The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group (tDMRG). This yields quantitatively reliable results for the current (with errors 3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model. Introduction.-A major open problem in the theoretical study of nanostructures such as quantum dots or nanowires is the reliable computation of the nonlinear conductance under conditions of nonequilibrium steadystate (NESS) transport. These are open quantum systems featuring strong local interactions, typically described by quantum impurity models such as the interacting resonant level model (IRLM), the Kondo model (KM) or the single-impurity Anderson model (SIAM). Much work has been devoted to studying the NESS properties of such models using a variety of methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], leading to a fairly good qualitative understanding of their behavior. The interplay of strong correlations, NESS driving and dissipative effects leads to a rich and complex phenomenology. In particular, for the KM and SIAM, the nonlinear conductance exhibits a striking zero-bias peak, the so-called Kondo peak, characterized by a small energy scale, the Kondo temperature T K , that weakens with increasing temperature and splits with increasing magnetic field, in qualitative agreement with experiments [16][17][18][19][20][21][22]. However, a full, quantitative description of the NESS behavior of such models under generic conditions has so far been unfeasible: none of the currently available approaches meet the threefold challenge of (i) treating interactions essentially exactly, (ii) resolving very small energy scales, and (iii) incorporating NESS conditions.

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.

We study the single-impurity Anderson model out of equilibrium under the influence of a bias voltage f and a magnetic field B. We investigate the interplay between the shift ( B w ) of the Kondo peak in the spin-resolved density of states (DOS) and the one ( B f ) of the conductance anomaly. In agreement with experiments and previous theoretical calculations we find that, while the latter displays a rather linear behavior with an almost constant slope as a function of B down to the Kondo scale, the DOS shift first features a slower increase reaching the same behavior as. Our auxiliary master equation approach yields highly accurate nonequilibrium results for the DOS and for the conductance all the way from within the Kondo up to the charge fluctuation regime, showing excellent agreement with a recently introduced scheme based on a combination of numerical renormalization group with time-dependent density matrix renormalization group.

When constructing a Wilson chain to represent a quantum impurity model, the effects of truncated bath modes are neglected. We show that their influence can be kept track of systematically by constructing an "open Wilson chain" in which each site is coupled to a separate effective bath of its own. As a first application, we use the method to cure the so-called mass-flow problem that can arise when using standard Wilson chains to treat impurity models with asymmetric bath spectral functions at finite temperature. We demonstrate this for the strongly sub-Ohmic spin-boson model at quantum criticality where we directly observe the flow towards a Gaussian critical fixed point.A quantum impurity model describes a discrete set of degrees of freedom, the "impurity", coupled to a bath of excitations. For an infinite bath this is effectively an open system. However, the most powerful numerical methods for solving such models, Wilson's numerical renormalization group (NRG) [1,2] and variational matrixproduct-state (VMPS) generalizations thereof [3][4][5][6], actually treat it as closed : The continuous bath is replaced by a so-called Wilson chain, a finite-length tight-binding chain whose hopping matrix elements t n decrease exponentially with site number n, ensuring energy-scale separation along the chain. This works well for numerous applications, ranging from transport through nanostructures [7,8] to impurity solvers for dynamical mean-field theory [9][10][11]. However, replacing an open by a closed system brings about finite-size effects. Wilson himself had anticipated that the effect of bath modes neglected during discretization might need to be included perturbatively "to achieve reasonable accuracy", but concluded that "this has proven to be unnecessary" for his purposes (see p. 813 of Ref.[1]). By now, it is understood that finite-size effects often do matter. They hamper the treatment of dissipative effects [12], e.g., in the context of nonequilibrium transport [13] and equilibration after a local quench [14]. Moreover, even in equilibrium, they may cause errors when computing the bath-induced renormalization of impurity properties [15][16][17]. Indeed, finite-size issues constitute arguably the most serious conceptual limitation of approaches based on Wilson chains.Here we set the stage for controlling finite-size effects by constructing "open Wilson chains" (OWCs) in which each site is coupled to a bath of its own. The resulting open system implements energy-scale separation in a way that, in contrast to standard Wilson chains (SWC), fully keeps track of all bath-induced dissipative and renormalization effects. The key step involved in any renormalization group (RG) approach, namely integrating out degrees of freedom at one energy scale to obtain a renormalized description at a lower scale, can then be performed more carefully than for SWCs. We illustrate this by focusing on renormalization effects, leaving a systematic treatment of dissipative effects on OWCs for the future.A SWC is constructed by logarithmicall...