Classification of Classical Spin Liquids: Topological Quantum Chemistry and Crystalline Symmetry

TL;DR

This paper extends the classification of classical spin liquids by incorporating crystalline symmetry, using a mathematical framework to analyze band representations and symmetry protection of gapless features.

Contribution

It introduces a novel framework for analyzing the role of crystalline symmetry in classifying CSLs, including the protection of gapless points and finer topological distinctions.

Findings

Framework for computing band representations of flat bands in CSLs

Identification of symmetry-protected gapless points in algebraic CSLs

Construction of a symmetry-protected pinch-line algebraic CSL

Abstract

Frustrated magnetic systems can host highly interesting phases known as classical spin liquids (CSLs), which feature {extensive} ground state degeneracy and lack long-range magnetic order. Recently, Yan and Benton et al. proposed a classification scheme of CSLs in the large-$\mathcal{N}$ (soft spin) limit [arXiv.2305.00155], [arXiv:2305.19189]. This scheme classifies CSLs into two categories: the algebraic CSLs and the fragile topological CSLs, each with their own correlation properties, low energy effective description, and finer classification frameworks. In this work, we further develop the classification scheme by considering the role of crystalline symmetry. We present a mathematical framework for computing the band representation of the flat bands in the spectrum of these CSLs, which extends beyond the conventional representation analysis. It allows one to determine whether the algebraic CSLs, which features gapless points on their bottom flat bands, are protected by symmetry or not. It also provides more information on the finer classifications of algebraic and fragile topological CSLs. We demonstrate this framework via concrete examples and showcase its power by constructing a pinch-line algebraic CSL protected by symmetry.