Real-time simulations of nonequilibrium transport in the single-impurity Anderson model

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.

Figure 1

(Color online) Particle-hole symmetric point, $V_{\text{g}}=-U/2$: Current $J\left(\tau \right)/V$ vs time $\tau$ for $U/\Gamma =2$ ($U=0.5t$, $t^{\prime }=0.3535t$, $V_{\text{g}}=-U/2$) for $V/U=0.4,0.6,\dots ,2$ [$N=96$ (thick lines); $N=64$ (dashed lines); $N=48$ (dot-dashed lines)]. The arrows indicate the time-interval used for the calculation of the steady-state current $J_{\text{steady}}={⟨J\left(\tau \right)⟩}_{\tau }$ for $V/U=1.2$ from a time average ${⟨\cdot ⟩}_{\tau }$ (Ref. 38). Inset: estimates of the transient time ${\tau }_{\text{trans}}$ as a function of bias for $U=t/2$ (squares, $\Gamma =t/4$) and $U/t=1$ (triangles up, $\Gamma =t/2$), both at $U/\Gamma =2$.

Figure 2

(Color online) Particle-hole symmetric point, $V_{\text{g}}=-U/2$: (a) Current-voltage curves for $U/\Gamma =2,4,6$ with $U=t/2$ (squares, stars, circles) as well as data for $U/\Gamma =2,8$ with $U=t$ (triangles down and up, respectively). Open symbols: $N=64$, full ones: $N=96$ sites, shaded ones: $N=128$, stars and triangles down: finite-size extrapolations in $1/N$ ($U/\Gamma =4,8$ only). Dash-dotted line: ${⟨J⟩}_{\tau }=2G_{0}V$. (b) Enlarged view of the low-bias region for $U/\Gamma =4$ and 8, ${⟨J⟩}_{\tau }/V$ vs. $V$. Solid lines: fRG results from Ref. 22; dotted lines: 4th-order perturbation theory results from Ref. 14. (from top to bottom). For $U/\Gamma =8$, the arrow in the inset indicates the difference between the fRG and tDMRG results on the one hand and those from Ref. 14 on the other hand. For NRG results for $T_{\text{K}}$, see, e.g., Fig. 6 in Ref. 38.

Figure 3

(Color online) Mixed-valence regime: Current $J\left(\tau \right)/V$ vs time $\tau$ for $U/\Gamma =2$ $(U=t\phantom{)}$, $t^{\prime }=0.5t$, $V_{\text{g}}=0$ and bias values $V/U=0.3,0.8,1,1.2$ [$N=96$ (thin lines), $N=64$ (thick lines)]. Arrows indicate the interval used for the averaging for $\Delta V=0.8U$ and $N=64$. The thin solid line for $V=1.2U$ is a fit of $a+b\text{cos}\left(\omega \tau +\varphi \right)$ to the corresponding $N=64$ tDMRG data (dotted line).

Figure 4

(Color online) Mixed-valence regime: Current-voltage curves for $U/\Gamma =2$ ($U=t$, symbols with dotted lines), $U/\Gamma =4$ [$U=t/2$, extrapolated data only, panels (b) and (d)], and several gate potentials: (a) $V_{\text{g}}=-U/4$; (b) $V_{\text{g}}=0$; (c) $V_{\text{g}}=U/4$; and (d) $V_{\text{g}}=U/2$. Solid diamonds are extrapolations of finite-size data in $1/N$. Inset in (c): $⟨n_{\text{dot}}\left(\tau =0\right)⟩$ vs $V_{\text{g}}/U$ for $U/\Gamma =2$ (circles) and 4 (squares).