Scaling law for topologically ordered systems at finite temperature

TL;DR

This paper investigates how topologically ordered systems behave at finite temperatures by analyzing the topological entanglement entropy and its extensions, revealing scaling laws and the interplay between temperature and topological order.

Contribution

It extends the understanding of topological entanglement entropy at finite temperature to Abelian and non-Abelian models, providing a scaling framework and numerical validation.

Findings

Topological entanglement entropy decreases with temperature in Abelian models.

Scaling behavior of mutual information can be characterized analytically.

Numerical results for the $D(S_3)$ model support the theoretical predictions.

Abstract

Understanding the behaviour of topologically ordered lattice systems at finite temperature is a way of assessing their potential as fault-tolerant quantum memories. We compute the natural extension of the topological entanglement entropy for T > 0, namely the subleading correction $I_{\textrm{topo}}$ to the area law for mutual information. Its dependence on T can be written, for Abelian Kitaev models, in terms of information-theoretic functions and readily identifiable scaling behaviour, from which the interplay between volume, temperature, and topological order, can be read. These arguments are extended to non-Abelian quantum double models, and numerical results are given for the $D(S_3)$ model, showing qualitative agreement with the Abelian case.