Dynamic Critical Behaviour of Wolff's Algorithm for $RP^N$ $\sigma$-Models

We study the performance of a Wolff-type embedding algorithm for $RP^N$ $\sigma$-models. We find that the algorithm in which we update the embedded Ising model \`a la Swendsen-Wang has critical slowing-down as $z_\chi \approx 1$. If instead we update the Ising spins with a perfect algorithm which at every iteration produces a new independent configuration, we obtain $z_\chi \approx 0$. This shows that the Ising embedding encodes well the collective modes of the system, and that the behaviour of the first algorithm is connected to the poor performance of the Swendsen-Wang algorithm in dealing with a frustrated Ising model.