Auxiliary master equation approach within stochastic wave functions: Application to the interacting resonant level model

We present further developments of the auxiliary master equation approach (AMEA), a numerical method to simulate many-body quantum systems in as well as out of equilibrium and apply it to the interacting resonant level model to benchmark the new developments. In particular, our results are obtained by employing the stochastic wave functions method to solve the auxiliary open quantum system arising within AMEA. This development allows us to reach extremely low wall times for the calculation of correlation functions with respect to previous implementations of AMEA. An additional significant improvement is obtained by extrapolating a series of results obtained by increasing the number of auxiliary bath sites, $N_B$, used within the auxiliary open quantum system formally to the limit of $N_B\to \infty$. Results for the current-voltage characteristics and for equilibrium correlation functions are compared with the one obtained by exact and matrix-product states–based approaches. Further, we complement this benchmark by the presentation of spectral functions for higher temperatures where we find different behaviors around zero frequency depending on the hybridization strength.

Figure 1

The stochastic wave function algorithm for the time evolution.

Figure 2

The stochastic wave function algorithm in the doubled Hilbert space which allows us to calculate correlation functions.

Figure 3

A sketch of the IRLM as lattice model and its mapping to the auxiliary open quantum system used within AMEA.

Figure 4

Comparison of the physical and auxiliary hybridization function at the boundary of the left bath, i.e., $r=-1$, and $T=0.025$. [(a) and (b)] Retarded-Keldysh part of the hybridization function for $L=19,\mu =2$. [(c) and (d)] Retarded-Keldysh part of the hybridization for $L=13,\mu =0$. The $L=19$ results where obtained with the ADAM routine from Sec. 2b1 while $L=13$ was optimized with PT. Solid lines represent the hybridization of the physical system, ${\mathrm{\Delta }}_{\text{ph}}$, and dashed lines that of the auxiliary system, ${\mathrm{\Delta }}_{\text{aux}}$. Panels (a) and (b) show a fit for $\mu \ne 0$ to exemplify the capability of representing a nonequilibrium situation. Panels (c) and (d) illustrate the fit used for the calculation of the equilibrium spectral functions in Fig. 6. The insets in panel (a) show a zoom onto the band edge and the region around the chemical potential, $\mu =1$, where the sudden occupation change in the Keldysh component typically induces oscillations in the retarded one.

Figure 5

Scaled steady-state current as function of the scaled bias voltage $V/V_c$. We plot the analytic solution for $T=0$ (solid black line), the extrapolated AMEA current (filled circles), and the current for $L=17$ and $L=19$ (open symbols). Shown are results for $J^{\prime }=0.2$ (red symbols) and $J^{\prime }=0.5$ (blue symbols). The arrows indicate the data points which correspond to the voltage $V=2$ for the two different considered $J^{\prime }$. The inset shows an example of the current vs. cost function $I\left(\chi \right)$ for $V=1.2,J^{\prime }=0.5$ (filled blue circles) and the corresponding linear fit (solid red line) as well as the extrapolated value at zero cost function (open red circle) together with the analytic result (filled black diamond). Other parameters are $T=0.025$ and $U=2$.

Figure 6

Equilibrium ($V=0$) spectral function at the impurity site, $r=0$, for different interaction strengths. We compare our results with Braun et al. [72] (obtained at $T=0$). Our parameters are $J^{\prime }=0.2,T=0.025$.

Figure 7

Equilibrium spectral function at the impurity site, $r=0$, for different temperatures and two different hybridization strengths. Panel (a) for $J^{\prime }=0.2$ and panel (b) for $J^{\prime }=0.5$.