Non-invertible translation from Lieb-Schultz-Mattis anomaly

TL;DR

This paper explores how lattice translation symmetry becomes non-invertible in systems with Lieb-Schultz-Mattis anomalies, revealing a deep connection between internal and crystalline symmetries in higher dimensions.

Contribution

It constructs explicit higher-dimensional lattice models showing translation symmetry becomes non-invertible after gauging internal symmetries, extending previous 1D results.

Findings

Translation becomes non-invertible after gauging internal symmetry.

Supports the result with anomaly-inflow and topological field theory.

Unifies higher-dimensional understanding of LSM anomalies.

Abstract

Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.