Generalization of the Haldane conjecture to SU(n) chains
department or school: Science, Faculty of
department or school: Physics and Astronomy, Department of
data provider: DSpace
degree campus: UBCV
creator: Wamer, Kyle Patrick
issued: 2021-04-09T22:21:40Z
issued: 2021
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: 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 mode…
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/77741
note: 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 asymptoticall…
has type: Thesis/Dissertation
date issued: 2021-05
is shown at: 10.14288/1.0396649
language: eng
degree discipline: Physics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/77741