Quantum propagating front and the edge of the Wigner function

TL;DR

This paper investigates the behavior of the Wigner function and quantum front propagation in one-dimensional fermionic systems, revealing connections to Airy functions and the Airy kernel, with implications for semiclassical limits and quantum dynamics.

Contribution

It introduces a detailed analysis of the Wigner function near the Fermi surf and demonstrates the emergence of the Airy kernel in quantum front dynamics after a potential quench.

Findings

Wigner function near the Fermi surf can be expressed with Airy functions.

The long-time limit of the quantum front exhibits the Airy kernel.

Power law decay of the potential causes anomalous diffusive spreading.

Abstract

In the first part of the article, we study one-dimensional noninteracting fermions in the continuum and in the presence of the repulsive inverse power law potential, with an emphasis on the Wigner function in the semiclassical limit. In this limit, the Wigner function exhibits an edge called the Fermi surf that depends only on the classical one-particle Hamiltonian. Around the Fermi surf, under a well-defined semiclassical limit, the Wigner function can be expressed in terms of Airy functions which yield a smooth matching between the two regions delimited by the Fermi surf. In the second part of the article, the system is prepared in the ground state of the inverse power law potential where only the left half line is filled with fermions. Then the potential is switched off, resulting in the emergence of a propagating quantum front. We show that the power law decay of the pre-quench potential that separates the left and right half systems leads to the emergence of the Airy kernel (well known in Random Matrix Theory) at the quantum front in the long-time limit. This also comes with anomalous diffusive spreading around the front.