Steady-state and quench-dependent relaxation of a quantum dot coupled to one-dimensional leads

We study the time evolution and steady state of the charge current in a single-impurity Anderson model, using matrix product states techniques. A nonequilibrium situation is imposed by applying a bias voltage across one-dimensional tight-binding leads. Focusing on particle-hole symmetry, we extract current-voltage characteristics from universal low-bias up to high-bias regimes, where band effects start to play a dominant role. We discuss three quenches, which after strongly quench-dependent transients yield the same steady-state current. Among these quenches we identify those favorable for extracting steady-state observables. The period of short-time oscillations is shown to compare well to real-time renormalization group results for a simpler model of spinless fermions. We find indications that many-body effects play an important role at high-bias voltage and finite bandwidth of the metallic leads. The growth of entanglement entropy after a certain time scale $\propto {\Delta }^{-1}$ is the major limiting factor for calculating the time evolution. We show that the magnitude of the steady-state current positively correlates with entanglement entropy. The role of high-energy states for the steady-state current is explored by considering a damping term in the time evolution.

Figure 1

Illustration of the three quenches performed for the SIAM: (i) QT I: quenching of both quantum dot-reservoir tunnelings $t_L^{\prime }$ and $t_R^{\prime }$, (ii) QT II: quenching the bias voltage $V_B$, and (iii) QT III: quenching the dot-lead tunneling $t_R^{\prime }$ to one lead.

Figure 2

Time dependence of the current [Eq. (A1)] at $U/\Delta =12$ for the three different QTs and for different bias voltages. The curves are plotted as solid lines up to the last reliable point in the TEBD calculation (see text). Larger times are plotted as dashed-dotted lines. Solid horizontal lines are fits to extract the steady-state currents. The time domain for these fits starts at $\tau \approx {\Delta }^{-1}$ and ends at a point which is identified as the last reliable data point (symbols, see text). Dashed horizontal lines indicate the uncertainty. The insets in the mid row show respective zooms onto short-time regions, which are not visible in the main part of the figure.

Figure 3

Period of the sinusoidal oscillations of the current in QT II for various values of interaction strength $U/\Delta =0,4,8,12,\text{and}20$ (symbols). Solid lines indicate the predicted form for the interacting resonant level model (Ref. 50).

Figure 4

Current-voltage characteristics of the quantum dot. The steady-state currents shown are obtained by a fit of the expectation value of the current operator within the steady-state plateau. Regions where only a likely upper bound for the steady-state current could be obtained are indicated by pedestals (see text).

Figure 5

Comparison of the current-voltage characteristics of the SIAM obtained with different methods in the low-bias regime. Some of the methods use a wide-band limit and others a semicircular reservoir DOS which (for equal $\Delta$) become comparable in the low-bias region shown. The methods are (1) diagrammatic QMC for $T=0$ in the wide-band limit (dQMC) (Ref. 36), (2) fourth-order Keldysh perturbation theory for $T=0$ in the wide-band limit (PT4) (Ref. 37), (3) time-dependent DMRG for $T=0$ using a semicircular DOS (tDMRG) (Ref. 14), (4) TEBD for $T=0$ using a semicircular DOS (TEBD, this work), (5) nonequilibrium fRG for $T=0$ using a wide-band limit (fRG) (Ref. 38), (6) nonequilibrium cluster perturbation theory for $T=0$ using a semicircular DOS (nCPT${}^{11}$) (Ref. 39), (7) nonequilibrium variational cluster approach for $T=0$ using a semicircular DOS (nCPT${}_T^7$) (Ref. 39), (8) imaginary-time QMC for $T=0.2\Delta$ in the wide-band limit (cQMC) (Ref. 40), (9) iterative summation of real-time path integrals for $T=0.2\Delta$ in the wide-band limit (ISPI) (Ref. 41), and (10) the linear response result for the Kondo regime $j_{\text{lin}}=2G_0{V}_B$ (lin. resp.).

