Vison Crystals in an Extended Kitaev Model on the Honeycomb Lattice

We introduce an extension of the Kitaev honeycomb model by including four-spin interactions that preserve the local gauge structure and hence the integrability of the original model. The extended model has a rich phase diagram containing five distinct vison crystals, as well as a symmetric $\pi$-flux spin liquid with a Fermi surface of Majorana fermions and a sequence of Lifshitz transitions. We discuss possible experimental signatures and, in particular, present finite-temperature Monte Carlo calculations of the specific heat and the static vison structure factor. We argue that our extended model emerges naturally from generic perturbations to the Kitaev honeycomb model.