Ultrathin XY-like magnetic films with 4-fold symmetry breaking crystal fields h4 show a universal magnetization exponent β≈0.23. This suggests that the films fall into the universality class of the isotropic XY model, which we have proposed to be characterized, in any large but finite system, by an effective exponent β = 3 π2/128 ≈0.23. This empirical conclusion is at first sight in disagreement with theoretical expectations, which would place the films in a class of nonuniversal critical behavior characteristic of h4, with β= J/8h4≫1. However, as h4 is a marginal perturbation to the 2D-XY model, crossover to critical behavior characteristic of the perturbation occurs only after rescaling to an exponentially large length scale. We argue that this is the reason for the fact that the nonuniversal behavior characteristic of the crystal fields has not yet been observed in experiment. We present Monte Carlo data to show that one should always observe the universal magnetization characteristic of the finite size 2D-XY model, although there is a possibility of observing a crossover to non-power law behavior characteristic of h4 at low temperature.

Bramwell, S. T., Holdsworth, P. C. W., & Rothman, J. (1997). Magnetization in Ultrathin Films: Critical Exponent β for the 2D XY Model with 4-Fold Crystal Fields. Modern Physics Letters B, 11(04), 139–148.