Hole probability for noninteracting fermions in a d-dimensional trap

¹ Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS, ENS & PSL University, Sorbonne Université, Université de Paris - 75005 Paris, France
² Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589 4 Place Jussieu, 75252 Paris Cedex 05, France

^(a) gregory.schehr@u-psud.fr (corresponding author)

Received: 3 September 2021
Accepted: 12 January 2022

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 sphere 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 where 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.