Possibility of a Topological Phase Transition in Two-dimensional $RP^3$ Model

We study by large-scale Monte Carlo simulation the $RP^3$ model, which can be regarded as an effective low-energy model of a triangular lattice Heisenberg antiferromagnet. $Z_2$ vortices appear as elementary excitations in the triangular lattice Heisenberg antiferromagnet. Such $Z_2$ vortices are ubiquitous in other frustrated Heisenberg spin systems that have noncollinear long-range orders. In this study, we investigate a possible topological phase transition driven by the binding--unbinding of $Z_2$ vortices. By extracting important degrees of freedom, we map a frustrated spin system to an effective $RP^3$ model. From large-scale Monte Carlo simulation, we obtain an order parameter and a correlation length of up to $L=16384$. Concerning the existence of a $Z_2$-vortex transition, by extrapolating the order parameter to the thermodynamics limit assuming the $Z_2$-vortex transition, we obtain a finite transition temperature as $T_v/\tilde{J} \simeq 0.25$. Our estimate of the correlation length at $T_v$ is much larger than $L=16384$, which is beyond the previous estimate obtained with the triangular lattice Heisenberg model.