Chen X S, Dohm V
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 May;67(5 Pt 2):056127. doi: 10.1103/PhysRevE.67.056127. Epub 2003 May 27.
We calculate finite-size effects of the Gaussian model in a Lx(d-1) box geometry with free boundary conditions in one direction and periodic boundary conditions in d-1 directions for 2<d<4. We also consider film geometry (L--> infinity ). Finite-size scaling is found to be valid for d<3 and d>3 but logarithmic deviations from finite-size scaling are found for the free energy and energy density at the Gaussian upper borderline dimension d*=3. The logarithms are related to the vanishing critical exponent 1-alpha-nu=(d-3)/2 of the Gaussian surface energy density. The latter has a cusplike singularity in d>3 dimensions. We show that these properties are the origin of nonscaling finite-size effects in the mean spherical model with free boundary conditions in d > or =3 dimensions. At bulk T(c), in d=3 dimensions we find an unexpected nonlogarithmic violation of finite-size scaling for the susceptibility chi approximately L3 of the mean spherical model in film geometry, whereas only a logarithmic deviation chi approximately L2 ln L exists for box geometry. The result for film geometry is explained by the existence of the lower borderline dimension d(l)=3, as implied by the Mermin-Wagner theorem, that coincides with the Gaussian upper borderline dimension d*=3. For 3<d<4 we find a power-law violation of scaling chi approximately L(d-1) at bulk T(c) for box geometry and a nonscaling temperature dependence chi(surface) approximately xi(d) of the surface susceptibility above T(c). For 2<d<3 dimensions we show the validity of universal finite-size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for T > or =T(c).
我们计算了高斯模型在Lx(d - 1)盒状几何结构中的有限尺寸效应，其中在一个方向上为自由边界条件，在d - 1个方向上为周期性边界条件，2 < d < 4。我们还考虑了薄膜几何结构（L趋于无穷大）。发现对于d < 3和d > 3，有限尺寸标度是有效的，但在高斯上临界维度d* = 3时，自由能和能量密度存在与有限尺寸标度的对数偏差。这些对数与高斯表面能密度的临界指数1 - α - ν = (d - 3)/2的消失有关。后者在d > 3维中有尖点状奇点。我们表明，这些性质是d≥3维中具有自由边界条件的平均球模型中无标度有限尺寸效应的起源。在体相临界温度T(c)时，在d = 3维中，我们发现薄膜几何结构中平均球模型的磁化率χ约为L3时，有限尺寸标度存在意外的非对数违反，而盒状几何结构中仅存在对数偏差χ约为L2lnL。薄膜几何结构的结果可由Mermin - Wagner定理所暗示的下临界维度d(l) = 3的存在来解释，它与高斯上临界维度d* = 3一致。对于3 < d < 4，我们发现在体相临界温度T(c)时，盒状几何结构的磁化率存在幂律违反χ约为L(d - 1)，并且在T(c)以上表面磁化率χ(surface)约为ξ(d)存在无标度温度依赖性。对于2 < d < 3维，我们展示了具有自由边界条件的平均球模型在盒状和薄膜几何结构中磁化率的通用有限尺寸标度的有效性，并计算了T≥T(c)时相应的通用标度函数。
