number of Holstein-Primakoff bosons at each site, captured byν( jγ) =∑α,γb†α( jγ)bα( jγ),32Again, we emphasize that n-site structure refers to the presence of a Zn symmetry, and not of a spontaneously brokenquantum ground state.815.3. Derivation of the Flag Manifold Sigma Modeland writes the Hamiltonian (5.5) in terms of these n(n − 1) bosons. In the large-p limit, p ν( jγ)and we expandS γγ( jγ) = p − ν( jγ),S αγ ( jγ) ≈√pb†α( jγ),S γα( jγ) ≈ √pbα( jγ),to findtr[S ( jγ)S ( jη)] = p[b†γ( jη)bγ( jη) + b†γ( jη)bη( jγ) + b†η( jγ)b†γ( jη) + bη( jγ)bγ( jη)]+ O(p0). (5.7)In terms of these degrees of freedom, the Hamiltonian (5.5) decomposes into a sumH =∑γ<ηHγη, (5.8)where Hγη is a Hamiltonian involving only the two bosons bγ( jη) and bη( jγ). In momentum space,this gives n(n−1)2 different 2 × 2 matrices, each of which can be diagonalized by a Bogoliubovtransformation:Hγ,γ+t = const. +∑kωt(k)2∑j=1(d†j,t(k)d j,t(k) +12), (5.9)whereωt(k) = 2p√Jt Jn−t∣∣∣∣∣sin nka2∣∣∣∣∣ , (5.10)and the d j,t are new boson operators. Therefore, the corresponding flavour-wave velocities arevt = np√Jt Jn−t t = 1, 2, . . . , n − 1. (5.11)When n is odd, there are n modes with each flavour-wave velocity. When n is even, this is trueexcept for the velocity v n2, which has only n2 modes. In each case, the number of modes adds up ton(n− 1). We note that for n > 3, there is no longer a unique velocity, and the emergence of Lorentzinvariance does not occur. Only for a specific fine-tuning of the couplings can Lorentz invariancebe restored. These tuned models were the ones considered by Bykov in [71] and [72].5.3 Derivation of the Flag Manifold Sigma ModelAs hinted in Section 4.3, much of our calculations from the symmetric SU(3) chains easily general-ize to SU(n) chains in the same representation. As such, we include only an abbreviated derivationof the sigma model; for further details, we refer the reader to Chapter 4. Since the classical ground825.3. Derivation of the Flag Manifold Sigma Modelstate has n-site order, with unit vectors φα ∈ Cn defined on each site, we first introduce a secondset of vectors, ϕα ∈ Cn, defined via a unitary matrix, U:Uαβ = ϕαβ . (5.12)This new basis can be used to describe fluctuations about the φα on each site, as in (4.15):φα =∑β1pLαβϕβ +√1 − µ(α)ϕα. (5.13)As before, Lαα = 0 (no sum), and p2µ(α) =∑β |Lαβ|2. Following Chapter 4 (which in turn followsChapter 2 for SU(2)), we let U and L vary uniformly from site to site, and write S ( jγ) in terms ofthem. After many simplifications, we findtr[S ( jγ)S ( jη)] = p2(η − γ)2tr[U∂xU†Λγ∂xUU†Λη] (5.14)+2(η − γ)p(Lηγ[∂xUU†]γη + Lγη[U∂xU†]ηγ) + 4|Lηγ|2 + const..The complete derivation of this result can be found in Appendix B. The next step is to construct thecoherent state path integral, and derive the Berry phase contribution to the Lagrangian. This pro-cedure was done for general n in Section 4.3.1. We repeat the result here, (4.36), for convenience:LB = −1n(ptr[Λα∂τUU†] + tr[{Λα, L}∂τUU†])+ O(p−1). (5.15)5.3.1 The complete field theorySince our approximated action is only quadratic in the L matrices, we may integrate out these modesto obtain an action in terms of the U matrices only. Combining (5.14) and (5.15), the Lagrangianterms involving a given matrix element Lαβ are:4(Jt + Jn−t)|Lαβ|2 − 2Lαβ([∂τUU†]βα + p((n − t)Jn−t − tJt)[∂xUU†]βα)(5.16)−2Lβα([∂τUU†]αβ − p((n − t)Jn−t − tJt)[∂xUU†]αβ),where t := |α − β|. The ∂τ-dependent terms have come from the Berry phase term (5.15), and the∂x-dependent terms have come fromJttr[S ( jα)S ( jβ)] + Jn−ttr[S ( jβ)S ( jn+α)] (5.17)in the Hamiltonian. Integrating over Lαβ, we are left with a real term,Lrealαβ =1n(Jt + Jn−t)tr[ΛαU∂τU†Λβ∂τUU†] − p2 [(n − t)Jn−t − tJt]2n(Jt + Jn−t)tr[ΛαU∂xU†Λβ∂xUU†],(5.18)835.3. Derivation of the Flag Manifold Sigma Modelas well as an imaginary termLimagαβ = p((n − t)Jn−t − tJt)n(Jt + Jn−t)([∂xUU†]αβ[∂τUU†]βα − [∂τUU†]βα[∂xUU†]αβ). (5.19)Above, there is no sum over the repeated α, β indices. The factor of n in the denominator comesfrom converting the sum over lattice sites within the n-site unit cell to an integral. To these terms,we must add the L-independent terms appearing in (5.14) and (5.15). They modify (5.18) toLrealαβ →1n(Jt + Jn−t)tr[ΛαU∂τU†Λβ∂τUU†] + p2Jn−t Jtn(Jt + Jn−t)tr[ΛαU∂xU†Λβ∂xUU†]. (5.20)Comparing the ratios of the prefactors of the spatial and imaginary temporal terms, we identify thevelocities of the theory asv2t = n2 p2Jn−t Jt. (5.21)This agrees with the flavour-wave velocities found in Section 5.2. Meanwhile, the terms in (5.15)modify (5.19) to produce the following pure-imaginary contribution to the Lagrangian:Limag = −µν∑α<βλ|α−β|tr[∂µUU†Λα∂νUU†Λβ] − S, (5.22)whereS := pn∑αtr[Λα∂τUU†] (5.23)andnλtp:=(n − t)Jn−t − tJtJt + Jn−t. (5.24)Using the identity tr[∂UU†] = 0, the integral of S can be shown to be a total derivative:iS = 2pipnn∑α=2(α − 1)Qα (5.25)whereQα :=12piiµν∫dxdτtr[∂µU∂νU†Λα]. (5.26)Relabelling S = −S top, and combining (5.22) with (5.20), we arrive at the following action:S =∑α<β∫dxdτ1g|α−β|(v|α−β|tr[ΛαU∂xU†Λβ∂xUU†] +1v|α−β|tr[ΛαU∂τU†Λβ∂τUU†])(5.27)−µν∑α<βλ|α−β|∫dxdτtr[∂µUU†Λα∂νUU†Λβ] + S top,845.3. Derivation of the Flag Manifold Sigma Modelwithgt =nvt(Jt + Jn−t). (5.28)Since the coupling constants and velocities satisfy gt = gn−t and vt = vn−t, we conclude that thereare bnc velocities and coupling constants. Moreover, we have thatn∑α=1Qα =12piiµν∫dxdτtr[∂µU∂νU†] =12piiµν∫dxdτ∂µtr[U∂νU†] = 0, (5.29)so that there are n − 1 independent topological charges. We note that the λ-terms appearing in(5.27) are not quantized, despite the fact that they are pure-imaginary in imaginary time. We givean interpretation of these terms below. In [72], these λ-terms were absent as a result of the samefine-tuning that ensured a unique velocity. Indeed, the choice Jt =√n−tt ensures that vt ≡ 1 forall t, and moreover that λt = 0 for all t. Note that this choice does not, however, equate all of thecoupling constants, gt.5.3.2 Gauge invarianceThe theory (5.27) is invariant under the gauge transformationsU(x, τ)→ D(x, τ)U(x, τ), (5.30)where D(x, τ) is a local, diagonal matrix. Since such matrices are generated by the n − 1 diagonalSU(n) generators, this corresponds to a [U(1)]n−1 gauge symmetry. Thus, we have obtained theSU(n)/[U(1)]n−1 flag manifold sigma model description of the SU(n) chain, as expected. Thetopological angle content of this model follows from the second cohomology group, which isH2(SU(n)/[U(1)]n−1) =n − 1 times︷ ︸︸ ︷Z × · · · × Z . (5.31)So, this model is characterized by n−1 topological charges, which is consistent with S top in (5.27).The coupling constants gt and λt correspond to the metric and torsion tensors on this manifold,respectively [79]. However, a unique metric cannot be defined, since the theory (5.27) lacks theLorentz invariance that is often assumed for sigma models. Thus, we have a non-Lorentz invariantflag manifold sigma model, just as was the case in Section 4.7, where self-conjugate SU(3) chainswere considered. In the following section, we will use the renormalization group to show that at lowenough energies, these distinct velocities in fact flow to a single value, so that Lorentz invarianceemerges after all.5.3.3 Embedding into complex projective spacesIn Chapter 3, it was explained how the complete flag manifold SU(n)/[U(1)]n−1 can be embeddedinto an n-fold product of CPn−1 spaces. In (6.7), this was restated explicitly for the case of SU(3).855.4. Renormalization Group AnalysisHere, the situation is no different. The n complex fields ϕα may be viewed as each living in its ownCPn−1 sector, and we may identify the flag manifold with its image under the following map:U(n)[U(1)]n↪→n copies︷ ︸︸ ︷CPn−1 × · · · × CPn−1 . (5.32)Explicitly, this embedding is determined by the metric parameters gt, the torsion parameters λt,as well as the condition of orthogonality between the fields. In this context, we may view thetopological charges Qα as the pull-backs to the flag manifold of the familiar topological chargesdefined onCPn−1. In Section 4.4.2, we further made use of this embedding to introduce U(1) gaugefields Aaµ, to rewrite the action as in (4.60). A similar rewriting proves to be much more difficult inSU(n), since now there are multiple coupling constants gt. Said differently, the metric is no longerdiagonal in the ϕα.5.3.4 Translation invarianceSince it will be critical below, we take a moment to observe the action of translation on the flagmanifold sigma model. Since translation shifts the chain by one site, it corresponds to the mappingϕα 7→ ϕα+1, (5.33)where the index α is defined modulo n. Therefore, translation manifests itself in the sigma model asa Zn action, generalizing the Z3 symmetry present in the symmetric SU(3) chains. We will returnto this fact when we discuss ’t Hooft anomalies below.5.4 Renormalization Group AnalysisRecently, the Lorentz invariant versions of the above flag manifold sigma models were studied ingreat detail in [79]. In particular, the renormalization group flow of both the gt and λt parameterswere determined for general n. Moreover, field theoretic versions of the LSMA theorem wereformulated, using the methods of ’t Hooft anomaly matching. For a review of this subject, seeSection 4.6.4. Here, we would like to apply these results to our SU(n) chains which lack Lorentzinvariance in general. To do so, we consider the differences of velocities occurring in (5.27), namely∆tt′ := vt − vt′ , (5.34)and ask how they behave at low energies. More precisely, we calculate the one-loop beta functionsof these ∆tt′ , to O(gt) and O(λt). We will find that each of the ∆tt′ flows to zero under renormaliza-tion. Moreover, we will show that this implies Lorentz invariance at our order of approximation.This is consistent with the fundamental SU(n) models with p = 1, where it is known by Betheansatz that Lorentz invariance is present [12, 84, 85]. Our calculations were motivated by a sim-ilar phenomenon in 2+1 dimensional systems, where an interacting theory of bosons and Weylfermions renormalizes to a Lorentz invariant model [133, 134].865.4. Renormalization Group Analysis5.4.1 Goldstone mode expansionIn the following, it will be useful to introduce dimensionless velocities, ut, defined according tout :=vtv¯v¯ =1bn2cb n2 c∑t=1vt, (5.35)and introduce new spacetime coordinates which both have units of (length · time)1/2:x→ x√v¯τ→ √v¯τ. (5.36)In these units, ∆tt′ = ut − ut′ . The coefficients gt appearing in (5.27) are dimensionless, and are allproportional to 1p . Since we’ve taken a large-p limit, we will expand all quantities in powers of thegt. As we will see below, the coefficients λt in (5.27) do not enter into our one-loop calculations,and so we will neglect them throughout.Since we are interested in the low energy dynamics of these quantum field theories, we makethe simplifying assumption that the matrices U are close to the identity matrix, and expand them interms of the SU(n) generators. In Appendix C, we prove that U may factorized according toU = DVD = eiωγTγV = eiωaTa . (5.37)Let us explain our notation. Here and throughout, we use Greek letters to index the diagonal SU(n)generators, lowercase Latin letters to index the off-diagonal ones, and uppercase Latin letters toindex the complete set. Repeated indices will always be summed over unless otherwise specified.Since D is diagonal, this factorization of U implies that D drops out from the traces occurring in(5.27):tr[U∂µU†Λα∂µUU†Λβ] = tr[V∂µV†Λα∂µVV†Λβ] (5.38)Therefore, when deriving the Lagrangian of the ωa, we may write U in terms of the off-diagonalgenerators only:U = eiωaTa = 1 + iωaTa − 12ωaωbTaTb + O(ω3). (5.39)We choose a convenient normalization in which the off-diagonal generators have entries 1 or ±i,and satisfy[Ta,Tb] = 2i fabCTC . (5.40)These generators are n × n matrices that have a very specific structure. There are n − 1 diagonalones, and n(n− 1) off-diagonal ones, that come in pairs. For each pair of integers {α, β} with α, β =1, . . . , n and α , β, there are exactly two generators with nonzero (α, β) entries. We define Iαβ tobe the set of two indices corresponding to the SU(n) generators with nonzero (α, β) entries. Forexample, in SU(3), the off-diagonal generators (in Gell-Mann’s notation) are T1,T2 with nonzero875.4. Renormalization Group Analysisentries in the (1, 2) positions; T4,T5 with nonzero entries in the (1, 3) positions; and T6,T7 withnonzero entries in the (2, 3) positions. They are provided explicitly in Appendix E. Then,I12 = {1, 2} I13 = {4, 5} I23 = {6, 7}. (5.41)With this notation, we now expand −tr[∂µUU†Λα∂µUU†Λβ] to O(ω4). We start with∂µUU† = i∂µωaTa +[∂µωaωb − 12∂µ(ωaωb)]TaTb (5.42)− i2[∂µωaωbωc − ∂µ(ωaωb)ωc + 13∂µ(ωaωbωc)]TaTbTc + O(ω4).(5.43)Since∂µωaωb − 12∂µ(ωaωb) =12[∂µωaωb − ωa∂µωb], (5.44)we have [∂µωaωb − 12∂µ(ωaωb)]TaTb =12∂µωaωb[Ta,Tb] = i∂µωaωb fabCTC . (5.45)Moreover, since∂µωaωbωc−∂µ(ωaωb)ωc+ 13∂µ(ωaωbωc) =13(∂µωaωbωc−ωa∂µωbωc)+ 13(ωaωb∂µωc−ωa∂µωbωc),(5.46)we also have[∂µωaωbωc − ∂µ(ωaωb)ωc + 13∂µ(ωaωbωc)]TaTbTc (5.47)= −13ωa∂µωbωc ([Ta,Tb]Tc + Ta[Tb,Tc])= −2i3ωa∂µωbωc fabD[TD,Tc]=43ωa∂µωbωc fabD fDcETE .Combining these facts, we have∂µUU† = i∂µωaTa + i∂µωaωb fabCTC − 2i3 fabD fDcEωa∂µωbωcTE + O(ω4). (5.48)This yields−tr[∂µUU†Λα∂µUU†Λβ] = ∂µωa∂µωbtr[TaΛαTbΛβ] (5.49)+[fbcE +23fbcD fDgEωg]∂µωa∂µωbωc(tr[TaΛαTEΛβ] + tr[TEΛβTaΛα])885.4. Renormalization Group Analysis+∂µωaωb∂µωcωd fabE fcdGtrΛαTEΛβTG.Now we want to simplify this by understandingtr[TaΛαTbΛβ] = [Ta]βα[Tb]αβ. (5.50)Since α , β, the above expression vanishes if either of a or b is a diagonal generator. In fact, wehavetr[TaΛαTbΛβ] + tr[TbΛαTaΛβ] =2δab a, b ∈ Iαβ0 else . (5.51)Returning to our calculation, we now have− tr[∂µUU†Λα∂µUU†Λβ] =∑a∈Iαβ[(∂µω2a) + 2 fbca∂µωa∂µωbωc (5.52)+43fbcE fEda∂µωa∂µωbωcωd + ∂µωeωb∂µωcωd feba fcda],where all repeated indices are summed over. Now, to obtain the full Lagrangian, we must sum overthe possible combinations of α and β. Sincen∑α<β∑a∈Iαβ:=n∑β=2β−1∑α=1∑a∈Iαβ=∑a, (5.53)where∑a again denotes a sum over all the off-diagonal generators of SU(n), the non-interactingLagrangian has the formL0 = 1ga[1ua(∂τωa)2 + ua(∂xωa)2], (5.54)wherega := g|α−β|∣∣∣∣Iαβ3aua := u|α−β|∣∣∣∣Iαβ3a. (5.55)and again, all repeated indices are summed over. We rescale the fields according toωa 7→√ga2ωa (5.56)to yieldL = 12[1ua(∂τωa)2 + ua(∂xωa)2]+√gagbgc√2ha(µ)gafbca∂µωa∂µωbωc (5.57)√gbgcgd4ha(µ)ga[√ge∂µωe∂µωbωcωd feca fbda +43fbcE fEda√ga∂µωa∂µωbωcωd]+ O(ω5),whereha(µ) = 1ua µ = τua µ = x . (5.58)This expression will be the starting point of our renormalization group calculations.895.4. Renormalization Group Analysis5.4.2 Dimensional regularizationIn Chapter 4, we introduced the theory behind the renormalization group and proceeded to useWilson’s formalism in order to calculate the beta functions of g and λ in the symmetric SU(3)chain. In this approach, the length scale of the theory is explicitly changed, and the beta functionsare determined by finding the altered values of g and λ at this new scale.An alternative approach is to use a regularized theory at a fixed energy scale.33 Consider thefollowing example Lagrangian density of a real scalar field:Lϕ4 = 12(∂ϕ)2 + m2ϕ2 + Xϕ4. (5.59)This theory is not regularized: a perturbative expansion of its correlation functions will lead todivergences, order by order. To resolve this, we use a procedure known as dimensional regulariza-tion, in which we continue the spacetime dimension away from an integer, rendering momentumloop integrals finite. We then subtract off these contributions by introducing counterterms into theLagrangian density, before continuing back to an integer dimension. This is performed at a givenenergy scale M, which we fix.For example, at one loop, the ϕ self energy receives a contribution proportional to∫dd p(2pi)d1(p2 + m2)∝ Γ(1 − d2), (5.60)where Γ(x) is the gamma function with poles at non-positive integers. For non-integer d, thisexpression is finite, and can be cancelled by introducing a countertermδZϕ12(∂ϕ)2 (5.61)into the Lagrangian, withδZϕ ∝ −Γ(1 − d2). (5.62)Repeating these steps for all divergences at a given order, we arrive at a renormalized Lagrangiandensity at scale M,Lϕ4,r = 12Zϕ(∂ϕr)2 + ZmM2m2rϕ2r + ZX M4−dXrϕ4r , (5.63)in terms of a renormalized field ϕr and renormalized coupling constants mr and Xr. The renormal-ization constants Zi contain the introduced counterterms δZi according toZi = 1 + δZi. (5.64)The explicit energy scale M enters to make the renormalized coupling constants dimensionless.Now, to extract the beta functions, we must relate the two Lagrangian densities (5.59) and (5.63).Matching kinetic terms, we findϕ =√Zϕϕr, (5.65)33The beginning of this discussion first appeared in [124].905.4. Renormalization Group Analysisand then rescaling, we findmr = mM−1ZϕZ−1m , (5.66)Xr = XM4−dZ2ϕZ−1X . (5.67)These equations are the dimensional regularized analogues of (4.79) in Chapter 4. Differentiatingwith respect to (− log M) generates the desired beta functions. The negative sign enters because anincrease in energy M corresponds to a decrease in length scale, b.Exactly how the counterterms in (5.64) are defined leads to further choice in renormaliza-tion scheme. If only the divergent parts of the loop diagram are included in the counterterm, thescheme is known as ‘minimal subtraction’. In our calculations, we use the more common ‘mod-ified minimal subtraction’ scheme, or MS , which adds to the counterterm the universal constantlog (eγE/4pi) that often occurs in Feynman diagrams. This is implemented by rescaling the energyscale M → M eγE4pi in (5.63) [130, 135].5.4.3 Renormalization group equationsIn order to derive the renormalization group equations for the model (5.57), we introduce a set ofrenormalization coefficients, Zµa and Zeµabcd, as follows. Since (5.57) has divergences at one-looporder, we rewrite the theory in terms of renormalized parameters, asL = 12[Zτa1ura(∂τωa)2 + uraZxa(∂xωa)2]+ Z(1)µabc√gragrbgrc√2hra(µ)grafbca∂µωa∂µωbωc (5.68)+√grbgrcgrd4hra(µ)gra[Z(2),eµabcd√gre∂µωe∂µωbωcωd feca fbda+Z(3)µabcd43fbcE fEda√gra∂µωa∂µωbωcωd]+O(ω5).The superscripts ‘r’ emphasize that the coupling constants and velocities appearing in (5.68) aredifferent from those appearing in (5.57) (and are not indices to be summed over). Each of the renor-malization coefficients has the form Z = 1 + δZ, where δZ is a one-loop counterterm regularizingany UV divergence. Below, we use dimensional regularization to calculate the δZ at a fixed energyscale M. Then, by rescalingωa →(1ZxaZτa)1/4(5.69)in (5.68), and comparing (∂xωa)2 terms in (5.57) and (5.68), we obtain the following equation forura:ua = ura√ZxaZτa. (5.70)The derivative of ura with respect to − log M,βua := −durad log M, (5.71)915.4. Renormalization Group Analysisis the beta function of ua, and describes the flow of ua as the energy scale, M, is changed. Thenegative sign is present, so as to be consistent with our definition of beta function in Chapter 4,which differentiated with respect to a length scale, log(b). It is important to note that since thisequation only depends on Zτa and Zxa , we are only tasked with calculating divergences of two-pointfunctions in our lowest-order regularization scheme.We use dimensional regularization to evaluate one-loop diagrams in d = 2 −  dimensions in(5.68). We drop all ‘r’ superscripts, and introduce the following compact notation:g(1)abcd(µ) :=M4√gagbgcgdhe(µ)face fbdege(5.72)g(2)abcd(µ) :=Mha(µ)3√gagbgcgdgafbcE fEda (5.73)Again, all indices refer to off-diagonal SU(n) generators, except for the uppercase letters, whichrefer to the complete set. We’ve introduced the renormalization scale M so that the coupling con-stants remain dimensionless. Since we are only tasked with calculating the Zµa , the only divergingdiagrams we must consider are those that correct the boson self energy. This immediately impliesthat the cubic interaction term occurring in (5.68) plays no effect at this order. The only contributingdiagram Πab(k), shown in Figure 5.2, equalsΠab(k) = −2∫ddq(2pi)d〈ωc(q)ωc(−q)〉[gabcc(µ)kµkµ + gccab(µ)qµqµ](5.74)wheregabcd = g(1)abcd + g(2)abcd. (5.75)In addition to UV divergences, there are also IR divergences occurring at zero momenta. Toremove these, we introduce a small mass m to the boson fields ωa, and take the limit m → 0 oncewe’ve extracted the UV divergence. A convenient mass term with the appropriate dimensions ism2v¯3uaω2a. Then, the free propagator is〈ωc(q)ωc(−q)〉 = ucω2 + u2c~q2 + m2u2c v¯3q = (ω, ~q) (5.76)and we have two integrals to consider:• Integral 1: ∫ddq(2pi)d〈ωc(q)ωc(−q)〉 = 12∫dd−1q(2pi)d−11√~q2 + m2v¯3=12pi+ O(0) (5.77)• Integral 2:∫ddq(2pi)d〈ωc(q)ωc(−q)〉qxqx = 12∫dd−1q(2pi)d−1~q2√~q2 + m2v¯3=12pim2v¯3 + O(0) (5.78)925.4. Renormalization Group