Topological order in a 3D toric code at finite temperature

TL;DR

This paper analyzes the stability of topological order in a 3D toric code at finite temperature, revealing a phase transition characterized by a partial loss of topological entropy and the role of fluctuating extended structures.

Contribution

It provides an exact calculation of topological entropy at finite temperature in a 3D toric code and interprets the phase transition in terms of fluctuating strings and membranes.

Findings

Topological entropy drops to half its zero-temperature value at any nonzero temperature.

A critical temperature Tc exists where topological order vanishes.

Topologically ordered phases can persist at finite temperatures under certain conditions.

Abstract

We study topological order in a toric code in three spatial dimensions, or a 3+1D Z_2 gauge theory, at finite temperature. We compute exactly the topological entropy of the system, and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature Tc, dropping to zero above Tc. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of fluctuating strings and membranes, and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature, and its quantum and classical aspects.