Entanglement entropy and hyperuniformity of Ginibre and Weyl-Heisenberg ensembles

We show that, for a class of planar determinantal point processes (DPP) $X$, the growth of the entanglement entropy $S\left(X\right(\Omega \left)\right)$ of $X$ on a compact region $\Omega \subset R^{2d}$, is related to the variance $V\left(X,\left(,\Omega ,\right)\right)$ as follows: $\begin{array}{c}V\left(X,\left(,\Omega ,\right)\right)\lesssim S\left(X(,\Omega ,)\right)\lesssim V\left(X,\left(,\Omega ,\right)\right).\end{array}$Therefore, such DPPs satisfy an area law $S\left(X_g,\left(,\Omega ,\right)\right)\lesssim \left(\partial ,\Omega \right)$, where $\partial \Omega$ is the boundary of $\Omega$ if they are of Class I hyperuniformity ($V\left(X,\left(,\Omega ,\right)\right)\lesssim \left(\partial ,\Omega \right)$), while the area law is violated if they are of Class II hyperuniformity (as $L\to \infty$, $V\left(X,\left(,L,\Omega ,\right)\right)\sim C_{\Omega }{L}^{d-1}logL$). As a result, the entanglement entropy of Weyl-Heisenberg ensembles (a family of DPPs containing the Ginibre ensemble and Ginibre-type ensembles in higher Landau levels), satisfies an area law, as a consequence of its hyperuniformity.