Wigner-Weyl calculus in Keldysh technique

TL;DR

This paper integrates Wigner-Weyl calculus with the Keldysh technique to analyze non-equilibrium condensed matter systems, revealing how Hall conductivity relates to topological invariants and interaction effects.

Contribution

It demonstrates the unification of Wigner-Weyl formalism with Keldysh technique for inhomogeneous systems, applied to Hall conductivity and topological invariance.

Findings

Hall conductivity expressed via Wigner-transformed Green's functions

Topological invariance holds at zero temperature in equilibrium

Finite temperature and out-of-equilibrium conditions break topological invariance

Abstract

We discuss the non-equilibrium dynamics of condensed matter/quantum field systems in the framework of Keldysh technique. In order to deal with the inhomogeneous systems we use the Wigner-Weyl formalism. Unification of the mentioned two approaches is demonstrated on the example of Hall conductivity. We express Hall conductivity through the Wigner transformed two-point Green's functions. We demonstrate how this expression is reduced to the topological number in thermal equilibrium at zero temperature. At the same time both at finite temperature and out of equilibrium the topological invariance is lost. Moreover, Hall conductivity becomes sensitive to interaction corrections.