  We construct an exactly soluble spin-$\frac{1}2$ model on a honeycomb
lattice, which is a generalization of Kitaev model. The topological phases of
the system are analyzed by study of the ground state sector of this model, the
vortex-free states. Basically, there are two phases, A phase and B phase. The
behaviors of both A and B phases may be studied by mapping the ground state
sector into a general p-wave paired states of spinless fermions with tunable
pairing parameters on a square lattice. In this p-wave paired state theory, the
A phase is shown to be the strong paired phase, an insulating phase. The B
phase may be either gapped or gapless determined by the generalized inversion
symmetry is broken or not. The gapped B is the weak pairing phase described by
either the Moore-Read Pfaffian state of the spinless fermions or anti-Pfaffian
state of holes depending on the sign of the next nearest neighbor hopping
amplitude. A phase transition between Pfaffian and anti-Pfaffian states are
found in the gapped B phase. Furthermore, we show that there is a hidden SU(2)
gauge symmetry in our model. In the gapped B phase, the ground state has a
non-trivial topological number, the spectral first Chern number or the chiral
central charge, which reflects the chiral anomaly of the edge state. We proved
that the topological number is identified to the reduced eta-invariant and this
anomaly may be cancelled by a bulk Wess-Zumino term of SO(3) group through an
index theorem in 2+1 dimensions.
