Hole probability for noninteracting fermions in a d-dimensional trap

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 P(R) proportional to e(-kappa d(kFR)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. Copyright (C) 2022 EPLA

Automated summary

The study focuses on the hole probability of N noninteracting fermions, obtaining a universal scaling function and a super-exponential tail for the probability P(R) of a sphere region. The results are in good agreement with existing numerical simulations. At the order of the radius of the Fermi gas, the hole probability is described by a large deviation form.