Full counting statistics for interacting trapped fermions

We study N 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 β . In the fermion model β β controls the strength of the interaction, β=2 β = 2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_D N D in a domain D D of macroscopic size in the bulk of the Fermi gas. We predict that for general β β the variance of N_D N D grows as Aβ log N + Bβ A β log N + B β for N 1 N ≫ 1 and we obtain a formula for A_β A β and B_β B β . This is based on an explicit calculation for β∈\{ 1,2,4\} β ∈ { 1 , 2 , 4 } and on a conjecture that we formulate for general β β . This conjecture further allows us to obtain a universal formula for the higher cumulants of N_D N D . 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 K = <mml: