Flag manifold sigma models from SU($n$) chains

TL;DR

This paper classifies SU(n) spin chains that map to flag manifold sigma models, analyzing their topological properties, dispersion relations, and symmetry features, leading to predictions about their ground state gaps or gaplessness based on representation theory.

Contribution

It identifies which SU(n) representations lead to flag manifold sigma models with specific topological and symmetry properties, extending Haldane's conjecture to new classes of models.

Findings

Certain SU(n) chains map to flag manifold sigma models with calculable topological angles.

The dispersion relations of fields depend on the representation, with some models having purely linear dispersion.

Predictions about gapless or gapped ground states based on representation properties and symmetry considerations.

Abstract

One dimensional SU($n$) chains with the same irreducible representation $\mathcal{R}$ at each site are considered. We determine which $\mathcal{R}$ admit low-energy mappings to a $\text{SU}(n)/[\text{U}(1)]^{n-1}$ flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each $\mathcal{R}$, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreducible representations that also possess a $\mathbb{Z}_n$ symmetry that acts transitively on the $\text{SU}(n)/[\text{U}(1)]^{n-1}$ fields. Such SU($n$) chains have an 't Hooft anomaly in certain cases, allowing for a generalization of Haldane's conjecture to these novel representations. In particular, for even $n$ and for representations whose Young tableaux have two rows, of lengths $p_1$ and $p_2$ satisfying $p_1\not=p_2$, we predict a gapless ground state when $p_1+p_2$ is coprime with $n$. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if $p_1+p_2$ is not a multiple of $n$.