Steady-State Currents through Nanodevices: A Scattering-States Numerical Renormalization-Group Approach to Open Quantum Systems

We propose a numerical renormalization group (NRG) approach to steady-state currents through nanodevices. A discretization of the scattering-states continuum ensures the correct boundary condition for an open quantum system. We introduce two degenerate Wilson chains for current carrying left- and right-moving electrons reflecting time-reversal symmetry in the absence of a finite bias $V$. We employ the time-dependent NRG to evolve the known steady-state density operator for a noninteracting junction into the density operator of the fully interacting nanodevice at finite bias. We calculate the differential conductance as function of $V$, $T$, and the external magnetic field.

Figure 1

The local $d$-orbital is expanded in left-moving and right-moving scattering states. Each contributions defines one fictitious local orbital $d_{\sigma \alpha }$ of the junction of the scattering-states NRG. The Coulomb repulsion introduces backscattering between left and right movers.

Figure 2

Nonequilibrium spectral function for (a) a symmetric junction $R=1$ at various values of finite bias voltage $V$, and (b) for a strongly asymmetric junction $R=1000$. The insets show the evolution of the Kondo-resonance. Parameters: $U=8$, ${ϵ}_f=-4$, and $T\to 0$.

Figure 3

The differential conductance $G=dI/dV$ as function of the bias voltage (a) for different asymmetry factors $R$, (b) for different magnetic field $H=0$, 0.1, 0.2, 0.4, and $R=1$. Parameters: as in Fig. 2. (c) Comparison between the results for $U=5$ from Ref. 16 and the NRG calculation at $T/\Gamma =0.04$ and $R=1$ using $z$-averaging over 4 $z$-values [20, 21].

Figure 4

The differential conductance $G$ as function of the bias voltage for different temperatures. Parameters $R=10$, ${ϵ}_f=-1.5$, and $U=12$.