Hole probability for noninteracting fermions in a $d$-dimensional trap

TL;DR

This paper analytically computes the probability that a spherical region is empty of particles in a system of noninteracting fermions, revealing universal scaling behaviors and large deviation properties in various trapping potentials.

Contribution

It introduces an exact analytical framework for the hole probability in fermionic systems, connecting it to random matrix ensembles and deriving universal and non-universal regimes.

Findings

Universal scaling function for hole probability in large N limit

Super exponential tail of hole probability with a universal amplitude

Exact large deviation formula for harmonic traps

Abstract

The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability $P(R)$ for a spherical region of radius $R$ in the case of $N$ noninteracting fermions in their ground state in a $d$-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large $N$ and in the bulk of the Fermi gas, $P(R)$ is described by a universal scaling function of $k_F R$, for which we obtain an exact formula ($k_F$ being the local Fermi wave-vector). It exhibits a super exponential tail $P(R)\propto e^{- \kappa_d (k_F R)^{d+1}}$ where $\kappa_d$ is a universal amplitude, in good agreement with existing numerical simulations. When $R$ is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.