Iterative real-time path integral approach to nonequilibrium quantum transport

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.

Figure 1

Keldysh contour: Every physical time has two contour representatives on the branches ±. The measurement time $t_m$ only has a single representative on the upper branch.

Figure 2

(Color online) Real part of the Keldysh GF component $G_{0\uparrow }^{--}(t-t^{\prime })$ for (a) ${ϵ}_0=0$, $B=0$, $T=\Gamma$, (b) ${ϵ}_0=0$, $B=0$, $T=0.1\Gamma$, (c) ${ϵ}_0=0$, $T=0.1\Gamma$, $eV=\Gamma$, $B=0$ and $B=3\Gamma$, and (d) $B=0$, $T=0.1\Gamma$, $eV=\Gamma$, ${ϵ}_0=\Gamma$ and ${ϵ}_0=5\Gamma$. The inset in (a) shows the absolute values of the same data in log-linear representation, highlighting the exponential decrease.

Figure 3

(Color online) Quadratic dependence of the Trotter error as obtained in the convergence procedure for $\delta t\to 0$ for a fixed memory time ${\tau }_c=0.5∕\Gamma$. Parameters are ${ϵ}_0=0$, $B=0$, $T=0.1\Gamma$. The extrapolated values are given in the figure.

Figure 4

(Color online) Dependence of the differential conductance $dI\left({\tau }_c\right)∕dV$ on the inverse memory time $1∕{\tau }_c$ for different values of $U$ after the Trotter error has been eliminated. Parameters are ${ϵ}_0=B=0$, $T=0.1\Gamma$. Solid lines correspond to a linear fit. The extrapolated values for $dI∕dV=dI(1∕{\tau }_c\to 0)∕dV$ are given in the figure.

Figure 5

(Color online) Time-dependent current $I\left(t\right)$ for two representative parameter sets: (1) $U=4\Gamma$, $eV=2\Gamma$, $T=0.1\Gamma$ (black circles) and (2) $U=0.5\Gamma$, $eV=0.6\Gamma$, $T=\Gamma$ (red squares). Solid line: analytical result for stationary $I$ from nonequilibrium Kondo theory applied to parameter set (1) (Refs. 12, 58) Symbols are ISPI results, dashed lines are guides to the eyes only, and ${ϵ}_0=B=0$.

Figure 6

(Color online) Stationary current as a function of ${ϵ}_0$ (a) and of the bias voltage $eV$ (b) for the noninteracting case $U=0$. Shown are the numerical (circles) and the exact analytical (dashed curve) results for (a) $eV=0.2\Gamma$, and (b) ${ϵ}_0=0$. In both cases, $T=0.1\Gamma$ and $B=0$.

Figure 7

(Color online) Interaction corrections $\delta I\left(V\right)=I\left(U\right)-I(U=0)$ from the numerical ISPI approach (red symbols) and from a second-order perturbative calculation (black solid lines), for $U=0.1\Gamma$ (squares) and $U=0.3\Gamma$ (circles). Parameters are ${ϵ}_0=B=0$, $T=0.1\Gamma$, and dotted lines are guides to the eyes only.

Figure 8

(Color online) Interaction correction $\delta I$ for the nonlinear current for $U=\Gamma$, comparing ISPI (red symbols) and $U^2$ perturbation theory (black solid curve). The inset shows the corresponding interaction corrections $\delta (dI∕dV)$ to the differential conductance. Other parameters are as in Fig. 7. Dashed lines are guides to the eyes only.

Figure 9

(Color online) Comparison of the ISPI method (red symbols) and the master equation approach (solid lines) for $U=\Gamma$. Main: corrections $\delta (dI∕dV)$ of the nonlinear conductance as a function of temperature for $eV=3\Gamma$. Inset: same for the linear conductance $\delta G$. For the remaining parameters, see Fig. 7.

Figure 10

(Color online) Interaction correction $\delta G$ as a function of ${ϵ}_0$, for $U=\Gamma$, different $B$, and $T=0.1\Gamma$. Inset: dependence of $\delta G$ on $U$ for $B=0$, on resonance (${ϵ}_0=0$, red circles) and away from resonance (${ϵ}_0=-\Gamma$, black diamonds).

Figure 11

(Color online) Linear conductance $G$ vs temperature $T$, for $U=0$ (dashed line) and $U=1.2,3,4\Gamma$ $({ϵ}_0=B=0)$. Symbols denote the ISPI results while the solid lines are those of Eq. (27).

Figure 12

(Color online) Same as Fig. 10 but for the nonlinear conductance, taken at $eV=3\Gamma$.

Figure 13

(Color online) Nonlinear differential conductance $dI∕dV$ vs temperature $T$, for $eV=2\Gamma$, ${ϵ}_0=B=0$, and $U=0,1.2,3\Gamma$. Inset: corresponding interaction corrections $\delta (dI∕dV)$. (The Kondo temperature for $U=3\Gamma$ is $T_K=0.38\Gamma$.)