Agreement of Capillary-Wave Theory with Exact Results for the Interface Profile of the Two-Dimensional Ising Model
Agreement of Capillary-Wave Theory with Exact Results for the Interface Profile of the Two-Dimensional Ising Model
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DOI：
10.1103/physrevlett.48.368
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发表时间：
1982-02
影响因子：
8.6
通讯作者：
M. Fisher;D. Fisher;J. Weeks
中科院分区：
文献类型：
--
作者：
M. Fisher;D. Fisher;J. Weeks
Abraham最近计算了二维Ising模型中界面的若干性质，特别是当系统尺寸趋于无穷大时的界面轮廓。他注意到他的精确结果采用了简单毛细波理论所预言的函数形式，但其中涉及界面张力的系数与毛细波的预言不一致。然而，通常的毛细波理论假定各向同性流体的界面张力与角度无关。这一假设不适用于低温下的伊辛模型，因为在低温下界面张力具有显著的角相关性。本文指出，如果采用广义毛细波表达式，并以适当的宏观方式定义界面张力（可能与角有关），则进入Abraham界面轮廓表达式的系数在所有温度下也都由广义毛细波理论精确地给出。因此，这一精确结果为毛细波理论给出的界面漂移的物理图象提供了额外的支持。“在二维伊辛模型等晶格中，界面张力y（θ）定义为平均角为~的壁的每单位笛卡尔长度的界面自由能，在低温下通常是各向异性的。“[当T-T_y（9）变得与8无关时，就像普通流体的情况一样。将在角度θ = 0（i.例如，平行于晶格轴），并且对于小角度，我们可以展开y（θ）= y（θ）+ Ty”（θ）9 '+.由于二维伊辛模型的粗糙化温度为零，因此在任何非零温度下都没有线性项。“如果我们有一个宽度为I-的宏观系统，并在水平界面上施加平均角度为0的倾斜，则界面的新长度为L sec 9，由于这种扭曲而引起的自由能变化为：
Abraham'has recently calculatedseveral prop-erties of interfaces in two-dimensional(2D) Ising models, in particular the interface profile as the system size tends to infinity. He observes that his exact results take the functional forms pre-dicted by simple capillary-wave theory'but with coefficients involving the interface tension in dis-agreement with the capillary-wave predictions. However, the usual capillary-wave theory'as-sumes an isotropic fluid with an interface tension independent of angle. This assumption does not apply to theIsing model at low temperatures for which the interface tension has a significantangu-lar dependence.'In this note we point out that if generalized capillary-wave expressions are used'with the (possibly angular dependent) interface tension de-fined in the proper macroscopic manner, then the coefficients entering Abraham's expressions for the interface profile are also given exactly by the generalized capillary-wave theory at all tempera-tures. Thisexact result thus provides additional support for the physical picture of interface wandering given by the capillary-wavetheory." In a latticesystem such as the 2D Ising model, the interface tension y (9), defined as the interfacial free energy per unit Cartesian length for a wall with average angle~, is at low temperatures generally anisotropic."[As T-T „y (9) becomes independent of 8, as is the case for an ordinary fluid.'] There will be a minimum at an angle 9= 0 (i. e., parallel to alattice axis), and for small angles we can expand y (9)= y (0)+ Ty"(0) 9'+.... There is no linear term in 0 at any nonzero temperature since the roughening temperature for the 2D Ising model is zero." If we have a macroscopic system of width I-and impose atilt with average angle 0 on a horizontal interface the new length of the interface is L sec9 and the change in free energy due to this distortion is