Topological order in a three-dimensional toric code at finite temperature

Abstract

We study topological order in a toric code in three spatial dimensions or a $3+1\text{D}$ ${\mathbb{Z}}_{2}$ 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 ${T}_{c}$, dropping to zero above ${T}_{c}$. 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.