Topological entanglement entropy

Abstract

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator \rho for the degrees of freedom in the interior. The von Neumann entropy S(\rho) of this density operator, a measure of the entanglement of the interior and exterior variables, has the form S(\rho)= \alpha L -\gamma + ..., where the ellipsis represents terms that vanish in the limit L\to\infty. The coefficient \alpha, arising from short wavelength modes localized near the boundary, is nonuniversal and ultraviolet divergent, but -\gamma is a universal additive constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for \gamma in terms of properties of the superselection sectors of the medium.