Full counting statistics for interacting trapped fermions

We study NN 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 \betaβ. In the fermion model \betaβ controls the strength of the interaction, \beta=2β=2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_DND in a domain DD of macroscopic size in the bulk of the Fermi gas. We predict that for general \betaβ the variance of N_DND grows as A_{\beta} \log N + B_{\beta}AβlogN+Bβ for N \gg 1N≫1 and we obtain a formula for A_\betaAβ and B_\betaBβ. This is based on an explicit calculation for \beta\in\left\{ 1,2,4\right\}β∈{1,2,4} and on a conjecture that we formulate for general \betaβ. This conjecture further allows us to obtain a universal formula for the higher cumulants of N_DND. 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/\betaK=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 \betaβ. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.