Abstract
We analyse theoretically a spin-1 Bose gas loaded into a three-dimensional optical lattice in order to emerge different superfluid phases with external magnetic field. To achieve this, we generalize the mean-field perturbation theory up to the sixth hopping order for the underlying spin-1 Bose–Hubbard model. Our results indicate that the condensate density can be valid deep in the superfluid phase in comparison with that of the fourth hopping order. At zero temperature, new deep superfluid phases for an anti-ferromagnetic interaction are determined in the presence of the external magnetic field. Moreover, we find the second-order quantum phase transition between Mott insulator and superfluid phase for both ferromagnetic and anti-ferromagnetic interactions. Within the anti-ferromagnetic interaction, the transitions between respective superfluid phases can be of the first order.
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