Auxiliary master equation approach to nonequilibrium correlated impurities

We present a numerical method for the study of correlated quantum impurity problems out of equilibrium, which is particularly suited to address steady-state properties within dynamical mean field theory. The approach, recently introduced by Arrigoni et al. [Phys. Rev. Lett. 110, 086403 (2013)], is based upon a mapping of the original impurity problem onto an auxiliary open quantum system, consisting of the interacting impurity coupled to bath sites as well as to a Markovian environment. The dynamics of the auxiliary system is governed by a Lindblad master equation whose parameters are used to optimize the mapping. The accuracy of the results can be readily estimated and systematically improved by increasing the number of auxiliary bath sites, or by introducing a linear correction. Here, we focus on a detailed discussion of the proposed approach including technical remarks. To solve for the Green's functions of the auxiliary impurity problem, a non-Hermitian Lanczos diagonalization is applied. As a benchmark, results for the steady-state current-voltage characteristics of the single-impurity Anderson model are presented. Furthermore, the bias dependence of the single-particle spectral function and the splitting of the Kondo resonance are discussed. In its present form, the method is fast, efficient, and features a controlled accuracy.

Figure 1

(a) Sketch of the quantum impurity model (1) consisting of an impurity with interaction $U$ coupled via hybridizations $t_{\lambda }^{\prime }$ to noninteracting leads at chemical potential ${\mu }_{\lambda }$ and temperature $T_{\lambda }$, $\lambda \in \{L,R\}$. (b) Illustration of the auxiliary open quantum system [Eq. (10a)] with single-particle parameters $E_{\mu \nu }$ and Lindblad dissipators ${\Gamma }_{\mu \nu }^{\kappa }$ consisting of the impurity at site $f=0$, $N_B$ bath sites ($N_B=4$ in the plot), as well as a Markovian environment (shaded areas). When evaluating linear corrections (see Appendix pp3), an additional site $N_B+1$ is used.

Figure 2

Comparison of $\text{Im}\left[{\Delta }^{\alpha }\left(\omega \right)\right]$ from (7) (black) with $\text{Im}\left[{\Delta }_{\text{aux}}^{\alpha }\left(\omega \right)\right]$ at the absolute minimum of the cost function (12) for auxiliary system sizes $N_B=2,4,\text{and}6$ bath sites (green, blue, and orange, respectively), and $\alpha =R$ (top) and $K$ (bottom). Results are shown for tight-binding leads (2) and (8) with $t=10{\Delta }_0$, and three different bias voltages $\varphi \in \{10,27,28\}{\Delta }_0$ from left to right.

Figure 3

Current $j$ vs voltage $\varphi$ for the model (1) with tight-binding leads and onsite interaction $U=12{\Delta }_0$ (left) and $U=20{\Delta }_0$ (right). Results for three different auxiliary systems with $N_B\in \{2,4,6\}$ are displayed and compared with reference data from TEBD (magenta dotted and $\times$) [77]. We plot the averaged mean values connected by lines together with error bars determined according to Sec. 3a and Appendix pp2. The additional data marks for $U=20{\Delta }_0$ are as follows: The circles for $N_B=6$ display $j\left(\varphi \right)$ when considering the absolute minimum in the fit (12). $N_B=2,\mathrm{lc}$ and $N_B=4,\mathrm{lc}$ present the results of a linear correction of the current values of the absolute minima as described in Appendix pp3. The inset displays the difference $\Delta j$ of the calculated currents to the TEBD results.

Figure 4

Single-particle spectral function at the impurity evaluated for $N_B=6$, different bias voltages $\varphi$, and $U=12{\Delta }_0$ (left) and $U=20{\Delta }_0$ (right). Data are obtained according to Sec. 3a and Appendix pp2. Other parameters are as in Fig. 3.

Figure 5

Single-particle spectral function for a constant DOS of the leads (3), with $D_{\text{WB}}=20{\Delta }_0$ and $U=16{\Delta }_0$ and different bias voltages $\varphi$ (in units of ${\Delta }_0$). Results are obtained for $N_B=6$ and at the absolute minimum of Eq. (12). For a comparison with SNRG [53] [Fig. 2 therein], note that their $\Gamma =2{\Delta }_0$.

Figure 6

Effects of the linear corrections of the Green's functions according to Apendix pp3 (solid blue lines). The dashed lines indicate the uncorrected $\underline{G}\left(\omega \right)$ with the same $N_B$, while solid orange and light blue lines display results for larger $N_B$ for comparison. Results are shown for a constant lead DOS [Eq. (3)] with $D_{\text{WB}}=20{\Delta }_0$, $U=16{\Delta }_0$, and $\varphi =4{\Delta }_0$.