Transport through a correlated interface: Auxiliary master equation approach

We present improvements of a recently introduced numerical method [E. Arrigoni et al., Phys. Rev. Lett. 110, 086403 (2013)] to compute steady-state properties of strongly correlated electronic systems out of equilibrium. The method can be considered as a nonequilibrium generalization of exact diagonalization based dynamical mean-field theory (DMFT). The key modification for the nonequilibrium situation consists in addressing the DMFT impurity problem within an auxiliary system consisting of the correlated impurity, $N_b$ uncorrelated bath sites, and two Markovian environments (sink and reservoir). Algorithmic improvements in the impurity solver allow to treat efficiently larger values of $N_b$ than previously in DMFT. This increases the accuracy of the results and is crucial for a correct description of the physical behavior of the system in the relevant parameter range including a semiquantitative description of the Kondo regime. To illustrate the approach, we consider a monoatomic layer of correlated orbitals, described by the single-band Hubbard model, attached to two metallic leads. The nonequilibrium situation is driven by a bias voltage applied to the leads. For this system, we investigate the spectral function and the steady-state current-voltage characteristics in the weakly as well as in the strongly interacting limit. In particular, we investigate the nonequilibrium behavior of quasiparticle excitations within the Mott gap of the correlated layer. We find for low-bias voltage Kondo-type behavior in the vicinity of the insulating phase. In particular, we observe a splitting of the Kondo resonance as a function of the bias voltage.

Figure 1

Schematic representation of the system, consisting of the correlated layer (red) with local Hubbard interaction $U$ and onsite energy ${\varepsilon }_c$, sandwiched between two semi-infinite metallic leads (blue), with onsite energies ${\varepsilon }_l$ and ${\varepsilon }_r$, respectively. The hopping between neighboring sites of the correlated layer is $t_c$, while the one for the left (right) lead is $t_l$ ($t_r$). Hybridization between the left (right) lead and the correlated layer is $v_l$ ($v_r$). A bias voltage $\mathrm{\Phi }={\mu }_l-{\mu }_r$ is applied between the leads.

Figure 2

Sketch of the auxiliary open quantum impurity problem consisting of the impurity site at position $i=0$ (red circle), $N_b=6$ bath sites (cyan circles), and two Markovian environments sink (blue) and reservoir (green). Parameters $E_{ij}$ and ${\mathrm{\Gamma }}_{ij}$ are explained in the main text.

Figure 3

Retarded $G^R$ and Keldysh $G^K$ Green's functions for the correlated layer, for Hubbard interaction $U=2$ and bias voltage $\mathrm{\Phi }=5$ (upper panel) and for $U=12$ and $\mathrm{\Phi }=10$ (lower panel). Other parameters are $t_c=1, t_l=t_r=2.5, v_l=v_r=0.5$ ($2v_l$ is our unit of energy). Solid, dashed, and dotted lines are obtained with $N_b=6$, 4, and 2 correspondingly.

Figure 4

Current $J$ vs bias voltage $\mathrm{\Phi }$ for different $U$. The solid red curve represents the $U=0$ result. The three curves peaked at about $\mathrm{\Phi }=4$ are for $U=2$ and the remaining three curves show the $U=12$ results. Solid, dashed, and dotted lines correspond to $N_b=6$, 4, and 2, respectively. Other parameters are as in Fig. 3.

Figure 5

Current $J$ vs interaction $U$ for two values of $\mathrm{\Phi }$. Calculations are performed for $N_b=4$. Other parameters are as in Fig. 3.

Figure 6

Single-particle spectral function for $N_b=4$, different values of $\mathrm{\Phi }, U=2$ (a), and $U=12$ (b). Other parameters are as in Fig. 3.

Figure 7

Single-particle spectral function for $N_b=6, U=2, t_c=0.1$ and for different values of $\mathrm{\Phi }$. Other parameters are as in Fig. 3.

Figure 8

Single-particle spectral function for $N_b=6, U=12, t_c=1$ and for different values of $\mathrm{\Phi }$. Other parameters are as in Fig. 3.

Figure 9

Single-particle spectral function for $N_b=4$, different values of $U, \mathrm{\Phi }=0$ (a), $\mathrm{\Phi }=3.5$ (b), and $\mathrm{\Phi }=15$ (c). Other parameters are as in Fig. 3.