We present a general scheme to address correlated nonequilibrium quantum impurity problems based on a mapping 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 are described by a Lindblad equation, which can then be exactly solved by numerical methods such as Lanczos or matrix-product states. The mapping becomes exponentially exact with increasing N B , and is already quite accurate for small N B . Due to the presence of the intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. While in previous work we put the focus on the manybody solution of the associated Lindblad problem, here we discuss the mapping scheme itself, which is an essential part of the overall approach. On the one hand, we provide technical details together with an in-depth discussion of the employed algorithms, and on the other hand, we present a detailed convergence study. The latter clearly demonstrates the above-mentioned exponential convergence of the procedure with increasing N B . Furthermore, the influence of temperature and an external bias voltage on the reservoirs is investigated. The knowledge of the particular convergence behavior is of great value to assess the applicability of the scheme to certain physical situations. Moreover, we study different geometries for the auxiliary system. On the one hand, this is of importance for advanced manybody solution techniques such as matrix product states which work well for short-ranged couplings, and on the other hand, it allows us to gain more insights into the underlying mechanisms when mapping non-Markovian reservoirs onto Lindblad-type impurity problems. Finally, we present results for the spectral function of the Anderson impurity model in and out of equilibrium and discuss the accuracy obtained with the different geometries of the auxiliary system. In particular, we show that allowing for complex Lindblad couplings produces a drastic improvement in the description of the Kondo resonance.
We present a general scheme to address correlated nonequilibrium quantum impurity problems based on a mapping 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 are described by a Lindblad equation, which can then be exactly solved by numerical methods such as Lanczos or matrix-product states. The mapping becomes exponentially exact with increasing N B , and is already quite accurate for small N B . Due to the presence of the intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. While in previous work we put the focus on the manybody solution of the associated Lindblad problem, here we discuss the mapping scheme itself, which is an essential part of the overall approach. On the one hand, we provide technical details together with an in-depth discussion of the employed algorithms, and on the other hand, we present a detailed convergence study. The latter clearly demonstrates the above-mentioned exponential convergence of the procedure with increasing N B . Furthermore, the influence of temperature and an external bias voltage on the reservoirs is investigated. The knowledge of the particular convergence behavior is of great value to assess the applicability of the scheme to certain physical situations. Moreover, we study different geometries for the auxiliary system. On the one hand, this is of importance for advanced manybody solution techniques such as matrix product states which work well for short-ranged couplings, and on the other hand, it allows us to gain more insights into the underlying mechanisms when mapping non-Markovian reservoirs onto Lindblad-type impurity problems. Finally, we present results for the spectral function of the Anderson impurity model in and out of equilibrium and discuss the accuracy obtained with the different geometries of the auxiliary system. In particular, we show that allowing for complex Lindblad couplings produces a drastic improvement in the description of the Kondo resonance.
We present a numerical method for the study of correlated quantum impurity problems out of equilibrium, which is particularly suited to address steady-state properties within dynamical mean field theory. The approach, recently introduced by Arrigoni et al. [Phys. Rev. Lett. 110, 086403 (2013)], is based upon a mapping of the original impurity problem onto an auxiliary open quantum system, consisting of the interacting impurity coupled to bath sites as well as to a Markovian environment. The dynamics of the auxiliary system is governed by a Lindblad master equation whose parameters are used to optimize the mapping. The accuracy of the results can be readily estimated and systematically improved by increasing the number of auxiliary bath sites, or by introducing a linear correction. Here, we focus on a detailed discussion of the proposed approach including technical remarks. To solve for the Green's functions of the auxiliary impurity problem, a non-Hermitian Lanczos diagonalization is applied. As a benchmark, results for the steady-state current-voltage characteristics of the single-impurity Anderson model are presented. Furthermore, the bias dependence of the single-particle spectral function and the splitting of the Kondo resonance are discussed. In its present form, the method is fast, efficient, and features a controlled accuracy.
Within the recently introduced auxiliary master equation approach it is possible to address steady state properties of strongly correlated impurity models, small molecules or clusters efficiently and with high accuracy. It is particularly suited for dynamical mean field theory in the nonequilibrium as well as in the equilibrium case. The method is based on the solution of an auxiliary open quantum system, which can be made quickly equivalent to the original impurity problem. In its first implementation a Krylov space method was employed. Here, we aim at extending the capabilities of the approach by adopting matrix product states for the solution of the corresponding auxiliary quantum master equation. This allows for a drastic increase in accuracy and permits us to access the Kondo regime for large values of the interaction. In particular, we investigate the nonequilibrium steady state of a single impurity Anderson model and focus on the spectral properties for temperatures T below the Kondo temperature T_K and for small bias voltages phi. For the two cases considered, with T=T_K/4 and T=T_K/10 we find a clear splitting of the Kondo resonance into a two-peak structure for phi close above T_K. In the equilibrium case (phi=0) and for T=T_K/4, the obtained spectral function essentially coincides with the one from numerical renormalization group.
