Lindblad-Driven Discretized Leads for Non-Equilibrium Steady-State Transport in Quantum Impurity Models: Recovering the Continuum Limit

TL;DR

This paper introduces Lindblad-driven discretized leads (LDDL) as a method to accurately model non-equilibrium steady states in quantum impurity models, bridging the gap between finite lead discretization and true open quantum systems, with analytical and numerical insights.

Contribution

The paper presents an analytical approach for quadratic models with Lindbladian dynamics and discusses how to recover the continuum limit of the LDDL approach for thermal reservoirs.

Findings

Analytical correlation functions for quadratic Lindbladian models.

Conditions for LDDL to accurately represent thermal reservoirs.

A reformulation of the Lindblad equation for local chain mapping.

Abstract

The description of interacting quantum impurity models in steady-state nonequilibrium is an open challenge for computational many-particle methods: the numerical requirement of using a finite number of lead levels and the physical requirement of describing a truly open quantum system are seemingly incompatible. One possibility to bridge this gap is the use of Lindblad-driven discretized leads (LDDL): one couples auxiliary continuous reservoirs to the discretized lead levels and represents these additional reservoirs by Lindblad terms in the Liouville equation. For quadratic models governed by Lindbladian dynamics, we present an elementary approach for obtaining correlation functions analytically. In a second part, we use this approach to explicitly discuss the conditions under which the continuum limit of the LDDL approach recovers the correct representation of thermal reservoirs. As an analytically solvable example, the nonequilibrium resonant level model is studied in greater detail. Lastly, we present ideas towards a numerical evaluation of the suggested Lindblad equation for interacting impurities based on matrix product states. In particular, we present a reformulation of the Lindblad equation, which has the useful property that the leads can be mapped onto a chain where both the Hamiltonian dynamics and the Lindblad driving are local at the same time. Moreover, we discuss the possibility to combine the Lindblad approach with a logarithmic discretization needed for the exploration of exponentially small energy scales.