Generalization of the Haldane conjecture to SU($n$) chains

TL;DR

This paper generalizes the Haldane conjecture to SU(n) chains, predicting gapless or gapped ground states based on the representation parameters, using sigma models, renormalization group analysis, and anomaly matching.

Contribution

It extends the Haldane conjecture to SU(n) chains in symmetric representations, providing a unified framework for predicting ground state properties.

Findings

Gapless excitations for coprime n and p.

Finite energy gap when p is multiple of n.

Ground state degeneracy related to gcd of n and p.

Abstract

Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-$p$ symmetric ones with $p$ not a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU($n$) chains in the symmetric representation. For a rank-$p$ symmetric representation with $n$ and $p$ coprime, we predict gapless excitations above the ground state. If $p$ is a multiple of $n$, we predict a unique ground state with a finite energy gap. Finally, if $p$ and $n$ have a greatest common divisor $1<q<n$, we predict a ground state degeneracy of $n/q$, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU($n$) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Shultz-Mattis-Affleck theorem, and extend it to SU($n$) models with longer range interactions.