Theoretical studies of electron and energy transport in nanostructures usually involve nonequilibrium quantum impurity models, which feature a locally interacting quantum system (impurity) coupled to noninteracting bosonic and/or electronic baths each at its own equilibrium. In these models, the interplay of strong correlation, nonequilibrium effects and dissipation leads to rich and complex phenomena. As a nonequilibrium strongly correlated many-body problem, these models, except at some rare cases, have no exact analytical solution, and hence their treatment heavily relies upon numerical techniques. The goal of this thesis is to develop numerically accurate and relatively low-cost techniques for treatment of strongly correlated open nonequilibirum quantum systems. Such impurity solvers are in high demand both as standalone methods for simulation of response in nanoscale junctions and as part of divide-and-conquer schemes such as dynamical mean field theory (DMFT).For nonequilibirum open quantum systems, two characteristic energy scales governing the physics are intra-system interactions and strength of the system-baths couplings. Thus, systems strongly coupled to their baths with weak intra-system interactions or weakly coupled systems with strong local interactions can be treated within developed perturbation theory approaches. My research focuses on development of nonequilibirum dual techniques, which are capable to accurately treat situations where perturbative techniques are inapplicable due to absence of a small parameter in the system and which reduce to standard perturbative expansions in the limits of weak coupling or weak intra-system interaction. \tDual techniques rely on ability to solve exactly a simplified reference system. The latter includes exactly all intra-system interactions and accounts approximately for system-bath couplings. Dual techniques (described below) allow to account for deviation from the correct hybridization function in a organized and controlled manner of superperturbative expansion. Here, I use Markovian Lindbald quantum master equation (QME) as a solver for the reference system. In the thesis I discuss possibility of mapping between true non-Markov dynamics and simpler Markov Lindblad QME (mapping between physical and reference systems), numerical methods for the mapping and for solution of the resulting Lindblad QME, and application of the mapping in novel dual formulations: auxiliary-QME dual fermion and dual boson approaches. \tFirst I analytically prove that the continuous fermionic baths that the impurity is coupled to can be mapped onto a set of dissipative auxiliary modes under Lindblad QME. Based on this mapping, complicated non-Markovian dynamics of the open fermionic system can be treated by a much simpler Markovian Lindblad QME. However, nice mapping generally requires many auxiliary modes, and numerically only a small number of modes can be exactly diagonalized due to exponential increase of the Hilbert space. To perturbatively treat the difference between the true bath and the small-size auxiliary one, we introduce dual-fermion method. It can directly target the steady state and avoid long-time propagation. Besides current-voltage relation, it can also provide spectral functions, indicating itself as a potential impurity solver for DMFT. Furthermore, we generalize dual-fermion method to consider additionally the effects of bosonic environment; hence it is called dual-boson method. It can be used for studying electron-phonon interaction and light-matter interaction in nanojunctions. By comparing with numerically exact results we argue that both methods are very accurate and relatively cheap. \tFinally, matrix product state representation of the Lindblad QME enables us to treat a large number of auxiliary modes, which provide an exponentially improved mapping. Adding counting field to the auxiliary QME, we can compute the generating function of time-dependent full-counting statistics of electron transport, which yields not only electronic current but also higher-order cumulants (e.g., zero-frequency noise).
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 present a solver for correlated impurity problems out of equilibrium based on a combination of the so-called auxiliary master equation approach (AMEA) and the configuration interaction expansion. Within AMEA one maps the original impurity model onto an auxiliary open quantum system with a restricted number of bath sites which can be addressed by numerical many-body approaches such as Lanczos/Arnoldi exact diagonalization (ED) or matrix product states (MPS). While the mapping becomes exponentially more accurate with increasing number of bath sites, ED implementations are severely limited due to the fast increase of the Hilbert space dimension for open systems, and the MPS solver typically requires rather long runtimes. Here, we propose to adopt a configuration interaction approach augmented by active space extension to solve numerically the correlated auxiliary open quantum system. This allows access to a larger number of bath sites at lower computational costs than for plain ED. We benchmark the approach with numerical renormalization group results in equilibrium and with MPS out of equilibrium. In particular, we evaluate the current, the conductance, as well as the Kondo peak and its splitting as a function of increasing bias voltage below the Kondo temperature TK. We obtain a rather accurate scaling of the conductance as a function of the bias voltage and temperature rescaled by TK for moderate to strong interactions in a wide range of parameters. The approach combines the fast runtime of ED with an accuracy close to the one achieved by MPS making it an attractive solver for nonequilibrium dynamical mean field theory.
We present improvements of a recently introduced numerical method [E. Arrigoni et al., Phys. Rev. Lett. 110, 086403 (2013)] to compute steady-state properties of strongly correlated electronic systems out of equilibrium. The method can be considered as a nonequilibrium generalization of exact diagonalization based dynamical mean-field theory (DMFT). The key modification for the nonequilibrium situation consists in addressing the DMFT impurity problem within an auxiliary system consisting of the correlated impurity, Nb uncorrelated bath sites, and two Markovian environments (sink and reservoir). Algorithmic improvements in the impurity solver allow to treat efficiently larger values of Nb than previously in DMFT. This increases the accuracy of the results and is crucial for a correct description of the physical behavior of the system in the relevant parameter range including a semiquantitative description of the Kondo regime. To illustrate the approach, we consider a monoatomic layer of correlated orbitals, described by the single-band Hubbard model, attached to two metallic leads. The nonequilibrium situation is driven by a bias voltage applied to the leads. For this system, we investigate the spectral function and the steady-state current-voltage characteristics in the weakly as well as in the strongly interacting limit. In particular, we investigate the nonequilibrium behavior of quasiparticle excitations within the Mott gap of the correlated layer. We find for low-bias voltage Kondo-type behavior in the vicinity of the insulating phase. In particular, we observe a splitting of the Kondo resonance as a function of the bias voltage.