Analysisq-kkcbaFigure 5.2: The diagram Πab(k), drawn using [136].In Integral 2, we’ve taken µ = x without loss of generality. It appears that such integrals willrenormalize the boson masses; however, since these contributions are proportional to the IRcutoff m, when we restore m→ 0, these poles will drop out of our calculations. See equation13.82 of [130] for a similar argument in the O(3) nonlinear sigma model.Returning to the process (5.74), we find thatΠab(k) = − 1pikµkµgabcc(µ) = −M√gagbpikµkµgc[14he(µ)face fbcege− ha(µ)3gafbcE facE]. (5.79)This result will contribute to the renormalization constants involving ωa and ωb.5.4.4 Beta functions of the Goldstone velocitiesIn order to simplify (5.79) further, we make use of two identities (F.1) and (F.12), proven in Ap-pendix F. Doing so, we conclude that (5.79) equalsΠab(k) = −Mgaδab2pikµkµ( n−1∑i=1i,thi(µ)2gig|t−i| − ha(µ)3ga[ga +12∑cgc]). (5.80)(no sum over a). Since∂µω∂µω ∼ −ω∂2µω ∼ +kµkµω(k)ω(−k) (5.81)935.4. Renormalization Group Analysiswe may read off from Πab(k) the renormalization group constants:Zτa = 1 +Mgaua2pi( n−1∑i=1i,t1uigig|t−i| − 23gaua[ga +12∑cgc])(5.82)Zxa = 1 +Mga2piua( n−1∑i=1i,tuigig|t−i| − 2ua3ga[ga +12∑cgc])(5.83)(no sum over a). Inserting these expressions into (5.71), and using (5.70), we find that for t =1, 2, . . . , q := b n2c,βut = −utgt4pin−1∑i=1i,tg|t−i|gi[utui− uiut]. (5.84)These are the beta functions for the different Goldstone velocities in the flag manifold sigma model.In the following subsection, we use (5.84) to show that these velocities flow to a common value atsmall energies.5.4.5 Renormalization of velocity differencesWe want to study the renormalization group flow of the velocity differences, ∆tt′ defined in (5.34).As mentioned above, the identity ut = un−t reduces the number of independent velocities to q = b n2c,and the relation∆tt′ = ∆t1 + ∆1t′ = ∆1t′ − ∆1t (5.85)shows that the number of independent velocity differences is q−1. To study their flow collectively,we introduce a (q − 1)-component vector, ∆, with components∆i := ∆1,i+1 i = 1, 2, . . . , q − 1. (5.86)If we assume that the velocities ut are initially close together, so that the SU(n) chain is approxi-mately Lorentz invariant, the vector ∆ will obey an equation of the form−β∆ = dd log M∆ = R∆ (5.87)for a (q− 1)× (q− 1) matrix R. The spectrum of R will reveal the low energy behaviour of the ∆tt′ :if the spectrum is strictly positive, we may conclude that all velocity differences flow to zero in theIR. Let us now determine R.Assuming the velocities ut are initially close together, we rewrite (5.84) to linear order in ∆t asβn=2qut = −gt2pi[ q−1∑i=1∆tigi(gi+t + g|t−i|)+g|t−q|gq∆tq]+ O(∆2), (5.88)945.4. Renormalization Group Analysisβn=2q+1ut = −gt2piq∑i=1∆tigi(gi+t + g|i−t|)+ O(∆2), (5.89)depending on the parity of n. (We’ve introduced a g0 := 0 for notational convenience). Herewe have used the fact that only q := b n2c velocities and coupling constants are unique. The betafunction for a component ∆i of ∆ is thenβn=2q∆t= − 12piq−1∑i=1∆igi[g1(gi+1 + g|1−i|) − gt (gi+t + g|t−i|) ] (5.90)−gt∆t2pi[ q−1∑i=11gi(gi+t + g|t−i|)+g|t−q|gq]− ∆q2pigq[−gtg|t−q| + g1g|1−q|]+ O(∆2),βn=2q+1∆t= − 12piq∑i=1∆igi[g1(gi+1 + g|i−1|) − gt (gi+t + g|i−t|) ] − gt∆t2piq∑i=11gi(gi+t + g|i−t|)+ O(∆2),(5.91)depending on the parity of n. Clearly, finding the eigenvalues of the R matrix in (5.87) is a difficulttask. As a first check, we consider the symmetric point where all couplings equal the same value, g(except for the artificial g0, which is always zero). In this case, we can clearly read off from (5.84)thatβ∆t = −g∆t2pi(n − 1) (5.92)so that the matrix beta equation is diagonal, with R having positive eigenvalues. Next, we considersmall values of n.• SU(4)In this case, there is a single velocity difference, β∆12 , withβ∆12 = −∆122pig2[g21 + 2g22]< 0. (5.93)• SU(5)In this case, there is again a single velocity difference, withβ∆12 ==1pig2∆12(g21 + g22) < 0. (5.94)• SU(6)In this case, there are three velocities, three coupling constants, and two unique velocitydifferences, ∆12 and ∆13. The eigenvalues of the 2 × 2 matrix R are both positive. Explicitly,they are{ 12pig1g2g3(g21g22 + g31g3 + g1g22g3 + g21g23 + g22g23),12pig1g2g3(g21g22 + 2g21g23 + 2g22g23)}.(5.95)955.5. General Phase DiagramUnable to find the eigenvalues of the R matrix explicitly, we resort to a numerical investiga-tion of its spectrum. We verify that the spectrum is positive definite by computing the minimaleigenvalue of R for fixed coupling constants. First, we choose the b n2c coupling constants randomlyfrom the interval (0, 1). In 10 000 trials, we find that the minimal eigenvalue is always strictlypositive, for SU(n) with n = 3, 4, . . . , 50. Next, we probe points in parameter space where differentcoupling constants have a common value, by choosing coupling constants from a discrete latticeon (0, 1)b n2 c. Since the dimension of the lattice increases with n, we choose a coarser discretizationas n increases, to keep the number of lattice points below 100 000. In this case, we find that forn = 3, 4, . . . , 16, the minimal eigenvalue of the R matrix is again strictly positive. This supports theconjecture that the spectrum of R is always positive, so that each velocity difference ∆tt′ flows tozero in the IR.So far, we have verified that the velocity differences ∆tt′ in these sigma models flow to zero atlow energies. However, we now claim that this is sufficient to conclude that the entire theory (5.27)is Lorentz invariant at low energies. We note that we are not required to restore the pure-imaginaryλ-terms occurring in (5.27), since they are proportional to µν∂µφ∂νφ, a Lorentz scalar. Indeed,since the interaction vertex receives no O(g) correction, the only spacetime dependence entersthrough the renormalization of ht(µ) and through the renormalization of the fields ωa themselves.Since the latter are independent of µ (see (5.82) and (5.83)), the Lorentz non-invariance of theinteractions is entirely captured by the ht(µ). Since ut − u′t → 0 implies u−1t − u−1t′ → 0 at O(g),we may use the results of the previous subsection to conclude that the ht(µ) all flow to a commonvalue h(µ), and thus Lorentz invariance of the entire model (5.27) is possible if the velocities areinitially close to each other.5.5 General Phase DiagramBased on the renormalization group analysis in the previous section, we conclude that at lowenough energies, SU(n) chains in the rank-p symmetric representation (without fine-tuning) willbe described by a Lorentz invariant flag manifold sigma modelL =∑α<β1g|α−β|tr[ΛαU∂µU†Λβ∂µUU†] − µν∑α<βλ|α−β|tr[∂µUU†Λα∂νUU†Λβ], (5.96)with topological termS top = iθn−1∑α=1αQα θ :=2pipn. (5.97)In [79], the renormalization group flow of the λt and the gt was determined, and given a geometricinterpretation. It was found that for n > 4, the gt flow to a common value in the IR, and that forn > 6, the λt flow to zero in the IR. Thus we may expect an S n (permutation group) symmetry toemerge at low enough energies, and for n > 6. It is known that in these S n-symmetric models, the965.5. General Phase Diagramunique coupling constant g obeys [79]βg = − dgd log M =n + 24pig2 > 0, (5.98)and the theory is asymptotically free. These facts suggest that a gapless phase, driven by topologicalterms, may be present in these types of SU(n) chains. In previous chapters, we learned that thisoccurred whenever the topological angles of the theory were nontrivial, i.e. for p not a multiple ofn. Should this phenomenon extend to all values of n?Before jumping to this conclusion, it is important to realize that SU(2) and SU(3) are special.Since 2 and 3 are prime, any value of p not a multiple of n is necessarily coprime with n, andleads to n − 1 nontrivial topological angles. And, by the LSMA theorem, this leads to either agapless phase or an n-fold degenerate ground state. However, for nonprime values of n, that havenontrivial greatest common divisors with p, another scenario is possible. For example, in thep = 2 representation of SU(4), the topological angle θ2 =pip2 becomes trivial; moreover, the LSMAtheorem implies only a 2-fold degenerate ground state, instead of an n-fold one. These observationsshould make one hesitate before extrapolating our results from the previous chapters.On the other hand, we do expect the prediction of a unique gapped ground state when p is amultiple of n to be robust. This follows directly from the fact that AKLT-like states exist for theserepresentations, as shown in Section 3.5.3.5.5.1 Strong coupling analysisBy taking the large coupling limit, gt → ∞, we are able to see that the ground state degeneracy ofthe flag manifold sigma model does in fact depend on gcd(n, p). In this limit, only the λ-terms andtopological terms survive. We will neglect the λ-terms: for n > 6 this is justified by the fact thatthe λt parameters flow to zero under renormalization; for n ≤ 6, this is an added assumption that ismade to simplify our analysis. In this case, with only the topological terms remaining, the partitionfunction is a path integral over the topological charge densities, with the extra constraints that thetopological charges take integer values, and that the topological charge densities on each plaquetteof the lattice sum to zero. We findZ(θ1, . . . , θn) =∑{m1,...,mn}z(θ1, θ2 + 2pim2, . . . , θn + 2pimn)V (5.99)where V is the spacetime volume, mi ∈ Z, andz(θ1, θ2, . . . , θn) =∫dk2pin∏α=12 sin(12 (k − θα))(k − θα) . (5.100)For our models, we have θα =2pipn (α − 1), so that the denominator of this expression does notdiverge. The term(s) with the largest value dominate the sum, so that the free energy density reads975.5. General Phase Diagramasf (θ1, . . . , θn) = − log(max{m2,m3,··· ,mn}z(θ1, θ2 + 2pim2, . . . , θn + 2pimn)). (5.101)Just as in SU(3), we may identify phase transitions in this model by identifying sets of {mα} wheremultiple terms in the free energy density dominate the logarithm. Indeed, as we pass from onephase to another, we shift from one dominant term to another, resulting in a cusp in f . The numberof dominant terms meeting at a given point will give the degeneracy of the phase transition. Thus,we search for distinct sets {mα} and {nα} that obeyz(θ1, θ2 + 2pim2, . . . , θn + 2pimn) = z(θ1, θ2 + 2pin2, . . . , θn + 2pinn), (5.102)which can happen if the angles on the right hand side are a permutation of the ones on the left, upto a constant shift. This fact follows from the explicit expression of z, given in (5.100). So, we lookfor permutations σ ∈ S n such thatmσ(α) = nα + ∆, (5.103)where ∆ is some constant shift. The solutions of this expression areσ(k) : α 7→ α + k k = 1, 2, . . . , n − 1, (5.104)where the right hand side is defined modulo n. The number of distinct solutions equals n/ gcd(n, p).Thus, we find that if n and p are coprime, there are n such solutions, leading to an n-fold groundstate degeneracy. Meanwhile, if gcd(n, p) = q > 1, we find an r-fold degeneracy, where r := n/q.5.5.2 ’t Hooft anomaly matchingUsing the notion of ’t Hooft anomaly matching, both [79] and [78] were able to formulate a field-theoretic version of the LSMA theorem for SU(n) chains. In short, the presence of an ’t Hooftanomaly signifies nontrivial low energy physics; in one dimension, this necessitates a gapless phaseso long as the symmetries of the SU(n) chain are not spontaneously broken (see Section 4.6.4 fora review). It was shown that in these models, an ’t Hooft anomaly is present so long as p is not amultiple of n. Explicitly, it is a mixed anomaly between the PSU(n) := SU(n)/Zn symmetry andthe Zn translation symmetry of the n-site-ordered classical ground state. It is a PSU(n) symmetry,and not an SU(n) symmetry, because of a Zn subgroup of SU(n) that acts trivially on each term inthe field theory. This is important because when one tries to gauge PSU(n), it leads to gauge fieldswith topological charge 1n , and forces one to include in the Lagrangian a counterterm that violatesthe Zn symmetry. When this anomaly is present, the gapped phase must have spontaneously brokentranslation or PSU(n) symmetry; the latter is ruled out by Coleman’s theorem. In the gapped phase,the ground state degeneracy is predicted to be ngcd(n,p) , which is consistent with the strong couplinganalysis of the previous subsection, as well as the LSMA theorem.The authors of [79] then argued that while an anomaly is present whenever p mod n , 0,an RG flow to an IR stable WZW fixed point is possible only when p and n have no nontrivialcommon divisor. In this case, the flow is to SU(n)1. Otherwise, the candidate IR fixed point985.5. General Phase Diagramis SU(n)q, where q = gcd(n, p), which is unstable and requires fine-tuning in order for the flagmanifold sigma model to flow there. Also, there is no possible flow from this unstable theoryto SU(n)1, since this would violate the anomaly matching conditions derived in [116] for genericSU(n) WZW models. There, it was found that the WZW model of level k′ can flow to level k onlyif k′ > k and gcd(n, k) = gcd(n, k′).5.5.3 A Haldane conjecture for SU(n) chainsBased on these arguments, we conclude that the rank-p symmetric SU(n) chains may flow to anSU(n)1 WZW model if p and n do not have a common divisor. In this case, we expect gaplessexcitations to appear in the excitation spectrum. For other values of p not a multiple of n, wepredict a gapped phase with n/ gcd(n, p)-fold ground state degeneracy, in accordance with theLSMA theorem. Finally, when p is a multiple of n, we predict a unique gapped ground state,such as the AKLT-like configuration given in Section 3.5.3. See Figure 5.3 for a simplified phasediagram of the SU(n) chain in the case when p and n are coprime. We can understand this phasetransition as being driven by the SU(n)-invariant interaction term∑a JaLJaR in the SU(n)1 WZWmodel, where JL and JR are the left and right moving currents, respectively. This operator ismarginal: for one sign of its coupling constant, it is marginally irrelevant and flows to zero; for theother sign it flows to large values [115]. Our claim about gapless behaviour translates to a statementabout the sign of this coupling constant: at weak flag manifold coupling g, we expect this term to bemarginally irrelevant. Then, at larger couplings, the term becomes marginally relevant, and thereis an RG flow to a gapped phase with n degenerate ground states that break translation symmetry.These predictions are a natural extension of our SU(3) Haldane conjecture from the previouschapter. We note that when p and n have a common divisor, at least one of the topological anglesoccurring in (5.97) is necessarily trivial. In the instanton gas picture of Haldane’s conjecture, eachtype of topological excitation must have a nontrivial topological angle in order to ensure totaldestructive interference in half-odd integer spin chains [137]. This might lead one to speculate thata similar mechanism is at play here in SU(n) chains. We will prove that this is indeed the case inChapter 7.ggapless gapped(n-fold degeneracy)SU(n)1Figure 5.3: A simplified phase diagram of the SU(n) chains we consider, as a function of couplingconstant g when p and n are coprime. A more accurate diagram would include b n2c different cou-pling constants; in this case, we predict a critical point described by SU(n)1 occurring somewherein this multidimensional space.995.6. ConclusionFinally, we may speculate on the size of the energy gap when p is not coprime with n. Ingeneral this is a difficult task, but in the special case of a theory with a unique coupling constant g,one may identify a characteristic length scale. This follows from the fact that g is described by thebeta function (5.98), which has a lowest-order solution of the formg(M) =g01 + n+24pi g0 log(M). (5.105)Here g0 is the bare value of g, and is a function of the interaction strengths Jt of the underlyingSU(n) chain. This equation shows that g becomes order 1 at an energy scale of∆ = Ce−4pi(n+2)g0 , (5.106)for some unknown prefactor C with units of energy. This expression defines a characteristic energyscale that should correspond in magnitude to the finite mass gap in our SU(n) chains. For SU(2)and SU(3) chains, this calculation leads to a direct prediction. In particular, we predict the Haldanegap in SU(3) chains with p a multiple of 3 to be of the order∆n=3 = Ce− 4pip√J1 J25(J1+J2) . (5.107)Of course, if one starts with a nearest-neighbour SU(3) model, then this expression is less predictivebecause of the difficulty in finding J2 as a function of J1. And even still, the prefactor C is unknown.In [77], a numerical estimate of this gap in the nearest-neighbour p = 3 model was found to be∆n=3/J ∈ (0.017, 0.046).For n > 3, the presence of multiple coupling constants leads to another complication. Whilewe know that each coupling is inversely proportional to p, the prefactors depends on the interactionstrengths Jt of the chain, and we do not have a beta function as simple as (5.98). Nonetheless, if theinteraction strengths are chosen so that all of the couplings gt are of the same magnitude gt ∼ c/p,then we may still expect the gap to scale like e−4picp(n+2) . For example, in SU(4) the simultaneous tuning√J1J3/(J1 + J3) ∼ c J2 ∼ c (5.108)leads to a Haldane gap prediction of order ∆n=4 = Ce−2pip3c . In any case, the prediction of a gap thatdecreases exponentially in p should hold for all values of n, even without fine-tuning.5.6 ConclusionIn this chapter, we have extended Haldane’s conjecture to SU(n) chains that have a rank-p sym-metric representation at each site. Our starting point was SU(n) flavour-wave theory, where wefirst realized that the property of having a unique excitation velocity, which is present in SU(2)and SU(3) chains, no longer is true for chains with n > 3. This fact inhibits the immediate emer-gence of Lorentz invariance that occurred in those simpler models. In Section 5.4, we came to1005.6. Conclusionthe same conclusion by deriving in detail the low energy sigma model description of these SU(n)chains. However, in Section 5.5, after extensive renormalization group calculations, we were ableto conclude that these distinct velocities in fact flow to a common value at low enough energies.Because of this, we argued that SU(n) chains in the rank-p symmetric representation do in factcorrespond to the (Lorentz-invariant) SU(n)/[U(1)]n−1 flag manifold sigma model, equipped withn−1 topological angles θα = 2pin pα. Then, by applying a combination of exact results, including theLSMA theorem, AKLT constructions, and ’t Hooft anomaly matching, we were able to formulatethe following conjecture:• For p and n coprime, we predict a gapless phase driven by topological angles to exist belowsome critical interaction strength. This phase is characterized by the SU(n)1 WZW model.Above this critical coupling, we predict spontaneously broken translation symmetry and n-fold degenerate ground states.• For 1 < gcd(n, p) < n, we predict that a gapless phase can never occur, so that translationsymmetry is always spontaneously broken by the LSMA theorem. The ground states aren/ gcd(n, p)-fold degenerate.