Exactly solvable Kitaev model in three dimensions

We introduce a spin-$\frac{1}{2}$ model in three dimensions which is a generalization of the well-known Kitaev model on a honeycomb lattice. Following Kitaev, we solve the model exactly by mapping it to a theory of noninteracting fermions in the background of a static ${\mathbb{Z}}_2$ gauge field. The phase diagram consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly, unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the $\mathbf{k}$ space. Furthermore, we show that the flux excitations of the gauge field, due to some local constraints, form looplike structures; such loops exist on a lattice formed by the plaquettes in the original lattice and is topologically equivalent to the pyrochlore lattice. Finally, we derive a low-energy effective Hamiltonian that can be used to study the properties of the excitations in the gapped phase.

Figure 1

The honeycomb lattice: the three types of links are labeled $x$, $y$, and $z$.

Figure 2

The 3D lattice: the four sites inside the loop (marked 1–4) constitute a unit cell; ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, and ${\mathbf{a}}_3$ are the basis vectors. Plaquette $p$ consists of sites marked 1–10.

Figure 3

In the $\mathbf{k}$ space, the contour on which the gap vanishes is projected on $k_3=0$ plane. $J_x=J_y=1$, and $J_z$ is varied between zero and two. Corresponding values of $J_z$ are shown next to the contours.

Figure 4

The phase diagram: it shows the plane defined by $J_x+J_y+J_z=1$. Point $A$ corresponds to $J_x=1$ and $J_y=J_z=0$, $B$ corresponds to $J_y=1$ and $J_z=J_x=0$, and $C$ corresponds to $J_z=1$ and $J_x=J_y=0$. The shaded region is the gapless phase.

Figure 5

[(a)–(d)] The four types of plaquettes. (e) Part of the lattice involving four such adjacent plaquettes; the corresponding operators give rise to a constraint. The ellipses, labeled $a\text{−}d$, respectively, represent each of the loops and form a “tetrahedron.”

Figure 6

The lattice $\mathfrak{L}$ formed by the plaquettes—the pyrochlore lattice. Loops such as the dashed one, which goes through six sites, are the shortest (apart from the basic triangles).

Figure 7

Representation of the flux operators in $H_{\text{eff}}$ associated with the four types of plaquettes.