Finite-temperature topological entanglement entropy for CSS codes

Understand

We consider topological entanglement entropy (TEE) at finite temperature for CSS codes, which include some ordinary topological-ordered systems such as the toric code and some fracton models such as the Haah's code and the X-cube model.

• We find, under the assumption that there is no extended critical phase, the finite-temperature TEE is a piecewise constant function of the temperature, with possible discontinuities only at phase transitions.
• We then consider phase transitions of CSS codes.
• We claim that there must exist a phase transition at zero temperature for any CSS codes in 2D and 3D with topological order.