Topological entanglement entropy of the three-dimensional Kitaev model

We calculate the topological entanglement entropy (TEE) for a three-dimensional hyperhoneycomb lattice generalization of Kitaev's honeycomb lattice spin model. We find that for this model TEE is not directly determined by the total quantum dimension of the system. This is in contrast to general two-dimensional systems and many three-dimensional models, where TEE is related to the total quantum dimension. Our calculation also provides TEE for a three-dimensional toric-code-type Hamiltonian that emerges as the effective low-energy theory for the Kitaev model in a particular limit.

Figure 1

The $3\mathrm{D}$ lattice. The four sites labeled $(1–4)$ constitute a unit cell, and ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, ${\mathbf{a}}_3$ are the basis vectors. $x$, $y$, and $z$ links are represented by dashed, dotted, and bold lines, respectively.

Figure 2

Bipartition scheme in which region $A$ (volume enclosed by the shaded surface) has the topology of a solid sphere. The dashed lines are the links on the boundary. $u_{ij}$ variables on the boundary links are transformed to $w_{A,n}$ and $w_{B,n}$, which are defined on the links $(a_{2n-1},a_{2n})$ and $(b_{2n-1},b_{2n})$, respectively.

Figure 3

Bipartition scheme in which region $A$ (volume enclosed by the solid surface) is a solid torus. The dashed lines are the links on the boundary.

Figure 4

Various regions considered for the calculation of TEE in the sphere (1–4) and torus (5–8) bipartition schemes.