  In the previous papers, we studied the bosonic t-J mode and derived an
effective field theory, which is a kind of quantum XY model. The bosonic t-J
model is expected to be realized by experiments of two-component cold atoms in
an optical lattice. In this paper, we consider a similar XY model that
describes phase diagram of the t-J model with a mass difference. Phase diagram
and critical behavior of the quantum XY model are clarified by means of the
Monte-Carlo simulations. Effective field theory that describes the phase
structure and low-energy excitations of the quantum XY model is derived.
Nambu-Goldstone bosons and the Higgs mode are studied by using the effective
field theory and interesting findings are obtained for the system with multiple
order, i.e., Bose-Einstein condensations and pseudo-spin symmetry. We also
investigate physical properties of the quantum XY model in an effective
magnetic field that is realized by rotating the optical lattice, etc. We show
that low-energy states of the system strongly depend on the strength of the
"magnetic field". For some specific strength of the magnetic field, vortex
lattice forms and the correlation function of the bosons exhibits solid like
behavior, which is a kind of Bose-Einstein condensation.
