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

We investigate the ground-state properies of the $K-Γ$ model on a honeycomb lattice using series expansions and numerical exact diagonalizations, where the model includes Kitaev ($K$) and symmetric off-diagonal ($Γ$) 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 $Γ$ 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 Kekulé dimerized state is unstable in the isotropic $K-Γ$ model.