Auxiliary master equation approach within matrix product states: Spectral properties of the nonequilibrium Anderson impurity model

Within the recently introduced auxiliary master equation approach it is possible to address steady state properties of strongly correlated impurity models, small molecules, or clusters efficiently and with high accuracy. It is particularly suited for dynamical mean field theory in the nonequilibrium as well as in the equilibrium case. The method is based on the solution of an auxiliary open quantum system, which can be made quickly equivalent to the original impurity problem. In its first implementation a Krylov space method was employed. Here, we aim at extending the capabilities of the approach by adopting matrix product states for the solution of the corresponding auxiliary quantum master equation. This allows for a drastic increase in accuracy and permits us to access the Kondo regime for large values of the interaction. In particular, we investigate the nonequilibrium steady state of a single-impurity Anderson model and focus on the spectral properties for temperatures $T$ below the Kondo temperature $T_K$ and for small bias voltages $\varphi$. For the two cases considered, with $T\approx T_K/4$ and $T\approx T_K/10$, we find a clear splitting of the Kondo resonance into a two-peak structure for $\varphi$ close above $T_K$. In the equilibrium case ($\varphi =0$) and for $T\approx T_K/4$, the obtained spectral function essentially coincides with the one from numerical renormalization group.

Figure 1

The upper part of the figure shows a schematic drawing of the auxiliary system in the superfermion representation, for $N_B$ bath sites and with the impurity located at the central site $f=N_B/2$. The upper chain corresponds to original sites and the lower chain to the additionally introduced “tilde” sites; see Sec. 2e1. In the chosen fit restriction, see Sec. 2d, the coupling terms of the Lindblad operator $L$, Eq. (21), represent a ladder geometry with cross links. ${\mathrm{\Gamma }}^{\left(1\right)}$ causes a directional hopping from the upper to the lower chain and for ${\mathrm{\Gamma }}^{\left(2\right)}$ it is vice versa. Moreover, ${\mathrm{\Gamma }}_{i,j}^{\left(2\right)}$ is nonzero only for $i,j<f$ and ${\mathrm{\Gamma }}_{i,j}^{\left(1\right)}$ only for $i,j>f$. In the lower part, in gray scale, we schematically depict the tensor network for TEBD (Sec. 2e2), where $L$ is decomposed in nearest neighbor terms $L_{i,i+1}$ which are applied in an alternating manner.

Figure 2

Temporal evolution of the bipartite entanglement entropy $S$ in a typical ${\mathrm{IM}}_{\mathrm{aux}}$ with $N_B=12$, representing the case $\varphi =1\mathrm{\Gamma }$. The system is in the steady state for $t<0$ and $c_{f\sigma }$ is applied to $|{\rho }_{\infty }\rangle$ at $t=0$. We show $S\left(t\right)$ where it is largest, namely for the innermost bonds at the impurity, as well as for the next ones to the outside.

Figure 3

Spectral function in equilibrium: plotted for different number of bath sites $N_B$ and compared with reference data from NRG [86]. Results are for $U=6\mathrm{\Gamma }$ and flat band leads, Eq. (11), with $D=10\mathrm{\Gamma }$ and $T=0.05\mathrm{\Gamma }$.

Figure 4

Bias-dependent spectral function (left) and retarded self-energy (right) for $T=0.05\mathrm{\Gamma }$. Solid lines correspond to calculations with $N_B=16$ and dash-dotted lines to $N_B=14$, but in many cases they cannot be distinguished. Bias voltage $\varphi$ is given in units of $\mathrm{\Gamma }$. Results are for $U=6\mathrm{\Gamma }$ and flat band leads, Eq. (11), with $D=10\mathrm{\Gamma }$.

Figure 5

Bias-dependent spectral function (left) and retarded self-energy (right) for $T=0.5\mathrm{\Gamma }$. Calculations are performed with $N_B=10$ and other parameters are the same as in Fig. 4.

Figure 6

Double occupancy (left) and transport properties (right) as a function of bias voltage $\varphi$. Current $j$ is depicted with solid lines and the differential conductance $\partial j/\partial \varphi$ with dash-dotted lines. The latter is calculated with three-point Lagrange polynomials, based on the data for $j$ as marked in the plot. Results are shown for $T=0.5\mathrm{\Gamma }$ with $N_B=10$, and for $T=0.05\mathrm{\Gamma }$ with $N_B=14$. Other parameters are as in Fig. 4. The linear response and equilibrium values for $\langle n_{f\uparrow }{n}_{f\downarrow }\rangle$ are from NRG [86].

Figure 7

Bias-dependent spectral function (left) and retarded self-energy (right) for $U=6\mathrm{\Gamma }, T=0.02\mathrm{\Gamma }$, and a Lorentzian density of states in the leads; see Eq. (24). Solid lines correspond to calculations with $N_B=16$ whereas dash-dotted lines to ones with $N_B=14$. The bias voltage $\varphi$ is in units of $\mathrm{\Gamma }$. The inset on the left is for $N_B=16$ and $\varphi =0$.

Figure 8

Hybridization function ${\underline{\mathrm{\Delta }}}_{\mathrm{aux}}\left(\omega \right)$ as obtained from minimizing the cost function Eq. (A1) with ${\omega }_c=15\mathrm{\Gamma }$ and $W\left(\omega \right)=1$, for the flat band model Eqs. (10), (11) with $\varphi =2\mathrm{\Gamma }$ and $T=0.05\mathrm{\Gamma }$. Results on the left are for $N_B=12$ and on the right for $N_B=14$.

Figure 9

Convergence of ${\underline{\mathrm{\Delta }}}_{\mathrm{aux}}\left(\omega \right)$ with increasing $N_B$. Results with $T=0.5\mathrm{\Gamma }$ and $T=0.05\mathrm{\Gamma }$ are for the flat band case Eqs. (10), (11), and the ones with $T=0.02\mathrm{\Gamma }$ are for the Lorentzian density of states Eq. (24). For the cost function $\mathcal{C}$, Eq. (A1), we chose $W\left(\omega \right)=1$ as well as ${\omega }_c=15\mathrm{\Gamma }$ for the flat band model and ${\omega }_c=5\mathrm{\Gamma }$ for the Lorentzian case. The normalization ${\mathcal{C}}_0$ refers to the value of $\mathcal{C}$ for ${\underline{\mathrm{\Delta }}}_{\mathrm{aux}}\left(\omega \right)\equiv 0$.

Figure 10

Convergence of the spectral function as depicted in Fig. 3 with increasing $N_B$, i.e., for the flat band model with $U=6\mathrm{\Gamma }$ and $T=0.05\mathrm{\Gamma }$.