• Finally, for p a multiple of n, we predict a unique, gapped ground state that is translationallyinvariant. The AKLT ground states presented in Section 3.5.3 correspond to this scenario.Our prediction that translation symmetry is necessarily broken when p and n have a nontrivialcommon divisor other than n is rather interesting. This was not a possibility for the chains withsmaller symmetry groups considered in Chapters 2 and 4, because n = 2 and n = 3 are both primenumbers. When this is the case, we see that some, but not all of the topological angles in thesigma model vanish. It appears then, that in order for a phase transition to be driven by topologicaleffects in these SU(n) chains, we require all n − 1 angles to be nontrivial. In Chapter 7, we presentan elegant argument for why this is the case. In essence, the finite gap in SU(n) chains can beunderstood as being generated by topological excitations. However, when all n − 1 topologicalangles are nonzero, these excitations destructively interfere with each other, and destroy the mass-generating mechanism.101Chapter 6Flag Manifold Sigma Models FromOther Representations of SU(n)6.1 IntroductionIn the previous chapters, we have clearly seen that much can be learned by establishing a mappingbetween SU(n) chains and flag manifold sigma models. In this chapter, we seek a more completeunderstanding of this correspondence. What this entails is addressing the following question: Whatrepresentations of SU(n) chains give rise to SU(n)/[U(1)]n−1 flag manifold sigma models? Oncethis has been answered, we should then further ask: What representations of SU(n) chains giverise to those flag manifold sigma models that possess ’t Hooft anomalies, and lend themselves to aHaldane-like prediction of gapless excitations in certain cases.In this chapter, we answer these two questions, ultimately classifying all SU(n) chains thatadmit such a flag manifold sigma model mapping. We have already set the stage in Section 3.4,where we classified the classical ground states for the most general representations of SU(n). Infact, we suggest to the reader that they recall much of Chapter 3 in order to motivate the presentdiscussion. Here, we will begin by summarizing various facts about the flag manifold sigma modelsin Section 6.2. Then we turn in Section 6.3 to the derivation of the dispersion relations of lowenergy modes in generic SU(n) chains. In Chapters 2, 4, 5, these were called the dispersion relationsof flavour-wave, or Goldstone mode excitations. We will see that generically, both quadratic andlinear dispersing modes are present in these models, which is a manifestation of the ferro- andantiferromagnetic order parameters that coexist. This phenomenon was previously observed inSection 4.7 for SU(3) chains; now we learn that it occurs for all n. We also speculate in this sectionabout how such coexisting order parameters may lead to an entire hierarchy of flag manifold sigmamodels, of different dimension, for a given value of n. Next, in Section 6.3, we restrict our attentionto the representations with linear dispersion only, and determine their topological angle content.In Section 4.7, this amounted to considering the self-conjugate SU(3) chains, in addition to thesymmetric ones. For n > 3, we will see that many other families of representations also possessthis property. Then, in Section 6.4, we summarize our calculations and determine which of therepresentations considered in the previous section also admit a mixed ’t Hooft anomaly. This willallow us to extend our SU(n) version of the Haldane conjecture from the previous chapter to newrepresentations of SU(n). Finally, our concluding remarks can be found in Section 6.6.1026.2. Flag Manifold Sigma Models6.2 Flag Manifold Sigma ModelsFor concreteness, let us write down the Lagrangian for the SU(n)/[U(1)]n−1 flag manifold sigmamodel, in a manner more formal than that found in previous chapters. We introduce n orthonor-malized fields ϕα ∈ Cn, which are each invariant under a separate U(1) symmetry,ϕα 7→ eiθϕα. (6.1)If we denote by gαβ the (symmetric) metric on the flag manifold, and bαβ the (antisymmetric)torsion, then the Lagrangian (5.97) can be rewritten asL =n∑α,β=1[gαβδµν + bαβµν](ϕα · ∂µϕ∗,β)(ϕβ · ∂νϕ∗,α) +Ltop (6.2)where Ltop is the topological termLtop =n∑α=1θα2piµν∂µϕα · ∂νϕ∗,α. (6.3)Only n − 1 of the angles θα are independent, since the theory is invariant under shifting all anglesby the same amount. This reflects the fact thatH2(SU(n)/[U(1)]n−1) = Zn−1. (6.4)In fact, all of the topological angles may be removed using the combined shiftsθα → θα + 2pibα + bαβ 7→ bαβ − bα − bβ (6.5)but this hides the 2pi periodicity of the θα [79]. Finally, the shift gαβ → gαβ − cα − cβ introduces thefamiliar CPn−1 kinetic terms into the Lagrangian:L → L +n∑α=1cα[|∂µϕα|2 − |ϕ∗,α · ∂µϕα|2]. (6.6)Based on this fact, it is useful to use think of the embeddingSU(n)[U(1)]n−1↪→ CPn−1 × · · · × CPn−1 (6.7)and visualize the field content as a set of orthogonal CPn−1 fields, coupled through the metric andtorsion terms. Since CPn−1 = G1,n, this embedding is a special case of (3.16).In most cases, the tensors gαβ and bαβ will admit additional, discrete symmetries. For example,a sigma model that arises from an SU(n) chain with a d-site unit cell in its classical ground statewill possess a Zd symmetry as a manifestation of the translation symmetry on the chain.1036.3. Dispersion RelationsHowever, what is also true is that in most cases, the SU(n) chain will not directly map to theabove Lorentz-invariant sigma model. There are two reasons for this. The first is that the fields ϕαare not guaranteed to propagate with the same velocity. Indeed, for the symmetric representationSU(n) chains, it was shown in Chapter 5 that only for a fine-tuned choice of SU(n) chain couplingconstants do these velocities become equal. However, we now know that at low enough energies,all of the velocity differences flow to zero under renormalization. In this chapter, we will assumethat this mechanism holds more generally, so that we may identify the various velocities of theCPn−1 fields.The second reason for Lorentz non-invariance is more of a hindrance. It follows from amismatch of terms arising from the coherent state path integral construction, ultimately lead-ing to some of the ϕα having quadratic dispersion. Mathematically, we say that the embeddingSU(n)/[U(1)]n−1 ↪→ [CPn−1]n is not Lagrangian in this case. Let us speculate a consequence ofthese quadratic modes. Since Coleman’s theorem does not apply in 1+1 dimensions to modes withquadratic dispersion, the generators associated with these modes may spontaneously order, result-ing in true Goldstone modes with quadratic dispersion [95–97, 138]. These Goldstone bosons willcouple to the remaining linear modes, which themselves form an SU(n′)/[U(1)]n′−1 sigma modelwith n′ < n. If a subgroup of the translation symmetry acts transitively on the n′ linear fields, thenit becomes possible for a novel ’t Hooft anomaly (mixed with the Zn′ subgroup) to exist in suchSU(n) chains.34 However, for now we will avoid these complications by focusing only on theo-ries with purely linearly dispersing modes. In the following section, we determine exactly whichrepresentations of SU(n) lead to such theories.6.3 Dispersion RelationsIn Section 3.4, we determined which SU(n) Hamiltonians were required in order to establish aflag manifold sigma model mapping for a given representation, R. In other words, we determinedhow many longer-range interaction terms had to be included in order to stabilize local zero modes.These results are summarized in Table 3.1, which is repeated here for convenience:With these results in hand, we now turn to the field theory mapping itself. Of course, a de-tailed derivation for each Hamiltonian would be a very tedious undertaking, and we do not pursuethis here. Instead, we focus on particular features, namely the topological angles and possibledispersion relations that exist in these theories, and refrain from determining the precise couplingconstants and velocities as a function of the interaction strengths Jr. Later on, we will explain howthese pieces of information, combined with a set of reasonable assumptions, will allow us to makepredictions about the ground state behaviour of certain SU(n) chains. But first, let us review the34Note that while mixed anomalies between PSU(n) and Zn′ do exist for n and n′ sharing a nontrivial common divisor,they will not allow for a renormalization group flow to the stable fixed point SU(n)1 [78].1046.3. Dispersion RelationsRepresentation Longest Interaction Translation Group Orderk = 1 Jn−1tr[S ( j)S ( j + n − 1)] nk = n − 1 J1tr[S ( j)S ( j + 1)] 2k = nλ , λ < n Jλtr[S ( j)S ( j + λ)] 2λn = 2λ + 1, k = 2 Jλtr[S ( j)S ( j + λ)] λ(λ + 1)n = kλ + 1, λ > 1, k > 2 Jλtr[S ( j)S ( j + λ)] lcm[2λ, (λ + 1)]n = kλ + c, c , 1, k − 1 Jλ+1tr[S ( j)S ( j + λ + 1)] 2λ(λ + 1)n = kλ + (k − 1) Jλ+1tr[S ( j)S ( j + λ + 1)] lcm[λ, 2(λ + 1)]Table 6.1: Classification results of all SU(n) representations satisfying pα , pβ for all nonzero pα.coherent state path integral construction for an arbitrary representation of SU(n). This generalizesthe discussion found in Section 4.3.1.6.3.1 Generalizing the coherent state path integral constructionIn order to construct a mapping from Hamiltonian to sigma model, we use coherent states to gener-ate a path integral of the ground state fluctuations. These coherent states are constructed as follows.For a representation with nonzero Young tableau rows p1, . . . , pk, we introduce k orthonormal fieldsφα ∈ Cn, and k n-component creation operators aα,†. We then define [99, 100]|Φ〉 :=k∑α=1[φα · aα]pα |0〉. (6.8)These are the coherent states of SU(n), and in order to construct a path integral, we insert thembetween thin time slices of the partition function:〈Φ(τi)|e−Hδτ|Φ(τi + δτ)〉 = 〈Φ(τi)|Φ(τi + δτ)〉e−Hδτ. (6.9)The right hand side can be approximated using〈Φ(τi)|Φ(τi + δτ)〉 ≈k∑α=1[1 + φ∗,α · ∂τφα]pα , (6.10)which follows from〈Φ(τ)|Φ(τ′)〉 =k∑α=1(φ∗,α(τ) · φα(τ′))pα . (6.11)1056.3. Dispersion RelationsBy taking the product over all time slices τi, we can then reexponentiate according to∏i〈Φ(τi)|Φ(τi + δτ)〉 = exp∑ilog〈Φ(τi)|Φ(τi + δτ)〉 ≈ exp∑i∑αpαφ∗,α · ∂τφα. (6.12)The Berry phase contribution to the path integral is obtained by adding up this contribution overeach site of the unit cell:LBerry = −1dd∑j=1k∑α=1pαφ∗,α( j) · ∂τφα( j). (6.13)Here d is the size of the unit cell, and φα( j) is the field φα evaluate at site j. Since we are derivinga field theory about a classical ground state, to lowest order φα( j) is the colour of node α at site j,which we have been representing using ϕα.To obtain the complete quantum field theory, one must add to LBerry a gradient expansion ofthe SU(n) lattice Hamiltonian, and this is where the lengthy calculations lie. However, if one isinterested only in time-derivatives, it suffices to restrict attention to LBerry, since the Hamiltonianis time-independent. The lowest-order expansion of LBerry, which amounts to replacing the φα( j)with their uniform counterparts ϕβ (where β depends on α and j), will indicate how many of thesigma model’s modes have linear dispersion. The next-order expansion, which takes into accountthe spatial fluctuations of the ϕα across the unit cell, will provide the topological angle content ofthe theory. It turns out that the non-uniform fluctuations of the φα, namely Lαβ as defined in (5.13),do not contribute to the topological term, and so in the following, we may safely replace φα withϕα in (6.13) throughout.Let us explain why this is the case. According to (5.13), in deriving a field theory we shouldwriteφα =∑βLαβϕβ +√1 − µ(α)ϕα, (6.14)where Lαα = 0 (no sum), and µ(α) =∑β |Lαβ|2. Unlike (5.13), we absorb the ‘large representation’parameter p−1 into the definition of the Lαβ terms. This is to avoid the notational complication ofdealing with multiple nonzero parameters pα in the expansion of the φα.Now, this expansion formula (6.14) will be inserted both into the Hamiltonian and the Berryphase term (6.13). What we seek to show here is that no term involving the Lαβ will contribute tothe topological terms µν∂µϕ∗,α · ∂νϕβ upon integrating out the Lαβ. Indeed, this follows from theGaussian integral formula35 ∫dLαβdL∗αβe−L∗12wL12+uL12+vL∗12 =piweuv/w, (6.15)which we used explicitly in Chapters 2, 4, and 5. Now, any term involving one time derivative andone space derivative on the right hand side of (6.15) will have come from expressions of the formuL12 = aL12[∂τUU†]12 + bL12[∂xUU†]12 (6.16)vL∗12 = aL∗12[∂τUU†]12 − bL∗12[∂xUU†]12, (6.17)35In the following, we restrict to the case of Lαβ = L12.1066.3. Dispersion Relationsfor real parameters a and b. This structure arises from the Hermitian requirement of the Hamilto-nian, which is the source of the spatial derivative terms. Meanwhile, the time-derivative terms areanti-Hermitian in imaginary time. This is a property generic to any integral over Lαβ appearing inour field theory derivation. From here, it is easy to see that the cross-terms involving ∂τ and ∂x willalways be of the formµνtr[Λα∂µUU†Λβ∂νUU†], (6.18)with α , β. These terms are not topological; instead, they are the so-called ‘torsion terms’, whichoccur in (6.2) with coefficients bαβ (in previous chapters, we referred to them as λ-terms). There-fore, we conclude that the Lαβ do not play a role in our discussion of topological angles below, andwe will henceforth omit them entirely.Dispersion Relation CalculationsFor each family of representations in Table 6.1, we determine the lowest-order contribution toLBerry. For each field ϕα that is present, this indicates the presence of (n−1) quadratically dispersingmodes. Only in the case of a vanishing LBerry at this order does linear dispersion occur for eachmode of the theory. However, even in this case, Lorentz invariance is not automatic since the fieldsϕα will generically propagate with different velocities, as we saw in the previous chapter.6.3.2 Representations with 1 rowFirst we consider representations with one row. The ground states are very simple in this case: onerow of n nodes, with each colour occurring once. For example, in SU(5) one such ground state is· · · · · · . (6.19)Therefore, we haveLBerry = − p1nn∑j=1ϕ∗,α( j) · ∂τϕα( j) + H.O. (6.20)where H.O. includes higher-order terms. Since each colour occurs once in the sum on the righthand side, the sum equals tr[U†∂τU] for a unitary matrix U, which vanishes by Appendix A, andso all modes have linear dispersion in the representations with k = 1 row.6.3.3 Representations with n − 1 rowsNext, we consider representations with k = n − 1 rows. According to the pattern of ground states,two of the colours occur once (in the first position of the columns), and the remaining n− 2 colours1076.3. Dispersion Relationsoccur twice, with reverse-ordering. For example, in SU(5), the ground state looks like· · · · · ·· · · · · ·· · · · · ·· · · · · · .Therefore we have, according to (6.13),LBerry = − p12 (ϕ∗,1 · ∂τϕ1 + ϕ∗,2 · ∂τϕ2) − 12n∑α=3(pα−1 + pn−α+2)ϕ∗,α( j) · ∂τϕα( j). (6.21)Using tr[U†∂τU] = 0, this can be rewritten asLBerry = −12n∑α=3(pα−1 + pn−α+2 − p1)ϕ∗,α( j) · ∂τϕα( j). (6.22)Now we have up to (n − 2) fields with quadratic dispersion. The exact number will depend onhow many of the conditions pα−1 + pn−α+2 − p1 = 0 are satisfied. Each constraint correspondsto two fields ϕα, except for the constraint 2p(n+1)/2 = p1 when n is odd, which corresponds to asingle field. The representations that satisfy every constraint, and thus give rise to sigma modelswith purely linear dispersion, correspond to the so-called self-conjugate representations of SU(n)(that don’t have equal row lengths in their tableaux). Indeed, in SU(3), the condition is 2p2 =p1, corresponding to the self-conjugate representations considered in Section 4.7. Similarly, inSU(4), the condition for linear dispersion is p2 + p3 = p1, which is equivalent to the self-conjugatecondition p1 − p2 = p3.6.3.4 Representations with row number dividing n.Now we consider representations with k rows, with nk = λ ∈ Z. In this case, the ground states haveorder of length 2λ, with each colour occurring twice. If k = n2 , the pattern looks like· · · · · ·· · · · · ·· · · · · · .For k < n2 , the pattern is modified, which can be seen in (3.20). Introducing the notationAα := ϕ∗,α · ∂τϕα, (6.23)and following these patterns as a guide, we find thatLBerry = − 12λk∑α=1λ∑j=1(pα + pk+1−α)Aα+( j−1)k), (6.24)1086.3. Dispersion Relationswhich can be rewritten using tr[U†∂τU] = 0 to yieldLBerry = − 12λk−1∑α=2λ∑j=1(pα + pk+1−α − p1 − pk)Aα+( j−1)k. (6.25)This suggests that there can be up to (k − 2)λ fields with quadratic dispersion. In order to removeall of these modes, the representation must satisfy certain constraints. When k is even, they arepα + pk+1−α = p1 + pk α = 2, . . . ,k2(6.26)and when k is odd, they arepα + pk+1−α = p1 + pk α = 2, . . . ,k − 12(6.27)2p k+12= p1 + pk (6.28)Each of the constraints (6.26), (6.27) corresponds to the dispersion of 2λ fields, and the constraint(6.28) corresponds to the dispersion relation of λ fields. In the special case of k = 2, whichcorresponds to Young tableaux with two rows, we have automatically that LBerry = 0 for all valuesof p1 and p2. The simplest representation for larger k is shown below, for SU(8):(6.29)This diagram satisfies p1 + p4 = p2 + p3.6.3.5 Remaining representationsRepresentations of SU(2λ + 1) with two rowsLet n = 2λ + 1, and consider representations with 2 rows. In this case the ground states have unitcells of length λ(λ + 1). For example, in SU(5) with k = 2, a candidate ground state is· · · · · ·· · · · · · .For all of these representations, the first row will have periodicity λ + 1, and the second row willhave periodicity λ. The Berry phase term is thenLBerry = − p1(λ + 1)λ+1∑α=1Aα − p2λλ∑β=1Aβ, (6.30)1096.3. Dispersion Relationswhich can be rewritten to giveLBerry = − 1λ(λ + 1)[p2(λ + 1) − p1λ] λ∑β=1Aβ. (6.31)For most values of p1 and p2, the corresponding sigma model will have λ fields with quadraticdispersion, and λ + 1 fields with linear dispersion.36 However, for the special representationssatisfyingλp1 = (λ + 1)p2 (6.32)a theory with purely linearly dispersing modes is achieved.Representations of SU(kλ + 1), with λ > 1, k > 2Now we let n = kλ+ 1, with λ > 1 and k > 2, and consider representations with k rows. Accordingto Table 6.1, we must further specify the parity of λ:• λ even:The ground state unit cell has length 2λ(λ + 1) in this case. The first row has (λ + 1)-siteorder, while the remaining k − 1 rows have 2λ-site order, with the coloured nodes exhibitingreverse-ordering. Therefore, the Berry phase term isLBerry = − p1(λ + 1)λ+1∑α=1Aα − 12λk∑β=2λ∑j=1(pβ + pk+2−β)Aβ+ j(k−1). (6.33)While simpler to rewrite in terms of the Aα with α > λ + 1, we will always choose to rewritethe Berry phase in terms of the least number of fields possible. This leads toLBerry = − 12λ(λ + 1)λ+1∑α=1[2λp1 − (λ + 1)(p2 + pk)] Aα− 12λk−1∑β=3λ∑j=1(pβ+pk+2−β−p2−pk)Aβ+ j(k−1).(6.34)• λ odd:In this case, the result found in (6.34) is slightly modified toLBerry = − 1λ(λ + 1)λ+1∑α=1[λp1 − (λ + 1)(p2 + pk)] Aα− 1λk−1∑β=3λ∑j=1(pβ+pk+2−β−p2−pk)Aβ+ j(k−1).(6.35)36As mentioned below (6.40), we can rewrite LBerry to have λ + 1 fields with quadratic dispersion, and λ fields withlinear dispersion.1106.3. Dispersion RelationsRepresentations of SU(kλ + c) with c , 1, k − 1Again, let k be the number of rows. The ground states in the case have unit cell order of length2λ(λ + 1). The Berry phase contribution isLBerry = − 12(λ + 1)c∑α=1λ+1∑j=1[pα + pc+1−α]Aα+( j−1)c− 12λk∑β=c+1λ∑j=1[pβ + pk−β+c+1]Ac(λ+1)+(β−c)+( j−1)(k−c).(6.36)Using the tr[U†∂U] = 0 identity, we can only remove 2(λ + 1) fields. We are left with:LBerry = − 12(λ + 1)c−1∑α=2λ+1∑j=1[pα + pc+1−α − (p1 + pc)] Aα+( j−1)c (6.37)− 12λ(λ + 1)k∑β=c+1λ∑j=1[(λ + 1)(pβ + pk−β+c+1) − λ(p1 + pc)]Ac(λ+1)+(β−c)+( j−1)(k−c).The number of independent constraints that must be satisfied in order to achieve a sigma modelwith purely linear dispersion is b c2c + b k−c2 c.Representations of SU(kλ + (k − 1))The remaining representations are for n = kλ + (k − 1). In this final case, we must again split intotwo subcases, based on the parity of λ:• λ even:According to Table 6.1, the ground state has λ(λ + 1)-site order. The first c − 1 rows of theground state have period 2(λ + 1), while the last row has period λ. The Berry phase term isLBerry = − 1(λ + 1)k−1∑α=1λ+1∑j=1[pα + pk−α]Aα+( j−1)(k−1) − 1λλ∑β=1pkA(k−1)(λ+1)+β (6.38)= − 1(λ + 1)k−2∑α=2λ+1∑j=1[pα + pk−α − p1 − pk−1]Aα+( j−1)(k−1)− 1λ(λ + 1)λ∑β=1[(λ + 1)pk − λ(p1 + pk−1)]A(k−1)(λ+1)+β.