Memory effects in nonequilibrium quantum impurity models

Memory effects play a key role in the dynamics of strongly correlated systems driven out of equilibrium. In this paper, we explore the nature of memory in the nonequilibrium Anderson impurity model. The Nakajima-Zwanzig-Mori formalism is used to derive an exact generalized quantum master equation for the reduced density matrix of the interacting quantum dot, which includes a non-Markovian memory kernel. A real-time path integral formulation is developed in which all diagrams are stochastically sampled in order to numerically evaluate the memory kernel. We explore the effects of temperature down to the Kondo regime, as well as the role of source-drain-bias voltage and bandwidth on the memory. Typically, the memory decays on time scales significantly shorter than the dynamics of the reduced density matrix itself, yet under certain conditions, it develops a low magnitude but long-ranged tail. In addition, we address the conditions required for the existence, uniqueness, and stability of a steady state.

Figure 1

Distinct nonzero memory-kernel elements for an initially unoccupied dot and several values of the interaction energy $U$. The remaining parameters are $\frac{{\mu }_L}{\Gamma }=-\frac{{\mu }_R}{\Gamma }=\frac{1}{2}$, $\frac{{\varepsilon }_c}{\Gamma }=20$, and $\beta \Gamma =1$.

Figure 2

Total population derivative from direct RT-PIMC data compared with the results of the memory-kernel formalism (left panels) and predicted dot populations (right panels) for an initially unoccupied dot for several values of the interaction energy $U$. Remaining parameters are the same as in Fig. 1. The cutoff time is $\frac{1.5}{\Gamma }$, shown on the right panels as a vertical dotted line.

Figure 3

The average absolute value of memory-kernel elements at $\frac{{\mu }_L}{\Gamma }=-\frac{{\mu }_R}{\Gamma }=\frac{1}{2}$, $\frac{{\varepsilon }_c}{\Gamma }=20$, $\beta \Gamma =1$, and different values of the interaction energy $U$.

Figure 4

The average absolute value of memory-kernel elements at ${\mu }_L={\mu }_R=0$, $\frac{{\varepsilon }_c}{\Gamma }=40$, $\frac{U}{\Gamma }=6$, and different temperatures $\beta$.

Figure 5

The predicted steady-state values of the diagonal density matrix elements at $\frac{{\mu }_L}{\Gamma }=-\frac{{\mu }_R}{\Gamma }=\frac{1}{2}$, $\frac{{\varepsilon }_c}{\Gamma }=20$, $\beta \Gamma =1$, and $\frac{U}{\Gamma }=6$.