Optimized auxiliary representation of a non-Markovian environment by a Lindblad equation

TL;DR

This paper introduces a scheme to accurately map non-Markovian quantum impurity problems onto small auxiliary open quantum systems using optimized Lindblad equations, with detailed analysis of geometric configurations affecting accuracy.

Contribution

The paper provides a detailed investigation of the mapping procedure, focusing on how auxiliary system geometry influences the accuracy of the Lindblad-based representation.

Findings

Two-chain geometry significantly improves accuracy over single-chain or star geometries.

Long-ranged and complex Lindblad parameters enhance the mapping precision.

The approach becomes exponentially exact as the number of auxiliary bath sites increases.

Abstract

We present a general scheme to map correlated nonequilibrium quantum impurity problems onto an auxiliary open quantum system of small size. The infinite fermionic reservoirs of the original system are thereby replaced by a small number $N_B$ of noninteracting auxiliary bath sites whose dynamics is described by a Lindblad equation. Due to the presence of the intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. With the help of an optimization scheme for the auxiliary Lindblad parameters, an accurate mapping is achieved, which becomes exponentially exact upon increasing $N_B$. The basic idea for this scheme was presented previously in the context of nonequilibrium dynamical mean field theory. In successive works on improved manybody solution strategies for the auxiliary Lindblad equation, such as Lanczos exact diagonalization or matrix product states, we applied the approach to study the nonequilibrium Kondo regime. In the present paper, we address in detail the mapping procedure itself, rather than the many-body solution. In particular, we investigate the effects of the geometry of the auxiliary system on the accuracy of the mapping for given $N_B$. Specifically, we present a detailed convergence study for five different geometries which, besides being of practical utility, reveals important insights into the underlying mechanisms of the mapping. For setups with onsite or nearest-neighbor Lindblad parameters we find that a representation adopting two separate bath chains is by far more accurate with respect to other choices based on a single chain or a commonly used star geometry. A significant improvement is obtained by allowing for long-ranged and complex Lindblad parameters. These results can be of great value when studying Lindblad-type approaches to correlated systems.