Master Equations Versus Keldysh Green's Functions for Correlated Quantum Systems Out of Equilibrium

Abstract The goal of these lecture notes is to illustrate connections between two
widely used, but often separately adopted approaches to deal with quantum systems
out of equilibrium, namely quantum master equations and nonequilibrium Green's
functions. For the paradigmatic case of the Anderson impurity model out of equilib-
rium we elaborate on these connections and map its description from one approach
to the other. At the end of this chapter, we will show how the "best of the two worlds"
can be combined to obtain a highly accurate solution of this model, which resolves
the nonequilibrium Kondo physics down to temperatures well below the Kondo scale.
As a training course, these lectures devote a large portion to an introduction to the
Lindblad quantum master equation based on standard treatments, as well as methods
to solve this equation. For nonequilibrium Green's functions, which are discussed
in the first part of the course, we only provide a summary of the most important
aspects necessary to address the topics of the present chapter. The relevant aspects
of these two topics are presented in a self-contained manner so that a background in
equilibrium many-body physics is sufficient to follow these notes.