The Nature of the Continuum Limit in the 2D RP^2 Gauge Model

The RP 2 gauge model which allows interpolation between the RP 2 and O(3) spin models is studied in 2D. We use Monte-Carlo renormalization techniques for blocking the mean spin-spin interaction, 〈A〉, and the mean gauge field plaquette, 〈P 〉. The presence of the O(3) renormalized trajectory is verified and is consistent with the known three-loop β-function. The first-order ‘vorticity’ transition observed by Solomon et al. [1] is confirmed, and the location of the terminating critical point is established. New scaling flows in (〈A〉, 〈P 〉) are observed associated with a large exponent κ in the range 4 ∼ 5. The scaling flows are found to give rise to a strong cross-over effect between regions of high and low vorticity and are likely to induce an apparent signal for scaling in the cross-over region which we propose explains the scaling observed for RP 2 and RP 3 models [2] and also in a study of the SO(4) matrix model [3]. We show that the signal for this ‘pseudo’ scaling will occur for the RP 2 spin model in the cross-over region which is precisely the region in which computer simulations are done. We find that the RP 2 spin model is in the same universality class as the O(3) spin model but that it is likely to require a very large correlation length before the true scaling of this class sets in. We conjecture that the scaling flows are due either to the influence of a nearby new renormalized trajectory or to the ghost of the Kosterlitz-Thouless trajectory in the associated XY model. In the former case it is argued that the ‘vorticity’ fixed point controlling the critical behaviour terminating the first-order line cannot be identified with the conjectured new renormalized trajectory. DAMTP 97-68 and HUB-EP-97/78