Topological Entanglement Entropy

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 of $\rho$, a measure of the entanglement of the interior and exterior variables, has the form $S(\rho )=\alpha L-\gamma +\cdots$, where the ellipsis represents terms that vanish in the limit $L\to \infty$. We show that $-\gamma$ is a universal 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.

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 $S_{\mathrm{topo}}=-2\gamma$.