Counting statistics for noninteracting fermions in a d -dimensional potential

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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 ND in a domain D of macroscopic size in the bulk of the support. We show that the variance of ND grows as N(d-1)/d(AdlogN+Bd) for large N, and obtain the explicit dependence of Ad,Bd 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.

Original languageEnglish
Article numberL030105
JournalPhysical Review E
Issue number3
StatePublished - 1 Mar 2021
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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