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We study the Kitaev-Heisenberg model with spin-1 local degree of freedom on atwo-dimensional honeycomb lattice numerically by density matrix renormalizationgroup method. By tuning the relative value of the Kitaev and Heisenbergexchange couplings, we obtain the whole phase diagram with two spin liquidphases and four symmetry broken phases. We identify that the spin liquid phasesare gapless by calculating the central charge at the pure Kitaev points withoutHeisenberg interaction. Comparing to its spin-1/2 counterpart, the position andnumber of gapless modes of the spin-1 case are quite different. Due to theapproximate $Z_2$ local conservations, the expectation value of Wilson loopoperator measuring the flux of each plaquette stays near to 1, and the staticspin-spin correlations remain short-range in the entire spin liquid phases.

Spin-1 Kitaev-Heisenberg model on a two-dimensional honeycomb lattice