String order parameters for symmetry fractionalization in an enriched toric code

TL;DR

This paper demonstrates that string order parameters effectively characterize symmetry fractionalization in a symmetry-enriched toric code model, showing their robustness and revealing how anyon condensation can induce symmetry breaking.

Contribution

It extends the use of string order parameters to characterize symmetry fractionalization beyond tensor network states and analyzes the impact of anyon condensation on symmetry and phase transitions.

Findings

String order parameters reliably detect symmetry fractionalization.

Symmetry breaking occurs when an anyon fractionalizing a symmetry condenses.

Phase diagram of the toric code is altered by anyon condensation and magnetic fields.

Abstract

We study a simple model of symmetry-enriched topological order obtained by decorating a toric code model with lower-dimensional symmetry-protected topological states. We show that the symmetry fractionalization in this model can be characterized by string order parameters, and that these signatures are robust under the effects of external fields and interactions, up to the phase transition point. This extends the recent proposal of [New Journal of Physics 21, 113016 (2019)] beyond the setting of fixed-point tensor network states, and solidifies string order parameters as a useful tool to characterize and detect symmetry fractionalization. In addition to this, we observe how the condensation of an anyon that fractionalizes a symmetry forces that symmetry to spontaneously break, and we give a proof of this in the framework of projected entangled pair states. This phenomenon leads to a notable change in the phase diagram of the toric code in parallel magnetic fields.