Flow equation calculation of transient and steady-state currents in the Anderson impurity model

Transient and steady-state currents through dc-biased quantum impurity models beyond the linear-response regime are of considerable interest, both from an experimental and a theoretical point of view. Here we present an analytical approach for the calculation of such currents based on the flow equation method (method of infinitesimal unitary transformations). Specifically, we analyze the Anderson impurity model in its mixed valence regime where the coupling to the leads is switched on suddenly at time $t=0$. We observe the real time buildup of the current until it reaches its steady-state limit.

Figure 1

Schematic representation of the parameters in the Anderson impurity model.

Figure 2

The current correction $I^{\left(c\right)}\left(t\right)$ due to interaction at particle-hole symmetry, ${ϵ}_d=-U/2$, for zero temperature. The interaction strength is $U=\Gamma$. Results for voltage bias $V_{sd}=\Gamma$ and $V_{sd}=2\Gamma$ are depicted. The main features of $I^{\left(c\right)}\left(t\right)$ are a vanishing derivative at $t=0$, followed by a sharp decrease and finally a smooth crossover toward its steady value. One also notices the onset of oscillations at large voltage bias $V_{sd}=2\Gamma$.

Figure 3

The current without interaction and for interaction strength $U=1.5\Gamma$ at voltage bias $V_{sd}=\Gamma$. The interaction suppresses the current. The inset shows the suppressed oscillation of the current.

Figure 4

The current without interaction and for interaction strength $U=1.5\Gamma$ at voltage bias $V_{sd}=2\Gamma$. The free current increases compared to $V_{sd}=\Gamma$ while its interaction suppression also becomes stronger due to shot-noise-induced decoherence. The inset shows suppressed current oscillation.