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Feynman-Vernon influence functional approach to quantum transport in interacting nanojunctions: An analytical hierarchical study

We present a nonperturbative and formally exact approach for the charge transport in interacting nanojunctions based on a real time path integral formulation of the reduced system dynamics. An expansion of the influence functional in terms of the number of tunneling transitions, and integration of the Grassmann variables between the tunneling times, allows us to obtain a still exact generalized master equation (GME) for the populations of the reduced density matrix (RDM) in the occupation number representation, as well as a formally exact expression for the current. By borrowing the nomenclature of the famous spin-boson problem, we characterize the two-state dynamics of such degrees of freedom on the forward and backward branches in terms of single four-state paths with alternating blips and sojourns. This allows a diagrammatic representation of the GME kernel and its parametrization in terms of sequences of blips and sojourns. We apply our formalism to the exactly solvable resonant level model (RLM) and to the the single impurity Anderson model (SIAM). We demonstrate a hierarchical diagrammatic structure of the exact GME kernel. While the hierarchy closes at the second-tier level for the RLM, this is not the case for the interacting SIAM. Upon inspection of the GME, known results from various perturbative and nonperturbative approximation schemes to quantum transport in the SIAM are recovered. Finally, a noncrossing approximation for the hierarchical kernel is developed, which enables us to systematically decrease temperature at each next level of the approximation. Analytical results are presented, both in equilibrium and nonequilibrium, including the case of an applied magnetic field.