Phase transitions and spin excitations of spin-1 bosons in optical lattice
For spin-1 bosonic system trapped in optical lattice, we investigate two main problems, including MI-SF phase transition and magnetic phase separations in MI phase, with extended standard basis operator (SBO) method. For both ferromagnetic (U2 < 0) and antiferromagnetic (U2 > 0) systems, we analytically figure out the symmetry properties in Mott-insulator and superfluid phases, which would provide a deeper insight into the MI-SF phase transition process. Then by applying self-consistent approach to the method, we include the effect of quantum and thermal fluctuations and derive the MI-SF transition phase diagram, which is in quantitative agreement with recent Monte-Carlo simulation at zero temperature, and at finite temperature, we find the underestimation of finite-temperature-effect in the mean-field approximation method. If we further consider the spin excitations in the insulating states of spin-1 system in external field, distinct spin phases are expected. Therefore, in the Mott lobes with n = 1 and n = 2 atoms per site, we give analytical and numerical boundaries of the singlet, nematic, partially magnetic and ferromagnetic phases in the magnetic phase diagrams.