Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

TL;DR

This paper analyzes the dynamics of a one-dimensional free fermionic chain with a variable link, providing exact formulas for full counting statistics and Loschmidt echo, revealing how the link's properties influence steady states and oscillations.

Contribution

It introduces an exact Fredholm determinant representation for time-dependent statistics and Loschmidt echo in a fermionic chain with a variable link, highlighting the impact of the link's hopping constant.

Findings

Local steady state forms when the link's hopping is smaller than the bulk.

Persistent oscillations occur when the link's hopping exceeds the bulk value.

Oscillation frequency relates to bound state energy and chemical potential bias.

Abstract

We consider an integrable system of two one-dimensional fermionic chains connected by a link. The hopping constant at the link can be different from that in the bulk. Starting from an initial state in which the left chain is populated while the right is empty, we present time-dependent full counting statistics and the Loschmidt echo in terms of Fredholm determinants. Using this exact representation, we compute the above quantities as well as the current through the link, the shot noise and the entanglement entropy in the large time limit. We find that the physics is strongly affected by the value of the hopping constant at the link. If it is smaller than the hopping constant in the bulk, then a local steady state is established at the link, while in the opposite case all physical quantities studied experience persistent oscillations. In the latter case the frequency of the oscillations is determined by the energy of the bound state and, for the Loschmidt echo, by the bias of chemical potentials.