Entanglement entropy and Berezin-Toeplitz operators

Understand

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.