Topological entanglement entropy in the quantum dimer model on the triangular lattice

A characterization of topological order in terms of bi-partite entanglement was proposed recently [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, ibid. 96, 110405 (2006)]. It was argued that in a topological phase there is a universal additive constant in the entanglement entropy, called the topological entanglement entropy, which reflects the underlying gauge theory for the topological order. In the present paper, we evaluate numerically the topological entanglement entropy in the ground states of a quantum dimer model on the triangular lattice, which is known to have a dimer liquid phase with ${\mathbb{Z}}_2$ topological order. We examine the two original constructions to measure the topological entropy by combining entropies on plural areas, and we observe that in the large-area limit they both approach the value expected for ${\mathbb{Z}}_2$ topological order. We also consider the entanglement entropy on a topologically nontrivial “zigzag” area and propose to use it as another way to measure the topological entropy.

Figure 1

Triangular lattices with periodic boundary conditions.

Figure 2

Circular areas centered at (a) a site or (b) an interior of a triangle. As examples, areas with $R=2.5$ and $R=2.47$ are shaded for (a) and (b), respectively.

Figure 3

(Color online) Entanglement entropy on circular areas with radii $R$ at the RK point. The ambiguity in $R$ is indicated by horizontal bars (only for $N=52$). The data for $N=52$ are fitted by lines using minimum or maximum radii. The resultant linear functions shown in the figure contains some numbers enclosed in parentheses, which indicates the standard errors in the last displayed digits.

Figure 4

Divisions of circular areas for the Kitaev-Preskill construction. (a) and (b): site-centered, $R=2.78$. (c) and (d): triangle-centered, $R=2.84$.

Figure 5

(Color online) Topological entanglement entropy from the Kitaev-Preskill construction (8) at the RK point. Examples of areas are shown in Fig. 4. Some explicit values for large radii are shown in Table .

Figure 6

(Color online) Kitaev-Preskill topological entropy (8) as a function of $v∕t$ for $N=36$. In the large-$R$ limit, $S_{\mathrm{topo}}^{\mathrm{KP}}$ is expected to jump from $\ln 2$ in ${\mathbb{Z}}_2$ liquid phase $0.82\left(3\right)\lesssim v∕t⩽1$, to some positive value in the VBC phase $v∕t\lesssim 0.82\left(3\right)$.

Figure 7

Division of annular areas for the Levin-Wen construction.

Figure 8

(Color online) Topological entanglement entropy from the Levin-Wen construction.

Figure 9

Zigzag area on a lattice with $T_1=l_x\mathbf{u}$ and $T_2=l_y\mathbf{v}$.

Figure 10

(Color online) Entanglement entropies on a zigzag area at the RK point. The upper line is a linear fit to $S\left[\mathrm{RK}\right]$. The lower one is a fit to $S[\mathrm{RK};p=++]$ when $l_x=l_y$ is a multiple of 4 and $S[\mathrm{RK};p=--]$ otherwise. The topological constant estimated from the latter fit is $\gamma =0.73\pm 0.01$. This particular choice of $p$ as a function of $l_x$ is motivated by the fact that, when $v∕t<1$, it corresponds to the ground-state sector. Explicit values of $S\left[\mathrm{RK}\right]-S[\mathrm{RK};p]$ are shown in Table .

Figure 11

The topological term $-\gamma$ as a function of $v∕t$, estimated by using zigzag areas for $l_x=4$ and 6.

Figure 12

Left (right): dimer configuration $c$ $\left(\overline{c}\right)$ on $\Omega$ $\left(\overline{\Omega }\right)$ and their “occupied sites” $\Lambda \left(c\right)$ $\left[\Lambda \left(\overline{c}\right)\right]$, marked with circles (squares). In this picture, $\Lambda \left(c\right)$ and $\Lambda \left(\overline{c}\right)$ are compatible and thus $(c,\overline{c})$ is physical.