On Landauer--B\"uttiker formalism from a quantum quench

TL;DR

This paper analyzes quantum transport in one-dimensional free fermionic systems after a local potential quench, deriving the full counting statistics and revealing how initial state information persists or vanishes over time depending on the presence of bound states.

Contribution

It provides a detailed computation of the full counting statistics for a quantum quench in fermionic systems and links the long-time behavior to the Landauer--Büttiker formalism, highlighting the role of bound states.

Findings

FCS expressed via Fredholm determinant depending on scattering data

Persistent oscillations occur if multiple bound states are present

Asymptotic behavior aligns with Landauer--Büttiker formalism when no bound states

Abstract

We study transport in the free fermionic one-dimensional systems subjected to arbitrary local potentials. The bias needed for the transport is modeled by the initial highly non-equilibrium distribution where only half of the system is populated. Additionally to that, the local potential is also suddenly changed when the transport starts. For such a quench protocol we compute the Full Counting Statistics (FCS) of the number of particles in the initially empty part. In the thermodynamic limit, the FCS can be expressed via the Fredholm determinant with the kernel depending on the scattering data and Jost solutions of the pre-quench and the post-quench potentials. We discuss the large-time asymptotic behavior of the obtained determinant and observe that if two or more bound states are present in the spectrum of the post-quench potential the information about the initial state manifests itself in the persistent oscillations of the FCS. On the contrary, when there are no bound states the asymptotic behavior of the FCS is determined solely by the scattering data of the post-quench potential, which for the current (the first moment) is given by the Landauer--B\"uttiker formalism. The information about the initial state can be observed only in the transient dynamics.