Anomaly-free symmetries with obstructions to gauging and onsiteability

TL;DR

This paper constructs examples of two-dimensional lattice symmetries that cannot be gauged or localized but are still anomaly-free, challenging the common association between non-gaugability and anomalies.

Contribution

It introduces a new class of anomaly-free symmetries that defy the usual link between gauging obstructions and anomalies, characterized by an index in cohomology.

Findings

Constructed explicit counterexamples of non-gaugeable, onsite symmetries

Demonstrated these symmetries admit symmetric, gapped Hamiltonians

Linked symmetries to an index in cohomology involving quantum cellular automata

Abstract

We present counterexamples to the lore that symmetries that cannot be gauged or made on-site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index $[\omega]\in H^2(G,\mathbb{Q}_+)$, where $\mathbb{Q}_+$ is the multiplicative group of rational numbers labeling one-dimensional quantum cellular automata.