Lindblad-driven discretized leads for nonequilibrium steady-state transport in quantum impurity models: Recovering the continuum limit

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.

Nonequilibrium Kondo effect in a magnetic field: auxiliary master equation approach

“…The auxiliary hybridization function aux D can be calculated for a general set of bath parameters by solving a noninteracting Lindblad problem, see, e.g. [24][25][26][27][28]. The determination of these parameters and thus the mapping to the physical system is carried out with a parallel tempering algorithm [25].…”

Optimized auxiliary representation of non-Markovian impurity problems by a Lindblad equation

“…One possible way to eliminate the dependence of the Lindblad couplings on the parameters of the central region is to use an intermediate auxiliary buffer zone (mesoreservoir) between the Lindblad couplings and the central region (see, e.g. [68][69][70][71]). The buffer zone consists of isolated discrete sites (levels), each one coupled to a Markovian environment described by Lindblad operators with the same T and μ as given in equations (2) and (3).…”

“…An analytic expression for the noninteracting steady-state retarded and Keldysh auxiliary Green's functions was derived in [86]. An alternative derivation, which does not rely on super-fermions, is given in [71]. For the retarded component, we get (see footnote 7)…”

Markovian treatment of non-Markovian dynamics of open Fermionic systems

“…While the mappings appear to be very useful and accurate, in most cases only semi-quantitative arguments to justify the mapping were presented with main supporting evidence being benchmarking versus numerically exact computational techniques. In particular, a justification for the mapping was put forward in [23][24][25] based upon the singular coupling derivation of the Lindblad equation.…”