Lieb-Schultz-Mattis constraints from stratified anomalies of modulated symmetries

TL;DR

This paper develops a formalism for stratified symmetry operators and anomalies in quantum lattice systems, revealing how they impose Lieb-Schultz-Mattis constraints, especially in systems with modulated and crystalline symmetries, with applications to SPT phases.

Contribution

It introduces stratified symmetry operators and anomalies, providing a new framework to analyze LSM constraints in systems with complex, modulated symmetries and crystalline structures.

Findings

Stratified anomalies can lead to LSM constraints in modulated symmetry systems.

Explicit classification of LSM anomalies using homology groups.

Construction of a stabilizer code model demonstrating the theory.

Abstract

We introduce stratified symmetry operators and stratified anomalies in quantum lattice systems as generalizations of onsite symmetry operators and onsite projective representations. A stratified symmetry operator is a symmetry operator that factorizes into mutually independent subsystem symmetry operators; its stratified anomaly is defined as the collection of anomalies associated with these subsystem operators. We develop a cellular chain complex formalism for stratified anomalies of internal symmetries and show that, in the presence of crystalline symmetries, they give rise to Lieb-Schultz-Mattis (LSM) constraints. This includes LSM anomalies and SPT-LSM theorems. We apply this framework to modulated $G$ symmetries, which are symmetries whose total symmetry group is ${G_\mathrm{tot} = G \rtimes G_\mathrm{s}}$, with $G_\mathrm{s}$ the crystalline symmetry group. Notably, a nonzero stratified anomaly within a fundamental domain of $G_\mathrm{s}$ (e.g., a unit cell) does not always imply an LSM anomaly for modulated symmetries. Instead, the existence of an LSM anomaly also depends on how $G_\mathrm{s}$ acts on $G$. When $G_\mathrm{s}$ is the lattice translation group, we find an explicit criterion for when a stratified anomaly causes an LSM anomaly, and classify LSM anomalies using homology groups of $G_\mathrm{s}$-invariant cellular chains. We illustrate this through examples of exponential and dipole symmetries with stratified anomalies, both in ${(1+1)}$D and ${(2+1)}$D, and construct a stabilizer code model of a modulated SPT subject to an SPT-LSM theorem.