Possible Failure of Asymptotic Freedom in Two-Dimensional $RP^2$ and $RP^3$ $σ$-Models

We have simulated the two-dimensional $RP^2$ and $RP^3$ $\sigma$-models, at correlation lengths up to about 220 (resp.\ 30), using a Wolff-type embedding algorithm. We see no evidence of asymptotic scaling. Indeed, the data rule out the conventional asymptotic scaling scenario at all correlation lengths less than about $10^{9}$ (resp.\ $10^{5}$). Moreover, they are consistent with a critical point at $\beta \approx 5.70$ (resp.\ 6.96), only 2\% (resp.\ 5\%) beyond the largest $\beta$ at which we ran. Preliminary studies of a mixed $S^{N-1}/RP^{N-1}$ model (i.e. isovector + isotensor action) show a similar behavior when $\beta_T \to \infty$ with $\beta_V$ fixed $\ltapprox 0.6$, while they are consistent with conventional asymptotic freedom along the lines $\beta_T/\beta_V$ fixed $\ltapprox 2$. Taken as a whole, the data cast doubt on (though they do not completely exclude) the idea that $RP^{N-1}$ and $S^{N-1}$ $\sigma$-models lie in the same universality class