具有非周期边界条件的薄膜几何尺寸效应：高斯模型和定维重整化群理论
Finite-size effects in film geometry with nonperiodic boundary conditions: Gaussian model and renormalization-group theory at fixed dimension
摘要：
在具有非周期边界条件的薄膜几何形状中，在具有各向同性和各向异性短程相互作用的高斯模型中研究了薄膜尺寸效应（b. c）。我们得到了1<d<4维中反周期、Neumann、Dirichlet和Neumann-Dirichlet混合b.c.的自由能和Casimir力的精确结果。对于Casimir力，发现有限尺寸标度对所有的b.c.都是有效的。对于自由能，有限尺寸标度在1<d<3和3<d<4维中对于反周期、Neumann和Dirichlet b.c.是有效的，但是对于Neumann和Dirichlet b.c.，在d=3维中存在与有限尺寸标度的对数偏差。这可以用边界尺寸d*=3来解释，其中高斯表面能密度的临界指数为零。对于诺依曼-狄利克雷公元前，当1<d<4时，在T_c以上强烈违反有限尺寸标度。我们的结果包括了体材料T_c附近的d维有限尺寸临界行为和T_c，薄膜（L）附近的（d-1）维临界行为之间的维度交叉的精确描述。这个维度的交叉被说明为比热的临界行为。当2<d<4时，高斯结果被重新表述为固定维数下φ ^4理论的单圈贡献，并与φ =4-d展开结果以及d=3的蒙特卡罗数据进行了比较。对于d=2，Casimir力标度函数的高斯结果进行了比较与那些为伊辛模型，意想不到的精确关系之间的高斯和伊辛标度函数。对于高斯和伊辛模型，它表明，各向异性耦合意味着非通用的Casimir力的标度函数。我们的高斯结果为研究具有非周期b.c.的平均球模型的有限尺寸效应提供了基础。
英文摘要：
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (b.c). We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed b.c. in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all b.c.. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet b.c., but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet b.c.. This is explained in terms of the borderline dimension d*=3, where the critical exponent of the Gaussian surface energy density vanishes. For Neumann-Dirichlet b.c., finite-size scaling is strongly violated above T_c for 1<d<4. Our results include an exact description of the dimensional crossover between the d-dimensional finite-size critical behavior near bulk T_c and the (d-1)-dimensional critical behavior near T_c,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. For 2<d<4, the Gaussian results are reformulated as one-loop contributions of the phi^4 theory at fixed dimension and are compared with the epsilon=4-d expansion results as well as with d=3 Monte Carlo data. For d=2, the Gaussian results for the Casimir force scaling function are compared with those for the Ising model; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the Gaussian and the Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model with nonperiodic b.c..
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