"Symmetry-from-Anomaly" in Condensed Matter related Constructions

TL;DR

This paper explores how noninvertible axial symmetries, derived from anomalies, manifest in condensed matter models across 1d and 3d systems, linking field theory concepts with physical observables and emphasizing the role of topological order.

Contribution

It introduces a new perspective on axial symmetry in condensed matter systems, highlighting the necessity of topological order and distinguishing it from recent noninvertible symmetry proposals.

Findings

Established connection between axial charge and physical quantities in models.

Demonstrated the necessity of topological order for defining axial symmetry.

Clarified differences between the proposed axial symmetry and recent noninvertible symmetries.

Abstract

The noninvertible axial symmetry constructed from the ABJ-anomaly has attracted enormous interest. We discuss the mechanism of "symmetry-from-anomaly" in condensed matter-related models in both 1d and 3d spaces (which correspond to (1+1)d and (3+1)d space-time). Within the models discussed here, we establish the connection between field theory quantities such as different versions of the axial charge, and quantities with simple physical meanings in our systems. In our models and likely a class of related constructions, the existence of a topological order is necessary for the purpose of properly defining the axial symmetry. But the proper axial symmetry we define, though requires a topological order, is different from the noninvertible axial symmetry discussed in recent proposals.