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In the first part of this paper, we study the spin-S Kitaev model using spinwave theory. We discover a remarkable geometry of the minimum energy surface inthe N-spin space. The classical ground states, called Cartesian or CN-groundstates, whose number grows exponentially with the number of spins N, form a setof points in the N-spin space. These points are connected by a network of flatvalleys in the N-spin space, giving rise to a continuous family of classicalground states. Further, the CN-ground states have a correspondence with dimercoverings and with self avoiding walks on a honeycomb lattice. The zero pointenergy of our spin wave theory picks out a subset from a continuous family ofclassically degenerate states as the quantum ground states; the number of thesestates also grows exponentially with N. In the second part, we present someexact results. For arbitrary spin-S, we show that localized Z_2 fluxexcitations are present by constructing plaquette operators with eigenvalues\pm 1 which commute with the Hamiltonian. This set of commuting plaquetteoperators 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, S. Mandal and R. Shankar, Phys. Rev. Lett. 98, 247201 (2007)]. Weintroduce a generalized Jordan-Wigner transformation for the case of generalspin-S, and find a complete set of commuting link operators, similar to thespin-1/2 model, thereby making the Z_2 gauge structure more manifest. TheJordan-Wigner construction also leads, in a natural fashion, to Majoranafermion operators for half-integer spin cases and hard-core boson operators forinteger spin cases, strongly suggesting the presence of Majorana fermion andboson excitations in the respective low energy sectors.

Spin-S Kitaev model: Classical Ground States, Order by Disorder and Exact Correlation Functions