Nonequilibrium self-energy functional theory

TL;DR

This paper extends self-energy functional theory to nonequilibrium dynamics of correlated lattice fermions, enabling causal, conserving, and non-perturbative approximations via a variational approach on the Keldysh contour.

Contribution

It reformulates SFT for real-time nonequilibrium systems using double-time Green's functions, allowing variational approximations with reference systems on the Keldysh contour.

Findings

Reduces to conventional SFT in equilibrium.

Ensures causality and conservation laws in approximations.

Enables numerical evaluation via time-propagation schemes.

Abstract

The self-energy functional theory (SFT) is generalized to describe the real-time dynamics of correlated lattice-fermion models far from thermal equilibrium. This is achieved by means of a reformulation of the original equilibrium theory in terms of double-time Green's functions on the Keldysh-Matsubara contour. We construct a functional which is stationary at the physical (nonequilibrium) self-energy and which yields the grand potential of the initial thermal state at the physical point. Non-perturbative approximations can be defined by specifying a reference system that serves to generate trial self-energies. These self-energies are varied by varying the reference system's one-particle parameters on the Keldysh-Matsubara contour. In case of thermal equilibrium, the new approach reduces to the conventional SFT. However, "unphysical" variations, i.e., variations that are different on the upper and the lower branch of the Keldysh contour, must be considered to fix the time-dependence of the optimal physical parameters via the variational principle. Functional derivatives in the nonequilibrium SFT Euler equation are carried out analytically to derive conditional equations for the variational parameters that are accessible to a numerical evaluation via a time-propagation scheme. Approximations constructed by means of the nonequilibrium SFT are shown to be inherently causal, internally consistent and to respect macroscopic conservation laws resulting from gauge symmetries of the Hamiltonian. This comprises the nonequilibrium dynamical mean-field theory but also dynamical-impurity and variational-cluster approximations that are specified by reference systems with a finite number of degrees of freedom. In this way, non-perturbative and consistent approximations can be set up, the numerical evaluation of which is accessible to an exact-diagonalization approach.