Universal anisotropic finite-size critical behavior of the two-dimensional Ising model on a strip and of d-dimensional models on films

摘要

Anisotropy effects on the finite-size critical behavior of a two-dimensional Ising model on a general triangularnlattice in an infinite-strip geometry with periodic, antiperiodic, and free boundary conditions (bc) in the finitendirection are investigated. Exact results are obtained for the scaling functions of the finite-size contributions tonthe free energy density. With ξ> the largest and ξ< the smallest bulk correlation length at a given temperaturennear criticality, we find that the dependence of these functions on the ratio ξ and on the angle parametrizingnthe orientation of the correlation volume is of geometric nature. Since the scaling functions are independentnof the particular microscopic realization of the anisotropy within the two-dimensional Ising model, our resultsnprovide a limited verification of universality. We explain our observations by considering finite-size scalingnof free energy densities of general weakly anisotropic models on a d-dimensional film (i.e., in an L×∞d−1ngeometry) with bc in the finite direction that are invariant under a shear transformation relating the anisotropicnand isotropic cases. This allows us to relate free energy scaling functions in the presence of an anisotropy to thosenof the corresponding isotropic system. We interpret our results as a simple and transparent case of anisotropicnuniversality, where, compared to the isotropic case, scaling functions depend additionally on the shape andnorientation of the correlation volume. We conjecture that this universality extends to cases where the geometrynand/or the bc are not invariant under the shear transformation and argue in favor of validity of two-scale factornuniversality for weakly anisotropic systems.