Pseudogap Anderson impurity model out of equilibrium: A master equation tensor network approach

We study equilibrium and nonequilibrium properties of the single-impurity Anderson model with a power-law pseudogap in the density of states. In equilibrium, the model is known to display a quantum phase transition from a generalized Kondo to a local moment phase. In the present work, we focus on the extension of these phases beyond equilibrium, i.e. under the influence of a bias voltage. Within the auxiliary master equation approach combined with a scheme based on matrix product states (MPS) we are able to directly address the current-carrying steady state. Starting with the equilibrium situation, we first corroborate our results by comparing with a direct numerical evaluation of ground state spectral properties of the system by MPS. Here, a scheme to locate the phase boundary by extrapolating the power-law exponent of the self energy produces a very good agreement with previous results obtained by the numerical renormalization group. Our nonequilibrium study as a function of the applied bias voltage is then carried out for two points on either side of the phase boundary. In the Kondo regime the resonance in the spectral function is splitted as a function of the increasing bias voltage. The local moment regime, instead, displays a dip in the spectrum near the position of the chemical potentials. Similar features are observed in the corresponding self energies. The Kondo split peaks approximately obey a power-law behavior as a function of frequency, whose exponents depend only slightly on voltage. Finally, the differential conductance in the Kondo regime shows a peculiar maximum at finite voltages, whose height, however, is below the accuracy level.

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

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.