Spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>Kitaev model: Classical ground states, order from disorder, and exact correlation functions

Abstract

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 $\ifmmode\pm\else\textpm\fi{}1$, 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.