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

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 VX(Ω) as follows VX(Ω)≲SX(Ω)≲VX(Ω).Therefore, such DPPs satisfy an area law SXg(Ω)≲∂Ω, where ∂Ω is the boundary of Ω if they are of Class I hyperuniformity (VX(Ω)≲∂Ω), while the area law is violated if they are of Class II hyperuniformity (as L→∞, VX(LΩ)∼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.