Interface localization transition in Ising films with competing walls: Ginzburg criterion and crossover scaling

In a simple fluid or Ising magnet in a thin film geometry confined between walls a distance D apart that exert opposing surface fields, an interface parallel to the walls is stabilized below the bulk critical temperature T_cb. While this interface is "delocalized" (i.e., freely fluctuating in the center of the film) for T_cb>T>T_c(D), below the "interface localization transition" temperature T_c(D) the interface is bound to one of the walls. Using the mean field description of Parry and Evans [Physica A 181, 250 (1992)], we develop a k^-1 Ginzburg criterion to show that the Ginzburg number scales exponentially with thickness, Gi∞exp( - κD/2), κ^-1 being the appropriate transverse length scale associated with the interface. Therefore, mean field theory is self-consistent for large D, thus explaining why recent Monte Carlo simulations observed Ising criticality only in a very close neighborhood of T_c(D). A crossover scaling description is used to work out the thickness dependence of the critical amplitudes in the Ising critical regime. Extending these concepts to consider finite size effects associated with the lateral linear dimension L, we reanalyze the Monte Carlo results of Binder, Landau, and Ferrenberg [Phys. Rev. B 51, 2823 (1995)]. The data are in reasonable agreement with the theory, provided one accepts the suggestion of Parry et al. [Physica A 218, 77 (1995); 218, 109 (1995)] that the length scale κ^-1=ξ_b(1 + ω/2), where ξ_b is the true correlation range in the bulk, and ω is the universal amplitude associated with the interfacial stiffness.