Weak-coupling quantum Monte Carlo calculations on the Keldysh contour: Theory and application to the current-voltage characteristics of the Anderson model

We present optimized implementations of the weak-coupling continuous-time Monte Carlo method defined for nonequilibrium problems on the Keldysh contour. We describe and compare two methods of preparing the system before beginning the real-time calculation: the “interaction quench” and the “voltage quench,” which are found to be suitable for large and small voltage biases, respectively. We also discuss technical optimizations which increase the efficiency of the real-time measurements. The methods allow the accurate simulation of transport through quantum dots over wider interaction ranges and longer times than have heretofore been possible. The current-voltage characteristics of the particle-hole symmetric Anderson-impurity model is presented for interactions $U$ up to ten times the intrinsic level width $\Gamma$. We compare the Monte Carlo results to fourth-order perturbation theory, finding that perturbation theory is accurate up to $U\approx 4\Gamma$ or for a voltage bias $V\gtrsim 4\Gamma$. The interplay of voltage and temperature and the Coulomb blockade conductance regime are studied.

Figure 1

(Color online) Illustration of the Keldysh contour for the interaction quench (top panel) and voltage quench (bottom panel). In an interaction quench starting from $U=0$, the imaginary-time branch of the contour is shifted to $t=-\infty$ and must not be explicitly considered in the Monte Carlo simulation. The red arrows represent auxiliary Ising spin variables. The top panel shows a Monte Carlo configuration corresponding to perturbation order $n_{+}=2$ and $n_{-}=2$, and the bottom panel a configuration corresponding to $n_{+}=3$, $n_{-}=2$, and $n_{\beta }=2$.

Figure 2

(Color online) Time evolution of the current for $V/\Gamma =4$ and different interaction strengths $\left(T=0\right)$. In the initial state, the current is given by the steady-state current through the noninteracting dot. At time $t=0$, the interaction is turned on. After a time of a few inverse $\Gamma$, the current saturates at the value corresponding to the steady-state current in the interacting dot.

Figure 3

(Color online) Time evolution of the current for different voltage biases and interaction strength $U/\Gamma =6\left(T=0\right)$. In the initial state, the current is given by the steady-state current through the noninteracting dot. At time $t=0$, the interaction is turned on.

Figure 4

(Color online) Interaction quench in the small voltage regime $\left(T=0\right)$: the top panel shows the ratio of interacting to noninteracting current for $U/\Gamma =4$ and $U/\Gamma =6$ and indicated voltage biases. For $V/\Gamma \lesssim 0.5$ the time needed to reach the steady state grows much beyond the largest time accessible in the Monte Carlo simulation. Bottom panel: comparison of $U$ quench estimates (symbols, red and blue online) for the steady-state current to the noninteracting current (thick black line) and fourth-order perturbation theory (Ref. 21) (light lines, red and blue online). For $V/\Gamma \ge 0.5$ the Monte Carlo data show the value of $I\left(t_{\text{max}}\right)$ with error bar ${\text{max}}_{t∊\left[t_{\text{max}}/2,t_{\text{max}}\right]}\left|I\left(t\right)-I\left(t_{\text{max}}\right)\right|$ ($t_{\text{max}}\Gamma =6$ for $U/\Gamma =4$ and $t_{\text{max}}\Gamma =4$ for $U/\Gamma =6$). For $V/\Gamma =0.25$, we use $I\approx \left[I\left(t_{\text{max}}\right)+I_0\right]/2$ with error bar of size $\left[I_0-I\left(t_{\text{max}}\right)\right]/2$.

Figure 5

(Color online) Temperature dependence of the ratio of interacting current at temperature $T$ to noninteracting current at temperature zero for indicated values of the voltage bias. The interaction strength is $U/\Gamma =6$. The symbols show Monte Carlo results, the black line the analytical curve for $V\to 0$ extracted from Ref. 28 and plotted for $T_K/\Gamma =0.24$.

Figure 6

(Color online) Effect of the smoothing parameter $\nu$ and cutoff ${\omega }_c$ for $\beta \Gamma =10$. The top panel shows the interacting $\left(U/\Gamma =4\right)$ and noninteracting current for a quench to $V/\Gamma =0.25$. The red line and red circles correspond to $\nu \Gamma =10$ and cutoff ${\omega }_c/\Gamma =10$, the blue lines and blue diamonds to $\nu \Gamma =3$ and cutoff ${\omega }_c/\Gamma =10$. A sharp band edge (large value of $\nu$) leads to oscillations in the current which make it difficult to estimate the steady-state value. The bottom panel plots noninteracting and interacting currents for $V/\Gamma =0.5$, $\nu \Gamma =3$, and $\beta \Gamma =10$, and indicated values of the cutoff. While the short-time behavior of the current is cutoff dependent, the steady-state value is essentially cutoff independent, as long as ${\omega }_c\gtrsim V$.

Figure 7

(Color online) Voltage dependence of the current for $U/\Gamma =4$ (top panel) and $U/\Gamma =6$ (bottom panel). Solid lined show the noninteracting current and symbols the interacting current. As the voltage reaches $V/\Gamma =1$ $\left(U/\Gamma =4\right)$ or $V/\Gamma =0.5$ $\left(U/\Gamma =6\right)$, the interacting current overshoots, resulting in a slow convergence to the steady state. For smaller voltages, however, the steady-state current can be computed on the basis of $V$ quenches. All results are for $\beta \Gamma =10$, ${\omega }_c/\Gamma =10$, and $\nu \Gamma =3$.

Figure 8

(Color online) Temperature effect on the current in the small voltage bias regime. Red circles, blue diamonds, and black diamonds show the interacting current $\left(U/\Gamma =4\right)$ for $V/\Gamma =0.25$ and $V/\Gamma =0.5$ while the red, blue, and black curves indicate the corresponding noninteracting currents. As the temperature is lowered, the interacting current increases toward the noninteracting value.

Figure 9

(Color online) Current-voltage characteristics of the single-orbital Anderson-impurity model in the small voltage regime for $U/\Gamma =4$ and 6 at $\beta \Gamma =10$. The blue circles and black diamonds show $V$-quench results. For comparison, we also plot $U$-quench data ($U∕\Gamma =6$, $\beta \Gamma =10$, green crosses).

Figure 10

(Color online) Current-voltage characteristics of the single-orbital Anderson-impurity model. The symbols show Monte Carlo data for $U/\Gamma =4$, 6, 8, and 10 while the lines correspond to fourth-order perturbation theory in $\Sigma$. The results have been obtained by means of $U$ quenches at $T=0$. Error bars are on the order of the symbol size.