Entanglement entropy and hyperuniformity of Ginibre and Weyl–Heisenberg ensembles

We show that, for a class of planar determinantal point processes (DPP) X X , the growth of the entanglement entropy S(X(Ω )) S ( X ( Ω ) ) of X\ X on a compact region Ω ⊂ R²ᵈ Ω ⊂ R 2 d , is related to the variance V( X(Ω )) V X ( Ω ) as follows: aligned V( X(Ω )) S( X( Ω )) V( X(Ω )) . aligned V X ( Ω ) ≲ S X ( Ω ) ≲ V X ( Ω ) . Therefore, such DPPs satisfy an area law S( Xg (Ω )) | ∂ Ω | S X g ( Ω ) ≲ ∂ Ω , where ∂ Ω ∂ Ω is the boundary of Ω Ω if they are of Class I hyperuniformity ( V( X (Ω )) | ∂ Ω | V X ( Ω ) ≲ ∂ Ω ), while the area law is violated if they are of Class II hyperuniformity (as \ L→ ∞ L → ∞ , V( X (LΩ )) ~ CΩLᵈ⁻¹log 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.