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.
Full Counting Statistics in a Propagating Quantum Front and Random Matrix Spectra
