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Generalization of the Haldane conjecture to SU(n) chains

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.

Text

2021-04-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77741

Physics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/