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We study the three-dimensional U(1) lattice gauge theory of a CP$^1$ spinon(Schwinger boson) field and a Higgs field. It is a bosonic $t$-$J$ model inslave-particle representation, describing the antiferromagnetic (AF) Heisenbergspin model with doped bosonic holes expressed by the Higgs field. The spinoncoupling term of the action favors AF long-range order, whereas the holonhopping term in the ferromagnetic channel favors Bose-Einstein condensation(BEC) of holons. We investigate the phase structure by means of Monte-Carlosimulations and study an interplay of AF order and BEC of holes. We considerthe two variations of the model; (i) the three-dimensional model at finitetemperatures, and (ii) the two-dimensional model at vanishing temperature. Inthe model (i) we find that the AF order and BEC coexist at low temperatures andcertain hole concentrations. In the model (ii), by varying the holeconcentration and the stiffness of AF spin coupling, we find a phase diagramsimilar to the model (i). Implications of the results to systems of cold atomsand the fermionic $t$-$J$ model of strongly-correlated electrons are discussed.

Magnetic order, Bose-Einstein condensation, and superfluidity in a bosonic t-J model of CP^1 spinons and doped Higgs holons