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Iterative path integral summation for nonequilibrium quantum transport



We have developed a numerically exact approach to compute real-time path integral expressions for quantum transport problems out of equilibrium. The scheme is based on a deterministic iterative summation of the path integral (ISPI) for the generating function of nonequilibrium observables of interest. We apply the scheme to compute the charge current or dynamical quantities of small strongly correlated quantum systems as a quantum dot or molecules that are tunnel coupled to leads. Self-energies due to the leads, being nonlocal in time, are fully taken into account within a finite memory time, thereby including non-Markovian effects. Numerical results are extrapolated first to vanishing (Trotter) time discretization and, second, to infinite memory time. The method is applied to nonequilibrium transport through a single-impurity Anderson dot in the first place. We benchmark our results in various regimes of the rich parameter space. In the respective regime of validity, iterative summation of real-time path integrals (ISPI) results are shown to match those of other state-of-the art methods. Especially, we have chosen the mixed valence regime of the Anderson model to compare ISPI to time dependent density matrix renormalization group (tDMRG) and functional RG calculations. Secondly, we determine the nonequilibrium current math formula through a molecular junction in presence of a vibrational mode. We have found an exact mapping of the single impurity Anderson–Holstein model to an effective spin-1 problem. In analytically tractable regimes, as the adiabatic phonon or weak molecule-lead coupling regime, we reproduce known perturbative results. Studying the crossover regime between those limits shows that the Franck–Condon blockade persists in the quantum limit. At low temperature, the Franck–Condon steps in the math formula characteristics are smeared due to nonequilibrium conditions. The third system under investigation here is the magnetic Anderson model which consists of a spinful single-orbital quantum dot with an incorporated quantum mechanical spin–1/2 magnetic impurity. Coulomb interaction together with the exchange coupling of the magnetic impurity with the electron spins strongly influence the dynamics. We investigate the nonequilibrium tunneling current through the system as a function of exchange and Coulomb interaction as well as the real-time impurity polarization. From the real-time evolution of physical observables, we are able to determine characteristics of the time-dependent nonequilibrium current and the relaxation dynamics of the impurity. These examples illustrate that the ISPI technique is particularly well suited for the quantum regime, when all time and energy scales are of the same order of magnitude.