Introduction to the Keldysh formalism and applications to time-dependent density-functional theory

TL;DR

This paper introduces the Keldysh formalism and demonstrates its application to time-dependent density functional theory, emphasizing Green functions, conservation laws, and derivation of the Kohn-Sham potential.

Contribution

It provides a systematic introduction to the Keldysh formalism and applies it to derive key quantities in time-dependent density functional theory, including the Kohn-Sham potential.

Findings

Derivation of the time-dependent Kohn-Sham potential from an action functional.

Application of the Keldysh contour to response functions ensuring causality.

Use of the formalism to propagate the Kadanoff-Baym equations with conserving approximations.

Abstract

This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.