Numerically exact path-integral simulation of nonequilibrium quantum transport and dissipation

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.

Figure 1

Polarization in the nonequilibrium spin-fermion model at different values of the bias voltage $\Delta \mu =0.6$ (full); $\Delta \mu =1.4$ (dashed); and $\Delta \mu =2$ (dotted), $B=0$, $\Delta =1$, $\lambda =0.2$, $\delta t=0.25$, and $N_s=8$. Inset: convergence with increasing correlation time at $\Delta \mu =0.6$, $N_s=3$ (dark full); $N_s=4$ (dashed-dotted); $N_s=7$ (dashed); $N_s=8$ (dotted); and $N_s=9$ (light full). Data were generated using 80 states per bath, which is sufficient to ensure convergence in the regime of parameters presented here.

Figure 2

(Color online) Polarization in the nonequilibrium spin-fermion model at different spin-bath couplings, $\lambda =0.1$ (dashed); $\lambda =0.2$ (full); and $\lambda =0.3$ (dashed-dotted). Here $\Delta \mu =0.6$ and $\delta t=0.25$, $N_s=10$. The dotted line was generated using a nonequilibrium version of the “noninteracting spin-blip approximation” (Refs. 31, 32).

Figure 3

(Color online) Resonant level dynamics at different values of the voltage bias, $\Delta \mu =0.6$ (dashed) and $\Delta \mu =0.4$ (full). $U=0.1$, ${\Gamma }_{\alpha }=0.025$, $E_d=0.3$, and ${\tau }_c=3.2$. The dotted lines show for reference the exact $U=0$ dynamics at $\Delta \mu =0.6$ and 0.4 (top to bottom). The circles are the respective Monte Carlo points. Calculations are performed at $T=0$ while Monte Carlo data utilizes $T=1/200$ which is effectively converged to the $T=0$ limit.

Figure 4

(Color online) Population of the resonant level in the Anderson model. The results for $U=0$ (full), $U=0.1$ (dashed), $U=0.3$ (dashed-dotted), and $U=0.5$ (dotted) are compared with the exact dynamics at $U=0$ (○) and Monte Carlo data (*, ◻, and $◁$). The physical parameters of the model are $D=1$, $\Delta \mu =0.4$, $E_d=0.3$, and ${\Gamma }_{\alpha }=0.025$. The numerical parameters used are $L=240$ lead states, ${\tau }_c=3.2$ with $N_s=4$ and $\delta t=0.8$. Note that convergence and thus agreement with Monte Carlo cannot be achieved for $t\ge 10$ if $\frac{U}{\Gamma }\ge 3$.

Figure 5

(Color online) Short time dynamics in the Anderson model (◻) compared with Monte Carlo data (○) for $U=0.5$, 0.3, and 0.1 (top to bottom). $D=1$, $\Delta \mu =0.4$, $E_d=0.3$, ${\Gamma }_{\alpha }=0.025$, $L=240$, and ${\tau }_c=3.2$ with $N_s=4$ and $\delta t=0.8$.

Figure 6

Convergence of the dot occupancy with increasing number of bath states $L$. $E_d=0.3$, ${\Gamma }_{\alpha }=0.025$, $N_s=4$, and ${\tau }_c=3.2$. Full lines (top to bottom); $U=0.5$, $L=20$, 40, 80, 120, and 240; dashed lines (top to bottom): $U=0.1$, $L=20$, 40, 80, 120, and 240. Inset: data as a function of $L^{-1}$ at $t=20$, $U=0.5$ (square) and $U=0.1$ (circle).

Figure 7

Convergence of dot occupancy reducing the time step $\delta t={\tau }_c/N_s$. $E_d=0.3$, $U=0.5$, ${\Gamma }_{\alpha }=0.025$, ${\tau }_c=3.2$, $L=120$, and $\delta t=1.6$ (full); $\delta t=1.07$ (dashed); $\delta t=0.8$ (dashed-dotted); and $\delta t=0.64$ (dotted). Inset: data as a function of ${\left(\delta t\right)}^2$ for three representative times.

Figure 8

(Color online) Convergence of dot occupancy with increasing memory size ${\tau }_c$. $E_d=0.3$, $U=0.1$, ${\Gamma }_{\alpha }=0.025$, $L=120$, and $\delta t=0.8$. $N_s=2$ (dashed-dotted); $N_s=3$ (×); $N_s=4$ (dashed); $N_s=5$ (full); $N_s=6$ (dotted); and $N_s=7$ $(◇)$ Inset: dot population vs ${\tau }_c$ at a specific time, $t=12$.