Possible Failure of Asymptotic Freedom in Two-Dimensional $R{P}^2$ and $R{P}^3$ $\sigma$-Models

Abstract

We have simulated the two-dimensional $R{P}^2$ and $R{P}^3$ $\sigma$-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 $S^{N-1}/R{P}^{N-1}$ model (i.e. isovector + isotensor action) show a similar behavior when ${\beta }_T\to \infty$ with ${\beta }_V$ fixed $\text{ltapprox}0.6$, while they are consistent with conventional asymptotic freedom along the lines ${\beta }_T/{\beta }_V$ fixed $\text{ltapprox}2$. Taken as a whole, the data cast doubt on (though they do not completely exclude) the idea that $R{P}^{N-1}$ and $S^{N-1}$ $\sigma$-models lie in the same universality class.

Comments: Talk given at the Lattice 92 Conference, latex, 5 postscript figures, uses epsf.sty (all included) IFUP-Th 58/92 and SNS 16/92