Counting statistics for noninteracting fermions in a d-dimensional potential

We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d > 1). In a confining potential, the Fermi gas is supported over a bounded domain. In d = 1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N-D in a domain D of macroscopic size in the bulk of the support. We show that the variance of N-D grows as N(d-1)/d (A(d) log N + B-d) for large N, and obtain the explicit dependence of A(d), B-d on the potential and on the size of D (for a spherical domain in d > 1). This generalizes the free-fermion results for microscopic domains, given in d = 1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d = 1.

Automated summary

A first-principles approach was developed to compute the counting statistics of N noninteracting spinless fermions in the ground state in arbitrary dimensions. The variance of N-D was shown to grow as N(d-1)/d for large N, with explicit dependence on the potential and size of D. Conjectures were made for similar asymptotics in entanglement entropy in any dimension.