Topological phases and topological entropy of two-dimensional systems with finite correlation length

We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\xi$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.

Figure 1

(Color online) (a) Allowed vertices in the classical eight-vertex model. (b) The phase diagram of the classical eight-vertex model. Phases I and II represent “antiferroelectric” phases dominated by $d$ and $c$ vertices, respectively. The middle D phase is disordered. The point $K$ is where $c=d=1$. In the quantum version of the model, the ground state wave function at this “Kitaev point” is the same as the ground state of Kitaev’s toric code (Ref. 3)

Figure 2

(Color online) (a) A pictorial representation of the boundary line $\Gamma$ and the dual lattice variables which have to remain frozen in the calculation of the entanglement entropy. (b) The dual lattice variables with respect to a lattice site of the direct lattice.

Figure 3

(Color online) (a) A part of the transition graph between a configuration $\mathcal{C}$ [light (red)] and a reference columnar configuration ${\mathcal{C}}_0$ [dark (blue)]. It is clear that the transition graph is composed typically of loops and isolated links. If the configuration $\mathcal{C}$ is changed locally, then either the number of loops changes, adding or removing isolated links from the loops or the relative positions of the loops and the links change (without any shape change). (b) Definition of region A with respect to region B. The light (green) dimers define the configuration in A and the dark (red) ones the configuration in B. More generally, any link that lies on the boundary line or inside of it belongs to A.

Figure 4

(Color online) (a) The positions of the arrows denote the direct lattice and the dots (located on the vertices) the corresponding dual lattice. The dashed line marks the boundary of region A and the thick bonds are the bonds on which the dual coupling ${\beta }^{*}$ has to reverse its sign when a correlation function $⟨\prod \tau ⟩$, a dual Wilson loop, is to be computed. (b) The string representation of a typical eight-vertex configuration.