Out-of-equilibrium dynamics in a quantum impurity model: Numerics for particle transport and entanglement entropy

We investigate the out-of-equilibrium properties of a simple quantum impurity model, the interacting resonant level model. We focus on the scaling regime, where the bandwidth of the fermions in the leads is larger than all the other energies, so that the lattice and the continuum versions of the model become equivalent. Using time-dependent density matrix renormalization group simulations initialized with states having different densities in the two leads, we extend the results of Boulat, Saleur, and Schmitteckert [Phys. Rev. Lett. 101, 140601 (2008)] concerning the current-voltage $\left(I\text{−}V\right)$ curves, for several values of the interaction strength $U$. We estimate numerically the Kondo scale $T_B$ and the exponent $b\left(U\right)$ associated to the tunneling of the fermions from the leads to the dot. Next, we analyze the quantum entanglement properties of the steady states. We focus in particular on the entropy rate $\alpha$, describing the linear growth with time of the bipartite entanglement in the system. We show that, as for the current, $\alpha /T_B$ is described by some function of $U$ and of the rescaled bias $V/T_B$. Finally, the spatial structure of the entropy profiles is discussed.

Figure 1

Schematics of the interacting resonant level model. The system is prepared at $t=0$ in the ground state of the model with an additional chemical potential equal to $+V/2$ in the left lead and equal to $-V/2$ in the right lead. For $t>0$, the system then evolves with the bias switched to zero. In all cases, the chemical potential is zero on the dot, hence the name “resonant”. Note that, by symmetry, the average fermion occupation number $\langle c_0^{†}{c}_0\rangle$ on the dot is equal to $\frac{1}{2}$.

Figure 2

Magnetization profile $S^z(r,t)$ at times $t=0,6,12,18,24$. Parameters of the model: $J^{\prime }=0.3,V=2.0,N=254$. Top panel: $U=2.0$, bottom: $U=0$. In both cases, the initial magnetization $m_0$ in the bulk of the left lead is indicated by a (red) horizontal line. The orange horizontal line marks the exact bulk stationary magnetization $m$ for $U=0$ [see Eq. (6)]. The data in the upper panel show some deviations from this noninteracting value. Note that some Friedel-type oscillations develop at long times in the vicinity of the dot, and these are much stronger in the noninteracting case.

Figure 3

Top panels: evolution of the entropy $S(r,t)$ profile for $U=0$ and 2. Bottom panels: evolution of the current profile $I(r,t)$ for $U=0$ and 2.0. Parameters of the model: $J^{\prime }=0.3,V=2.0,N=254$, as in Fig. 2.

Figure 4

Top panel: current $I\left(t\right)$. Bottom panel: entanglement entropy $S\left(t\right)$ between the left lead and the rest of the system. The horizontal line in the top panel is the exact result at the self-dual point (see text), derived in Ref. [7]. Parameters: $U=2.0,J^{\prime }=0.3,V=2.0,N=254$.

Figure 5

Stationary current $I$ as a function of $V$ for $J^{\prime }=0.08$, for a few selected values of $U$. A log-log scale is used to show the power-law behavior of the current at large $V/T_B$ $\left(I\sim V^{-b}\right)$, and to extract the associated exponent $b\left(U\right)$. The values $b\left(U\right)$ extracted by these fits are displayed in Fig. 6.

Figure 6

Exponent $b\left(U\right)$ as a function of the interaction strength $U$, obtained by fitting the steady current $I$ to $I\sim V^{-b}$, at large $V/T_B$ (see Fig. 5). The full (black) line is $b=2U/\pi$, the result of some small-$U$ expansion [29, 30].

Figure 7

Rescaled current $I/T_B$ as a function of the rescaled bias $V/T_B$, for different values of $U$. The colors label the values of $U$, while the symbol shapes encode $J^{\prime }$ (see the legend of Fig. 8 for details). For a given $U$, the results obtained for different values of $J^{\prime }$ approximately collapse onto a single curve, as expected in the scaling regime. The red line is the theoretical result for the self-dual point [7]. The numerical data for $U=2.0$ appear to be in very good agreement with this theoretical curve. The black line is the exact result for $U=0$ [Eq. (B7)]. $T_B$ is defined as $T_B={\left(J^{\prime }\right)}^{\frac{1}{1-h\left(U\right)}}$ where the exponent $h\left(U\right)$ is determined from the behavior of $I$ at large $V/T_B$ (see text and Fig. 6).

Figure 8

Same as Fig. 7 (rescaled current $I/T_B$ as a function of the rescaled bias $V/T_B)$, with a zoom on the low-bias region. The colors label the values of $U$, while the symbol shapes encode $J^{\prime }$ (see the legend).

