The modular S-matrix as order parameter for topological phase transitions

TL;DR

This paper introduces the modular S-matrix as a non-local order parameter to characterize topological phase transitions in discrete gauge theories, supported by lattice simulations of non-abelian anyon models.

Contribution

It demonstrates how the broken S-matrix reveals sector splitting, identification, and confinement, providing a comprehensive method to analyze phase structure in topological quantum field theories.

Findings

The modular S-matrix acts as an effective order parameter for topological phase transitions.

Monte Carlo simulations confirm the predicted phase structure and confinement phenomena.

Various phase transitions and their orders are identified in non-abelian gauge theories.

Abstract

We study topological phase transitions in discrete gauge theories in two spatial dimensions induced by the formation of a Bose condensate. We analyse a general class of euclidean lattice actions for these theories which contain one coupling constant for each conjugacy class of the gauge group. To probe the phase structure we use a complete set of open and closed anyonic string operators. The open strings allow one to determine the particle content of the condensate, whereas the closed strings enable us to determine the matrix elements of the modular $S$-matrix, also in the broken phase. From the measured broken $S$-matrix we may read off the sectors that split or get identified in the broken phase, as well as the sectors that are confined. In this sense the modular $S$-matrix can be employed as a matrix valued non-local order parameter from which the low-energy effective theories that occur in different regions of parameter space can be fully determined. To verify our predictions we studied a non-abelian anyon model based on the quaternion group $H=\bar{D_2}$ of order eight by Monte Carlo simulation. We probe part of the phase diagram for the pure gauge theory and find a variety of phases with magnetic condensates leading to various forms of (partial) confinement in complete agreement with the algebraic breaking analysis. Also the order of various transitions is established.