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

TL;DR

This work introduces stratified symmetry operators and stratified anomalies as a real-space generalization of onsite symmetry representations, enabling a cellular-chain description of how subsystem anomalies inflow to higher dimensions. It develops a framework that couples stratified anomalies with crystalline and modulated symmetries, showing how LSM constraints arise from anomaly inflows and how SPT-LSM theorems can occur when stratified anomalies are compatible with SPTs. The authors derive explicit criteria, via a translation-invariant cellular chain complex and the Atiyah–Hirzebruch spectral sequence, for when stratified anomalies in modulated symmetries yield LSM constraints, and they classify LSM anomalies using Td-invariant homology groups. They illustrate the framework with concrete (1+1)D and (2+1)D examples (dipole and exponential symmetries) and construct a stabilizer-code model of a modulated SPT subject to an SPT-LSM theorem, highlighting how stratified anomalies can obstruct trivial SPT phases even in translation-symmetric settings. Overall, the work provides a versatile, topological toolkit for understanding how spatial modulation of internal symmetries interacts with lattice translations to constrain quantum phases of matter.

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.