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Quantum phase transition from the N\'eel to the dimer states in anantiferromagnetic(AF) Heisenberg model on square lattice is studied. Weintroduce a control parameter $\alpha$ for the exchange coupling which connectsthe N\'eel ($\alpha=0$) and the dimer ($\alpha=1$) states. We employ the $CP^1$(the Schwinger boson) representation of the $s={1\over 2}$ spin operator andintegrate out the half of the $CP^1$ variables at odd sites and we obtain a$CP^1$ nonlinear $\sigma$ model. The effective coupling constant is a functionof $\alpha$ and at $\alpha=0$ the $CP^1$ model is in the ordered phase whichcorresponds to the N\'eel state of the AF Heisenberg model. A phase transitionto the dimer state occurs at a certain critical value of $\alpha_C$ as $\alpha$increases. In the N\'eel state, the dynamical composite U(1) gauge field in the$CP^1$ model is in a Higgs phase and low-energy excitations are gapless spinwave. In the dimer phase, a confinement phase of the gauge theory is realizedand low-energy excitations are $s=1$ magnons. For the critical point, we arguethat a deconfinement phase, which is similar to the Coulomb phase in 3 spatialdimensions, is realized and $s={1\over 2}$ spinons appear as low-energyexcitations

Néel-dimer transition in an antiferromagnetic Heisenberg model and deconfinement of spinons at the critical point