# 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.

## Key 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.