Memory truncated Kadanoff-Baym equations

TL;DR

This paper introduces a method to truncate the memory kernel in Kadanoff-Baym equations, significantly reducing computational costs and enabling long-time simulations of strongly correlated systems.

Contribution

It presents a systematic approach to incorporate memory kernel truncation into the KBE solver, improving long-time simulation capabilities for the Hubbard model.

Findings

Memory truncation reduces computational complexity from cubic to linear.

Longer simulation times are achievable, up to two orders of magnitude.

The method is effective in both weak and strong coupling regimes.

Abstract

The Keldysh formalism for nonequilibrium Green's functions is a powerful theoretical framework for the description of the electronic structure, spectroscopy, and dynamics of strongly correlated systems. However, the underlying Kadanoff-Baym equations (KBE) for the two-time Keldysh Green's functions involve a memory kernel which results in a high computational cost for long simulation times $t_\text{max}$, with a cubic scaling of the computation time with $t_\text{max}$. Truncation of the memory kernel can reduce the computational cost to linear scaling with $t_\text{max}$, but the required memory times will depend on the model and the diagrammatic approximation to the self-energy. We explain how a truncation of the memory kernel can be incorporated into the time-propagation algorithm to solve the KBE, and investigate the systematic truncation of the memory kernel for the Hubbard model in different parameter regimes, and for different diagrammatic approximations. The truncation is easier to control within dynamical mean-field solutions, where it is applied to a momentum-independent self-energy. Here, simulation times up to two orders of magnitude longer are accessible both in the weak and strong coupling regime, allowing for a study of long-time phenomena such as the crossover between pre-thermalization and thermalization dynamics.