Modified logarithmic Sobolev inequalities for Abelian quantum double models

TL;DR

This paper proves rapid mixing and modified logarithmic Sobolev inequalities for 2D Abelian quantum double models at all positive temperatures, advancing understanding of their thermalization properties.

Contribution

It establishes a Dobrushin-Shlosman type condition and links it to modified logarithmic Sobolev inequalities for these quantum models.

Findings

Rapid mixing for Davies Markov semigroups at any positive temperature

Verification of a strong martingale condition for local expectations

Dobrushin-Shlosman condition implies modified logarithmic Sobolev inequality

Abstract

We establish rapid mixing for Davies Markov semigroups associated with 2D Abelian quantum double models at any positive temperature. A condition of Dobrushin-Shlosman (DS) type holds at any temperature, and we show that the latter implies a modified logarithmic Sobolev inequality for the Davies Lindbladian. A key step in the argument is to verify a strong martingale condition for the local conditional expectations of the Davies semigroup in the regime of validity of the DS condition.