Abstract
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 ⊂ R2d, is related to the variance V(X,(,Ω,)) as follows: [Formula: see text] Therefore, such DPPs satisfy an area law S(Xg,(,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,)) ∼ CΩLd-1logL). 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.