Figure 9

Rescaled entropy rate, $\alpha /T_B$, as a function of $V/T_B$, for several values of $U$. The colors label the values of $U$, while the symbol shapes encode $J^{\prime }$ (see the legend of Fig. 8 for details). The black line is the exact free-fermion result [Eq. (B16)].

Figure 10

Same as Fig. 9, with a zoom on the low-bias region. Top panel: $U=-0.1$, 0, and 3. Bottom panel: $U=0.2$, 1, and 2. The colors encode the values of $U$, while the symbol shapes label the values of $J^{\prime }$, as indicated in the legend of Fig. 8. The black line (top panel) is the exact free-fermion result in the limit $J^{\prime }\to 0$ [Eq. (B16)].

Figure 11

Top: entanglement entropy profiles for $U=0$, plotted as a function of the rescaled distance $r/\xi$, with $\xi =T_B^{-1}$ and $T_B=J^{\prime 2}$. Some constant value has been subtracted from each profile, to allow the data to (approximately) fall onto a single time-independent curve. We only show here the right part of the profile $(r>0)$, the other side $\left(r<0\right)$ being symmetric. Bottom: same data plotted as a function of the bare distance $r$. System size is $N=6000$ and time is $t=1200$, values for $V\left(J^{\prime }\right)$: $V\left(0.25\right)=0.5,V\left(0.1\right)=0.08,V\left(0.075\right)=0.045,V\left(0.05\right)=0.02$ with ratio $T_B^{-1}\left(0.05\right):T_B^{-1}\left(0.075\right):T_B^{-1}\left(0.1\right):T_B^{-1}\left(0.25\right)=25:4:2.25:1$. In all cases, the rescaled bias is $V/T_B=8.0$.

Figure 12

Same as Fig. 11, but for $U=2.0$. System size is $N=402$ and $t=60$. Values for $V\left(J^{\prime }\right)$: $V\left(0.12\right)=1.0,V\left(0.07\right)=0.5,V\left(0.04\right)=0.25,V\left(0.02\right)=0.1$ with ratio $T_B^{-1}\left(0.02\right):T_B^{-1}\left(0.04\right):T_B^{-1}\left(0.07\right):T_B^{-1}\left(1.2\right)=10:4:2:1$. In all cases, the rescaled bias is $V/T_B=17.1$.

Figure 13

Absolute value of the error for the expectation value of the current flowing through the dot, for $N=6,U=0,V=1.0,t=100,J=0.5,J^{\prime }=0.5$. The error is computed by comparing the MPS simulations with an exact free-fermion calculation. For small $\tau$ the main source of error is no longer the finite $\tau$, but the successive truncations in the Schmidt decompositions (SVD). This can be seen by comparing the data with cutoff parameter $\delta ={10}^{-10}$ to those without any truncation $\left(\delta =0\right)$. When the error becomes extremely small, of the order of ${10}^{-10}$, it stops decreasing with $\tau$. This is due to the finite floating point precision of the machine.

Figure 14

Top panel: entanglement entropy of the left lead as a function of time for different values of the SVD truncation parameter $\delta$ which defines the maximum discarded weight at each time step. Middle: maximum MPS bond dimension. Bottom: current flowing through the dot. Parameters of the model: $N=200,U=2,J^{\prime }=0.2$, and $V=1$. The (red) horizontal dotted line indicates the exact value of the steady current at the self-dual point [7]. For these parameters, the upper panel shows that a truncation value as low as $\delta ={10}^{-7}$ is required to obtain some accurate value for the entanglement entropy, leading to large bond dimensions (above 1000). In contrast, $\delta ={10}^{-6}$ appears to be enough to estimate the current correctly. Simulations performed with Trotter step $\tau =0.2$.

Figure 15

Same as Fig. 14, for $J^{\prime }=0.05$. For these parameters, a truncation value as low as $\delta ={10}^{-8}$ is required to obtain some accurate value for the entanglement entropy, up to $t=60$. One observes some small amplitude oscillations in $S\left(t\right)$, with the same period $T_{\mathrm{osc}}=4\pi /V$, as for $I\left(t\right)$.

Figure 16

Rescaled steady current as a function of the rescaled voltage in the free case $\left(U=0\right)$, using $T_B=J^{\prime 2}$. The symbols represent the DMRG data for three different values of $J^{\prime }$, and the full lines are, respectively, the exact result [Eq. (B7)] for $J^{\prime }=0.5$ (green), for $J^{\prime }=0.35$ (blue), for $J^{\prime }=0.2$ (orange), and the limit when $J^{\prime }$ is small [Eq. (B7), black]. The DMRG data are in perfect agreement with the exact results, but since chosen $J^{\prime }$ are not very small one observes some deviations from Eq. (B7).