Topological order in a 3D toric code at finite temperature

Understand

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.