Spin-1 Kitaev-Heisenberg model on a honeycomb lattice

We study the Kitaev-Heisenberg model with spin-1 local degree of freedom on a honeycomb lattice numerically by the infinite density matrix renormalization group method on a cylinder geometry. By tuning the relative value of the Kitaev and Heisenberg exchange couplings, we obtain the phase diagram with two spin liquid phases and four symmetry-broken phases. We identify that the spin liquid phases are gapless by calculating the central charge at the pure Kitaev points without the Heisenberg interactions. Comparing to its spin-1/2 counterpart, the position and number of gapless modes of the spin-1 case are quite different. Due to the approximate Z(2) local conservations, the expectation value of the Wilson loop operator measuring the flux of each plaquette stays near to 1, and the static spin-spin correlations remain short range in the entire spin liquid phases.