Full counting statistics for interacting trapped fermions

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

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We study N spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index β. In the fermion model β controls the strength of the interaction, β = 2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions ND in a domain D of macroscopic size in the bulk of the Fermi gas. We predict that for general β the variance of ND grows as Aβ log N + Bβ for N 1 and we obtain a formula for Aβ and Bβ. This is based on an explicit calculation for β ∈ {1, 2, 4} and on a conjecture that we formulate for general β. This conjecture further allows us to obtain a universal formula for the higher cumulants of ND. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/β, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter β. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

Original languageEnglish
Article number110
JournalSciPost Physics
Volume11
Issue number6
DOIs
StatePublished - 1 Dec 2021

ASJC Scopus subject areas

  • Physics and Astronomy (all)

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