Phase diagram of S=1 XXZ chain with next-nearest-neighbor interaction

The one dimensional S = 1 XXZ model with next-nearest-neighbor interaction alpha and Ising-type anisotropy Delta is studied by using a numerical diagonalization technique. We discuss the ground state phase diagram of this model numerically by the twisted-boundary-condition level spectroscopy method and the phenomenological renormalization group method, and analytically by the spin wave theory. We determine the phase boundaries among the XY phase, the Haldane phase, the ferromagnetic phase and the Neel phase, and then we confirm the universality class. Moreover, we map this model onto the nonlinear sigma model and analyze the phase diagram in the alpha << -1 and Delta similar to 1 region by using the renormalization group method.