Counting statistics for non-interacting fermions in a $d$-dimensional potential

TL;DR

This paper develops a method to compute the counting statistics of non-interacting fermions in arbitrary dimensions and potentials, revealing how quantum fluctuations scale with particle number and potential shape, and conjecturing similar behavior for entanglement entropy.

Contribution

It introduces a first-principle approach for counting statistics in higher-dimensional fermion systems and derives explicit variance scaling laws, extending known 1D results to general dimensions.

Findings

Variance of fermion number grows as N^{(d-1)/d} (A_d log N + B_d)

Explicit dependence of coefficients on potential and domain size

Conjecture of similar asymptotics for entanglement entropy in all dimensions

Abstract

We develop a first-principle 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 ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the support. We show that the variance of ${\cal N}_{\cal D}$ 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 ${\cal D}$ (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 $\cal{D}$, in any dimension, supported by exact results for $d=1$.