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

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 equilibrium 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.