Kastening Boris
Institute for Materials Science, Technische Universität Darmstadt, D-64287 Darmstadt, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Oct;86(4 Pt 1):041105. doi: 10.1103/PhysRevE.86.041105. Epub 2012 Oct 2.
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 ξ(>) the largest and ξ(<) the smallest bulk correlation length at a given temperature near criticality, we find that the dependence of these functions on the ratio ξ(<)/ξ(>) and on the angle parametrizing the orientation of the correlation volume is of geometric nature. 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×∞(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 weakly anisotropic systems.
研究了在无限条带几何结构中，具有周期、反周期和自由边界条件（bc）的一般三角形晶格上二维伊辛模型的有限尺寸临界行为的各向异性效应。得到了自由能密度有限尺寸贡献的标度函数的精确结果。在临界温度附近给定温度下，用ξ(>)表示最大体关联长度，ξ(<)表示最小体关联长度，我们发现这些函数对ξ(<)/ξ(>)的比值以及对描述关联体积取向的角度的依赖具有几何性质。由于标度函数与二维伊辛模型内各向异性的具体微观实现无关，我们的结果对普遍性提供了有限的验证。我们通过考虑在有限方向上具有bc的d维薄膜（即L×∞(d - 1)几何结构）上一般弱各向异性模型的自由能密度的有限尺寸标度来解释我们的观察结果，这些bc在将各向异性和各向同性情况联系起来的剪切变换下是不变的。这使我们能够将存在各向异性时的自由能标度函数与相应各向同性系统的自由能标度函数联系起来。我们将我们的结果解释为各向异性普遍性的一个简单而透明的例子，与各向同性情况相比，标度函数还额外依赖于关联体积的形状和取向。我们推测这种普遍性扩展到几何结构和/或bc在剪切变换下不不变的情况，并支持弱各向异性系统的双标度因子普遍性的有效性。
