N\'eel-Dimer Transition in Antiferromagnetic Heisenberg Model and   Deconfinement of Spinons at the Critical Point
  Quantum phase transition from the N\'eel to the dimer states in an antiferromagnetic(AF) Heisenberg model on square lattice is studied. We introduce a control parameter $\alpha$ for the exchange coupling which connects the 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 and integrate 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 function of $\alpha$ and at $\alpha=0$ the $CP^1$ model is in the ordered phase which corresponds to the N\'eel state of the AF Heisenberg model. A phase transition to 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 spin wave. In the dimer phase, a confinement phase of the gauge theory is realized and low-energy excitations are $s=1$ magnons. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in 3 spatial dimensions, is realized and $s={1\over 2}$ spinons appear as low-energy excitations. 
Daisuke Yoshioka, Gaku Arakawa, Ikuo Ichinose, Tetsuo Matsui