• λ odd:Now the ground state has 2λ(λ + 1) order, which changes the result in (6.38) to= − 12(λ + 1)k−2∑α=2λ+1∑j=1[pα + pk−α − p1 − pk−1]Aα+( j−1)(k−1) (6.39)1116.4. Topological Angles− 12λ(λ + 1)λ∑β=1[2(λ + 1)pk − λ(p1 + pk−1)]A(k−1)(λ+1)+β.6.3.6 Summary of calculationsWe have compiled these results in Table G.1 in Appendix G. Each row corresponds to a family ofSU(n) representations. The ‘Min #’ column counts the minimum number of Cn fields ϕα that havelinear dispersion in the corresponding SU(n)/[U(1)]n−1 sigma model. The larger, right hand columnlists the conditions that the representation parameters pα must satisfy in order for additional fields toacquire linear dispersion. Each condition is accompanied by a number in parentheses: this dictateshow many fields ϕα become linearly dispersing when this condition is satisfied. For example, thesecond row of Table G.1 corresponds to representations with n − 1 rows in their Young tableaux.These representations will always have at least two linearly dispersing fields in their sigma model.In order to have more linearly dispersing fields, we must start to satisfy conditions. When n iseven, these conditions are pα + pn−α+1 = p1, for α = 2, . . . , n2 . Each satisfied condition adds 2 morelinearly dispersing fields to the sigma model. It is amusing to note that when all of these conditionsare satisfied, we obtain the set of self-conjugate representations of SU(n) (that don’t have two rowsof the same length).It is important to note that there is some ambiguity in the number of linear vs. quadratic modes,which follows from the trace identity tr[U†∂τU] = 0. This expression allows us to rewrite a partialsum∑α∈A ϕ∗,α · ∂τϕα in terms of the ϕβ that do not occur in the sum:∑α∈Aϕ∗,α · ∂τϕα = −∑β<Aϕ∗,β · ∂τϕβ. (6.40)To be consistent, we will always choose to write the Berry phase contribution in terms of the leastnumber of fields possible. However, of primary interest to us in this thesis are theories that onlyhave linearly dispersing modes; in this case, the counting becomes uniquely defined.6.4 Topological AnglesIn addition to the dispersion relations, we may also extract the topological angle content from theBerry phase terms. This requires taking into account the spatial variation of the fields ϕα in eachof the terms found in the previous section.As we have already seen, each field is associated with some condition on the Young tableauxparameters pα. When determining the set of topological angles, it will be important to keep trackof these conditions; ultimately, this will lead to a list of angles for each of the conditions appearingin Table G.1. Our motivation for this bookkeeping is so that we can understand which topologicalangles correspond to linearly dispersing fields. Since quadratic modes correspond to ferromagneticorder parameters, it is conceivable that by ‘turning on’ a subset of the pα conditions, we may be ableto effectively reduce the symmetry of our sigma model from SU(n)/[U(1)]n−1 to SU(n′)/[U(n′)]n′−11126.4. Topological Anglesfor some n′ < n. It will be essential to keep track of which topological angles survive in the smallertheory.To begin, we recall (6.13):LBerry = −1dd∑j=1k∑α=1pαφα,∗( j) · ∂τφα( j) (6.41)where d is the unit cell length. The lowest-order terms were analyzed in the previous section, whichamounted to replacing φα with ϕα. Now, we take into account spatial fluctuations of the φα. Again,the non-uniform fluctuations Lαβ do not contribute at this order, so in this case, we may write theleading order correction to (6.41) asLBerry = · · · + µν 1dd∑j=1( j − 1)k∑α=1pα∂µϕ∗,x(α, j) · ∂νϕx(α, j) + H.O.. (6.42)Here all of the terms are evaluated at the same lattice site, and · · · hides the terms from the previoussection. The notation x(α, j) reminds us that for each field in the sum, we must consult the groundstate structure (see Section 3.4), and use both the row (α) and column ( j) to determine the index x.Using this, we may rewrite this contribution from the Berry phase term asLBerry = 12piin∑α=1θαqα, (6.43)whereqα := µν∂µϕ∗,α · ∂νϕα (6.44)is a total derivative. From here, we are able to read off the topological angles, θα. We will carry outthis procedure for each family of SU(n) representation listed in the previous section.6.4.1 Representations with one rowIn the case of one-row representations, the Berry phase term reduces toLBerry = µν p1nn∑α=1(α − 1)∂µϕ∗,α · ∂νϕα, (6.45)so thatθα =2pip1n(α − 1). (6.46)Since there are no quadratically dispersing fields when k = 1, these angles do not correspond to anontrivial condition on the pα.1136.4. Topological Angles6.4.2 Representations with n − 1 rowsNow we consider representations with k = n − 1 rows. Starting from (6.42), it is clear that we onlyhave to focus on a single column in the coloured ground state diagram:LBerry = · · · + 12µνn−1∑α=1pα∂µϕ∗,α · ∂νϕα. (6.47)The topological angles are thenθα = pipα. (6.48)According to the conditions in Table G.1, two fields are always linearly dispersing, correspondingto θ1 = p1pi and θn = 0. The remaining n − 2 angles correspond to fields that must satisfy condi-tions on the pα. The exact relationship between angle and nontrivial condition is given below inTable 6.2, making use of (6.48):Subcase Condition Anglesn even pα + pn−α+1 = p1 θα, θn−α+1 α = 2, . . . ,n2n oddpα + pn−α+1 = p12p n+12= p1θα, θn−α+1θ n+12α = 2, . . . , n−12Table 6.2: The relation between topological angles and condition on the pα, for representations ofSU(n) with n− 1 rows. For each condition, one or two fields ϕα becomes linear dispersing, and hastopological angle provided in the right-most column.6.4.3 Representations with row number dividing nNow we consider representations with k rows, with nk = λ ∈ Z. In this case, the correction to LBerryisLBerry = 12λµνk∑α=1λ∑j=1[( j − 1)pα + (λ + j − 1)pk+1−α]∂µϕ∗,α, j · ∂νϕα, j, (6.49)so that the topological angles areθα, j =piλ(pα + pk+1−α)( j − 1) + pipk+1−α. (6.50)1146.4. Topological AnglesThe range of these indices can be read off from the sums in (6.49). Here we use two indices toenumerate the fields. Since 2λ of the fields always have linear dispersion, the angles θ1, j and θk, jare always present. Meanwhile, the remaining (k−2)λ angles are associated with certain conditions,according to the following table Table 6.3.Subcase Condition Anglesk even pα + pk+1−α = p1 + pk θα, j, θk−α+1, j α = 2, . . . ,k2 ; j = 1, . . . , λk oddpα + pk−α+1 = p1 + pk2p k+12= p1 + pkθα, j, θk−α+1, jθ k+12 , jα = 2, . . . , k−12 , j = 1, . . . , λTable 6.3: The relation between topological angles and conditions on the pα for representationswith k = nλ rows in their Young tableaux.6.4.4 Remaining representationsRepresentations of SU(2λ + 1) with 2 rowsSuppose n = 2λ + 1, and consider representations with 2 rows. We again refer to Table G.1. FromLBerry = µν p1λ(λ + 1)λ∑j=1λ+1∑t=1(t − 1 + ( j − 1)(λ + 1))∂µϕ∗,t · ∂νϕt (6.51)+µνp2λ(λ + 1)λ∑j=1λ+1∑t=1( j − 1 + (t − 1)λ)∂µϕ∗, j+λ+1 · ∂νϕ j+λ+1,we see that there are two families of angles:θt =2pip1(λ + 1)(t − 1) + pip1(λ − 1) (6.52)for t = 1, . . . , λ + 1, andθ˜ j =2pip2λ( j − 1) + pip2λ (6.53)for j = 1, . . . , λ. The θt angles correspond to fields that are always linearly dispersing. Theremaining angles are associated to the single condition λp1 = (λ + 1)p2.1156.4. Topological AnglesRepresentations of SU(kλ + 1) with k > 2.The representations of SU(n), with n = kλ + 1, may be divided into two cases, according to theparity of λ:• Case 1: λ even.Since the first row of the ground state has (λ+1)-site order, and the unit cell order is 2λ(λ+1),we may write LBerry as two parts:LBerry = L(1)Berry +L(2)Berry. (6.54)The first part isL(1)Berry = µνp12λ(λ + 1)λ+1∑t=12λ∑j=1(t − 1 + ( j − 1)(λ + 1))∂µϕ∗,t · ∂νϕt, (6.55)which gives rise to the following topological angles:θt =pip1λ(λ + 1)2λ∑j=1(t − 1 + ( j − 1)(λ + 1)) (6.56)for t = 1, 2, . . . , λ + 1. These angles can be further simplified toθt =2pip1(λ + 1)(t − 1) + pip1. (6.57)The second part of the Berry phase corresponds to the lower k−1 rows of the classical groundstate. It readsL(2)Berry =µν2λ(λ + 1)k∑α=2λ∑j=1λ+1∑t=1[pα( j − 1 + (t − 1)2λ) (6.58)+ pk+2−α( j − 1 + λ + (t − 1)2λ)]∂µϕ∗,α, j · ∂νϕα, j.The associated topological angles areθα, j =piλ(λ + 1)λ+1∑t=1[pα( j − 1 + 2λ(t − 1)) + pk+2−α( j − 1 + λ + (t − 1)2λ)](6.59)for α = 2, . . . , k and j = 1, . . . , λ. Again, the angles simplify toθα, j =piλ(pα + pk+2−α)( j − 1) + pipα, (6.60)where we used the fact that λpipα ≡ 0 since λ is even. The correspondence between angle andcondition on the pα is provided in the following table, Table 6.41166.4. Topological AnglesSubcase Condition Anglesk evenpα + pk+2−α = p2 + pk(λ + 1)(p2 + pk) = 2λp1θα, j,θk+2−α, jθtα = 3, . . . ,k + 12;j = 1, . . . , λ; t = 1, . . . , λ + 1k oddpα + pk+2−α = p2 + pk2p k+22= p2 + pk(λ + 1)(p2 + pk) = 2λp1θα, j,θk+2−α, jθ k+22 , jθtα = 3, . . . ,k2;j = 1, . . . , λt = 1, . . . , λ + 1Table 6.4: The relation between topological angle and condition on pα for representations of SU(n),with n = kλ + 1, with k > 2 and λ > 1 odd.• Case 2: λ odd.Assume λ > 1 so that this case is distinct from the representations considered previously.Now that λ is odd, the order of the unit cell has changed to λ(λ + 1). The two parts of LBerryfrom the previous subsection are modified toL(1)Berry = µνp1λ(λ + 1)λ+1∑t=1λ∑j=1(t − 1 + ( j − 1)(λ + 1))∂µϕ∗,t · ∂νϕt, (6.61)andL(2)Berry =µνλ(λ + 1)k∑α=2λ∑j=1λ+12∑t=1[pα( j − 1 + (t − 1)2λ) (6.62)+ pk+2−α( j − 1 + λ + (t − 1)2λ)]∂µϕ∗,α, j · ∂νϕα, j.The angles are thenθt =2pip1(λ + 1)(t − 1) (6.63)andθα, j =piλ(pα + pk+2−α)(j − 1 + λ(λ − 1)2)+ pipα. (6.64)The range of these indices can be read off from the sums in (6.62). The same correspondencebetween condition and angle found in the previous case applies here as well, so long as theslightly modified conditions for λ odd are used (which can be found in Table G.1).1176.4. Topological AnglesRepresentations of SU(kλ + c) with c , 1, k − 1Write n = kλ+ c, for c , 1, k−1, and consider representations with k rows. In this case, the groundstate order is 2λ(λ + 1). Again, we split the Berry phase contribution into two pieces. The first crows contribute the following term:L1Berry =µν2λ(λ + 1)λ∑j=1λ+1∑t=1c∑α=1[pα(t − 1 + ( j − 1)2(λ + 1)) (6.65)= +pc−α+1(t − 1 + λ + 1 + ( j − 1)2(λ + 1))]∂µϕ∗,α,t · ∂νϕα,t,which gives the topological anglesθα,t =pi(pα + pc−α+1)(λ + 1)(t − 1) + piλpc−α+1 + pi(λ − 1)pα. (6.66)The range of these indices can be read off from the sums in (6.65). Meanwhile, the remaining k− crows contribute the termL2Berry =µν2λ(λ + 1)λ∑j=1λ+1∑t=1k∑β=c+1[pβ( j − 1 + (t − 1)2λ) (6.67)+ pk−β+c+1( j − 1 + (t − 1)2λ + λ)]∂µϕ˜∗,β, j · ∂νϕ˜β, jwhich gives the topological anglesθ˜β, j =pi(pβ + pk−β+c+1)λ( j − 1) + pi(λ + 1)pk−β+c+1 + piλpβ. (6.68)The range of these indices can be read off from the sums in (6.67). Here, θ˜ and ϕ˜ have been used inorder to differentiate the two families of fields and topological angles, so as to simplify our notation.The relationships between topological angle and condition on the pα are given in Table 6.5.Representations of SU(kλ + (k − 1))Finally, we must consider the representations with n = kλ + (k − 1), with k boxes. Here we alsohave two subcases, according to the parity of λ:• Case 1: λ oddThe ground state has 2λ(λ+ 1)-site order. The first k− 1 rows have the following Berry phasecontribution:L(1)Berry =µν2λ(λ + 1)k−1∑α=1λ∑j=1λ+1∑t=1[pα(t − 1 + ( j − 1)2(λ + 1))+ (6.69)= pk−α(t − 1 + λ + 1 + ( j − 1)2(λ + 1))]∂µϕ∗,α,t · ∂νϕα,t.1186.4. Topological AnglesSubcase Condition Anglesk, c evenpα + pc+1−α = p1 + pc(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc)θα,t, θc−α+1,tθ˜β, j, θ˜k−β+c+1, jα = 2, . . . ,c2; t = 1, . . . , λ + 1β − c = 1, . . . , k − c2; j = 1, . . . , λk oddc evenpα + pc+1−α = p1 + pc(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc)2(λ + 1)p k+c+12= λ(p1 + pc)θα,t, θc−α+1,tθ˜β, j, θ˜k−β+c+1, jθ˜ k+c+12 , jα = 2, . . . ,c2; t = 1, . . . , λ + 1β − c = 1, . . . , k − c − 12j = 1, . . . , λk, c oddpα + pc+1−α = p1 + pc2p c+12= p1 + pc(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc)θα,t, θc−α+1,tθ c+12 ,tθ˜β, j, θ˜k−β+c+1, jα = 2, . . . ,c − 12; t = 1, . . . , λ + 1β − c = 1, . . . , k − c − 12j = 1, . . . , λc oddk evenpα + pc+1−α = p1 + pc2p c+12= p1 + pc(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc)2(λ + 1)p k+c+12= λ(p1 + pc)θα,t, θc−α+1,tθ c+12 ,tθ˜β, j, θ˜k−β+c+1, jθ˜ k+c+12 , jα = 2, . . . ,c − 12; t = 1, . . . , λ + 1β − c = 1, . . . , k − c − 12j = 1, . . . , λTable 6.5: The relation between topological angle and condition on the pα for representations ofSU(n) with n = kλ + c, with c , 1, k − 1.The corresponding topological angles areθt,α =pi(pα + pk−α)(λ + 1)(t − 1) + pipk−α. (6.70)As before, the range of α and t can be read off from the sums in (6.69). Meanwhile, the lastrow of the ground state contributes the termL(2)Berry =µν2λ(λ + 1)pkλ∑j=12(λ+1)∑t=1[( j − 1) + (t − 1)λ] ∂µϕ∗, j · ∂νϕ j (6.71)giving rise to the anglesθ j =2pipkλ( j − 1) + pkpi, (6.72)for j = 1, . . . , λ. The correspondence between condition on the pα and topological angle canbe found in Table 6.6.1196.4. Topological AnglesSubcase Condition Anglesk oddpα + pk−α = p1 + pk−1(λ + 1)pk = λ(p1 + pk−1)θα,t,θk−α,tθ jα = 2, . . . ,k − 12;j = 1, . . . , λ; t = 1, . . . , λ + 1k evenpα + pk−α = p1 + pk−12p k2= p1 + pk−1(λ + 1)pk = λ(p1 + pk−1)θα,t,θk−α,tθ k2 ,tθ jα = 2, . . . ,k − 22;t = 1, . . . , λ + 1j = 1, . . . , λTable 6.6: The relation between topological angle and condition on the pα for representations ofSU(n) with n = kλ + (k − 1), with k boxes.• Case 2: λ evenThis is the final case. Now that λ is even, the order of the unit cell has changed to λ(λ + 1).The two parts of LBerry are modified toL(1)Berry =µνλ(λ + 1)k−1∑α=1λ2∑j=1λ+1∑t=1[pα(t − 1 + ( j − 1)2(λ + 1)) (6.73)+ pk−α−2(t − 1 + λ + 1 + ( j − 1)2(λ + 1))]∂µϕ∗,α,t · ∂νϕα,tandL(2)Berry =µνλ(λ + 1)pkλ∑j=1(λ+1)∑t=1[( j − 1) + (t − 1)λ] ∂µϕ∗, j · ∂νϕ j, (6.74)and the angles are modified toθα,t → pi(pα + pk−α)(λ + 1) (t − 1) + pipk−α +pi(pα + pk−α)2(λ − 2) (6.75)andθ j → 2pipkλ( j − 1). (6.76)The correspondence between angles and conditions on the pα follows the same pattern as theprevious case, with a slight modification of the conditions themselves, according to Table G.1.1206.5. A New Generalization of Haldane’s ConjectureSummary of calculationsIn Table G.2, found in Appendix G, we collect the results of this section, by listing all possibletopological angles for each representation of SU(n). The relationships between angle and condi-tions on the pα were provided explicitly in the various tables above.6.5 A New Generalization of Haldane’s ConjectureThe lengthy analysis of the previous sections makes clear the fact that most representations ofSU(n) chains do not lead to linearly dispersing sigma models like those of the symmetric SU(n)chains considered in Chapters 2,4, and 5. This is already apparent in SU(3): Any representation thatis neither self-conjugate nor completely symmetric has at least one (and at most two) CP2 fieldsϕα with quadratic dispersion. In order to achieve a purely linearly dispersing theory, a series ofconstraints on the Young tableaux parameters pα must be satisfied. In the special case of represen-tations with all pα nonzero and distinct, these constraints lead to the self-conjugate representationsof SU(n).Here, we restrict to representations of SU(n) that satisfy the various constraints listed in Ta-ble G.1. This ensures that the corresponding sigma models have linear dispersion only. Beforeproceeding further, we must reflect on what we are hoping to achieve with this classification. Ulti-mately, we are interested in the possible gapless phases in SU(n) chains, and how one might extendHaldane’s conjecture to novel representations. Based on our understanding of the symmetric mod-els, we know that this task can be recast in terms of ’t Hooft anomaly matching. The recipe is asfollows:• Step 1: Map an SU(n) chain to a (linear-dispersing) flag manifold sigma model at low ener-gies.• Step 2: Identify the ’t Hooft anomalies of the sigma model. When such an anomaly is present,we may conclude that the ground state either exhibits spontaneously broken symmetry, orgapless excitations.As we’ve mentioned several times already, in [78] and [79] it was shown that an ’t Hooft anomalymay occur in the SU(n)/[U(1)]n−1 sigma model when an additionalZn discrete symmetry is present.It is a mixed anomaly between the physical PSU(n) symmetry of the chain, and the discrete Znsymmetry. Actually, in [78] it was shown that mixed anomalies occur between PSU(n) and Zn′whenever n and n′ have a nontrivial common divisor. However, in this latter case, the correspond-ing sigma model cannot be embedded into a WZW theory that flows to level k = 1. Therefore, herewe focus on chains that have this Zn symmetry only. Moreover, this symmetry must act on the ncomplex fields transitively according toZn : ϕα 7→ ϕα+1. (6.77)In the following chapter, this gapless property of SU(n) chains with ’t Hooft anomalies willbe reinterpreted in terms of fractional instantons. We will show that in the SU(n)/[U(1)]n−1 sigma1216.5. A New Generalization of Haldane’s Conjecturemodel, topological excitations exist that generate a finite energy gap above the ground state, muchin the same way that vortices drive the familiar Kosterlitz-Thouless transition [90, 91]. For acertain set of topological angles {θ(◦)α }, these excitations destructively interfere with each other andthe mass-generating mechanism breaks down, thus leading to a gapless ground state. It turns outthat when an ’t Hooft anomaly is present, the topological angle content in the sigma model isprecisely {θ(◦)α }. Ultimately, this follows from the form of the action (6.77). Therefore, in additionto concerning ourselves with linear dispersion, we also restrict our focus to SU(n) representationswhose translational symmetry group Zd = Zn, and acts transitively on the fields ϕα.Of course, it is important to acknowledge this is by no means an exhaustive classification ofgapless phases in SU(n) chains. We do not attempt to classify all possible ’t Hooft anomalies inthese models, and so we are limited to the current list of known anomalies, and apply this knowl-edge to our theories. Moreover, we must also remember that the absence of an anomaly teachesus nothing: we are unable to predict any ground state properties when this is the case. However,we do have the LSMA theorem (see Section 3.5.2), which predicts either a gapless ground state orspontaneously broken symmetry for an SU(n) chain whenever the sum p :=∑α pα is not a multipleof n.6.5.1 Representations that admit transitive Zn actionsHaving made these remarks, we are now in a position to seek out representations of SU(n) thatmay be amenable to a generalized Haldane conjecture. We assume that all of the constraints onthe Young tableaux parameters pα have been satisfied, so that all of the n fields ϕα are linearlydispersing. For each class of representation occurring in Table 6.1, we record when it is possiblefor the translation group to equal Zn, and act transitively on the set of fields.• Case 1: k = 1.Since Zd = Zn, this is possible for all n.• Case 2: k = n − 1Since Zd = Z2, this is possible only in SU(2) (which reduces to Case 1).• Case 3: k = nλ , λ < nSince d = 2λ, and λ ≤ n2 , this is possible only when k = 2. See the ground state below for anexample in SU(4):· · · · · ·· · · · · · .In other words, when n is even, Young tableaux with two rows (of differing lengths) give riseto flag manifold sigma models with an additional Zn symmetry. According to Table G.1, suchrepresentations are always linearly dispersing, so no other assumption on the row lengths pαis required. Note that the angles in this case are (see Table G.2)θα =2pin(p1 + p2)(α − 1) α = 1, 2, . . . , n, (6.78)1226.6. Conclusionwhere we have shifted each angle by the constant pip1.• Case 4: n = 2λ + 1, k = 2Since n cannot equal d = λ(λ + 1), no such representations give rise to a Zn symmetry.• Case 5: n = kλ + 1, k > 2, λ > 1.In this case, the Zd symmetry does not act transitively on the set of n fields: Some of the fieldslie in an orbit of order λ + 1, while the remaining fields lie in orbits of size 2λ.• Case 6: n = kλ + c, c , 1, k − 1.Similar to Case 5, the fields do not lie in a single orbit under the action of Zd. So while itis possible for Zd = Zn, the fields do not transform under the necessary action (6.77). Thesimplest example of this is SU(12) with k = 5 rows in a Young diagram. Under the Zn action,the fields partition into three orbits of size 6,4 and 2, and the anomaly classification is nolonger applicable.