  We microscopically derive a lattice abelian mutual Chern-Simons gauge theory
for a honeycomb Kitaev model subjected to $(001)$ Zeeman and three-spin scalar
spin chirality perturbations. We identify the nature of topological orders,
emergent excitations and ground state degeneracy (GSD), topological
entanglement entropy ($\gamma$), and chiral central charge ($c$) in different
field regimes for both ferromagnetic (FM) and antiferromagnetic (AFM) sign of
the Kitaev interaction. A nonabelian Ising topological order (ITO) exists at
low fields in both cases, with $\gamma=\ln 2,$ $c=1/2,$ and GSD$=3,$ where the
nonabelian anyon, a twist defect, is an intrinsic bulk excitation. For AFM
Kitaev interactions, further increase of the field causes a transition from ITO
to an intermediate trivial topological phase with central charge $c=1/2,$
implying half-quantized thermal Hall response in both phases with no change of
sign. At sufficiently high fields there is a first order transition to a
polarized paramagnetic phase with $\gamma=c=0.$ For the FM case, there is a
direct transition from ITO to the polarized phase.
