Non-interacting fermions in hard-edge potentials

TL;DR

This paper studies the spatial fluctuations of non-interacting Fermi gases in non-smooth potentials, revealing new 'hard edge' kernels that differ from the classical 'soft edge' kernels, and connects these findings to random matrix theory.

Contribution

It introduces and characterizes 'hard edge' kernels for fermions in non-smooth potentials, extending the understanding of quantum correlations near boundaries and singularities.

Findings

Hard edge kernels depend on dimension and temperature.

Kernels interpolate between hard and soft edge behaviors.

Connections established between 1D fermion correlations and random matrix ensembles.

Abstract

We consider the spatial quantum and thermal fluctuations of non-interacting Fermi gases of $N$ particles confined in $d$-dimensional non-smooth potentials. We first present a thorough study of the spherically symmetric pure hard-box potential, with vanishing potential inside the box, both at $T=0$ and $T>0$. We find that the correlations near the wall are described by a "hard edge" kernel, which depend both on $d$ and $T$, and which is different from the "soft edge" Airy kernel, and its higher $d$ generalizations, found for smooth potentials. We extend these results to the case where the potential is non-uniform inside the box, and find that there exists a family of kernels which interpolate between the above "hard edge" kernel and the "soft edge" kernels. Finally, we consider one-dimensional singular potentials of the form $V(x)\sim |x|^{-\gamma}$ with $\gamma>0$. We show that the correlations close to the singularity at $x=0$ are described by this "hard edge" kernel for $1\leq\gamma<2$ while they are described by a broader family of "hard edge" kernels known as the Bessel kernel for $\gamma=2$ and, finally by the Airy kernel for $\gamma>2$. These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the $1d$ fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent Letter, EPL 120, 10006 (2017).