• Case 7: n = kλ + (k − 1).Similar to the previous two cases, the fields do not lie in a single orbit under the action of Zd.In summary, we find only one new family of SU(n) representations that give rise to a linearlydispersing SU(n)/[U(1)]n−1 flag manifold sigma model with the Zn symmetry (6.77). It is the setof representations with two rows (of different lengths) in their Young tableaux, when n is even. Thecorresponding topological angles in this theory areθα =2pin(p1 + p2)α α = 1, 2, . . . , n (6.79)so that p1 + p2 plays the role of p1 in the symmetric models. This means that an ’t Hooft anomalyis present whenever p1 + p2 is not a multiple of n. This is also consistent with the LSMA theorem,mentioned above. Moreover, based on the classification of SU(n) WZW flows discussed in Sec-tion 5.5.2, we may further conclude that only when (p1 + p2) is coprime with n is a stable gaplessphase possible. Otherwise, if p1 + p2 shares a nontrivial common divisor with n, then the theory isnecessarily gapped with spontaneously broken symmetry.On the other hand, when p1 + p2 is a multiple of n (and the LSMA theorem does not apply), itshould be possible to have a unique, translationally invariant ground state with a finite energy gap.This statement is supported by the fact that when p1 + p2 = n, it is straightforward to write down atranslationally invariant AKLT state. This is explained in Section 3.5.3.6.6 ConclusionIn this chapter, we have attempted to classify all SU(n) chains that admit a mapping to the SU(n)/[U(1)]n−1sigma model at low energies. Our motivation for doing so was the presence of mixed ‘t Hooftanomalies between the physical PSU(n) = SU(n)/Zn symmetry of the sigma model, and a tran-sitive Zn action. Such anomalies are an indicator of nontrivial low energy physics, and so byestablishing such mappings, we may uncover novel gapless phases in SU(n) chains.1236.6. ConclusionWe have found that unless two rows of the Young tableaux have the same length, it seemspossible to realize such a mapping for any irreducible representation of SU(n). When two rowlengths are degenerate, the SU(n) matrices S lie in a different (partial) flag manifold that can noteasily be embedded into the (complete) flag manifold SU(n)/[U(1)]n−1.However, even when no row lengths are degenerate, we find that the corresponding sigmamodel will have complex fields with both quadratic and linear dispersion relations. One conse-quence of this is that Lorentz invariance is likely never to emerge at low energies, as it does for thesymmetric SU(n) chains that possess only linearly dispersing fields. We have classified which rep-resentations lead to only linearly dispersing models, and have determined the topological angles ineach case. Moreover, within this subset of representations, we have further classified which chainsalso admit a Zn symmetry that acts transitively on the SU(n) fields, so that the possibility for an ‘tHooft anomaly exists.In the end, we have found that only the SU(n) representations with even n and two rows in theirYoung tableaux, with lengths p1 , p2, satisfy all of these properties (in addition to the symmetricrepresentations considered previously). As a result, we have made the following modest extensionof the SU(n) generalization of Haldane’s conjecture for even n: when p1 + p2 is coprime with n, agapless ground state is predicted; otherwise, a gapped ground state is expected, with spontaneouslybroken symmetry if p1 + p2 is not a multiple of n. The magnitude of the gap can be predicted in thesame way as in Section 5.5.3, and should generally decrease exponentially as a function of p1 + p2.In Table 6.7, we summarize these findings by listing which representations of SU(n) we predict tohave a gapless phase driven by topological terms.Representation Conditionp boxes (p, n) = 1p1 boxesp2 boxes(p1 + p2, n) = 0 and n evenTable 6.7: The set of SU(n) representations that exhibit a gapless phase at weak coupling.124Chapter 7Fractional Instantons and the HaldaneGap in SU(n) Chains7.1 IntroductionIn the past few chapters, we have followed the program of extending Haldane’s conjecture to moreand more general SU(n) chains. In the end, we have come to the conclusion that a gapless phaseshould only occur for representations that admit a mapping to the SU(n)/[U(1)]n−1 sigma modelwith n − 1 nontrivial topological angles. Indeed, this statement summarizes our findings for SU(2)in Chapter 2, for SU(3) in Chapter 4, for the symmetric representations of SU(n) in Chapter 5, andfinally for the most general representations of SU(n) in Chapter 6. Here, in this final chapter, weseek to develop an alternative argument that explains why these gapless phases arise. Instead ofmaking use of formal ‘t Hooft anomaly matching arguments, we will establish a physical pictureof Haldane’s conjecture in terms of a gas of interacting fractional topological excitations.This work generalizes an old result of Affleck that interpreted Haldane’s original conjecturein terms of interacting merons, or ‘half-instantons’ [137]. Merons behave like planar vortices farfrom their centers, but lift out of the plane near their cores, and form finite action configurationswhen a collection of them has net vorticity zero. Remarkably, when the CP1 sigma model has atopological angle of pi, corresponding to a half-odd integer spin chain, the merons that point up attheir origin destructively interfere with the merons that point down at their origin (also known asanti-merons), and the mass generating mechanism of the model is no longer effective. While thisisn’t the full story, since larger-action configurations such as double-merons do not destructivelyinterfere at θ = pi, it offers an intuitive picture in favour of Haldane’s conjecture.In Chapters 4, 5 and 6, we have generalized this paradigm of mapping spin chains to relativisticquantum field theories to SU(n) chains in various representations. When a transitive Zn action ispresent, the corresponding sigma model has n − 1 independent topological angles θα = 2pipn α.Throughout, we will treat p ∈ Z+ as given, labelling a particular representation of SU(n) thatadmits such topological angle content. From the previous chapters, we know that there are twopossibilities for p:• Case 1: The SU(n) representation has one row, of length p, in its Young tableau.• Case 2: The SU(n) representation has two rows, of lengths p1 and p2 in its Young tableau.Then p = p1 + p2. This case is possible only for even n.According to our generalized Haldane conjectures, these SU(n) chains have a gapless ground1257.2. Mass Generation in the O(3) Nonlinear Sigma Modelstate whenever p and n are coprime: gcd(n, p) = 1. This is in accordance with the LSMA theorem(see Section 3.5.2), which predicts either a gapless phase or spontaneously broken translation sym-metry when p is not a multiple of n [80, 81]. A subtle feature of this result, which can only occurfor n ≥ 4, is that when p and n have a nontrivial divisor, and p is a not a multiple of n, translationsymmetry is always spontaneously broken. This follows from an ’t Hooft anomaly that forbids theflow of the above sigma model to SU(n)1, the only stable SU(n) fixed point [115, 116].In this chapter, we will offer an intuitive explanation of this gap dependence on p and n, basedon a mass generation mechanism due to fractionalized topological excitations. We begin by re-viewing the Kosterlitz-Thouless transition in Section 7.2, and explain how it relates to the massgeneration mechanism in SU(2). In Section 7.3, we add anisotropic potentials to the SU(n) chainthat break the continuous global symmetry down to U(1), and classify the topological excitationsthat have finite action. Next, in Section 7.4, we show how these excitations, so-called fractionalinstantons which have topological charge 1n , are responsible for generating a mass gap in the sigmamodel, in a similar fashion as the vortices in the Kosterlitz-Thouless transition [90, 91]. However,this mechanism is highly dependent on the topological angles of the model, which are determinedby the representation p of SU(n). This dependence is discussed in Section 7.5, where we ultimatelyconclude that mass generation by fractional instantons occurs except when p and n are coprime, inwhich case a gapless phase is possible.7.2 Mass Generation in the O(3) Nonlinear Sigma Model7.2.1 The Kosterlitz-Thouless transitionSince we hope to relate (the absence of) a finite energy gap in our sigma models to some formof topological excitation, let us begin by reviewing how a similar mechanism works in a veryfamous example. We are referring to the continuous phase transition in the ‘XY model’, whichwas described independently by Berezinskii in 1971 [90] and Kosterlitz and Thouless in 1973 [91].This work contributed to Kosterlitz and Thouless sharing the Nobel prize with Haldane in 2016.The XY model describes the physics of a classical spin restricted to a two-dimensional plane(which we can think of as corresponding to the ‘X’ and ‘Y’ directions in real space, hence thename). Explicitly, its Hamiltonian isH = −J∑〈i, j〉~si · ~s j, (7.1)where∑〈i, j〉 denotes summation over all nearest-neighbour sites in the two-dimensional lattice, and~si ∈ R2 is the classical spin vector at site i ∈ Z2. Note that the interaction is ferromagnetic, i.e.the prefactor of the spin-spin interaction is negative, unlike all of the SU(n) chain Hamiltoniansthat we have considered thus far. If we write the spin vectors in polar coordinates according to~si = s cos(ρi), and rescale J by s−2, we are left withH = −J∑〈i, j〉cos(ρi − ρ j). (7.2)1267.2. Mass Generation in the O(3) Nonlinear Sigma ModelIn the continuum limit, the differences ρi − ρ j are replaced with partial derivatives, and we mayTaylor expand the cosine, so long as we assume the classical spins vary smoothly from site to site.We end up withH ≈ E0 + J2∫d2x(∇ρ)2 (7.3)where ~x ∈ R2, and (∇ρ)2 = (∂xρ)2 + (∂yρ)2. The thermodynamics of this system is obtained from apartition function, which may be written as a path integral as followsZ = e−βE0∫D[ρ]e−β J2∫d2 x(∇ρ)2 . (7.4)We may now identify the integrand in the exponent as a Lagrangian,LXY = 12g (∇ρ)2. (7.5)with g = 1βJ . This is the low temperature, or weak coupling, form of the XY model. To arrive atit, we had to assume that higher derivatives of ρ could be ignored in the expansion of the cosinein (7.2). Such terms do not reemerge upon renormalization, since it may be shown that all of thehigher derivative terms are irrelevant.This rewriting (7.5) will be useful later when we approach LXY from our flag manifold sigmamodels through a limit of theories, which restrict the target space from SU(n)/[U(1)]2 first downto [U(1)]n−1, and then down to U(1). At first glance, it appears that LXY is simply a theory of afree boson, ρ, and will thus exhibit gapless excitations for all values of β. However, Kosterlitz andThouless realized that this conclusion overlooks the crucial fact that ρ is a periodic variable. Inother words, the target space of ρ is U(1), which is equivalent to the circle S 1, and is topologi-cally nontrivial. Accounting for this, they found that a finite mass will be generated for the bosonρ at sufficiently high temperatures (equivalently, at strong enough coupling), and that this massgeneration can be explained in terms of topologically nontrivial field configurations.To understand this mass generation, let us estimate Z using a saddle point approximation(which is exact in the zero temperature limit). This amounts to solving the Euler-Lagrange equa-tions for LXY , and identifying extremal configurations, ρ∗, i.e. solving∇2ρ∗ = 0. (7.6)Of course, the most obvious solution to these saddle point equations is the trivial configuration,ρ∗ = const.. However, the story does not end here. Since ρ lives in U(1)  S 1, there also existsolutions ρκ that obey the following equation:∮Γdl · ∇ρ = 2piκ κ ∈ Z. (7.7)Here Γ is a closed contour integral in the XY plane. For nonzero κ, this equation requires that ρκ hasa singularity within the contour; consequently, it cannot be continuously deformed into the trivial1277.2. Mass Generation in the O(3) Nonlinear Sigma Modelconfiguration, ρ∗ = const.. In other words, ρκ is a topologically distinct saddle point, and mustalso be included in the path integral over all field configurations in (7.4). The integer κ is calledthe topological charge of ρκ, and the simplest configurations ρ1 and ρ−1 are called the vortex, andanti-vortex configuration, respectively.Let us now determine the classical action of these configurations. Due to the U(1) symmetryof the problem, ρκ should only depend on the radius, r, so that by (7.7), |∇ρ| = κr . Then, we haveS [ρκ] =κ22g∫d2x1r2. (7.8)Unless κ = 0, we see that the action of ρκ exhibits both an infrared and an ultraviolet divergence.The ultraviolet divergence can be resolved by introducing a lattice spacing a, which we will discussshortly, but the infrared divergence is problematic. If all topologically nontrivial configurationshave infinite action, they will not enter the saddle point approximation, and will not be responsiblefor a mass generating mechanism, as predicted by KT. How then, do we proceed?The resolution to this issue is found by considering configurations that are sums of vortices,with net topological charge zero. To see this more easily, define the dual field ρ˜ byµν∂νρ˜ = ρ, (7.9)where µν is the antisymmetric tensor with nonzero components 01 = −10 = 1. Then the topolog-ical charge equation (7.7) can be rewritten as follows:∫Σd2x∇2ρ˜κ = −2piκ. (7.10)Here Σ is the oriented surface whose boundary is the contour Γ. This equation shows that the mostgeneral saddle point we can write down should obey the following Poisson equation:∇2ρ˜ = −2pi∑jκ jδ(~x − ~x j). (7.11)In other words, the Euler-Lagrange equation ∇2ρ = 0 is modified by singularities on the righthand side. Now, using (∇ρ)2 = (∇ρ˜)2, one may show that the action remains finite so long as∑j κ j = 0, i.e. the net topological charge vanishes. In particular, the least-action configurationsthat are topologically nontrivial consist of a vortex/anti-vortex pair, with separation distance R.Using (7.8), we see thatS [ρ1 + ρ−1] =12glog(Ra)< ∞. (7.12)It is these configurations that KT used to explain the generation of a mass gap in LXY .The simplest, and most famous explanation for this mass gap follows from energy and entropyconsiderations. It can readily be shown that below temperature gc = pi/2, the free energy for anindividual vortex diverges, while above this temperature, it is favourable for individual vortices to1287.2. Mass Generation in the O(3) Nonlinear Sigma Modelform.37 Thus, we expect a phase transition to occur at this temperature, in which the vortex/anti-vortex pairs unbind, leading to so-called ‘vortex proliferation’.To see this mechanism at play more explicitly, one may derive an effective Lagrangian thatdescribes fluctuations about the vortex/anti-vortex pairs in the low temperature phase. To this end,we write ρ˜ = ρ˜c + ρ˜′, where ρ˜c solves (7.11), and ρ˜′ are small deviations. The action for ρ˜ thenreadsS [ρ˜] =12g∫d2x(∇ρ˜)2 + 4piiρ˜∑jκ j , (7.13)where on the right hand side we’ve shifted ρ˜′ → ρ˜. Now, returning to (7.4), we must sum over allpossible configurations of ρ˜, which amounts to summing over all choices of {κ j}. It is here that wemake the dilute gas approximation, allowing us to assume the individual vortices are far apart, anddo not interact with each other. Then, we may writee−S = e−12g∫d2 x(∇ρ˜)2 1n!n∏j=1∫d2x je− 2piig ρ˜(~x j) 1m!m∏k=1∫d2xke2piig ρ˜(~xk). (7.14)Here j and k label vortices and anti-vortices, respectively. Now, using the dilute gas approximation,we may sum over configurations before integrating over ρ˜. This yieldse−S = e−12g∫d2 x(∇ρ˜)2eγ2a∫d2 x(e−2piiρ˜g +e2piiρ˜g), (7.15)for some constant γ, which depends on our choice of regularization. In the context of dilute instan-ton gases, γ is called the fugacity. This equation reveals that in the continuum limit, the XY modelis characterized by the following Lagrangian, which is known as the sine-Gordon model:LS G = 12g (∇ρ˜)2 − γa2 cos(2piρ˜g). (7.16)From here, the predictions of Kosterlitz and Thouless of a vortex unbiding phase transition are man-ifest. Indeed, at low temperatures, corresponding to g→ 0, the sine-Gordon model corresponds toa free boson, and gapless behaviour is predicted. We should think of the vortex/anti-vortex pairs asbeing tightly bound in this phase. Meanwhile, for large values of g, we may expand the interactionterm to obtain (after rescaling ρ˜)LS G = 12(∇ρ˜)2 +2pi2a2γgρ˜2 + O(g3/2), (7.17)demonstrating that a boson mass mρ := (2api)2γ/g is generated. In this phase, the vortex/anti-vortexpairs are unbound, and unpaired vortices proliferate. A transition between these two phases willoccur precisely when the scaling dimension of the interaction term turns relevant, which is known37When discussing entropies, we set Boltzmann’s constant, kB, to 1.1297.2. Mass Generation in the O(3) Nonlinear Sigma Modelin the sine-Gordon model to occur at gc = pi/2. This is the same temperature quoted above thatfollowed from an energy vs. entropy argument.As a final observation, we stress the fact that in the low temperature (weak coupling) regime,the XY model exhibits quasi-long range order. That is, correlation functions in the gapless phaseexhibit power-low (i.e. algebraic) decay, and the U(1) symmetry is not spontaneously broken.This is the same kind of order that we predict for the topologically driven gapless phases of ourSU(n) chains. Of course, true long-range order, which necessitates the spontaneous breaking of the(continuous) U(1) symmetry, would be in direct violation of Coleman’s theorem.7.2.2 Mass generation in the CP1 sigma modelNow that we’ve reviewed the explicit mass generating mechanism of the XY model, we are readyto relate this phenomenon to spin chains. Historically, this was first done by Affleck in the languageof the O(3) nonlinear sigma model [137]. Here, we will repeat this derivation in CP1, as this willset the stage for our generalization to larger flag manifolds later on.Our strategy is quite simple. Since the U(1) symmetry of the XY model can be embedded intothe larger CP1 target space of the antiferromagnet, we will introduce a potential that restricts thefields ϕα ∈ C2 to a U(1) subspace of CP1. In the limit where this potential is infinitely large, wehave precisely the XY model, and we have an explicit mass generating mechanism at hand, namelyvortex proliferation. Then, we will weaken this potential, and see how this mechanism persiststhrough a family of models, all the way back to the isotropic theory of CP1.Our starting point isL = 12g(|∂µϕ|2 − |ϕ∗ · ∂µϕ|2)+Ltop. (7.18)with g = 1/s. For simplicity, we first consider the case of integer spin chains, so that we mayneglect Ltop. In the following subsection, we will restore it, and see remarkably how it destroys themass generating mechanism, and leads to the closing of the Haldane gap.Next, let us add to (7.18) the following potential,V = m(|ϕ1|2 − |ϕ2|2)2. (7.19)In terms of the spin vectors ~S ∈ R3, this corresponds to adding to the Heisenberg Hamiltonian theterm∑j S z( j)S z( j). Clearly, in the limit m → ∞, this potential restricts ~S to lie in the XY plane.Or, to relate things to the O(3) nonlinear sigma model, parametrized by ~n ∈ S 2, it forces the form~n =(cos ρ sin ρ 0)T, (7.20)which relates us back to (7.5). In the complex notation, we see that V restricts ϕ to lie in a U(1)subspace of CP1, parametrized by the angle σ:ϕ =1√2(1eiσ). (7.21)1307.2. Mass Generation in the O(3) Nonlinear Sigma ModelIn this form, we have explicitly chosen a gauge by setting the first component of ϕ to be real andpositive. This reflects the fact that CP1 is a coset space, SU(2)/U(1).Now, we know from the previous subsection that a mass is generated in this theory whenm → ∞, due to vortices. However, we stress that this argument necessitated the introduction of aUV cutoff, a−1. In terms of a classical XY model on the lattice, a−1 naturally arises as the inverselattice spacing. But here, in the continuum CP1 sigma model, a−1 must arise from somewhere else.It is perhaps not surprising that the UV cutoff now comes from m, the explicit (large) mass scalethat we have inserted by hand, via V .While this observation is satisfying, it does not help us relate back to the CP1 sigma model,which corresponds to m = 0. As we reduce m, how do we realize UV-finite vortex configurations?The answer to this question is that the vortices become nonplanar as we reduce m. In other words,since ~S is no longer restricted to the XY plane for finite m, it becomes energetically favourable fora vortex configuration in ρ˜ to lift off the plane and point in the ±zˆ direction at the vortex core. Inthe small-m limit, such configurations will have actionS [ρ˜]→ 2piglog(mL), (7.22)where L is the system length. As before, the infrared divergence associated with L→ ∞ is remediedby considering configurations with net topological charge of zero.Let us see what these nonplanar vortices look like in terms of the complex ϕ. Far from its core,a configuration with topological charge +1 will have the formϕ =1√2(1eiω)(7.23)where ω is the polar angle in the XY plane. In the infinite-m limit, such a configuration has asingularity in ω at the origin. However, for finite m, this singularity may be avoided by sending oneof the two components of ϕ to zero (which corresponds to n3 → ±1 in the O(3) nonlinear sigmamodel):ϕ→(10)or ϕ→(0eiω). (7.24)The phase remains in the second case due to our gauge choice; however, at the configuration’s core,it can be removed by a pure gauge transformation, and thus does not lead to a true UV singularity.Such configurations are known as ‘merons’, or fractional instantons. Specifically, we will referto the first as a meron, and to the second as an anti-meron. The word instanton makes reference tothe fact that one of the two spatial dimensions in the CP1 theory is actually Euclidean time. Meronrefers to the half-quantized charge of these configurations, which we now demonstrate.Since the topological charge, Q, of ϕ is the integral of a total derivative, we may rewrite Q asthe difference of two contour integrals, one around the origin and one around infinity:Q =12pii[∮0dxµϕ · ∂µϕ∗ −∮∞dxµϕ · ∂µϕ∗]. (7.25)1317.2. Mass Generation in the O(3) Nonlinear Sigma ModelThe contour is necessary to avoid any gauge singularity that may be present