Competing phases of S=1 XXZ model on Kagome lattice

We numerically study the S=1 XXZ model on the kagome lattice with the Density matrix renormalization group. We focus on the two types of expected magnetic order, $Q=0$ and $\sqrt{3}\times \sqrt{3}$. As a function of the coupling $\Delta$ for the $S_zS_z$ terms, we find two possible phase transitions. For $\Delta < \Delta_1$, we find that the $Q=0$ state has lower energy, but the difference between these two magnetic ordered phase is very small. For $\Delta_1< \Delta < \Delta_2$, $\sqrt{3}\times \sqrt{3}$ magnetic ordered state is the ground state. For $\Delta > \Delta_2$, we find that the magnetic ordered phases disappears, entering a magnetic disordered phase. We find a close competition between a trimerized phase and a hexagon singlet phase, which is consistent with recent numerical studies of S=1 Heisenberg model on the Kagome lattice.