Entanglement Entropy and Berezin–Toeplitz Operators

Abstract

We consider Berezin-Toeplitz operators on compact Kähler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a twoterm 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.

Entanglement spectroscopy of chiral edge modes in the quantum Hall effect

“…up to a constant that does not depend on θ, and subleading corrections in 1/ √ N . In this result the first term is the area law contribution [11], the second is the CFT term, the third stems from the corner near the origin. We note that similar corner terms have been studied numerically in Ref.…”

“…The problem of extracting the edge contribution to the EE in the QHE thus boil down to the computation of the variance κ. If we denote by {φ i , i ∈ I} the occupied one-body states, then the particle variance in region A is simply given by [11,19]…”

“…The second, proportional to 1/l B is the area law contribution of Ref. [11]. The logarithmic term is the edge CFT contribution, and is new.…”

“…It can be deduced [18] from the FCS, as explained in the text. However, a more transparent approach is to establish, in the spirit of [11], the slightly more general result gives the Rényi entropies. In the latter case the bulk term is absent since h(1) = 0, as expected.…”

“…It is perhaps the simplest setup in which topological order can emerge, and thus serves as a paradigmatic system. The scaling of the EE in the insulating 'bulk' of a quantum Hall liquid is well understood [9,10], and the area law in this context is even a mathematical theorem [11]. A distinctive feature of topological phases is that while they are insulators in the bulk, they display exotic, topologically protected metallic states at their surfaces/edges.…”