Transient dynamics of the Anderson impurity model out of equilibrium

We discuss the transient effects in the Anderson impurity model that occur when two fermionic continua with finite bandwidths are instantaneously coupled to a central level. We present results for the analytically solvable noninteracting resonant-level system first and then consistently extend them to the interacting case using the conventional perturbation theory and recently developed nonequilibrium Monte Carlo simulation schemes. The main goal is to gain an understanding of the full time-dependent nonlinear current-voltage characteristics and the population probability of the central level. We find that, contrary to the steady state, the transient dynamics of the system depends sensitively on the bandwidth of the electrode material.

Figure 1

Population of the noninteracting dot as a function of time measured in units of ${\Gamma }^{-1}$ for $V=0$, $\Delta /\Gamma =8$, and different bandwidths: for ${ϵ}_c/\Gamma =8$, 20, and $\infty$. Curves are the analytic results, symbols represent the MC simulation data (see Sec. 5).

Figure 2

DQMC data for the left, right, and total current $I_{L,R}\left(t\right)$, and $I\left(t\right)$, respectively, for $U=\Delta =0$, $V=20\Gamma$, and $T=0.4\Gamma$. Circles, triangles down, and triangles up refer to cut-off energies of ${ϵ}_c=20\Gamma$, $40\Gamma$, and $100\Gamma$, respectively. The inset shows the difference in $I\left(t\right)$ due to charge neutrality. The solid lines refer to the current calculated in the WFB limit via Eq. (31).

Figure 3

DQMC data for the displacement current $I_{\text{disp}}\left(t\right)$ (open symbols) and results according to the perturbation theory (lines) for $V=0$, ${ϵ}_c=20\Gamma$, and $T=0.4\Gamma$. Circles, squares, diamonds, and triangles (solid, dashed, dotted, and dotted-dashed lines) refer to $U=-2\Delta =0$, $\Gamma$, $2\Gamma$, and $4\Gamma$, respectively.

Figure 4

Time-dependent dot occupation for different interaction strengths $U$ ($V=0$, $\Delta =-U/2$, ${ϵ}_c/\Gamma =10$, and $T=0.2\Gamma$). Curves show the results from the first-order perturbation calculation in $U$. The symbols represent the MC data.

Figure 5

Change of the time-dependent dot occupation due to interactions, $\delta n_{\sigma }\left(U\right)=n_{\sigma }\left(U\right)-n_{\sigma }\left(U=0\right)$, for several values of the interaction strength. $V=0$, $\Delta =-U/2$, ${ϵ}_c/\Gamma =10$, and $T=0.2\Gamma$. Curves show the results from the first-order perturbation calculation in $U$. The symbols represent the MC data.

Figure 6

DQMC data for the total current $I\left(t\right)$ (symbols) and results according to Eqs. (52, 53) (lines) for $V=10\Gamma$, $T=0$, and ${ϵ}_c=10\Gamma$ and $40\Gamma$ (open and filled symbols, respectively). A WFB was assumed for the first-order perturbation calculation. Circles, squares, diamonds, and triangles (facing down and up) refer to $\Delta =-U/2$ and $U=0$, $\Gamma$, $2\Gamma$, $4\Gamma$, and $8\Gamma$, respectively, while solid, dashed, and dotted lines refer to $U=0$, $\Gamma$, and $2\Gamma$.