Counting statistics for noninteracting fermions in a rotating trap

We study the ground state of N≫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 Re. We calculate the variance of the number of fermions, NR, 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) ω/ω<1 and (ii) 1-ω/ω=O(1/N), where ω is the angular frequency of the oscillator. In the first regime we find that VarNR≃(AlnN+B)N and we calculate A and B as functions of R/Re, ω, and ω. We also predict the higher cumulants of NR 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 VarNR is a discontinuous piecewise linear function of ∼(R/Re)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 NR 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 VarNR near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the z direction.