Transport through an Anderson impurity: Current ringing, nonlinear magnetization, and a direct comparison of continuous-time quantum Monte Carlo and hierarchical quantum master equations

We give a detailed comparison of the hierarchical quantum master equation (HQME) method to a continuous-time quantum Monte Carlo (CT-QMC) approach, assessing the usability of these numerically exact schemes as impurity solvers in practical nonequilibrium calculations. We review the main characteristics of the methods and discuss the scaling of the associated numerical effort. We substantiate our discussion with explicit numerical results for the nonequilibrium transport properties of a single-site Anderson impurity. The numerical effort of the HQME scheme scales linearly with the simulation time but increases (at worst exponentially) with decreasing temperature. In contrast, CT-QMC is less restricted by temperature at short times, but in general the cost of going to longer times is also exponential. After establishing the numerical exactness of the HQME scheme, we use it to elucidate the influence of different ways to induce transport through the impurity on the initial dynamics, discuss the phenomenon of coherent current oscillations, known as current ringing, and explain the nonmonotonic temperature dependence of the steady-state magnetization as a result of competing broadening effects. We also elucidate the pronounced nonlinear magnetization dynamics, which appears on intermediate time scales in the presence of an asymmetric coupling to the electrodes.

Figure 1

(a) Graphical representation of an Anderson impurity, which is realized by a quantum dot (QD). The dot is coupled to a left (L) and a right electrode (R). (b) Single-particle levels of the quantum dot junction depicted in (a). The shaded areas depict the occupied states in the electrodes, which, for simplicity, are assumed to provide Lorentzian-shaped conduction bands. Applying a bias voltage to such a system means that the corresponding chemical potentials ${\mu }_{\text{L/R}}$ are shifted and the electrodes become charged. (c) Single-particle levels of a molecular junction. In contrast to the quantum dot realization [(a) and (b)], the conduction bands are shifted in the same way as the chemical potentials. Thus, applying a bias voltage, the electrodes do not become charged.

Figure 2

Sketch of the areas in simulation time and temperature where HQME and CT-QMC are useful. The dashed lines represent the exponential growth of the numerical effort with the simulation time in CT-QMC (black line) and with the number of coefficients in HQME (gray line) that increases (at worst exponentially) with the inverse temperature. High temperatures and short simulation times are accessible by both methods (white area). Very long simulation times are accessible only by HQME (yellow area). The low-temperature regime is reserved for CT-QMC (blue area). At low temperatures and if long simulation times are required, both CT-QMC and HQME cannot be used. For the given problem, the exponential walls are located around the Kondo temperature for HQME and time scales $\sim 10/\mathrm{\Gamma }$ for CT-QMC (cf. Sec. 3b and Refs. [43, 44, 80]).

Figure 3

Symmetrized current $I=(I_{\text{L}}-I_{\text{R}})/2$ flowing through the impurity at $k_{\text{B}}T=5\mathrm{\Gamma }$ as a function of time $t$ for a sequence of equally spaced bias voltages $\mathrm{\Phi }=\mathrm{\Gamma },3\mathrm{\Gamma },\cdots ,19\mathrm{\Gamma }$ where the conduction bands are not shifted with the bias voltage ($S=0$). The model parameters used to obtain this data are summarized in Table 1. After a linear increase, the current saturates to a stationary value on a voltage-independent time scale $0.2/\mathrm{\Gamma }$. The HQME (blue lines) and CT-QMC methods (orange lines) give identical results to within the numerical resolution of the data.

Figure 4

Symmetrized current $I=(I_{\text{L}}-I_{\text{R}})/2$ flowing through the impurity at $k_{\text{B}}T=\mathrm{\Gamma }$ as a function of time $t$ for a sequence of equally spaced bias voltages $\mathrm{\Phi }=\mathrm{\Gamma },3\mathrm{\Gamma },\cdots ,19\mathrm{\Gamma }$ where the conduction bands are not shifted with the bias voltage ($S=0$). The model parameters used to obtain this data are summarized in Table 1. After a linear increase, the current level slightly oscillates before it reaches its stationary value. The corresponding time scales are given by the dynamical phases of the problem and the inverse temperature $1/\left(k_{\text{B}}T\right)=1/\mathrm{\Gamma }$, respectively.

Figure 5

Symmetrized current $I=(I_{\text{L}}-I_{\text{R}})/2$ flowing through the impurity at $k_{\text{B}}T=\mathrm{\Gamma }/5$ as a function of time $t$ for a sequence of equally spaced bias voltages $\mathrm{\Phi }=\mathrm{\Gamma },3\mathrm{\Gamma },..,19\mathrm{\Gamma }$ where the conduction bands are not shifted with the bias voltage ($S=0$). The model parameters used to obtain this data are summarized in Table 1. The oscillations of the current level that appear right after the initial linear increase become more pronounced at lower temperatures. The corresponding time scales are given by the dynamical phases of the problem and the inverse temperature $1/\left(k_{\text{B}}T\right)=\mathrm{\Gamma }/5$, respectively. Deviations between the HQME (blue lines) and CT-QMC results (orange lines) are consistent.

Figure 6

Symmetrized current $I=(I_{\text{L}}-I_{\text{R}})/2$ flowing through the impurity at $k_{\text{B}}T=\mathrm{\Gamma }$ as a function of time $t$ for a sequence of equally spaced bias voltages $\mathrm{\Phi }=\mathrm{\Gamma },3\mathrm{\Gamma },\cdots ,11\mathrm{\Gamma }$ and $\mathrm{\Phi }=19\mathrm{\Gamma }$ where the conduction bands are shifted with the bias voltage ($S=1$). The model parameters used to obtain this data are summarized in Table 1. In contrast to Figs. 3, 4, 5, the current level is initially not linearly increasing with the time $t$ when the conduction bands are shifted with the applied bias voltage ($S=1$).

Figure 7

Magnetization of an Anderson impurity that is symmetrically coupled to the electrodes as a function of temperature and bias voltage ($S=0$), and for three different times (top row: $t=0.5/\mathrm{\Gamma }$; middle row: $t=5/\mathrm{\Gamma }$; bottom row: steady state). The left column depicts the full HQME result. The middle and the right column has been obtained by truncating the hierarchy (7) at the second and the first tier, respectively. The magnetization along the red dashed line in the lower left plot is also depicted in Fig. 8.

Figure 8

Magnetization of an Anderson impurity that is symmetrically coupled to the electrodes as a function of temperature in the steady state and applied bias voltages $\mathrm{\Phi }=\pm 13\mathrm{\Gamma }, \pm 15\mathrm{\Gamma }$, and $\pm 17\mathrm{\Gamma }$ ($S=0$). Note that the kinks originate from a simple linear interpolation between the data points.

Figure 9

Magnetization of an Anderson impurity asymmetrically coupled to the electrodes as a function temperature and bias voltage ($S=0$) and for three different times (top row: $t=0.5/\mathrm{\Gamma }$; middle row: $t=5/\mathrm{\Gamma }$; bottom row: steady state). The left column depicts the full HQME result. The middle and the right columns have been obtained by truncating the hierarchy (7) at the second and the first tier, respectively.

Figure 10

Convergence analysis of the symmetrized current $I=(I_{\text{L}}-I_{\text{R}})/2$ that is flowing through the impurity at $k_{\text{B}}T=\mathrm{\Gamma }$ and $\mathrm{\Phi }=5\mathrm{\Gamma }$ ($S=0$) as a function of time $t$. (a) shows the convergence behavior for $U=8\mathrm{\Gamma }$. The behavior for $U=0$ is shown in (b). The model parameters used to obtain this data are summarized in Table 1.