  We present a solution of Kitaev's spin model on the honeycomb lattice and of
related topologically ordered spin models. We employ a Jordan-Wigner type
fermionization and find that the Hamiltonian takes a BCS type form, allowing
the system to be solved by Bogoliubov transformation. Our fermionization does
not employ non-physical auxiliary degrees of freedom and the eigenstates we
obtain are completely explicit in terms of the spin variables. The ground-state
is obtained as a BCS condensate of fermion pairs over a vacuum state which
corresponds to the toric code state with the same vorticity. We show in detail
how to calculate all eigenstates and eigenvalues of the model on the torus. In
particular, we find that the topological degeneracy on the torus descends
directly from that of the toric code, which now supplies four vacua for the
fermions, one for each choice of periodic vs. anti-periodic boundary
conditions. The reduction of the degeneracy in the non-Abelian phase of the
model is seen to be due to the vanishing of one of the corresponding candidate
BCS ground-states in that phase. This occurs in particular in the fully
periodic vortex-free sector. The true ground-state in this sector is exhibited
and shown to be gapped away from the three partially anti-periodic
ground-states whenever the non-Abelian phase is gapped.
