# Generalization of the Haldane conjecture to SU(3) chains

## TL;DR

This paper extends the Haldane conjecture to SU(3) chains, predicting gapped or gapless phases based on the representation parameter, and confirms these predictions through field theory and Monte Carlo simulations.

## Contribution

It generalizes the Haldane conjecture to SU(3) chains and demonstrates the phase behavior using both theoretical mapping and numerical simulations.

## Key findings

- Models are gapped for p=3m
- Models are gapless for p=3m±1
- Monte Carlo confirms the phase predictions

## Abstract

We apply field theory methods to $\mbox{SU}(3)$ chains in the symmetric representation, with $p$ boxes in the Young tableau, mapping them into a flag manifold non-linear $\sigma$-model with a topological angle $\theta =2\pi p/3$. Generalizing the Haldane conjecture, we argue that the models are gapped for $p=3m$ but gapless for $p=3m\pm 1$ (for integer $m$), corresponding to a massless phase of the $\sigma$-model at $\theta =\pm 2\pi /3$. We confirm this with Monte Carlo calculations on the $\sigma$-model.