Many-body wavefunctions for quantum impurities out of equilibrium. II. Charge fluctuations

We extend the general formalism discussed in the previous paper (A. B. Culver and N. Andrei, [Phys. Rev. B 103, 195106 (2021)]) to two models with charge fluctuations: the interacting resonant level model and the Anderson impurity model. In the interacting resonant level model, we find the exact time-evolving wavefunction and calculate the steady-state impurity occupancy to leading order in the interaction. In the Anderson impurity model, we find the nonequilibrium steady state for small or large Coulomb repulsion $U$, and we find that the steady-state current to leading order in $U$ agrees with a Keldysh perturbation theory calculation.

Figure 1

Left: the steady-state occupancy $n_d\equiv {\langle n_d\rangle }_{\text{s.s.}}$ at zero temperature in the two-lead IRL, either as a function of dot potential ${\epsilon }_d$ or voltage $V$. The leads are held at chemical potentials ${\mu }_1=0$ and ${\mu }_2=-V$. The equilibrium ($V=0$) curves are given by Eq. (3.27) and in fact are independent of the number of leads, in agreement with the Bethe ansatz answer from the literature (Appendix pp5). The nonequilibrium ($V\ne 0$) curves are given by Eq. (3.28). In both cases, we compare the noninteracting occupancy ($\rho U=0$) with the weakly interacting occupancy (first order in $\rho U=0.1$), where $\rho =1/\left(2\pi \right)$ is the density of states per unit length. Right: the weakly interacting case with the noninteracting occupancy subtracted, i.e., $\delta n_d\equiv n_d-n_d{|}_{U=0}$. In equilibrium, $\delta n_d$ reaches finite limits as ${\epsilon }_d/T_K\to \pm \infty$. Out of equilibrium, $|\delta n_d|$ grows logarithmically as $V/T_K\to \pm \infty$, indicating that some resummation of the series in $\rho U$ is needed to make sense of the extremely large voltage regime.