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arXiv:0802.3374 · Physical Review B

Iterative real-time path integral approach to nonequilibrium quantum transport

S. Weiss, Jens Eckel, Michael Thorwart, Reinhold Egger

Published May 15, 2008• 211 citations
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Abstract

We have developed a numerical 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 for the generating function of the nonequilibrium current. 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, and numerical results are extrapolated both to vanishing (Trotter) time discretization and to infinite memory time. This extrapolation scheme converges except at very low temperatures, and the results are then numerically exact. The method is applied to nonequilibrium transport through an Anderson dot.

DOI
10.1103/physrevb.77.195316

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We have developed a numerical 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 for the generating function of the nonequilibrium current. 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, and numerical results are extrapolated both to vanishing (Trotter) time discretization and to infinite memory time. This extrapolation scheme converges except at very low temperatures, and the results are then numerically exact. The method is applied to nonequilibrium transport through an Anderson dot.
We develop an iterative, numerically exact approach for the treatment of nonequilibrium quantum transport and dissipation problems that avoids the real-time sign problem associated with standard Monte Carlo techniques. The method requires a well-defined decorrelation time of the nonlocal influence functional for proper convergence to the exact limit. Since finite decorrelation times may arise either from temperature or from a voltage drop at zero temperature, the approach is well suited for the description of the real-time dynamics of single-molecule devices and quantum dots driven to a steady state via interaction with two or more electron leads. We numerically investigate two nontrivial models: the evolution of the nonequilibrium population of a two-level system coupled to two electronic reservoirs, and quantum transport in the nonequilibrium Anderson model. For the latter case, two distinct formulations are described. Results are compared to those obtained by other techniques.
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 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 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.
We present a detailed comparison of three different methods designed to tackle non-equilibrium quantum transport, namely the functional renormalization group (fRG), the time-dependent density matrix renormalization group (tDMRG) and the iterative summation of real-time path integrals (ISPI). For the non-equilibrium single-impurity Anderson model (including a Zeeman term at the impurity site), we demonstrate that the three methods are in quantitative agreement over a wide range of parameters at the particle–hole symmetric point as well as in the mixed-valence regime. We further compare these techniques with two quantum Monte Carlo approaches and the time-dependent numerical renormalization group method.
The nonequilibrium time evolution of an Anderson quantum dot is investigated. The quantum dot is coupled between two leads forming a chemical-potential gradient. We use Kadanoff-Baym dynamic equations within a nonperturbative resummation of the sss-channel bubble chains. The effect of the resummation leads to the introduction of a frequency-dependent four-point vertex. The tunneling to the leads is taken into account exactly. The method allows the determination of the transient as well as stationary transport through the quantum dot, and results are compared with different schemes discussed in the literature (functional renormalization group, iterative real-time summation of the path integral, time-dependent matrix renormalization group, and quantum Monte Carlo methods).
Memory effects play a key role in the dynamics of strongly correlated systems driven out of equilibrium. In this paper, we explore the nature of memory in the nonequilibrium Anderson impurity model. The Nakajima-Zwanzig-Mori formalism is used to derive an exact generalized quantum master equation for the reduced density matrix of the interacting quantum dot, which includes a non-Markovian memory kernel. A real-time path integral formulation is developed in which all diagrams are stochastically sampled in order to numerically evaluate the memory kernel. We explore the effects of temperature down to the Kondo regime, as well as the role of source-drain-bias voltage and bandwidth on the memory. Typically, the memory decays on time scales significantly shorter than the dynamics of the reduced density matrix itself, yet under certain conditions, it develops a low magnitude but long-ranged tail. In addition, we address the conditions required for the existence, uniqueness, and stability of a steady state.
Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.