Lieb-Schultz-Mattis Anomalies and Anomaly Matching

TL;DR

This review explores Lieb-Schultz-Mattis anomalies as symmetry-based constraints in quantum many-body systems, covering their implications in various dimensions, disordered systems, and topological phases.

Contribution

It provides a comprehensive overview of LSM anomalies, including recent developments in disordered and fermionic systems, and their role in anomaly matching.

Findings

LSM anomalies impose constraints on quantum spin chains and higher-dimensional systems.

Disordered systems can preserve lattice symmetries on average, affecting anomaly considerations.

Symmetric short-range entangled states can be nontrivial SPT phases due to LSM anomalies.

Abstract

Lieb-Schultz-Mattis (LSM) anomalies are powerful symmetry-based constraints on the correlation, entanglement and dynamics of quantum many-body systems. In this review, we discuss various LSM anomalies and anomaly matching. We start with a pedagogical introduction to these subjects in quantum spin chains, and then generalize the discussion to higher dimensions and other systems. Besides covering the topics related to the standard LSM anomalies, we also review LSM anomalies in disordered systems where the lattice symmetries are only preserved on average, fermionic systems, and systems where the symmetric short-range entangled states are possible but must be nontrivial symmetry-protected topological phases.