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arXiv:1001.3773 · New Journal of Physics

Comparative study of theoretical methods for non-equilibrium quantum transport

Jens Eckel, Fabian Heidrich‐Meisner, Severin G. Jakobs, Michael Thorwart, Mikhail Pletyukhov, Reinhold Egger

Published Apr 28, 2010• 126 citations
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Abstract

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.

DOI
10.1088/1367-2630/12/4/043042

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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.
We study time-dependent electron transport through an Anderson model. The electronic interactions on the impurity site are included via the self-energy approximations at Hartree-Fock (HF), second Born (2B), GW, and TTT-matrix levels as well as within a time-dependent density functional (TDDFT) scheme based on the adiabatic Bethe-ansatz local density approximation (ABALDA) for the exchange-correlation potential. The Anderson model is driven out of equilibrium by applying a bias to the leads, and its nonequilibrium dynamics is determined by real-time propagation. The time-dependent currents and densities are compared to benchmark results obtained with the time-dependent density matrix renormalization group (tDMRG) method. Many-body perturbation theory beyond HF gives results in close agreement with tDMRG, especially within the 2B approximation. We find that the TDDFT approach with the ABALDA approximation produces accurate results for the densities on the impurity site, but overestimates the currents. This problem is found to have its origin in an overestimation of the lead densities, which indicates that the exchange-correlation potential must attain nonzero values in the leads.
One of the main open problems in the field of transport in strongly interacting nanostructures is the understanding of currents beyond the linear response regime. In this work, we consider the single-impurity Anderson model and use the adaptive time-dependent density matrix renormalization group method to compute real-time currents out of equilibrium. We first focus on the particle-hole symmetric point where Kondo correlations are the strongest and then extend the study of the nonequilibrium transport to the mixed-valence regime. As a main result, we present accurate data for the current-voltage characteristics of this model.
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.
The numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems (QISs) with unprecedented accuracy by a tailored finite size representation. Out of equilibrium, the steady-state density operator is not of the Boltzmannian form but one that is determined by the imposed boundary conditions. We extend the NRG to the nonequilibrium setting by augmenting each Wilson site with a reservoir, whose coupling functions are calculated via a continuous fraction expansion in order to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enters through the Bloch-Redfield tensor (BRT), whose zero eigenvector gives the steady-state density operator. We used the resulting open chain full density matrix (OC-FDM) approach with an effective single lead description to investigate the charge and spin transport through a quantum dot (QD) under finite bias and temperature gradient. The influence of lead asymmetry and an external magnetic field on the transport properties are also studied. For completeness, we have also investigated local properties of the QD, such as charge fluctuations and find excellent agreement with real-time quantum Monte Carlo (RT-QMC) data. The OC-FDM approach was able to explore Kondo energy scales as low as TK/D≈10−8T_K/D \approx 10^{-8}TK​/D≈10−8 in the non-equilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full density matrix numerical renormalization group (FDM-NRG) and Fermi-liquid scaling at small bias and temperature. Owing to the effective single lead description, a single OC-FDM data point takes orders of magnitude less time on a standard laptop, compared to other state-of-the-art numerical methods.
The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors ≲3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.
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.