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

TL;DR

This work generalizes Haldane's conjecture to SU($n$) spin chains with rank-$p$ symmetric representations, predicting gapless behavior when $\gcd(n,p)=1$ and gapped phases with ground-state degeneracies otherwise. A non-Lorentz-invariant flag-manifold sigma model arises as the low-energy description, with an emergent Lorentz invariance in the IR via velocity-flow homogenization, and a topological term with angle $\theta=2\pi p/n$ guiding anomaly-matching arguments. The analysis combines LSMA and AKLT-type exact results, flavour-wave theory, and a strong-coupling partition-function approach to relate infrared physics to the integer $q=\gcd(n,p)$ and the SU($n$)$_1$ WZW fixed point in the coprime case. The results yield precise predictions for ground-state degeneracies and phase transitions across $n$ and $p$, including explicit phase diagrams and degenerate-transition points, with additional insights from explicit SU(3)–SU(6) examples. The framework connects symmetry, topology, and RG flow to deliver a cohesive generalization of Haldane physics to SU($n$) chains and motivates future numerical verification.

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.

Figures (12)

• Figure 1: Young tableau for irreps of SU($n$). Left: the rank-$p$ symmetric irrep considered in this paper, and also in [LajkoNuclPhys2017] for SU(3). Right: the $(p,p)$ self-conjugate irrep, considered in [Wamer2019] for SU(3).
• Figure 2: Pictorial representation of the nearest (blue), next-nearest (red), and next-next-nearest (green) neighbour interactions occurring in (\ref{['1:2']}), for the case $n=4$.
• Figure 3: A valence bond construction for the predicted two-fold degenerate ground state of SU(4) with $p=2$. Each node represents a fundamental $p=1$ irrep of SU(4). Each link represents an antisymmetrization between two nodes, and the antisymmetrization of four neighbouring nodes results in a singlet.
• Figure 4: Valence bond constructions for SU(6). The left subfigure corresponds to $p=3$, and has a 2-fold degenerate ground state. The right subfigure corresponds to $p=2$, and has a 3-fold degenerate ground state. Singlets are constructed out of 6 nodes, each of which represents a fundamental irrep in SU(6).
• Figure 5: AKLT constructions in SU(3). Left: When $p\not=n$, multiple valence bond solids can be formed. The ground state is not translationally invariant and degenerate. Right: When $p=n$, a unique, translationally invariant ground state can be constructed, by projecting on to the symmetric-$p$ representation at each site.