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

Over the past few decades, there have been spectacular experimental developments in manipulating cold atoms (bosons or fermions) [1, 2], which allow one to probe quantum many-body physics, both for interacting and noninteracting systems. In this talk we focus on the noninteracting Fermi gas, for which a general theoretical framework has been developed over the recent years [3,4].
We consider a generic model of N non-interacting spinless fermions in d dimensions confined by a general trapping potential (we assume a central potential for d>1), in the ground-state. In d=1, for specific potentials, this system is related to classical random matrix ensembles. We develop a theoretical framework for studying the quantum fluctuations of the number of fermions N_D in a domain D of macroscopic size in the bulk of the Fermi gas (in d>1 we assume that D is a spherical domain). We show that the variance of N_D grows as N^((d-1)/d) * (A log(N) + B) for N>>1, and obtain the explicit dependence of A,B on the potential. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, which agrees with exact results for d=1.
The talk is based on the recent work [5].

[1] I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80 885 (2008).
[2] S. Giorgini, L. P. Pitaevski and S. Stringari, Rev. Mod. Phys. 80 1215 (2008).
[3] D. S. Dean, P. Le Doussal, S. N. Majumdar, G. Schehr, Phys. Rev. A 94, 063622 (2016).
[4] D. S. Dean, P. Le Doussal, S. N. Majumdar, G. Schehr, J. Phys. A: Math. Theor. 52 144006 (2019).
[5] N. R. Smith, P. Le Doussal, S. N. Majumdar, G. Schehr, arXiv:2008.01045.