Spin-$S$ Kitaev model: Classical ground states, order from disorder, and exact correlation functions

In the first part of this paper, we study the spin-$S$ Kitaev model using spin-wave theory. We discover a remarkable geometry of the minimum-energy surface in the $N$-spin space. The classical ground states, called Cartesian or CN-ground states, whose number grows exponentially with the number of spins $N$, form a set of points in the $N$-spin space. These points are connected by a network of flat valleys in the $N$-spin space giving rise to a continuous family of classical ground states. Further, the CN-ground states have a correspondence with dimer coverings and with self-avoiding walks on a honeycomb lattice. The zero-point energy of our spin-wave theory picks out a subset from a continuous family of classically degenerate states as the quantum ground states; the number of these states also grows exponentially with $N$. In the second part, we present some exact results. For arbitrary spin $S$, we show that localized $Z_2$ flux excitations are present by constructing plaquette operators with eigenvalues $\pm 1$, which commute with the Hamiltonian. This set of commuting plaquette operators leads to an exact vanishing of the spin-spin correlation functions beyond nearest-neighbor separation found earlier for the spin-1/2 model [G. Baskaran et al., Phys. Rev. Lett. 98, 247201 (2007)]. We introduce a generalized Jordan-Wigner transformation for the case of general spin $S$ and find a complete set of commuting link operators similar to the spin-1/2 model, thereby making the $Z_2$ gauge structure more manifest. The Jordan-Wigner construction also leads, in a natural fashion, to Majorana fermion operators for half-odd-integer spin cases and hard-core boson operators for integer spin cases strongly suggesting the presence of Majorana fermion and boson excitations in the respective low-energy sectors. Finally, we present a modified Kitaev Hamiltonian, which is exactly solvable for all half-odd-integer spins; it is equivalent to an exponentially large number of copies of spin-1/2 Kitaev Hamiltonians.

Figure 1

Schematic picture of the Kitaev model on a honeycomb lattice indicating the three kinds of bonds, $x$, $y$, and $z$. A hexagon with sites marked 1–6 is shown; the corresponding plaquette operator $W_p$ is defined in Eqs. (16, 18).

Figure 2

Pictures of two self-avoiding walks, marked as 1, which are related to each other by a slide. The walk in (a) has only nonvertical dimers, while the walk in (b) has only vertical dimers. The dimers are shown by solid lines.

Figure 3

Picture of a set of self-avoiding walks (dotted lines); one of the walks is marked as 1. The dimers are shown by solid lines.

Figure 4

Picture of a ground state in which the self-avoiding walks (dotted lines) form hexagons; three of these are marked as 1–3. The dimers are shown by solid lines.