A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases

TL;DR

This paper introduces a mathematical framework called the symmetry breaking long exact sequence (SBLES) to classify defects in symmetry breaking phases, linking anomalies, topological protection, and higher Berry phases.

Contribution

It develops the SBLES to classify defect anomalies, relate them to symmetry anomalies, and analyze obstructions in symmetry breaking phases, advancing the theory of higher Berry phases.

Findings

Classified symmetry breaking phases via defect anomalies.

Derived an obstruction criterion for local defect phases.

Connected higher Berry phases with bulk-boundary correspondence.

Abstract

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is studied in a companion paper. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases.