Topological entanglement entropy

TL;DR

The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho) = alphaL - gamma + ..., where the ellipsis represents terms that vanish in the limit L --> infinity.

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 ρfor the degrees of freedom in the interior. The von Neumann entropy S(ρ) of this density operator, a measure of the entanglement of the interior and exterior variables, has the form S(ρ)= αL -γ+ ..., where the ellipsis represents terms that vanish in the limit L\to\infty. The coefficient α, arising from short wavelength modes localized near the boundary, is nonuniversal and ultraviolet divergent, but -γis 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 γin terms of properties of the superselection sectors of the medium.

Figures (3)

• Figure 1: ($a$) The plane is divided into four regions, labeled $A, B, C, D$, that meet at double and triple intersections. ($b$) Moving the triple intersection where $B,C,D$ meet deforms the regions as shown.
• Figure 2: The planar medium is glued at spatial infinity to its time-reversal conjugate, and wormholes are attached that connect the two conjugate media at the locations of the triple intersections, creating a sphere with four handles. Each region, together with its image, becomes a sphere with three punctures, and each union of two regions, together with its image, becomes a sphere with four punctures. The punctures carry charges labeled $a,b,c,d$. Anyons that wind around a cycle enclosing a wormhole throat as shown detect a trivial charge.
• Figure 3: If the regions $A,B,C$ have the topology shown, then $S_{\rm topo}= -2\gamma$.