Negativity and topological order in the toric code

In this manuscript we study the behaviour of the entanglement measure dubbed negativity in the context of the toric code model. Using a method introduced recently by Calabrese, Cardy and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the non-local correlations present in a topologically ordered state reflect in the behaviour of the negativity of the system. We find that the negativity has a leading area-law contribution, if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically non-trivial in both directions on a torus. We also show that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered 8-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.