# Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems

## TL;DR

This paper develops a topological framework to understand Lieb-Schultz-Mattis theorems in quantum spin systems, linking symmetry, topology, and ground state properties to identify conditions for gapless or exotic gapped phases.

## Contribution

It introduces a general criterion for LSM-type theorems applicable to systems with complex symmetries and spin-orbit coupling, connecting to topological phase classification and anomalies.

## Key findings

- Provides a unified topological criterion for LSM theorems
- Connects LSM conditions to topological phase classification
- Enables computation of anomalies related to LSM theorems

## Abstract

The Lieb-Schultz-Mattis (LSM) theorem states that a spin system with translation and spin rotation symmetry and half-integer spin per unit cell does not admit a gapped symmetric ground state lacking fractionalized excitations. That is, the ground state must be gapless, spontaneously break a symmetry, or be a gapped spin liquid. Thus, such systems are natural spin-liquid candidates if no ordering is found. In this work, we give a much more general criterion that determines when an LSM-type theorem holds in a spin system. For example, we consider quantum magnets with arbitrary space group symmetry and/or spin-orbit coupling. Our criterion is intimately connected to recent work on the general classification of topological phases with spatial symmetries and also allows for the computation of an "anomaly" associated with the existence of an LSM theorem. Moreover, our framework is also general enough to encompass recent works on "SPT-LSM" theorems where the system admits a gapped symmetric ground state without fractionalized excitations, but such a ground state must still be non-trivial in the sense of symmetry-protected topological (SPT) phases.