Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: Exact results for the entanglement entropy and number variance

We establish an exact mapping between the positions of $N$ noninteracting fermions in a two-dimensional rotating harmonic trap in its ground state and the eigenvalues of the $N\times N$ complex Ginibre ensemble of random matrix theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite $N$, the Rényi entanglement entropy and the number variance of a disk of radius $r$ in the ground state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large $N$, we provide exact expressions for the scaling functions associated with these two observables.

Figure 1

Plot of the ratio $\frac{S_q(N,r)}{{\alpha }_q\mathrm{Var}\left(N_r\right)}$, for $N=100$ fermions, $q=2$ (dashed blue) and $q=4$ (orange) as a function of the rescaled radius $\zeta =r/\sqrt{N}$. The entropy $S_q(N,r)$ is given in Eq. (10), the variance $\mathrm{Var}\left(N_r\right)$ is given in Eq. (38) of Ref. [39], and ${\alpha }_q$ is a constant defined below Eq. (9). This ratio is constant and equal to 1 for all $q\ge 1$ in the extended bulk (ii), while it is neither the case in the deep bulk (i) nor at the edge (iii). Inset: Typical configuration of the eigenvalues obtained by diagonalizing an $N\times N$ Ginibre complex matrix, with $N={10}^3$, exhibiting a sharp edge at $r=\sqrt{N}$. The shaded red disk corresponds to a subsystem of radius $r$.