Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions

TL;DR

This paper analyzes entanglement entropy and number fluctuations of fermions in a rotating 2D harmonic trap, revealing a proportionality in the bulk and different behaviors at the edge, with exact finite-N results.

Contribution

It provides exact finite-N formulas for entanglement entropy and number cumulants, and uncovers their proportionality in the bulk for large N.

Findings

Entanglement entropy proportional to number variance in the bulk.

Exact formulas for finite N for entropy and cumulants.

Different scaling behaviors at the edge of the fermion density.

Abstract

We consider $N$ non-interacting fermions in a $2d$ harmonic potential of trapping frequency $\omega$ and in a rotating frame at angular frequency $\Omega$, with $0<\omega - \Omega\ll \omega$. At zero temperature, the fermions are in the non-degenerate lowest Landau level and their positions are in one to one correspondence with the eigenvalues of an $N\times N$ complex Ginibre matrix. For large $N$, the fermion density is uniform over the disk of radius $\sqrt{N}$ centered at the origin and vanishes outside this disk. We compute exactly, for any finite $N$, the R\'enyi entanglement entropy of order $q$, $S_q(N,r)$, as well as the cumulants of order $p$, $\langle{N_r^{p}}\rangle_c$, of the number of fermions $N_r$ in a disk of radius $r$ centered at the origin. For $N \gg 1$, in the (extended) bulk, i.e., for $0 < r/\sqrt{N} < 1$, we show that $S_q(N,r)$ is proportional to the number variance ${\rm Var}\,(N_r)$, despite the non-Gaussian fluctuations of $N_r$. This relation breaks down at the edge of the fermion density, for $r \approx \sqrt{N}$, where we show analytically that $S_q(N,r)$ and ${\rm Var}\,(N_r)$ have a different $r$-dependence.