# Self-conjugate representation SU(3) chains

## TL;DR

This paper investigates SU(3) chains with self-conjugate representations, demonstrating they are always gapped and exhibit spontaneous parity symmetry breaking for odd p, extending previous results on symmetric irreps.

## Contribution

It extends the understanding of SU(3) chains by analyzing self-conjugate irreps and showing they are always gapped with parity symmetry breaking for odd p.

## Key findings

- Self-conjugate SU(3) chains are always gapped.
- Parity symmetry is spontaneously broken for odd p.
- The results generalize previous findings on symmetric irreps.

## Abstract

It was recently argued that SU(3) chains in the $p$-box symmetric irreducible representation (irrep) exhibit a "Haldane gap" when $p$ is a multiple of 3 and are otherwise gapless [Nucl. Phys. B $\boldsymbol{924}$, 508 (2017)]. We extend this argument to the self-conjugate irreps of SU(3) with $p$ columns of length 2 and $p$ columns of length 1 in the Young tableau ($p=1$ corresponding to the adjoint irrep), arguing that they are always gapped but have spontaneously broken parity symmetry for $p$ odd but not even.