Deconfined quantum criticality of the O(3) nonlinear $\sigma$ model in   two spacial dimensions: A renormalization group study
  We investigate the quantum phase transition of the O(3) nonlinear $\sigma$ model {\it without Berry phase} in two spacial dimensions. Utilizing the $CP^{1}$ representation of the nonlinear $\sigma$ model, we obtain an effective action in terms of bosonic spinons interacting via compact U(1) gauge fields. Based on the effective field theory, we find that the bosonic spinons are deconfined to emerge at the quantum critical point of the nonlinear $\sigma$ model. It is emphasized that the deconfinement of spinons is realized {\it in the absence of Berry phase}. This is in contrast to the previous study of Senthil et al. [Science {\bf 303}, 1490 (2004)], where the Berry phase plays a crucial role, resulting in the deconfinement of spinons. It is the reason why the deconfinement is obtained even in the absence of the Berry phase effect that the quantum critical point is described by the XY ("neutral") fixed point, not the IXY ("charged") fixed point. The IXY fixed point is shown to be unstable against instanton excitations and the instanton excitations are proliferated. {\it At the IXY fixed point it is the Berry phase effect that suppresses the instanton excitations, causing the deconfinement of spinons}. {\it On the other hand, the XY fixed point is found to be stable against instanton excitations because an effective internal charge is zero at the neutral XY fixed point}. As a result the deconfinement of spinons occurs at the quantum critical point of the O(3) nonlinear $\sigma$ model in two dimensions. 
Ki-Seok Kim