Abstract
In this paper we study the lattice ${\text{CP}}^{1}$ model in $(3+1)$ dimensions coupled with a dynamical compact $\mathrm{U}(1)$ gauge field. This model is an effective field theory of the $s=\frac{1}{2}$ antiferromagnetic Heisenberg spin model in three spatial dimensions at zero temperature. By means of Monte Carlo simulations, we investigate its phase structure. There exist the Higgs, Coulomb and confinement phases, and the parameter regions of these phases are clarified. We also measure the magnetization of $O(3)$ spins, the energy gap of spin excitations, and the mass of gauge boson. Then we discuss the relationship between these three phases and magnetic properties of the high-${T}_{\text{c}}$ cuprates, in particular the possibility of deconfined-spinon phase. Effect of dimerlike spin exchange coupling and ring-exchange coupling is also studied.
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