Gibbs variational formula for thermal equilibrium states in terms of quantum relative entropy density

Abstract

We prove the Gibbs variational formula in terms of quantum relative entropy density that characterizes translation invariant thermal equilibrium states in quantum lattice systems. It is a natural quantum extension of the similar statement established by Föllmer for classical systems. In this work we advocate the viewpoint: Avoid the unique-phase assumption if it is not essential.

Eq (31) (32) are incorrect. Hence the given statements are not proved