Realization of the Exactly Solvable Kitaev Honeycomb Lattice Model in a Spin Rotation Invariant System

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.