Negativity and topological order in the toric code

In this paper we study the behavior of the entanglement measure dubbed negativity in the context of the toric code model. Using a replica method introduced recently by Calabrese, Cardy, and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the nonlocal correlations present in a topologically ordered state reflect in the behavior 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 nontrivial in both directions on a torus. We further confirm by explicit calculation that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered eight-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.

Figure 1

Examples of tripartitions of the system into $\mathcal{S}={\mathcal{A}}_1\cup {\mathcal{A}}_2\cup \mathcal{B}$. The top panel illustrates a topologically trivial choice for ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$, which corresponds to vanishing negativity. Although subsystems ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are no longer trivial in the middle panel, as they wind around the system in the vertical direction, the negativity remains zero. The bottom panel illustrates a choice of partitions with finite logarithmic negativity between ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$, where $\mathcal{E}$ exhibits both a boundary as well as a topological contribution.