at the origin (seebelow). For both possibilities in (7.24), the contour about infinity contributes a charge of 12 . Mean-while, the contribution from the contour about the origin is trivial in the first possibility, and -1 inthe second, leading to Q = ±12 , which proves the half-quantized charge of the meron. At the end ofthis subsection, we will comment on the gauge invariance of this topological charge.The arguments from the previous subsection carry over directly to these nonplanar topologicalconfigurations. As above, we may perform a dilute instanton gas summation, and arrive at a sine-Gordon Lagrangian. Before, the coupling constant g depended on the ultraviolet cutoff, a; now, itdepends on the anisotropic parameter, m. And before, we realized the KT transition by increasingg at fixed a. However, we may equally well increase g at a fixed bare coupling constant, bylowering the energy scale, m. This follows from the asymptotic freedom of the parameter g, whichis manifest in the following equation [137]geff ' g1 + (g/2pi) log(ma) . (7.26)Therefore, as we reduce m, we push the sine-Gordon model’s temperature closer and closerto the transition temperature. Eventually, as m approaches zero, we will necessarily be above anycritical value of bare g, and be in the gapped phase with condensed vortex configurations. Thus,we conclude that the CP1 sigma model, in the absence of any topological term, will be in a massivephase for any value of g, and that this mass is generated by topological excitations with half-integercharge.Gauge Invariance of the Topological ChargeBefore carrying on, let us reflect on the structure of (7.25). There, we rewrite the topologicalcharge Q as a difference of two contour integrals. It is interesting to note that while their sum isnecessarily invariant under the U(1) gauge symmetries ϕβ → eiϑβϕβ, an individual contour integralis not. Indeed, under ϕ 7→ eiϑϕ,∮dxµϕ · ∂µϕ∗ 7→∮dxµϕ · ∂µϕ∗ + i∮dxµ∂µϑ. (7.27)The second term, which breaks the gauge invariance of the contour, is necessary in order to ac-count for gauge transformations that alter the U(1) winding number of ϕ along the contour. Thisfeature implies that it is not meaningful to ask where the topological charge of a configuration ϕis localized. For instance, two gauge-equivalent versions of the SU(2) meron have the followingbehaviours at infinity:ϕ =1√2(1eiω)ϕ′ =1√2(e−iω1). (7.28)Their behaviour at zero is given by the first possibility in (7.24). The contour around infinity in(7.25) equals − 12 for the case of ϕ, while it equals + 12 for the case of ϕ′. Meanwhile, the contouraround zero is trivial in the case of ϕ, and equals +1 for the case of ϕ′. The full expression1327.3. Reducing the Global Continuous Symmetry to U(1)in (7.25) yields Q = 12 in both cases, as it must. Finally, we note that by choosing the gaugeϕ′ = 1√2(e−iω/2, eiω/2)T, the contour around infinity vanishes, which is what one finds in the O(3)version of the meron [137].7.2.3 The breakdown of mass generation at θ = pi.We are now in a position to interpret Haldane’s conjecture in terms of fractional instantons in theCP1 sigma model. Let us restore the topological angle θ = pi, and study how the mass generatingmechanism changes. For large m, we are in the U(1) model and the θ-term does not play a role;rather it is not well defined. However, as m is lowered towards zero, the fugacity in the sine-GordonLagrangian is modified as follows:γ → γ cos(θ/2). (7.29)This is because both vortices and anti-vortices are now weighted by the factor eiθ/2 if they corre-spond to merons, and by the factor e−iθ/2 if they correspond to anti-merons. Recall that we mustsum over all possible saddle point configurations.Thus, we see that γ vanishes at topological angle θ = pi, which corresponds to half-odd integerspin chains. This seems to suggest that the non-planar vortex/anti-vortex pairs will never unbind,and a massless phase will persist at all temperatures. However, this is not the whole story. Higher-order configurations, such as double vortices with κ = ±2 will have topological charge 0 or ±1, anddo not lead to a vanishing fugacity. Nonetheless, these configurations effectively raise the criticalvalue of g, below which occurs a gapless phase. Thus, we may at least conclude that the criticalvalue of m is smaller at θ = pi than at θ = 0. Based on the overwhelming evidence in favour ofHaldane’s conjecture, we expect that this critical value is driven all the way to m = 0, so that agapless phase does indeed occur.7.3 Reducing the Global Continuous Symmetry to U(1)Now that we have reviewed in great detail how fractional instantons generate a mass gap in theCP1 sigma model, we are ready to study how this mechanism generalizes to flag manifold sigmamodels. Our strategy is the same: we break the symmetry of the flag manifold down to U(1), wherea phase transition is well understood in terms of vortex proliferation. Our complete Lagrangian is(6.2), which we repeat here:L =n∑α,β=1[gαβδµν + bαβµν](ϕα · ∂µϕ∗,β)(ϕβ · ∂νϕ∗,α) +Ltop. (7.30)As in the previous section, we will for now neglect the topological Lagrangian. It will be restoredlater on, once we have established mass generation in the case of θα = 0. To this Lagrangian, we addan anisotropic potential V1 that breaks the SU(n) symmetry down to [U(1)]n−1, while preserving1337.3. Reducing the Global Continuous Symmetry to U(1)the discrete Zn symmetry and the [U(1)]n−1 gauge structure:38V1 = mn∑α=1∑β<γ(|ϕαβ |2 − |ϕαγ |2)2. (7.31)This is the SU(n) generalization of adding the term∑j S z( j)S z( j) to the SU(2) Hamiltonian. In-deed, for the symmetric SU(n) chains, it is straightforward to rewrite V1 in terms of the SU(n)generators on the chain. We have|ϕαβ |2 =1pS ββ( jα), (7.32)so thatV1 =mp2n∑α=1∑β<γ(S ββ( jα) − S γγ( jα))2. (7.33)To obtain the corresponding lattice terms, we simply multiply the right hand side of these expres-sions by n, and sum over j. It is interesting to note that in the case of SU(3), the potential V1 isproportional to3∑α=1[T3( jα)T3( jα) + T8( jα)T8( jα)], (7.34)where T3 and T8 are the two diagonal generators of SU(3) in the Gell-Mann basis. Now, in thelimit m → ∞, the potential V1 restricts each of the ϕα to lie on an n − 1 torus, parameterized byn − 1 angles, ςβ = ςβ(α):ϕα =1√n(eiς1 eiς2 · · · eiςn−1 eiςn)T. (7.35)Throughout, we fix a gauge by choosing ϕα1 to be real and positive, for all α. Since the number offree parameters equals the number of orthogonality constraints (both equal n(n− 1)/2)), one mightconclude that there is a unique configuration of the ϕα in this limit (up to permutations). However,the orthogonality constraints are invariant under the following transformations,ϕαβ → eiθβϕαβ ∀α, β, (7.36)which are true symmetries for each of the n − 1 un-gauge-fixed directions, β = 2, 3, . . . , n. There-fore, by introducing the potential (7.31), the continuous symmetry group of the sigma model hasbeen broken down to [U(1)]n−1. A typical configuration of the ϕα in this submanifold can beconstructed from the nth root of unity, ζ := ei2pin :ϕαβ =1√nζαβeiσβ σβ ∈ [0, 2pi]. (7.37)38Based on the previous chapters, we recognize how important the Zn symmetry is to generalizing Haldane’s conjec-ture. It would be a serious matter to break it explicitly now.1347.4. Mass Generation by Fractional InstantonsIndeed, orthonormality of these states follows from the identity∑nα=1 ζρα = 0, which holds for allρ , 0 mod n. Since the Lagrangian (7.30) couples all of the n − 1 copies of U(1) together, it willbe useful to break the symmetry down further. To this end, we introduce a second potential V2,given byV2 = −mn∑α=1n−1∑β=2((ϕα1 [ϕαβ ]∗)n − ([ϕα1 ]∗ϕαβ )n)2. (7.38)In terms of the SU(n) matrices, this potential takes the formV2 = −n∑jmp2n∑α=1n−1∑β=2((S 1β( jα))n − (S β1( jα))n)2. (7.39)The factors of n in the exponent are necessary in order to preserve the Zn symmetry, which corre-sponds to translational invariance in the underlying lattice model.39 Since V2 is still minimized bythe configurations (7.37), it can be rewritten in terms of the U(1) fields, σβ, yieldingV2 = 4mn∑α=1n−1∑β=2sin2(n(σ1 − σβ)). (7.40)It is clear that the effect of V2 is to equate all but one of the U(1) fields. In our fixed gauge, thisamounts to setting σβ = 0 for β , n, resulting in a theory that involves σn only. This is equivalentto the XY model. By inserting this restricted form of ϕα into (7.30), it is easy to show that the bαβterms vanish, and the resulting O(2) coupling constant is g, defined byg−1 =∑α,βg−1αβ. (7.41)7.4 Mass Generation by Fractional InstantonsIn this section, we would like to extend the discussion of non-planar vortices in the CP1 sigmamodel to the larger SU(n)/[U(1)]n−1 sigma model. With any luck, we should find that a similarmass generating mechanism is at play here, and that it breaks down when particular topologicalangles are restored.Recall that a non-planar vortex, (which we’ve already referred to as a meron or a fractionaltopological instanton) has the following form in terms of the CP1 field, ϕ. Far from its center, itreadsϕ =1√2(1eiω). (7.42)Meanwhile, near its center, it tends to either (1, 0)T or (0, eiω)T, in order to avoid an ultravioletsingularity.39Since ϕ has dimension 0 in 1+1 dimensions, an operator with arbitrarily high powers of ϕ is still renormalizable.1357.4. Mass Generation by Fractional InstantonsExtending this calculation to SU(n) is straightforward. Now, the topological charge is a vectorQ = (Q1,Q2, . . . ,Qn), corresponding to the n fields ϕα. A configuration far from its core hasϕαn =1√neiω for all α, and receives a contribution to the topological charge of 1n from the contour atinfinity. Meanwhile, in order for the configuration to be UV finite at its core, the phase eiω in eachof the nth components of the ϕα must either vanish or become a pure gauge transformation. Thisleads to an n-fold family of topological excitations, distinguished by which one of the ϕα tends to(0, 0, 0, . . . , eiω)T. For that particular ϕα, a contribution of -1 is added to its topological charge fromthe contour about the origin in (7.25), while the remaining n − 1 fields receive no contribution, asthey are topological trivially there. See the example below for an explicit solution in the case ofn = 3. Thus, we obtain n species of topological configuration, with chargesQα =1n(1, 1, . . . ,position α︷ ︸︸ ︷−(n − 1), . . . , 1). (7.43)And so for all n, we find fractionalized topological excitations, or fractional instantons, in oursymmetry broken model.7.4.1 Example: An explicit fractional instanton in SU(3)Here, we prove the existence of a fractional instanton in SU(3), with topological charge Q1 =(−23 , 13 , 13 ). While the most general complex unit vector in C3 depends on five real parameters, wemake a symmetric ansatz that reduces this number down to two, which is the minimum number ofparameters necessary to construct a fractional instanton. For our three orthogonal vectors ϕα, wetake:ϕ1 =1√2sinϑsinϑ√2 cosϑeiω ϕ2 = 12cosϑ + icosϑ − i−√2 sinϑeiω ϕ3 = 12cosϑ − icosϑ + i−√2 sinϑeiω . (7.44)Here ω is the polar angle coordinate, and ϑ ∈ [0, pi/2] depends only on the radial coordinate:ϑ = ϑ(r). Inserting this ansatz into the Lagrangian (7.30), one obtainsL = 1g[(∂rϑ)2 +1r2U1(ϑ) + mgU2(ϑ)], (7.45)whereU1(ϑ) = sin2 ϑ cos2 ϑ +14sin4 ϑ, (7.46)andU2(ϑ) =52[cos2 ϑ − 12sin2 ϑ]2+1128cos2 ϑ(3 sin4 ϑ − 4 cos2 ϑ)2. (7.47)1367.5. Destructive Interference in the Presence of Topological AnglesNote that there is a unique coupling constant g = g12 = g13 = g23 in the SU(3)/[U(1)]2 sigmamodel, and the terms proportional to bαβ have vanished. This is the classical Lagrangian for aparticle at position ϑ at time r, in a time-dependent effective potentialVeff = − 1r2 U1(ϑ) − mgU2(ϑ). (7.48)Now, at radii r such that r2mg 1, Veff is maximized by minimizing U1(ϑ) on [0, pi/2], whichis achieved with ϑ = 0. On the other hand, for r → ∞, the effective potential is maximizedby minimizing U2(ϑ), which is done with ϑ = ϑ = arccos(1/√3). Since ddϑU1 ≥ 0 on [0, ϑ],and ddϑU2 ≤ 0 on the same interval, we may conclude that a solution exists with ϑ(r) increasingmonotonically from 0 at r = 0 to ϑ at r = ∞. As expected, the vectors ϕα in (7.44) tend to (7.37),up to a gauge transformation, as r → ∞. Moreover, as r → 0, the UV singularities eiω in ϕ2 andϕ3 vanish, and ϕ1 → (0, 0, eiω)T. This completes the example.7.4.2 Multiple copies of the sine-Gordon modelWhile the number of such topological configurations has increased, the original argument fromCP1 for mass generation carries over to this more general case: for each species of topologicalexcitation in this n-fold family, we have a species of particle in the Coulomb gas formalism [137].That is, each particle has a partition function that is represented (at large distances) by its ownsine-Gordon model,LS G = 12(∂µσ)2 + γ cos(2pigσ), (7.49)which is identical for each of the n species. As above, γ represents the fugacity of the fractionalinstanton gas. Since all of the n species arise from the same action, each will have the same fugacityand critical g, so that the above model (7.49) is merely copied n times, and the SU(2) analysisfrom Section 7.2.2 can be applied directly. As m is lowered, the effective critical temperature isincreased until the topological excitations condense and a mass gap is produced. Thus, we concludethat fractional instantons are responsible for generating a mass gap in the SU(n)/[U(1)]n−1 sigmamodel (7.30), in the absence of topological angles.7.5 Destructive Interference in the Presence of Topological AnglesWe now restore the topological angles θα, and study how the mass generating mechanism changes.As before, when m is large, our theory reduces to the U(1) sigma model where the θ-terms do notplay a role. However, as m is lowered towards zero, the fugacity γ in the sine-Gordon model ismodified toγn∑α=1eiθ·Qα, (7.50)1377.5. Destructive Interference in the Presence of Topological Angleswhere the sum is over the n species of fractional instanton, and θα = αθ, with θ =2pipn . Using(7.43), one easily finds that this sum equals γ∑k ζpk. So long as p is not a multiple of n, this sumvanishes. In other words, for such values of p, the fugacity of the sine-Gordon model vanishes,suggesting that the Coulomb gas is always in its massless phase.At first glance, this appears to be inconsistent with our results from the previous chapters, whichalso predict a gap when p is not a multiple of n, but has a nontrivial shared divisor with n. Thisdiscrepancy is resolved by considering higher-order topological excitations. Indeed, while objectsthat have winding number greater than ±1 in ω have larger action, they too must be considered, anddo not necessarily lead to a vanishing fugacity. A similar calculation shows that a configurationwith winding number κ has charge Qα,κ, withQα,κβ =κn− κδα,β, (7.51)and modifies the fugacity to γ∑k ζpkκ. If κ is a multiple of n, this will lead to a nonzero fugacity.This is also true in the case of SU(2): recall the double meron configurations discussed above.However, the conclusion arrived at before also applies here: since these higher winding objectshave larger action, we can at least conclude that the critical value of m, above which occurs agapless phase, is lower at θ = 2pipn than at θ = 0.Now, if gcd(n, p) , 1, n, then not only do events with winding number n contribute to thefugacity, but so do events with winding number d := n/ gcd(n, p). Since these contributions willhave a much smaller action than the n-winding events, this shows that whenever p and n have anontrivial common divisor, the critical value m is larger than at θ = 2pin (although still lower thanat θ = 0). Thus, it is possible to interpret our generalized Haldane conjecture in terms of thesefractional topological excitations: When p and n are coprime, mass generation only starts to occurfor configurations that have winding number ±n. In SU(2), these events are not strong enough toopen a gap at the isotropic point m = 0, and we predict that this holds for general n. In otherwords, we are claiming that the critical value of m is zero when p and n are coprime. When p and nhave a nontrivial common divisor different from n, configurations that have much less action beginto contribute to mass generation, starting with objects that have winding number d. The simplestexample of this is in SU(4), with p = 2. In this case, configurations that have winding number ±2successfully generate a mass gap, while for p = 1 these configurations destructively interfere, anda mass gap is not generated by the subleading configurations, with winding ±4. Finally, when p isa multiple of n, the least-action configurations that have winding ±1 produce a mass gap, just likethe merons in SU(2).While we haven’t offered a rigorous argument as to why the critical value of m is indeed fixedat zero in the case of p and n coprime, we can make the following observation. According tothe LSMA theorem, a finite energy gap above the ground state implies spontaneously broken Znsymmetry whenever p is not a multiple of n. In the sine-Gordon model (7.49), this Zn symmetrycorresponds to the following transformation:σ 7→ σ − 2pin. (7.52)1387.6. ConclusionsWhile this transformation acts nontrivially on most fractional instantons, it has no effect on config-urations with winding number n, which are the elementary excitations in the sine-Gordon modelwhen p and n are coprime. Thus, our prediction of a gapless phase in this case is perhaps not toounreasonable, since any finite gap would necessitate the spontaneous breaking of (7.52), which actstrivially on the Coulomb gas of n-winding events.7.6 ConclusionsIn this chapter, we have proposed a mass generating mechanism by fractional instantons in thecomplete flag manifold sigma models, with target space SU(n)/[U(1)]n−1. These theories arisenaturally from SU(n) chains in particular representations, as we have seen many times throughoutthis thesis. In particular, we have focused on theories with n − 1 nontrivial topological angles,θα =2piαn p. These correspond to either the rank-p symmetric representations of SU(n), or thetwo-row representations for even n, with p = p1 + p2.For p a multiple of n, we have shown that topological configurations with charge 1n producea finite gap above the ground state, much in the same way that vortices produce a gap in theKosterlitz Thouless transition of the XY model. When p and n have a nontrivial common divisordifferent from n, we’ve shown that a mass gap is still produced, but now it is due to larger-actionconfigurations that have charge 1gcd(n,p) . Finally, when p and n are coprime, we have argued that noenergy gap is produced by topological configurations. Together, these three results offer an intuitiveexplanation of the generalizations of the Haldane conjecture to SU(n) chains that have appeared inthis thesis.139Chapter 8Summary and OutlookIn this thesis, we have succeeded in achieving our original goal of extending Haldane’s famousconjecture about antiferromagnetic chains to systems with SU(n) symmetry. To summarize ourmain result succinctly, we have found that there are two families of SU(n) chains that exhibit agapless phase at low energies. The first family consists of those chains with a completely symmetricrank-p representation at each site, with p a positive integer whose greatest common divisor withn is 1. In the case of n = 2, these chains are precisely those with half-odd integer spin, i.e. thespin chains predicted by Haldane to be gapless. The second family of SU(n) chains that we haveidentified are also labeled by a positive integer p satisfying the same coprime constraint with n,but are only gapless for even n. These representations are not completely symmetric: instead, theyhave two rows in their Young tableaux, of lengths p1 and p2 that sum to p. The simplest chaincorresponding to this second family has SU(4) symmetry, with the representation (p1 = 2, p2 = 1)at each site.While the above classification is elegant, and was the source of motivation for much of the workappearing in this thesis, it is by no means the only noteworthy result that has been obtained. Indeed,in our attempt to push the conceptual and technical limitations of Haldane’s original calculations,we have made numerous discoveries that will have a lasting impact on the field of SU(n) spinsystems.In Chapter 3, we learned early on that the nearest-neighbour Heisenberg model used by Hal-dane is inadequate for deriving a low energy field theory of chains with larger symmetry groups.This can be understood by considering how the Young tableau parameters pα, which generalizethe notion of spin in the antiferromagnet, determine the target space manifold of the chain’s matrixdegree of