Figure 6

Comparison of the current-voltage characteristics of a noninteracting, resonant level device with onsite potential ${\epsilon }_f=-\frac{U}{2}$ (solid lines) with the TEBD data for the interacting quantum dot (symbols). Both devices have the same specifications with only the interaction $U$ missing in the first case. The comparison is done for four values of interaction strengths resp. onsite potentials: $\frac{U}{\Delta }=\{4,8,12,20\}$ resp. $\frac{{\epsilon }_f}{\Delta }=\{-2,-4,-6,-10\}$ (blue/circles, green/triangles, red/stars, cyan/squares, respectively). In addition, we show the $U=0$ result (black/no symbols). The dashed-dotted lines indicate data for a noninteracting device in the wide-band limit.

Figure 7

(Top) Parameter regions in the $U-V_B$ of short (I) and long (II) physical relaxation scales as well as a regime of more complex behavior III. Data from the TEBD calculation are indicated with black and gray markers. For region II, pedestals are shown in Fig. 4. (Bottom) Single-particle DOS and single-particle dot level in a Hubbard-I-type picture at $U=20\Delta$, for (a) $V_B=6\Delta$, (b) $V_B=20\Delta$, and (c) $V_B=36\Delta$. The electronic DOS of the left (right) lead is shown in red (blue) and their overlap in brown. The single-particle level of the quantum dot is indicated at $-\frac{U}{2}$ in magenta.

Figure 8

Effects of damping $\Gamma$ of high-energy modes on the time evolution of the current ($U=0,{\chi }_{\text{TEBD}}=500$, QT I). Data shown are obtained for very low-bias voltage ($V_B=2\Delta$, group of gray curves in lower part of figure) and high-bias voltage ($V_B=32\Delta$, group of orange curves in upper part of figure). For each bias voltage, we compare data obtained by a standard ($\Gamma =0$) time evolution (full lines), data using an (empirically) optimally damped time evolution ($\Gamma =\Delta$, dashed-dotted lines) as well as for an overdamped evolution ($\Gamma =10\Delta$, dashed lines).

Figure 9

Convergence of the current with respect to several auxiliary numerical parameters. Left: Solid lines denote results obtained evaluating the expectation value of the current operator (A1), while dashed lines indicate data obtained by evaluating the time derivative of the expectation value of the particle number (A2) ($U=12\Delta$, $L=150$, ${\chi }_{\text{TEBD}}=2000$, QT I). Center: Matrix sizes ${\chi }_{\text{TEBD}}=250\text{(dotted}\text{line)},500\text{(dashed-dotted}\text{line)},2000\text{(dashed}\text{line),}\text{and}4000\text{(solid}\text{line)}$ are presented ($U=20\Delta$, $L=150$, QT I). Right: We show system sizes $L=20,40,60,80,100,120,\text{and}150$ (dotted, dashed-dotted, dashed, dashed-dashed-dotted-dotted, long-dashed-short-dashed, dashed-gap-dashed, and solid lines) at $U=0$, $L=150$, ${\chi }_{\text{TEBD}}=2000$ for QT I. The constant solid lines indicate the exact steady-state currents of the respective thermodynamic system.

Figure 10

Exact results for the noninteracting system. Comparison of the TEBD current (dashed lines) to an exact time evolution (solid black lines) for $U=0\Delta$, $L=150,{\chi }_{\text{TEBD}}=2000$, QT I. We show results for $V_B=4\Delta$ (cyan), $V_B=28\Delta$ (magenta), and $V_B=36\Delta$ (green). The respective maximum reliable simulation times (see Sec. 4a for definition) are indicated as triangles.

Figure 11

Maximum simulation time reachable (QT I, ${\chi }_{\text{TEBD}}=2000$) due to accumulation of entanglement entropy (left) and steady-state current (right). The left plot shows the time until a truncated weight of ${\epsilon }_c=5\times {10}^{-5}$ is reached at any bond of the chain for the first time. The right figure corresponds to the data in Fig. 4. Note the inverted color scale in the left image; dark regions correspond to low values of the maximum time reachable.