The present work is motivated by the controversy on the nature of the phase transition on the Heisenberg stacked triangular antiferromagnet (STA). In particular, the renormalization group with $4-\epsilon$ expansion suggests a new universality class, while the renormalization group with $2+\epsilon$ expansion using a nonlinear $\sigma$ (NLS) model shows that the transition, if not mean-field tricritical or first order, is of the known $O(4)$ universality class. In order to verify this conjecture, we study here an equivalent system obtained from the STA by imposing the local rigidity as has been used in the NLS model. The results show that none of the scenarios predicted by the NLS is found. The critical exponent $\nu=0.48\pm0.05$ is quite different from the original STA without local rigidity, indicating that the local rigidity changes the nature of the transition. It is also different from that of $O(4)$. It means that successive transformations used to buildup the NLS model from the original STA may lead to the $O(4)$ universality class.[pdf]

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