Nature of phase transition in magnetic thin films

We study the critical behavior of magnetic thin films as a function of the film thickness. We use the ferromagnetic Ising model with the high-resolution multiple histogram Monte Carlo (MC) simulation. We show that though the 2D behavior remains dominant at small thicknesses, there is a systematic continuous deviation of the critical exponents from their 2D values. We observe that in the same range of varying thickness the deviation of the exponent $\nu$ is very small from its 2D value, while exponent $\beta$ suffers a larger deviation. Moreover, as long as the film thickness is fixed, i. e. no finite size scaling is done in the $z$ direction perpendicular to the film, the 3D values of the critical exponents cannot be attained even with very large (but fixed) thickness. The crossover to 3D universality class cannot therefore take place without finite size scaling applied in the $z$ direction, in the limit of numerically accessible thicknesses. From values of exponent $\alpha$ obtained by MC, we estimate the effective dimension of the system. We conclude that with regard to the critical behavior, thin films behave as systems with effective dimension between 2 and 3.