In this paper we study the lattice CP 1 model in ͑3+1͒ dimensions coupled with a dynamical compact U͑1͒ gauge field. This model is an effective field theory of the s = 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 O͑3͒ 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 c cuprates, in particular the possibility of deconfined-spinon phase. Effect of dimerlike spin exchange coupling and ring-exchange coupling is also studied.The CP N spin model plays an important role in various fields of physics not only as a tractable field-theory model that has interesting phase structure, but also as an effective field theory for certain systems in condensed matter physics and beyond. In particular, the CP 1 model corresponds to the Schwinger-boson representation of the s = 1 2 antiferromagnetic ͑AF͒ quantum spin model, i.e., the AF Heisenberg model. 1 The CP 1 model is much more tractable than the original AF Heisenberg model, and its phase structure and critical behavior have been investigated both analytically and numerically. The system intrinsically contains compact U͑1͒ gauge degrees of freedom, and their dynamics determines the low-energy excitations in AF magnet. That is, if the gauge dynamics is in the deconfined-Coulomb phase, the lowenergy excitations are the s = 1 2 spinons. On the other hand, the Higgs phase corresponds to the Néel state with a longrange AF order, and the confinement phase is a valence-bond solid ͑VBS͒ state in which spin-triplet low-energy excitations appear. Most of the previous studies exploring a possible deconfined-spin-liquid phase have considered the twodimensional ͑2D͒ ͑doped͒ AF Heisenberg model at zero temperature ͑T =0͒ or its path-integral representation, the three-dimensional ͑3D͒ CP 1 model. In these cases, the Coulomb phase may be possible if there exist a sufficient number of gapless matter fields that couple to the gauge field. 2 In the present paper, we shall consider a 4D CP 1 model coupled with a dynamical U͑1͒ gauge field. This 4D CP 1 +U͑1͒ gauge model is viewed as an effective field theory of the 3D AF Heisenberg model at T = 0. From the gaugetheoretical point of view, the deconfinement nature is enhanced in ͑3+1͒ D case because the Coulomb phase exists even in the pure 4D U͑1͒ gauge system that involves no matter fields in contrast to the pure 3D U͑1͒ gauge system that has only the confinement phase. Therefore, it is interesting to study the phase structure of this 4D CP 1 gauge model. We shall first consider the CP 1 model for the 3D AF Heisenberg model with uniform nearest-neighbor spin coupling and then the CP 1 model for the 3D AF Heisenberg model with nonuniform dimerlike coupling and ring-...

In this paper we shall introduce a lattice model of unconventional superconductors (SC) like d-wave SC in order to study quantum phase transition at vanishing temperature (T ). Finite-T counterpart of the present model was proposed previously with which SC phase transition at finite T was investigated. The present model is a noncompact U(1) lattice-gauge-Higgs model in which the Higgs boson, the Cooper-pair field, is put on lattice links in order to describe d-wave SC. We first derive the model from a microscopic Hamiltonian in the path-integral formalism and then study its phase structure by means of the Monte Carlo simulations. We calculate the specific heat, monopole densities and the magnetic penetration depth (the gauge-boson mass). We verified that the model exhibits a second-order phase transition from normal to SC phases. Behavior of the magnetic penetration depth is compared with that obtained in the previous analytical calculation using XY model in four dimensions. Besides the normal to SC phase transition, we also found that another second-order phase transition takes place within the SC phase in the present model. We discuss physical meaning of that phase transition.

In this paper we shall study quantum critical behavior of lattice model of unconventional superconductors (SC) that was proposed in the previous papers. In this model, the Cooper-pair (CP) field is defined on lattice links in order to describe d-wave SC. The CP field can be regarded as a U (1) lattice gauge field, and the SC phase transition takes place as a result of the phase coherence of the CP field.Effects of the long-range Coulomb interactions between the CP's and fluctuations of the electromagnetic field are taken into account. We investigate the phase structure of the model and the critical behavior by means of the Monte Carlo simulations. We find that the parameter, which controls the fluxes (vortices) of the CP, strongly influences the phase structure. In three-dimensional case, the model has rich phase structure. In particular there is a "monopole proliferation" phase transition besides the SC phase transition. Depending on the parameters, this transition exists within the SC phase or takes place simultaneously with the SC transition. This new type of transition is relevant for unconventional SC's with strong spatial three-dimensionality and to be observed by experiments.