Three-dimensional generalized Kitaev models

We generalize Kitaev's honeycomb lattice spin model to a gamma matrix model on three-dimensional cubic octahedron and pyrochlore lattices. We find the ground state ${Z}_2$ flux configuration, reducing the problem to free Majorana fermion hopping. For the cubic octahedron lattice, which has reflection planes, the ground states must obey Lieb's theorem, i.e. the ${Z}_2$ fluxes are reflection symmetric. By adding flux-flux interaction terms, a variety of interesting phases can be stabilized, including metallic, semimetallic, and both trivial and topological insulating phases.