Entanglement entropy and hyperuniformity of Ginibre and Weyl-Heisenberg ensembles.

We show that, for a class of planar determinantal point processes (DPP) , the growth of the entanglement entropy of on a compact region , is related to the variance as follows: Therefore, such DPPs satisfy an , where is the boundary of if they are of (), while the if they are of (as , ). 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.