  The exactly solvable Kitaev honeycomb lattice model is realized as the low
energy effect Hamiltonian of a spin-1/2 model with spin rotation and
time-reversal symmetry. The mapping to low energy effective Hamiltonian is
exact, without truncation errors in traditional perturbation series expansions.
This model consists of a honeycomb lattice of clusters of four spin-1/2
moments, and contains short-range interactions up to six-spin(or eight-spin)
terms. The spin in the Kitaev model is represented not as these spin-1/2
moments, but as pseudo-spin of the two-dimensional spin singlet sector of the
four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin
correlations in the Kitaev model are mapped to dimer correlations or
spin-chirality correlations in this model. This exact construction is quite
general and can be used to make other interesting spin-1/2 models from spin
rotation invariant Hamiltonians. We discuss two possible routes to generate the
high order spin interactions from more natural couplings, which involves
perturbative expansions thus breaks the exact mapping, although in a controlled
manner.
