Entanglement entropy and hyperuniformity of Ginibre and Weyl–Heisenberg ensembles

Abstract

AbstractWe show that, for a class of planar determinantal point processes (DPP) $$ {\mathcal {X}}$$ X , the growth of the entanglement entropy $$S({\mathcal {X}}(\Omega )) $$ S ( X ( Ω ) ) of $${\mathcal {X}}\ $$ X on a compact region $$\Omega \subset {\mathbb {R}}^{2d}$$ Ω ⊂ R 2 d , is related to the variance $${\mathbb {V}}\left( {\mathcal {X}}(\Omega )\right) $$ V X ( Ω ) as follows: $$\begin{aligned} {\mathbb {V}}\left( {\mathcal {X}}(\Omega )\right) \lesssim S\left( \mathcal {X(} \Omega \mathcal {)}\right) \lesssim {\mathbb {V}}\left( {\mathcal {X}}(\Omega )\right) . \end{aligned}$$ V X ( Ω ) ≲ S X ( Ω ) ≲ V X ( Ω ) . Therefore, such DPPs satisfy an area law$$S\left( {\mathcal {X}}_{g} \mathcal {(}\Omega \mathcal {)}\right) \lesssim \left| \partial {\Omega } \right| $$ S X g ( Ω ) ≲ ∂ Ω , where $$\partial {\Omega }$$ ∂ Ω is the boundary of $$\Omega $$ Ω if they are of Class I hyperuniformity ($${\mathbb {V}}\left( {\mathcal {X}} (\Omega )\right) \lesssim \left| \partial {\Omega }\right| $$ V X ( Ω ) ≲ ∂ Ω ), while the area law is violated if they are of Class II hyperuniformity (as $$\ L\rightarrow \infty $$ L → ∞ , $${\mathbb {V}}\left( {\mathcal {X}} (L\Omega )\right) \sim C_{\Omega }L^{d-1}\log L$$ 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.