Crossover from O(3) to O(4) behavior in weakly frustrated antiferromagnets

We consider an anisotropic version of the $C{P}^1$ model which describes frustrated quantum antiferromagnets with incommensurate spin correlations. We extend the two-component spinon field, describing lattice spins, to the $M$-component complex vector, and show, in the $\frac{1}{M}$ expansion, that for arbitrary small incommensurability longitudinal and transverse stiffnesses tend to the same value as the system approaches the quantum critical point. For physical spins ($M=2\mathrm{}$), this yields O(4) critical behavior. However, if the spin structure is commensurate, the longitudinal stiffness is identically zero. In this case, the critical behavior is the same as in the O(3) $\sigma$ model. We show how the critical exponents interpolate between O(3) and O(4) values near the transition. We also show that the competition between these two fixed points leads to a confinement-deconfinement transition at a finite temperature.