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

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 NB 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 NB. 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 manybody solution. In particular, we investigate the effects of the geometry of the auxiliary system on the accuracy of the mapping for given NB. 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.

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

“…We here present a short sketch of the AMEA used in this paper. For more details, we refer to [21,[24][25][26]. The idea is to map the physical system described by (1) to a finite and open auxiliary system that has almost the same hybridization at the impurity as the original one (12) and thereby maintains the impurity physics, which we are interested in.…”

“…The auxiliary system consists of a small number of N B bath sites connected to Markovian environments and its dynamics is governed by a Lindblad master equation. The parameters in this equation are determined to achieve a corresponding auxiliary hybridization function aux w D ( )such that aux w w D » D ( ) ( )as accurately as possible, see [25]. The physical hybridization function D is calculated from the given lead DOS, equation (5), using equations (12)- (14) and the Kramers-Kronig relation that links the real and imaginary part of a Green's function.…”

“…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].…”

Thermoelectric properties of a strongly correlated layer

“…It will be determined self-consistently by solving the SIAM problem with the same local parameters U and ε c as the original model. More specifically we employ the auxiliary master equation approach (AMEA) [33][34][35][36] to determine the self-energy Σ(ω) in the equilibrium and nonequilibrium case. The key point of AMEA is to map the infinite SIAM problem to an auxiliary one with a finite number of bath sites N b and two Markovian environments (sink and reservoir), which are crucial to achieve a steady state in a finite system.…”

“…36,59 In practice N b = 4, 6 is sufficient to obtain reliable results for the current. The self consistence cycle proceeds as follows: Starting out with an initial guess for the self energy, we determine the physical hybridization function in Eq.…”