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

TL;DR

This paper investigates how anisotropy influences the finite-size critical behavior of the 2D Ising model and related d-dimensional models, revealing geometric dependence of scaling functions and proposing an extension of universality to anisotropic systems.

Contribution

It provides exact scaling functions for anisotropic 2D Ising models on strips and extends the concept of universality to anisotropic finite-size systems through shear transformations.

Findings

Scaling functions depend on the shape and orientation of the correlation volume.

Results are independent of microscopic anisotropy details, supporting a form of universality.

The approach relates anisotropic and isotropic free energy scaling functions via shear transformations.

Abstract

Anisotropy effects on the finite-size critical behavior of a two-dimensional Ising model on a general triangular lattice in an infinite-strip geometry with periodic, antiperiodic, and free boundary conditions (bc) in the finite direction are investigated. Exact results are obtained for the scaling functions of the finite-size contributions to the free energy density. With xi_> the largest and xi_< the smallest bulk correlation length at a given temperature near criticality, we find that the dependence of these functions on the ratio xi_< / xi_> and on the angle parameterizing the orientation of the correlation volume is of geometric rather than dynamic origin. Since the scaling functions are independent of the particular microscopic realization of the anisotropy within the two-dimensional Ising model, our results provide a limited verification of universality. We explain our observations by considering finite-size scaling of free energy densities of general weakly anisotropic models on a d-dimensional film, i.e., in an L x infinity^(d-1) geometry, with bc in the finite direction that are invariant under a shear transformation relating the anisotropic and isotropic cases. This allows us to relate free energy scaling functions in the presence of an anisotropy to those of the corresponding isotropic system. We interpret our results as a simple and transparent case of anisotropic universality, where, compared to the isotropic case, scaling functions depend additionally on the shape and orientation of the correlation volume. We conjecture that this universality extends to cases where the geometry and/or the bc are not invariant under the shear transformation and argue in favor of validity of two-scale factor universality for anisotropic systems.