Green’s Functions from Real-Time Bold-Line Monte Carlo Calculations: Spectral Properties of the Nonequilibrium Anderson Impurity Model

The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real-time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model’s spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a coupling or gate quench are studied. This representation is shown to contain additional information about the dot’s population dynamics. Further, we show that the voltage-dependent differential conductance gives a reasonable qualitative estimate of the equilibrium spectral function, but significant qualitative differences are found including incorrect trends and spurious temperature dependent effects.

Figure 1

The experimental auxiliary lead setup (top left) and the double probe scheme (top right) are illustrated. Below, the steady state spectral function $A(\omega )$ is shown at several voltages. The results are obtained from bold-CTQMC calculations using the double probe auxiliary lead formalism at $\mathrm{\Gamma }t=10$. Error bars estimate statistical Monte Carlo errors.

Figure 2

The time evolution of the spectral function $A_{\mathrm{aux}}(\omega )$ shown at several voltages, obtained from bold-CTQMC calculations using the double-probe auxiliary lead formalism, with $\eta ={10}^{-3}\mathrm{\Gamma }$.

Figure 3

The time evolution of the real part of the retarded Green’s function $\Re \{G^r(t,t-t^{\prime })\}$ (left panels) and the spectral function $A(\omega )$ (right panels) at voltages indicated, as calculated from two time correlation functions within bold-CTQMC calculations.

Figure 4

The steady state spectral function $A(\omega )$ (upper panel) compared to the steady state differential conductance (middle panel) as a function of half the voltage at two different inverse temperatures $\beta$. Both observables are obtained from bold-CTQMC at $\mathrm{\Gamma }t=10$. The area between the curves are shaded according to the maximal value and vertical dotted lines mark crossing points. The bottom panel displays all results: the solid lines are the spectral function and the dashed curves are the differential conductance.