Counting statistics for noninteracting fermions in a rotating trap

Abstract

We study the ground state of $N\ensuremath{\gg}1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $\mathrm{\ensuremath{\Omega}}>0$. The support of the density of the Fermi gas is a disk of radius ${R}_{e}$. We calculate the variance of the number of fermions, ${\mathcal{N}}_{R}$, inside a disk of radius $R$ centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and interesting behaviors in two different scaling regimes, (i) $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}<1$ and (ii) $1\ensuremath{-}\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1/N)$, where $\ensuremath{\omega}$ is the angular frequency of the oscillator. In the first regime we find that $\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}\ensuremath{\simeq}(AlnN+B)\sqrt{N}$ and we calculate $A$ and $B$ as functions of $R/{R}_{e}$, $\mathrm{\ensuremath{\Omega}}$, and $\ensuremath{\omega}$. We also predict the higher cumulants of ${\mathcal{N}}_{R}$ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime, the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling $k$ successive Landau levels, as found in previous studies. Here, we show that $\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}$ is a discontinuous piecewise linear function of $\ensuremath{\sim}(R/{R}_{e})\sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any $k$. We argue that a similar piecewise linear behavior extends to all the cumulants of ${\mathcal{N}}_{R}$ and to the entanglement entropy. We show that these results match smoothly at large $k$ with the above results for $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of $\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}$ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the $z$ direction.