Signatures of broken parity and time-reversal symmetry in generalized string-net models

TL;DR

This paper extends string-net models to include broken parity and time-reversal symmetries, introducing new graphical tools and indicators to detect such symmetry breaking in topological phases.

Contribution

It generalizes string-net models with an extra degree of freedom and non-trivalent graphs, enabling detection of symmetry breaking via higher Frobenius-Schur indicators.

Findings

Higher Frobenius-Schur indicators effectively detect symmetry breaking.

Generalized string-net models incorporate symmetry-breaking features.

Graphical calculus is extended to non-trivalent graphs.

Abstract

We study indicators of broken time-reversal and parity symmetries in gapped topological phases of matter. We focus on phases realized by Levin-Wen string-net models, and generalize the string-net model to describe phases which break parity and time-reversal symmetries. We do this by introducing an extra degree of freedom into the string-net graphical calculus, which takes the form of a branch cut located at each vertex of the underlying string-net lattice. We also work with string-net graphs defined on arbitrary (non-trivalent) graphs, which reveals otherwise hidden information about certain configurations of anyons in the string-net graph. Most significantly, we show that objects known as higher Frobenius-Schur indicators can provide several efficient ways to detect whether or not a given topological phase breaks parity or time-reversal symmetry.