  In this paper, we consider the bosonic t-J model, which describes
two-component hard-core bosons with a nearest-neighbor (NN) pseudo-spin
interaction and a NN hopping. To study phase diagram of this model, we derive
effective field theories for low-energy excitations. In order to represent the
hard-core nature of bosons, we employ a slave-particle representation. In the
path-integral quantization, we first integrate our the radial degrees of
freedom of each boson field and obtain the low-energy effective field theory of
phase degrees of freedom of each boson field and an easy-plane pseudo-spin.
Coherent condensates of the phases describe, e.g., a "magnetic order" of the
pseudo-spin, superfluidity of hard-core bosons, etc. This effective field
theory is a kind of extended quantum XY model, and its phase diagram can be
investigated precisely by means of the Monte-Carlo simulations. We then apply a
kind of Hubbard-Stratonovich transformation to the quantum XY model and obtain
the second-version of the effective field theory, which is composed of fields
describing the pseudo-spin degrees of freedom and boson fields of the original
two-component hard-core bosons. As application of the effective-field theory
approach, we consider the bosonic t-J model on the square lattice and also on
the triangular lattice, and compare the obtained phase diagrams with the
results of the numerical studies. We also study low-energy excitations rather
in detail in the effective field theory. Finally we consider the bosonic t-J
model on a stacked triangular lattice and obtain its phase diagram. We compare
the obtained phase diagram with that of the effective field theory to find
close resemblance.
