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 ( X ( Ω ) ) of X on a compact region Ω ⊂ R 2 d , is related to the variance V X ( Ω ) as follows: V X ( Ω ) ≲ S X ( Ω ) ≲ V X ( Ω ) . Therefore, such DPPs satisfy an area law S X g ( Ω ) ≲ ∂ Ω , where ∂ Ω is the boundary of Ω if they are of Class I hyperuniformity ( V X ( Ω ) ≲ ∂ Ω ), while the area law is violated if they are of Class II hyperuniformity (as L → ∞ , V X ( L Ω ) ∼ C Ω L d - 1 log L ). 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.