Full counting statistics for interacting trapped fermions

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