Ground-state properties of the K−Γ model on a honeycomb lattice

We investigate the ground-state properties of the $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a honeycomb lattice using series expansions and numerical exact diagonalizations, where the model includes Kitaev ($K$) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) interactions. Starting from the weakly interacting dimers on the specific bond, we strengthen the interdimer interactions to the isotropically interacting system. We show that depending on $\mathrm{\ensuremath{\Gamma}}$ and $K$, the dimer state survives up to the isotropically interacting system, where the phase transition occurs or obeys a phase transition to a magnetically ordered state at an anisotropic interaction. The results are summarized in the phase diagram. We also show that the Kekul\'e dimerized state is unstable in the isotropic $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model.