Stationary time correlations for fermions after a quench in the presence of an impurity

TL;DR

This paper analyzes the long-time behavior of non-interacting fermions after a quench in the presence of an impurity, revealing how correlations evolve and depend on impurity properties, with exact results for large time limits.

Contribution

It provides exact large-time multi-time correlation functions for fermions after a quench with a finite-size impurity, including effects of bound states and memory.

Findings

Correlations become time translationally invariant at large times.

In the non-equilibrium steady state, correlations are independent of initial reflection coefficients.

Bound states induce oscillations and memory effects in correlations.

Abstract

We consider the quench dynamics of non-interacting fermions in one dimension in the presence of a finite-size impurity at the origin. This impurity is characterized by general momentum-dependent reflection and transmission coefficients which are changed from ${\sf r}_0(k), {\sf t}_0(k)$ to ${\sf r}(k), {\sf t}(k)$ at time $t=0$. The initial state is at equilibrium with ${\sf t}_0(k)=0$ such that the system is cut in two independent halves with ${\sf r}_0^R(k)$, ${\sf r}_0^L(k)$ respectively to the right and to the left of the impurity. We obtain the exact large time limit of the multi-time correlations. These correlations become time translationally invariant, and are non-zero in two different regimes: (i) for $x=O(1)$ where the system reaches a non-equilibrium steady state (NESS) (ii) for $x \sim t$, i.e., the ray-regime. For a repulsive impurity these correlations are independent of ${\sf r}_0^R(k)$, ${\sf r}_0^L(k)$, while in the presence of bound states they oscillate and memory effects persist. We show that these nontrivial relaxational properties can be retrieved in a simple manner from the large time behaviour of the single particle wave functions.