freedom, S . For SU(2), this manifold is always CP1, and one may readily derive thefamiliar CP1 sigma model. But for n > 2, the space CP1 is promoted to a flag manifold, and furthercare must be taken when deriving a low energy description. Throughout Chapter 3, we developed aprescription for obtaining a well defined quantum theory, which amounted to adding longer-rangeinteraction terms to the Hamiltonian. These interaction terms serve to restrict the target space ofS to be an orthogonalized product of CPn−1 models. Then, by summing over the unit cell (whichis determined by the classical ground state of the chain), we obtain n orthogonalized CPn−1 mod-els, which together constitute a sigma model whose target space is the complete flag manifold,SU(n)/[U(1)]n−1. As a result, we were able to answer in Chapter 3 the obvious question posed bymany seeking to extend Haldane’s conjecture beyond SU(2): what is the appropriate generaliza-tion of the CP1 sigma model that describes the antiferromagnet at low energies? It turns out thatCP1 should be promoted to the complete flag manifold of SU(n). As a tangential development in140Chapter 8. Summary and Outlookthis chapter, we also extended the exact results of Lieb, Schultz, Mattis and Affleck (the LSMAtheorem) to the longer-range Hamiltonians that we introduced.In Chapter 4, in deriving an explicit expression for the sigma model of the SU(3) chain in thesymmetric representation, we made three important realizations. First, unlike the CP1 Lagrangianthat is characterized by a single parameter g, for n > 2 there is an additional λ parameter thatcorresponds to a two-derivative interaction term that couples the different CPn−1 sectors together.When we interpret the Lagrangian as a metric on the flag manifold target space, this parametercorresponds to a torsion, or antisymmetric tensor. Such terms were unanticipated, as torsion doesnot exist in the simpler case ofCP1. Using the renormalization group, we were able to conclude thatλ is a relevant parameter, and should be included in the low energy limit. The second realizationwe made concerned the topological angle content of these theories. Since flag manifolds may beembedded into a product of CPn−1 models, it is possible to define multiple topological charges, bypulling back the topological term from each CPn−1 sector. Mathematically, this is seen by notingthat the second cohomology group of the flag manifold is Zn−1. In the symmetric SU(3) chains,we found that this led to two topological angles θ1 =2pip3 and θ2 =4pip3 . Since 3 is prime, theseangles both are nontrivial so long as p is not a multiple of 3; however, we began to speculate thatfor larger values of n, it might be possible for some, but not all of the topological angles to betrivial. How would Haldane’s conjecture generalize in this case? Finally, the third realization wemade in Chapter 4 arose from considering other (non-symmetric) representations. In spin chains,it was well understood that for translationally invariant systems, the classical model would alwaysexhibit antiferromagnetic order for a positive interaction strength, J. However, for n > 2, welearned that it is possible to realize ferromagnetic and antiferromagnetic order simultaneously (inthe classical limit). This led to the fascinating consequence of having Goldstone bosons withtwo types of dispersion relation: linear dispersion for those corresponding to antiferromagneticorder, and quadratic dispersion for those corresponding to ferromagnetic order. And all of thesebosons arise from the same parent group, SU(n). We concluded this chapter by restricting to thoserepresentations with antiferromagnetic order only. In addition to the symmetric ones, this led alsoto the self-conjugate SU(3) chains. We found that these chains did not exhibit Lorentz invarianceat low energies – a surprising feature, but one that would prove to ubiquitous in the remainingchapters of this thesis.In Chapter 5, we were forced to confront the absence of Lorentz invariance head-on. Here, ourgoal was to extend our derivation of the flag manifold sigma model from the symmetric SU(3) chainto symmetric chains with generic SU(n) symmetry. Armed with our newfound understanding fromthe previous two chapters, we hoped to obtain a model with target space SU(n)/[U(1)]n−1, thatdepended on two parameters g and λ. But yet again, we were confronted with the unexpected.Instead of a single g and a single λ, we instead found b n2c different coupling constants of eachtype. We now understand that this is consistent with the underlying symmetry of the model: themost generic metric and torsion tensors on the complete flag manifold sigma model of SU(n)each have b n2c free parameters. But more importantly, we also found that for n > 3, our theorywas not characterized by a single velocity v (which corresponded to the velocity of spin-waves inthe antiferromagnet). This feature of multiple velocities meant that for all values of n > 3, the141Chapter 8. Summary and Outlooksymmetric SU(n) chain seemed not to have a Lorentz invariant low energy description. It was atthis point that we engaged in the arduous task of calculating the renormalization group flow of thesedistinct velocities. Our findings, which we believe are rather consequential to the study of SU(n)chains, were that at large length scales, these velocities all converge to a common value. As a result,we may write down a Lorentz invariant theory for the SU(n) chain after all. This observation wascrucial in establishing an SU(n) version of Haldane’s conjecture, which relies in part on ’t Hooftanomaly matching arguments of Lorentz invariant quantum field theories. We concluded that therank-p symmetric SU(n) chain corresponds to the relativistic complete flag manifold of SU(n),with topological angle content θα =2pipαn for α = 1, 2, . . . , n− 1. Moreover, we concluded that onlywhen p and n share no common divisor, may this theory exhibit a gapless phase at weak coupling.As we alluded to above, only in this scenario are all n − 1 topological angles nontrivial: if p and nhave common divisor q, then θn/q ≡ 0. This observation motivated our efforts in Chapter 7Before investigating the consequence of a vanishing subset of topological angles, we saw itappropriate in Chapter 6 to classify all remaining (i.e. nonsymmetric) SU(n) chains. Having n − 1distinct topological angles seemed to be a necessary condition for realizing a gapless phase, andso we set out to list all representations that would map to a sigma model with this property. It washere that the phenomenon of linear vs. quadratic dispersion that appeared in SU(3) chains returnedin full force. Throughout this chapter, we learned that linear dispersion is in fact quite a specialproperty of SU(n) chains. For most choices of representation parameters pα, there will at least besome corresponding Goldstone bosons with quadratic dispersion relations. In a series of tables, wewere able to record exactly what relations must be satisfied between the pα to guarantee that thisdoes not occur. Having done this, we then turned to the classification of topological angle contentfor all of these purely linearly dispersing theories. Again, we were met with a surprising result: inaddition to the symmetric chains, there is only one other type of SU(n) chain that has the appropriatedispersion relations and topological angles to allow for a mapping to the complete flag manifold ofSU(n) with θα =2pipαn . This type of chain occurs only for even n, and has the representation withtwo rows of lengths p1 and p2 at each site, with p = p1 + p2 and p1 , p2. Using arguments similarto those presented in Chapter 5, we concluded that when p has no nontrivial common divisor withn, the corresponding SU(n) chain will be in a gapless phase at weak coupling.Finally, in Chapter 7, we were able to provide a physical and intuitive argument as to why theseflag manifold sigma models exhibit a gapless phase only when all n− 1 topological angles are non-trivial. Arguably this chapter is one of most important of this thesis, as it recasts our generalizedHaldane conjecture in the familiar Coulomb gas formalism, and explains why certain represen-tations of SU(n) are so special. This work drew from an early result of Affleck that explainedHaldane’s original claims in terms of half-quantized topological excitations, or merons. Here,we generalized these notions to SU(n), and promoted the merons to excitations with topologicalcharges ± 1n . First, we showed that when θα = 0 for all α, the finite energy gap in the flag manifoldsigma model can be understood as being generated by a proliferation of these topological objects.This is the same mechanism that drives a finite gap at the Kosterlitz Thouless phase transition inthe XY model when vortex pairs unbind. Then, by turning on the θα, we were able to observe howinference effects come into play. In short, the fugacity in the Coulomb gas of topological excita-142Chapter 8. Summary and Outlooktions is modified by the θα, and vanishes precisely when all n − 1 values θα = 2pipαn are nonzero!With a vanishing fugacity, the mass-generating mechanism breaks down, so that a gapless phaseoccurs. Actually, what we observed is that the fugacity vanishes at lowest order for all values ofp , n (including those with a nontrivial common divisor), but when topological excitations withgreater charge (and correspondingly greater action) are considered, only those with gcd(n, p) = 1lead to a gapless phase. That this Coulomb gas picture correctly describes our generalized Haldaneconjecture is a very satisfying result.Let us now close by commenting on future research directions. The most pressing issue leftunsolved by the work in this thesis has to do with SU(n) chains that admit Goldstone modes withboth linear and quadratic dispersion relations. This already occurs in SU(3) for any representa-tion that is neither symmetric nor self-conjugate. How might one proceed in order to characterizesuch chains? One idea is that according to Coleman’s theorem, quadratically dispersing Goldstonebosons are perfectly valid in one spatial dimension (at least at zero temperature). Therefore, it isconceivable that the subgroup of the SU(n) symmetry corresponding to ferromagnetic order mightspontaneously order, leaving true quadratic Goldstone bosons coupled to a smaller theory. For in-stance, if only m < n of the n copies of CPn−1 have linear dispersion, then perhaps the low energytheory of such a chain will be that of a complete SU(m) flag manifold sigma model, coupled ton − m quadratically dispersing CPn−1 sectors. In anticipating such a possibility, already in Chap-ter 6 we have recorded which topological angles correspond to the linearly dispersing CPn−1 fieldsin such a scenario. If this description were true, then what we would have is a remarkable hier-archy of flag manifold sigma models: For each n, by altering the number of satisfied conditionson the pα representation labels, we could realize a nested set of flag manifold sigma models ofSU(m1), . . . ,SU(mk), with m1 < m2 < · · · < mk = n. Moreover, based on our topological Coulombgas picture (and likewise the ‘t Hooft anomaly matching picture), an SU(n) chain with discrete Zmsymmetry (arising from an m-site classical unit cell), might now realize a gapless phase, character-ized by an SU(m)1 WZW point. Again, this scenario is only possible because quadratic Goldstonebosons are not forbidden by Coleman’s theorem. This exciting possibility motivates our currentresearch efforts into understanding these more general SU(n) chain systems.In addition to this direct extension of our work, there are a variety of other research avenues thathave now been opened up thanks to this thesis. In particular, it is now a major numerical challengeto verify our generalization of the Haldane conjecture. We expect that this task will spur the devel-opment of new techniques, just as the challenge of verifying Haldane’s original conjecture led togreat numerical progress, including the invention of the celebrated density matrix renormalizationgroup. Already we have seen this occur in SU(3), with Gozel et. al. establishing the existence ofa finite mass gap ∆/J ∈ (0.017, 0.044) in the symmetric SU(3) chain with p = 3 [77]. Antiferro-magnetic SU(n) spin ladders have also been considered [139–141], as have two-dimensional SU(n)Heisenberg models [142–144].Furthermore, our predictions about SU(n) chains also serve to challenge the experimentalphysics community. Over the past decade, cold atom physicists have continued to make advancesin optical trapping technology, with the most recent publication being [66]. Our predicted finiteenergy gaps for particular representations should now be observable in such set-ups. For example,143Chapter 8. Summary and Outlookthe p = 3 representation of SU(3) should exhibit a nondegenerate ground state with a finite gap, andthe p = 2 representation of SU(4) should exhibit a gap above two-fold degenerate ground states.Since p > 1 in these chains, one must devise a way to achieve multiple sets of n particle species ateach site, either by trapping multiple atoms or by making use of an additional orbital degeneracy,such as the nearly degenerate clock states of alkaline Earth atoms. Of course, such experimentspresent a much greater challenge than those that originally measured the finite gap of the spin-1chain, since in that case an actual quasi-one dimensional material was readily available.Finally, a third research direction stemming from this thesis has to do with flag manifold sigmamodels in general. While these models have always been of interest to mathematical physicists,they have recently garnered more attention in the theoretical physics community thanks to theirapplicability to SU(n) spin chains. In particular, in [145] Bykov has since investigated integrableflag manifold sigma models, which generalize the older notions of integrability in the CP1 sigmamodel put forth by the Zamolodchikov brothers, among others. In fact, we will soon publishwith Bykov and Affleck an extensive review article on flag manifold sigma models, which shouldfurther encourage cross-disciplinary research efforts from mathematicians and physicists alike [6].We believe that this review, combined with our thesis, will inspire additional research efforts intothe rich structure of flag manifolds themselves, ultimately bringing new insights into the SU(n)chains that we have focused on here.144Bibliography[1] Miklo´s Lajko´, Kyle Wamer, Fre´de´ric Mila, and Ian Affleck. 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We willmake use of this identity frequently throughout this thesis. To begin, we write an arbitrary unitarymatrix U asUαβ = ϕαβ , (A.1)where the ϕα are the rows of U, and are orthonormal complex vectors: ϕ∗,α · ϕβ = δαβ. We canexpress the nth row-vector, ϕn, in terms of first n − 1 rows using the antisymmetric unit tensor, as:ϕ∗,nαn = α1,α2...αnϕ1α1ϕ2α2 . . . ϕn−1αn−1 . (A.2)This follows becauseϕ∗,n · ϕβ = α1,α2···αnϕ1α1ϕ2α2 . . . ϕn−1αn−1ϕβαn (A.3)andα1α2···αnϕβαβϕβαn = 0. (A.4)for β = 1, 2, 3, . . . , n − 1. Now, we use the identityα1α2,...αnβ1β2...βn−1αn =∑{a1,a2,...an−1}sgn{a1, a2, . . . aan−1}δα1βa1δα2βa2 . . . δαn−1βan−1 . (A.5)Here the sum is over all permutations of a1, a2, . . . an−1. Now, orthogonality and (A.5) imply|ϕn|2 = 1, (A.6)andϕn · ∂ϕ∗,n = α1α2···αnϕ∗,1α1 ϕ∗,2α2 · · ·ϕ∗,n−1αn−1 β1β2···βn−1αn((∂ϕ1β1)ϕ2β2 . . . ϕn−1βn−1 + · · · ). (A.7)Here the · · · is a sum over derivatives of each factor. Now we useα1α2...αnϕ∗,1α1ϕ∗,2α2 · · ·ϕ∗,n−1αn−1 β1β2···βn−1αn(∂ϕ1β1)ϕ2β2 · · ·ϕn−1βn−1 = ϕ∗,1 · ∂ϕ1, (A.8)which follows from orthogonality and (A.5). Soϕn · ∂ϕ∗,n =n−1∑α=1ϕ∗,α · ∂ϕα, (A.9)which implies,n∑α=1∂ϕα · ϕ∗,α = tr[∂UU†] = 0. (A.10)156Appendix BLow Energy Expansion of theSymmetric SU(n) HamiltonianIn this appendix, we provide some details from the derivation of the low energy Hamiltonian of theSU(n) chain in the rank-p symmetric representation. These results are used both in Chapter 4 forthe SU(3) chain, and more generally in Chapter 5 for the SU(n) chain. In particular, we simplifythe following Heisenberg interaction termtr[S ( jγ)S ( jη)], (B.1)using (4.23). We writetr[S ( jγ)S ( jη)] =2∑i=1Xi(γ, η) +4∑i=3(Xi(γ, η) + Xi(η, γ)) + O(p−2), (B.2)withX1(γ, η) := p2tr[U†( jγ)ΛγU( jγ)U†( jη)ΛηU( jη)] (B.3)X2(γ, η) := tr[U†( jγ){L( jγ),Λγ}U( jγ)U†( jη){L( jη),Λη}U( jη)] (B.4)X3(γ, η) := ptr[U†( jγ){L( jγ),Λγ}U( jγ)U†( jη)ΛηU( jη)] (B.5)X4(γ, η) := tr[U†( jγ)L( jγ)U( jγ)U†( jη)ΛηU( jη)] (B.6)Since the matrices U, L,L are evaluated at different sites, we Taylor expand. For example,U( jγ) = U(n j + (γ − 1)) = U( jη) + (η − γ)∂xU( jη) + 12(η − γ)2∂2xU( jη) + · · · . (B.7)We assume the derivative is uniform, ∂xU( jη) = ∂xU( j′λ), and consider each of the above termsseparately. Since L characterizes a fluctuation, we treat it as the same order as ∂U. Finally, wesuppress the argument jγ of each matrix throughout. Then:• Term 1:X1(γ, η) ≈ p2tr[ΛγΛη + (η − γ)2(U∂xU†Λγ∂xUU†Λη − ΛγΛη∂xU∂xU†)]. (B.8)Since ΛγΛη = 0 for γ , η, this simplifies toX1(γ, η) ≈ p2(η − γ)2tr[U∂xU†Λγ∂xUU†Λη]. (B.9)157Appendix B. Low Energy Expansion of the Symmetric SU(n) Hamiltonian• Term 3:X3(γ, η) ≈ ptr[{L,Λη}Λγ]+(η−γ)ptr[{L,Λη}U∂xU†Λγ]+(η−γ)tr[{L,Λη}Λγ∂xUU†] (B.10)Since the first term is a product of a diagonal and an off-diagonal matrix, its trace vanishes.What remains is a commutator:X3(γ, η) = (η − γ)ptr[[{L,Λη},Λγ]∂xUU†], (B.11)which simplifies toX3(γ, η) = (η − γ)p(Lγη[U∂xU†]ηγ + Lηγ[∂xUU†]γη). (B.12)Note that X3(γ, η) = X3(η, γ).• Term 4: Since L contains two powers of L, we only have to expand U to zeroth order. WefindX4(γ, η) = |Lγη|2 = X4(η, γ). (B.13)• Term 2: A similar calculation shows thatX2(γ, η) = 2|Lγη|2 = X2(η, γ). (B.14)Finally, combining the results of these five calculations, we findtr[S ( jγ)S ( jη)] = p2(η − γ)2tr[U∂xU†Λγ∂xUU†Λη] (B.15)+2(η − γ)p(Lηγ[∂xUU†]γη + Lγη[U∂xU†]ηγ)+ 4|Lηγ|2 + const.which proves (4.27).158Appendix CFactorization of SU(n) MatricesIn this appendix, we prove a factorization identity for SU(n) matrices. Let Greek letters index thediagonal generators of SU(n), lowercase Latin letters index the off-diagonal generators of SU(n),and uppercase Latin letters index the full set of generators. That is,∑ATA =∑aTa +∑γTγ. (C.1)Then, given U = eiωATA ∈ SU(n), we may factorize it as follows:U = eiωγTγeiωaTa . (C.2)We will prove this identity to third order in the ωγ and ωa, but mention how the proof extendsto every order in perturbation theory. Using the Baker-Campbell-Hausdorff formula,log(eXeY ) = X + Y +12[X,Y] +112([X, [X,Y]] − [Y, [X,Y]]) + · · · , (C.3)we havelog eiωγTγeiωaTa = iωATA − 12ωγωa[Tγ,Ta] −i12(ωγωβωa[Tγ, [Tβ,Ta]] − ωγωaωb[Ta, [Tγ,Tb])(C.4)to order O(ω4), which equals= i[ωa − ωγωb fγbA + 13(ωγωβωb fβbC fγCA − ωγωdωb fγbC fdCA) ]TA + O(ω4). (C.5)The formula for the higher-order terms occurring in (C.3) and (C.4) are quite complicated, butalways involve nested commutators. This important fact allows us to reduce every term in theexpansion to one that is linear in the generators, TA. A term that is ∼ ωn will involve n − 1 nestedcommutators, leading to a contribution that is proportional to a product of n − 1 structure factorsfabc, multiplied by a single SU(n) generator TA. Therefore, order-by-order, we may construct amapping between the ωA and the ωa:ωA = ωa − ωγωb fγbA + 13(ωγωβωb fβbC fγCA − ωγωdωb fγbC fdCA)+ O(ω4). (C.6)159Appendix C. Factorization of SU(n) MatricesTo prove the factorization identity, we must be able to invert this formula. This is done by repeat-edly insertingωa = ωA + ωγωb fγbA − 13(ωγωβωb fβbC fγCA − ωγωdωb fγbC fdCA)+ O(ω4). (C.7)into each of the terms on its right hand side We find:ωa = ωA + ωγωb fγbA +23ωγωβωb fβbe fγeA +13ωγωdωb fγbC fdCA + O(ω4). (C.8)Thus, for any SU(n) matrix U = eiωATA , we may perform this transformation to obtain the factorizedform occurring above.160Appendix DPolyakov’s Renormalization of the O(n)nonlinear Sigma ModelIn this appendix, we review Polyakov’s calculation of the beta function for the O(n) nonlinearsigma model [126]. In Section D.1, we use Polyakov’s original notation. Then in Section D.2,we demonstrate how our calculations can equivalently be done in the SU(2)/U(1) sigma modellanguage.D.1 Renormalization in the Real-Space Vector LanguageIn this section, we start with the following LagrangianL = 12g|∂µ~n|2. (D.1)The idea is to construct a ‘slow’ unit vector ~ns out of ~n’s momentum modes below b−1Λ, whereΛ is a reduced cutoff, and b & 1. The remaining modes of ~n can then be written in terms of anorthonormal basis {~ea}, orthogonal to ~ns:~n = ~ns(1 − φ2)1/2 +n−1∑i=1φi~ei, (D.2)where φ2 =∑n−1i=1 φ2i . The fields φi consist entirely of ‘fast’ modes, with momentum lying in theWilson shell [b−1Λ,Λ). Integration over the shell is then equivalent to integrating out the fields φi.Inserting this expansion (D.2) into L gives (to quadratic order in φi)2gL = (∂µ~ns)2(1 − φ2) + (∂µφi)2 + φiφ j∂µ~ei · ∂µ~e j + 2φi∂µφ j∂µ~ei · ~e j − 2φi∂2µ~ns · ~ei . (D.3)A naive argument would claim that the term linear in φi can be neglected, since ~ns and ~ei containslow modes only. However, their product will generically have modes lying in the Wilson shell;a better argument is presented in Section 4.5.2. Dropping linear terms, we are left to evaluate theGaussian integral∫D[φ] exp(− 12g∫d2x((∂µ~ns)2(1 − φ2) + (∂µφi)2 + φiφ j∂µ~ei · ∂µ~e j + 2φi∂µφ j∂µ~ei · ~e j) ). (D.4)161D.1. Renormalization in the Real-Space Vector LanguageUsing ∫D[φ] =∫ ∏b−1Λ<k<ΛDφ(k)Dφ(−k), (D.5)the effective Lagrangian is2gLeff = (∂µ~ns)2 + g tr[log(δi j −Gi jδi j(∂µ~ns)2 + Gi j∂µ~ei · ∂µ~e j + 2∂µGi j∂µ~ei · ~e j)] , (D.6)where Gi j(x) is the Green’s function of the fields φi. Since terms involving more than two deriva-tives of slow fields are irrelevant, we can expand the trace-logarithm:tr[log(δi j −Gi jδi j(∂µ~ns)2 + Gi j∂µ~ei · ∂µ~e j + 2∂µGi j∂µ~ei · ~e j)](D.7)= −∫d2x Gi j(0)δi j(∂µ~ns)2 +∫d2x Gi j(0)(∂µ~ei · ∂µ~e j) + 2∫d2x ∂µGi j(0)∂µ~ei · ~e j−124∫d2xd2y ∂µG(x − y)[∂µ~ei · ~e j](x), ∂νG(y − x)[∂ν~e j · ~ei](y) + irrelevant,where we defined G(x) by Gi j(x) = δi jG(x).) The third term vanishes since∂µG(0) = −i∫d2k(2pi)2kµk2= 0. (D.8)Rewriting G(0) =∫b−1Λ<k<Λd2k(2pi)2δi jk2 , the expansion in (D.7) reduces to=∫b−1Λ<k<Λd2k(2pi)21k2∫d2x(− (N − 1)(∂µ~ns)2 + (∂µ~ei · ∂µ~ei))(D.9)− 2∫d2kd2q(2pi)4kµk2(q + k)ν(q + k)2[∂µ~ei · ~e j](q)[∂ν~e j · ~ei](−q) + irrelevant,where we Fourier transformed the last term, and both ~k and ~k + ~q lie in the Wilson shell [b−1Λ,Λ).Now we expand ∫b−1Λ<|~q+~k|<Λ(q + k)ν(q + k)2(D.10)in powers of q, and keep the zeroth order term only, since terms with more powers of q willcorrespond to irrelevant operators:∫d2kd2q(2pi)4kµk2(q + k)ν(q + k)2=∫d2kd2q(2pi)4kµkνk4+ irrelevant =12∫d2kd2q(2pi)41k2+ irrelevant. (D.11)Inserting this expansion into (D.9), and integrating over k, the effective Lagrangian becomes2gLeff = (∂µns)2 − g log bpi[(n − 1)(∂µ~ns)2 − ∂µ~ei · ∂µ~ei + (∂µ~ei · ~e j)2]. (D.12)162D.2. Renormalization in the Flag Manifold LanguageFinally, we insert a complete set of states to obtain the identity(∂µ~ei)2 = (∂µ~ei · ~e j)2 + (∂µ~ns)2 (D.13)since ~ns · ~ei = 0. Thus,Leff = 12g (∂µ~ns)2[1 − g (n − 2)2pilog b]. (D.14)From this, we concludeβ(g) =dgd log b=(n − 2)2pig2. (D.15)D.2 Renormalization in the Flag Manifold LanguageIn the previous section, the renormalization calculation relied on an expansion of the field ~n interms of ‘slow’ and ‘fast’ components:~n = ~ns(1 − φ2)1/2 +2∑i=1φi~ei. (D.16)We would like to generalize this notion to matrix field theories, by writingU = U f Us (D.17)for U,U f ,Us ∈ SU(2) , where U f contains the fast modes of U, and Us the slow modes. Rewritingthe O(3) nonlinear sigma model in terms of SU(2) matrices,~n · ~σ = U†σzU U ∈ SU(2), (D.18)we prove that (D.17) is equivalent to (D.2) in the SU(2) case. We expand the fast matrix asU f = I + iωaσa − ωaωbσaσb + O(ω3), (D.19)and only keep terms up to quadratic order in ω (higher-order terms will correspond to diagramsbeyond one loop). NowU†σzU = U†s(I − iωaTa − 12ωaωbTaTb)σz(I + iωaTa − 12ωaωbTaTb)Us (D.20)= U†sσzUs − ωaωbU†sσzTaTbUs + iωaU†s [σz,Ta]Us −12ωaωbU†s {TaTb, σz}Us (D.21)= U†SσzUs(1 − 2ω2) + −2ωxU†sσyUs + 2ωyU†sσxUs. (D.22)Defining~e1 = tr12~σU†sσyUs, ~e2 =12tr~σU†sσxUs, (φ1, φ2) = (2ωy,−2ωx), (D.23)163D.2. Renormalization in the Flag Manifold Languagewe find~σ · ~n = ~σ · ~ns(1 − 12φ2) + φ1U†sσyUs + φ2U†sσxUs. (D.24)To read off the components of ~n, we useni =12tr[σi~σ · ~n] = 12tr[σiU†σzU]. (D.25)Applying 12 tr[σ] to the above expression, we find~n = ~ns(1 − 12φ2) + φ112tr[~σU†sσyUs] + φ212tr[~σU†sσxUs] = ~ns(1 − φ2)1/2 + φi~ei + O(φ3). (D.26)Finally, we check that we’ve found an orthonormal basis:~e1 · ~e2 = 12tr[(~e1 · ~σ)(~e2 · ~σ)] =12tr[U†sσyUsU†sσxUs] = 0, (D.27)~ns · ~ei = 12tr[(σ · ~ns~σ · ~ei)] =12tr[U†sσzUsU†sσiUs] = 0. (D.28)Therefore, our expansion (D.17) is equivalent to Polyakov’s expansion (D.2).164Appendix EIdentities Involving fABC and TA inSU(3)In this appendix, we prove various identities involving the SU(3) generators TA, and their structurefactors fABC . We label the 8 generators of SU(3), the so-called Gell-Mann matrices, according toT1 =0 1 01 0 00 0 0 T2 =0 −i 0i 0 00 0 0 T3 =1 0 00 −1 00 0 0 T4 =0 0 10 0 01 0 0 (E.1)T5 =0 0 −i0 0 0i 0 0 T6 =0 0 00 0 10 1 0 T7 =0 0 00 0 −i0 i 0 T8 = 1√31 0 00 1 00 0 −2These matrices obey the su(3) algebra[TA,TB] = 2i fABCTC (E.2)for completely antisymmetric fABC . Explicitly, the nontrivial values of fABC aref123 = 1 f345 = f376 = f147 = f156 = f257 =12f458 = f678 =√32. (E.3)Throughout we use the following index notation: Lowercase Latin letters index the off-diagonalGell-Mann matrices, lowercase Greek letters index the diagonal Gell-Mann matrices, uppercaseLatin letters index all eight matrices, and repeated indices are summed over. By construction, theGell-Mann matrices satisfytr[TATB] = 2δAB, (E.4)as well as the completeness relation[TA]i j[TA]kl = 2δilδk j − 23δi jδkl. (E.5)E.1 Structure Factor IdentitiesIn this section, we first prove two specific identities involving the fABC that we will require in ourRG calculations in Chapter 4. Then we define the following composite objects, given a unitarymatrix U:MA = U†TAU Nµ = ∂µUU†. (E.6)165E.1. Structure Factor IdentitiesWith these new objects, we prove six additional identities.E.1.1 Identity 1Here we provefcγa fcγb = δab. (E.7)We first writefcγa fcγb = fc3a fc3b + fc8a fc8b. (E.8)Then using the explicit values of fABC given in (E.3), we see the first term equalsfc3a fc3b = δa2δb2 + δa1δb1 +14δa5δb5 +14δa4δb4 +14δa6δb6 +14δa7δb7 (E.9)and the second term equals:fc8a fc8b =34(δa5δb5 + δa4δb4 + δa6δb6 + δa7δb7) . (E.10)Adding (E.9) and (E.10) completes the proof.E.1.2 Identity 2Here we provefabC fabD =δCD if C = c is off diagonal,3δCD if C = γ is diagonal. (E.11)First, we establishfABC fABD = 3δCD. (E.12)Using (E.4), we havetr[[TA,TB][TA,TC]]= tr[2TATBTATC − T 2ATCTB − T 2ATBTC]= −8 fABD fACD. (E.13)Then using (E.5) on the middle term, this becomestr[2TATBTATC − T 2ATCTB − T 2ATBTC]= 2[TA]i j[TB] jk[TA]kl[TC]li − [TA]i j[TA] jk[TCTB + TBTC]ki= −43tr[TBTC] − 6tr[TCTB + TBTC] + 23tr[TCTB + TBTC]= −12tr[TBTC]= −24δBC , (E.14)where in the last step we used (E.4). This proves (E.12). Now we expand it as follows:3δCD = fABC fABD = fabC fabD + 2 faβC faβD. (E.15)If C = γ is a diagonal index, the second term vanishes, and the second case of (E.11) follows. IfC is off-diagonal, the second term gives 2δCD according to (E.7), which proves the first case of(E.11).166E.1. Structure Factor IdentitiesE.1.3 Identity 3Here we prove ∑Atr[(∂µMA)2] = −6∑A(tr[NµTA])2. (E.16)Using (E.5), we have[MB]i j[MB]kl = [U†s ]in[TB]nm[Us]m j[U†s ]kp[TB]pq[Us]ql (E.17)= [U†s ]in[Us]ql[Us]m j[U†s ]kp(2δnqδpm − 23δnmδpq)= 2δilδk j − 23δi jδkl.Also, using (4.92), we havetr[MB∂µMA] = 2i fBACtr[∂µUsU†s TC]. (E.18)Now, we use these two expressions to simplify the following equation in two different ways. Onone hand, we have∑A,B(tr[MB∂µMA])2 =∑A,B[MB]i j[∂µMA] ji[MB]kl[∂µMA]lk (E.19)=∑A[∂µMA] ji[∂µMA]lk(2δilδk j − 23δi jδkl)= 2∑Atr[(∂µMA)2] − 23(tr[∂µMA])2= 2∑Atr[(∂µMA)2]since tr[∂µMa] = ∂µtr[Ta] = 0. On the other hand,∑A,B(tr[MB∂µMA])2 = −4∑A,B,C,DfBAC fBADtr[NµTC]tr[NµTD] = −12∑Ctr[(∂µUsU†s TC)2]. (E.20)This proves (E.16).E.1.4 Identity 4Here we prove−2∑c(tr[NµTc])2 =∑γtr[(∂µMγ)2]. (E.21)167E.1. Structure Factor IdentitiesWe repeat the proof of (E.16), but reduce the summation over A to a summation over diagonalGell-Mann matrices only. On the one hand,∑γ,B(tr[MB∂µMγ])2 =∑γ,B[MB]i j[∂µMγ] ji[MB]kl[∂µMγ]lk (E.22)=∑γ[∂µMγ] ji[∂µMγ]lk(2δilδk j − 23δi jδkl)(E.23)= 2∑Atr[(∂µMγ)2] − 23(tr[∂µMγ])2 = 2∑γtr[(∂µMγ)2]. (E.24)On the other hand,∑γ,B(tr[MB∂µMγ])2 = −4∑γ,B,C,DfBγc fBγdtr[NµTc]tr[NµTd] = −4∑c(tr[∂µUsU†s Tc])2 (E.25)where we used (E.7). This proves (E.21).Identity 5Here we provetr[NνTa (T3TbT8 − T8TbT3)] = 2itr[NνTa ( f3bcTcT8 − f8bcTcT3)]. (E.26)This follows directly from:T3TaT8 = [T3,Ta]T8 + TaT3T8 = 2i f3acTcT8 +1√3TaT3. (E.27)Identity 6Here we provef3bctr[Nν(TaTcT8 − T8TaTc)] + f8bctr[Nν(T3TaTc − TaTcT3)] (E.28)+ f3actr[Nν(T8TbTc − TbTcT8)] + f8actr[Nν(TbTcT3 − T3TbTc)]= 2itr[NνTdTc] ( f3bc fa8d + f8bc f3ad + f3ac f8bd + f8ac fb3d) .First, we rewrite the left hand side of (E.28) asf3bctr[Nν[TaTc,T8]]+ f8bctr[Nν[T3,TaTc]]+ f3actr[Nν[T8,TbTc]]+ f8actr[Nν[TbTc,T3]](E.29)which can further be rewritten asf3bctr[Nν (Ta[Tc,T8] + [Ta,T8]Tc)]+ f8bctr[Nν ([T3,Ta]Tc + Ta[T3,Tc])](E.30)+ f3actr[Nν ([T8,Tb]Tc + Tb[T8,Tc])]+ f8actr[Nν (Tb[Tc,T3] + [Tb,T3]Tc)].168E.1. Structure Factor IdentitiesReplacing commutators with structure constants gives2i[f3bctr[Nν (Ta fc8dTd + fa8dTdTc)] + f8bctr[Nν ( f3adTdTc + Ta f3cdTd)] (E.31)+ f3actr[Nν ( f8bdTdTc + Tb f8cdTd)] + f8actr[Nν (Tb fc3dTd + fb3dTdTc)]],which can be reorganized into2i[f3bc fc8dtr[NνTaTd] + f3bc fa8dtr[NνTdTc] + f8bc f3adtr[NνTdTc] + f8bc f3cdtr[NνTaTd] (E.32)+ f3ac f8bdtr[NνTdTc] + f3ac f8cdtr[NνTbTd] + f8ac fc3dtr[NνTbTd] + f8ac fb3dtr[NνTdTc]].Now, terms of the form f3bc fc8d vanish unless b = d. This means the first and fourth terms cancel.Likewise, the sixth and seventh terms cancel. What remains is2i[f3bc fa8dtr[NνTdTc] + f8bc f3adtr[NνTdTc] + f3ac f8bdtr[NνTdTc] + f8ac fb3dtr[NνTdTc]],which proves (E.28).Identity 7Here we proveµνtr[NνTdTc](fabEtr[NµTE] + fabγtr[NµTγ])(f8bc f3ad + f3ac f8bd)(E.33)= −µν√3(tr[NµT1]tr[NνT2] − tr[NµT4]tr[NνT5] + tr[NµT6]tr[NνT7]). (E.34)To begin, we check explicitly that the termfabE(f8bc f3ad + f3ac f8bd)(E.35)vanishes unless E is diagonal. In fact, it is either zero or ±√34 . We find the left hand side of (E.33)equals=√34µν(tr[NµT1]tr[Nν({T4,T7} − {T5,T6})] + tr[NµT2]tr[Nν({T4,T6} + {T5,T7})] (E.36)+ tr[NµT4]tr[Nν({T1,T7} + {T2,T6})] + tr[NµT5]tr[Nν({T2,T7} − {T1,T6})]+ tr[NµT6]tr[Nν({T2,T4} − {T1,T5})] + tr[NµT7]tr[Nν({T1,T4} + {T5,T2})]).Using{T5,T6} = −{T4,T7} = T2, {T4,T6} = {T5,T7} = T1, (E.37){T1,T7} = {T2,T6} = T5, {T1,T6} = −{T2,T7} = T4, (E.38){T1,T5} = −{T2,T4} = T7, {T1,T4} = {T5,T2} = T6, (E.39)169E.2. A Rewriting of the Sigma Model Lagrangianthe left hand side of (E.33) becomes=√32µν(− tr[NµT1]tr[NνT2] + tr[NµT2]tr[NνT1] + tr[NµT4]tr[NνT5] − tr[NµT5]tr[NνT4] (E.40)− tr[NµT6]tr[NνT7] + tr[NµT7]tr[NνT6]).The µν tensor allows us to combine these terms, proving (E.33).Identity 8Finally, we prove in this subsection thatµν(tr[NµT1]tr[NνT2] − tr[NµT4]tr[NνT5] + tr[NµT6]tr[NνT7])= −i√3tr[NµT8NνT3]. (E.41)First note thattr[NµT1] = [Nµ]i j[T1] ji = [Nµ]21 + [Nµ]12, (E.42)tr[NνT2] = [Nν]i j[T2] ji = −i[Nν]21 + i[Nν]12, (E.43)soµνtr[NµT1]tr[NνT2] = iµν[[Nµ]21 + [Nµ]12][−[Nν]21 + [Nν]12] (E.44)= iµν([Nµ]21[Nν]12 − [Nµ]21[Nν]21 + [Nµ]12[Nν]12 − [Nµ]12[Nν]21)(E.45)= 2iµν[Nµ]21[Nν]12. (E.46)Similar results hold for T4,T5,T6,T7. Therefore the left hand side of (E.41) is2iµν[[Nµ]21[Nν]12 − [Nµ]31[Nν]13 + [Nµ]32[Nν]23]= 2iµν3∑α=1tr[NµΛαNνΛα+1]. (E.47)Finally, using the results of Appendix E.2, we see that this equals−i√3µνtr[NµT8NνT3], (E.48)which proves (E.41).E.2 A Rewriting of the Sigma Model LagrangianIn this section, we obtain an equivalent form of the SU(3)/[U(1)]2 sigma model’s Lagrangian. Werewrite Λα in terms of T3,T8 and I, and expand each term occurring in (4.42) separately.170E.2. A Rewriting of the Sigma Model LagrangianReal part:We start withL = −3∑α=1tr[∂µUU†Λα∂µUU†Λα+1]. (E.49)This can be rewritten asL = −(1/2)3∑α=1tr[∂µ(U†ΛαU)∂µ(U†Λα+1U)] (E.50)since tr[∂µU∂µU†ΛαΛα+1] = 0. Note that6Λ1 =√3T8 + 2I + 3T3, (E.51)6Λ2 =√3T8 + 2I − 3T3, (E.52)3Λ3 = I −√3T8, (E.53)where I is the identity matrix. Substituting this into (E.50), we may drop the I terms since∂µ(U†U) = 0. Thus36L = − (1/2)tr[∂µ(U†(√3T8 + 3T3)U)∂µ(U†(√3T8 − 3T3)U)] (E.54)− tr[∂µ(U†(√3T8 − 3T3)U)∂µ(U†(−√3T8)U)] (E.55)− tr[∂µ(U†(−√3T8)U)∂µ(U†(√3T8 + 3T3)U)]. (E.56)Collecting terms, we findL = (1/8)tr[∂µ(U†TγU)∂µ(U†TγU)]. (E.57)This result doesn’t depend on how we choose the diagonal Gell-Mann matrices. Note that thediagonal matrix elements of the two diagonal Gell-Mann matrices together with√2/3I form acomplete orthogonal set of real vectors with norm√2. Thus:∑γ[Tγ]ii[Tγ] j j = 2δi j − 23 , (E.58)and we may also write:L = (1/8)∂µ([U†]i j[U] jk)∂µ([U†]k j[U] ji). (E.59)The 2/3 term can be dropped because it gives a term containing ∂µ(U†U). The same result is ob-tained with any basis of diagonal Gell-Mann matrices which obey the same completeness conditionand the same normalization.171E.2. A Rewriting of the Sigma Model LagrangianImaginary part:We start with−µν3∑α=1tr[∂µUU†Λα∂νUU†Λα+1] (E.60)and define Nµ := ∂µUU†. Then we have36tr[NµΛ1NνΛ2] = tr[Nµ(√3T8 + 2I + 3T3)Nν(√3T8 + 2I − 3T3)] (E.61)= 2√3tr[NµNνT8] + 3√3tr[NµT3NνT8] + 2√3tr[NµT8Nν] (E.62)+ 6tr[NµT3Nν] − 3√3tr[NµT8NνT3] − 6tr[NµNνT3] (E.63)= − 12tr[NµNνT3] − 6√3tr[NµT8NνT3] (E.64)and36tr[NµΛ2NνΛ3] = 2tr[Nµ(√3T8 + 2I − 3T3)Nν(I −√3T8)] (E.65)= 2√3tr[NµT8Nν] − 6tr[NµT3Nν] (E.66)− 4√3tr[NµNνT8] + 6√3tr[NµT3NνT8] (E.67)= 6tr[NµNνT3] − 6√3tr[NµNνT8] − 6√3tr[NµT8NνT3] (E.68)and36tr[NµΛ3NνΛ1] = 2tr[Nµ(I −√3T8)Nν(√3T8 + 2I + 3T3)] (E.69)= 6√3tr[NµNνT8] + 6tr[NµNνT3] − 6√3tr[NµT8NνT3]. (E.70)Taking the sum of all three, we find36µν3∑α=1tr[∂µUU†Λα∂νUU†Λα+1] = −18√3tr[∂µUU†T8∂µUU†T3]. (E.71)172Appendix FTwo Identities Involving SU(n)Structure FactorsIn this appendix, we prove two identities involving the SU(n) structure factors.F.1 Identity 1Here we provegche(µ)geface fbce = δabn−1∑i=1i,thi(µ)gig|t−i| (F.1)where t := |α − β|∣∣∣∣Iαβ3aand gx := gx mod n for x > n. Since a, c, e all correspond to off-diagonalgenerators, face will vanish unlessIa , Ic , Ie , Ia. (F.2)Moreover, for a and e fixed, there is a unique value of c such that face , 0. Calling this value c∗,we then havegche(µ)geface fbce =14δabgc∗he(µ)ge(no sum over e) (F.3)since fbc∗e = 0 unless a = b, and all purely off-diagonal structure factors in SU(n) have magnitude12 . Moreover, one can verify explicitly that for a ∈ Iαβ and e ∈ Iγδ, with Iαβ ∩ Iγδ = ∅,gc∗ = δαγg|β−δ| + δαδg|β−γ| + δβγg|α−δ| + δβδg|α−γ|. (F.4)Note that if {α, β}∩ = {γ, δ} = ∅, [Ta,Te] = 0. Therefore, writing ∑e = ∑γ<δ ∑e∈Iγδ , the left handside of (F.1) isgche(µ)geface fbce =14δabn∑γ<δhe(µ)ge∑e∈Iγδe<Iαβ[δαγg|β−δ| + δαδg|β−γ| + δβγg|α−δ| + δβδg|α−γ|](F.5)=δab2n∑γ<δIγδ,Iαβh|δ−γ|(µ)g|δ−γ|[δαγg|β−δ| + δαδg|β−γ| + δβγg|α−δ| + δβδg|α−γ|].173F.2. Identity 2We simplify each of these four terms. Let t := β−α > 0 (we assume without loss of generality thatα < β). Then:•n∑γ<δIγδ,Iαβh|δ−γ|(µ)g|δ−γ|δαγg|β−δ| =n∑δ=α+1δ,βh|δ−α|(µ)g|δ−α|g|β−δ| =n−α∑i=1i,thi(µ)gig|t−i| (F.6)•n∑γ<δIγδ,Iαβh|δ−γ|(µ)g|δ−γ|δαδg|β−γ| =α−1∑γ=1h|α−γ|(µ)g|α−γ|g|β−γ| =n−1∑i=n−α+1hi(µ)gi(µ)g|i−t| (F.7)•n∑γ<δIγδ,Iαβh|δ−γ|(µ)g|δ−γ|δβγg|α−δ| =n∑δ=β+1h|δ−β|(µ)g|δ−β|g|α−δ| =n−β∑i=1hi(µ)gig|t+i| (F.8)•n∑γ<δIγδ,Iαβh|δ−γ|(µ)g|δ−γ|δβδg|α−γ| =β−1∑γ=1γ,αh|β−γ|(µ)g|β−γ|g|α−γ| =n−1∑i=n−β+1i,n−thi(µ)gig|i+t| (F.9)where it is understood that gx := gx mod n for x > n. In (F.7) and (F.9), we used the fact that gi = gn−iand hi = hn−i in the last equations. Combining these results, we havegche(µ)geface fbce =12δabn−1∑i=1i,thi(µ)gig|t−i| +n−1∑i=1i,n−thi(µ)gig|t+i| . (F.10)Finally, replacing i → n − i in the second sum, we see that these two terms are in fact equal.Therefore, we arrive atgche(µ)geface fbce = δabn−1∑i=1i,thi(µ)gig|t−i|, (F.11)which completes the proof.F.2 Identity 2Here we provegc fbcE facE =12δabga + 12 ∑c gc . (F.12)174F.2. Identity 2We first writegc fbcE facE =n∑γ<δ∑c∈Iγδgc fbcE facE . (F.13)If c ∈ Ia, then fbcE facE vanishes unless b = a, and in this case equalsδab∑E[faa¯E]2= δab, (F.14)where a¯ is the unique index satisfying a¯ ∈ Ia with a¯ , a. Indeed, for a, a¯ ∈ Iαβ, we have[Ta, T¯a] = ±2i(Λα − Λβ). (F.15)Since Λα−Λβ generate the traceless diagonal Hermitian matrices, we may take them as the diagonalSU(n) generators. In this case, faa¯E = 0 unless E corresponds to (Λα−Λβ), where it equals 1. Now,if c < Ia, then facE will vanish except for a unique value e∗, with e∗ < Ia ∪ Ic. The term fbce forcesa = b, too. Since | fabc| = 12 for purely off-diagonal generators, we havegc fbcE facE = δabga +14δab∑c<Iagc. (F.16)Finally, noting that12ga +14∑c<Iagc =14∑cgc (F.17)completes the proof.175Appendix GDispersion Relations For SU(n) ChainsIn this appendix, we record the results of Section 6.3 and Section 6.4 in two tables.G.1 Dispersion RelationsThe following table lists the possible dispersion relations for any SU(n) representation satisfyingpα , pβ for all nonzero pα. The second column counts the minimum number of complex fields ϕαthat will have linear dispersion in the corresponding sigma model. The third column lists conditionsthat the pα must satisfy in order to achieve maximal linear dispersion (all n complex fields). Eachcondition is paired with a number in parentheses, (x), indicating how many fields become lineardispersing when that condition is satisfied. Additional comments are also included on the righthand side of the third column.Representation Min # Conditionsk = 1 n nonek = n − 1 2pα + pn−α+1 = p1 (2) n even; α = 2, . . . , n2pα + pn−α+1 = p1 (2)2p n+12= p1 (1) n odd; α = 2, . . . , n−12n = kλ 2λpα + pk+1−α = p1 + pk (2λ) k even; α = 2, . . . , k2pα + pk+1−α = p1 + pk (2λ)2p k+12= p1 + pk (λ) k odd; α = 2, . . . , k−12n = 2λ + 1, k = 2 λ + 1 λp1 = (λ + 1)p2 (λ)Continued on next page176G.1. Dispersion Relationsn = kλ + 12λpα + pk+2−α = p2 + pk (2λ)(λ + 1)(p2 + pk) = 2λp1 (λ + 1) k odd; α = 3, . . . ,k+12λ = even, k > 2pα + pk+2−α = p2 + pk (2λ)2p k+22= p2 + pk (λ)(λ + 1)(p2 + pk) = 2λp1 (λ + 1)k even; α = 3, . . . , k2n = kλ + 12λpα + pk+2−α = p2 + pk (2λ)(λ + 1)(p2 + pk) = 2λp1 (λ + 1) k odd; α = 3, . . . ,k+12λ = odd, k > 2pα + pk+2−α = p2 + pk (2λ)2p k+22= p2 + pk (λ)(λ + 1)(p2 + pk) = λp1 (λ + 1)k even; α = 3, . . . , k2n = kλ + cc = evenc , k − 1k > 12(λ + 1)pα + pc+1−α = p1 + pc (2(λ + 1))(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc) (2λ)2(λ + 1)p k+c+12= λ(p1 + pc) (λ)k oddα = 2, . . . ,c2β − c = 1, . . . , k − c − 12pα + pc+1−α = p1 + pc (2(λ + 1))(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc) (2λ)k even;α = 2, . . . ,c2β = c + 1, . . . , c +k − c2Continued on next page177G.2. Topological Anglesn = kλ + cc = oddc , k − 1k > 12(λ + 1)pα + pc+1−α = p1 + pc (2(λ + 1))2p c+12= p1 + pc (λ + 1)(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc) (2λ)2(λ + 1)p k+c+12= λ(p1 + pc) (λ)k evenα = 2, . . . ,c − 12β − c = 1, . . . , k − c − 12pα + pc+1−α = p1 + pc (2(λ + 1))2p c+12= p1 + pc (λ + 1)(λ + 1)(pβ + pk−β+c+1) = λ(p1 + pc) (2λ)k odd;α = 2, . . . ,c − 12β = c + 1, . . . , c +k − c2n = λk + (k − 1) 2(λ + 1)pα + pk−α = p1 + pk−1 (2(λ + 1))(λ + 1)pk = λ(p1 + pk−1) (λ) k odd;α = 2, . . . ,k−12λ evenpα + pk−α = p1 + pk−1 (2(λ + 1))(λ + 1)pk = λ(p1 + pk−1) (λ)2p k2= p1 + pk−1 (λ + 1)k even;α = 2, . . . , k−22n = λk + (k − 1) 2(λ + 1)pα + pk−α = p1 + pk−1 (2(λ + 1))(λ + 1)pk = λ(p1 + pk−1) (λ) k odd;α = 2, . . . ,k−12λ oddpα + pk−α = p1 + pk−1 (2(λ + 1))2(λ + 1)pk = λ(p1 + pk−1) (λ)2p k2= p1 + pk−1 (λ + 1)k even;α = 2, . . . , k−22G.2 Topological AnglesThe following table lists the possible topological angles for various representations of SU(n) chains.The index j runs from 1 to λ and the index t runs from 1 to λ + 1. These angles can often besimplified by using the freedom of shifting each angle by the same constant. For the relationshipsbetween angle and conditions on the pα, refer to Section 6.4 in the main text.178G.2. Topological AnglesRepresentation Topological Anglesk = 1 θα =2pip1n (α − 1) α = 1, 2, . . . , nk = n − 1 θα = pipα α = 1, 2, . . . , nk = nλ θα, j =pi(pα+pk+1−α)λ ( j − 1) + pipk+1−α α = 1, . . . , kn = 2λ + 1, k = 2θt =2pip1λ + 1(t − 1) + pip1(λ − 1)θ˜ j =2pip2λ( j − 1) + pip2λn = kλ + 1, k > 2λ = even, λ > 1θα, j =pi(pα + pk+2−α)λ( j − 1) + pipαθt =2pip1λ + 1(t − 1) + pip1 α = 2, . . . , kn = kλ + 1, k > 2λ = odd, λ > 1θα, j =pi(pα + pk+2−α)λ( j − 1) + pipα + pi(pα + pk+2−α)2 (λ − 1)θt =2pip1λ + 1(t − 1) α = 2, . . . , kn = kλ + c, c , 1, k − 1θα,t =pi(pα + pc−α+1)λ + 1(t − 1) + piλpc−α+1 + pi(λ − 1)pαθ˜β, j =pi(pβ + pk−β+c+1)λ( j − 1) + pi(λ + 1)pk−β+c+1 + piλpβα = 1, . . . , cβ = c + 1, . . . , kn = kλ + (k − 1)λ = oddθα,t =pi(pα + pk−α)λ + 1(t − 1) + pipk−αθ j =2pipkλ( j − 1) + pkpi α = 1, . . . , k − 1n = kλ + (k − 1)λ = evenθα,t =pi(pα + pk−α)λ + 1(t − 1) + pipk−α + pi(pα + pk−α)2 (λ − 2)θ j =2pipkλ( j − 1) α = 1, . . . , k − 1179
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Physics
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Generalization of the Haldane conjecture to SU(n) chains
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