Lehmann representation of the nonequilibrium self-energy

TL;DR

This paper establishes a unique Lehmann representation for the non-equilibrium self-energy of lattice fermion models and introduces a numerically efficient scheme to compute it, enabling longer time simulations and better understanding of non-equilibrium dynamics.

Contribution

It provides a novel Lehmann representation for the non-equilibrium self-energy and an explicit, practical scheme to construct it from Green's functions, improving long-time numerical simulations.

Findings

Efficient scheme extends maximum propagation time in Dyson's equation solutions.

Demonstrates applicability to Hubbard model quench dynamics on a 10x10 lattice.

Shows moderate violation of conservation laws and captures prethermalization phenomena.

Abstract

It is shown that the non-equilibrium self-energy of an interacting lattice-fermion model has a unique Lehmann representation. Based on the construction of a suitable non-interacting effective medium, we provide an explicit and numerically practicable scheme to construct the Lehmann representation for the self-energy, given the Lehmann representation of the single-particle non-equilibrium Green's function. This is of particular importance for an efficient numerical solution of Dyson's equation in the context of approximations where the self-energy is obtained from a reference system with a small Hilbert space. As compared to conventional techniques to solve Dyson's equation on the Keldysh contour, the effective-medium approach allows to reach a maximum propagation time which can be several orders of magnitude longer. This is demonstrated explicitly by choosing the non-equilibrium cluster-perturbation theory as a simple approach to study the long-time dynamics of an inhomogeneous initial state after a quantum quench in the Hubbard model on a 10 x 10 square lattice. We demonstrate that the violation of conservation laws is moderate for weak Hubbard interaction and that the cluster approach is able to describe prethermalization physics.