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Numerical Methods for Nonequilibrium Impurity Models


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.

Dual 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.

First 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.

Finally, 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).

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