Entanglement entropy and Berezin-Toeplitz operators

TL;DR

This paper establishes a Weyl law for Berezin-Toeplitz operators with characteristic function symbols on compact Kahler manifolds, linking boundary geometry to entanglement entropy and point process statistics.

Contribution

It proves a two-term Weyl law for these operators, connecting boundary volume to spectral asymptotics, and applies results to quantum Hall states and determinantal point processes.

Findings

Weyl law with boundary term proportional to boundary volume

Area law for entanglement entropy in quantum Hall states

Asymptotic normality of point counts in determinantal processes

Abstract

We consider Berezin-Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the Riemannian volume of the boundary. As a consequence, we deduce the area law for the entanglement entropy of integer quantum Hall states. Another application is for the determinantal processes with correlation kernel the Bergman kernels of a positive line bundle : we prove that the number of points in a smooth domain is asymptotically normal.