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Counting statistics for non-interacting fermions in a rotating trap

We study the ground state of $N \gg 1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $Ω>0$. 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 ${\cal N}_R$ inside a disk of radius $R$ centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and interesting behaviours in two different scaling regimes: (i) $Ω/ ω<1 $ and (ii) $1 - Ω/ ω= O(1/N)$, where $ω$ is the angular frequency of the oscillator. In the first regime (i) we find that ${\rm Var}\,{\cal N}_{R}\simeq\left(A\log N+B\right)\sqrt{N}$ and we calculate $A$ and $B$ as functions of $R/R_e$, $Ω$ and $ω$. We also predict the higher cumulants of ${\cal N}_{R}$ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime (ii), 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 ${\rm Var}\,{\cal N}_{R}$ is a discontinuous piecewise linear function of $\sim (R/R_e) \sqrt{N}$ 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 ${\cal N}_{R}$ and to the entanglement entropy. We show that these results match smoothly at large $k$ with the above results for $Ω/ω=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of ${\rm Var}\,{\cal N}_{R}$ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the $z$ direction.