Dantchev Daniel, Diehl H W, Grüneberg Daniel
Fachbereich Physik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jan;73(1 Pt 2):016131. doi: 10.1103/PhysRevE.73.016131. Epub 2006 Jan 25.
We consider systems confined to a d-dimensional slab of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L --> infinity and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as bx-(d+sigma) as x --> infinity, with 2<sigma<4 and 2<d+sigma< or =6, on the Casimir effect at and near the bulk critical temperature Tc,infinity, for 2<d<4. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged--i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force Fc. We generalize the phenomenological scaling Ansätze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form LdFc/kBt approximately Xi0(L/xi infinity) + g omegaL -omega Xi omega (L/Xi infinity) + g sigma L -omega sigma Xi sigma (L/Xi infinity). Here Xi0, Xi omega, and Xi sigma are universal scaling functions; g omega and g sigma are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; omega and omega sigma = sigma + eta - 2 are the associated correction-to-scaling exponents, where eta denotes the standard bulk correlation exponent of the system without long-range interactions; xi infinity is the (second-moment) bulk correlation length (which itself involves corrections to scaling). The contribution proportional variant g sigma decays for T not = Tc,infinity algebraically in L rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and L. We derive exact results for spherical and Gaussian models which confirm these findings. In the case d + sigma = 6, which includes that of nonretarded van der Waals interactions in d = 3 dimensions, the power laws of the corrections to scaling proportional to b of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy omega = omega sigma = 4 - d that occurs for the spherical model when d + sigma = 6, in conjunction with the b dependence of g omega.
我们考虑限制在具有宏观横向延伸和有限厚度(L)的(d)维平板中的系统，该系统在(L\to\infty)的极限下经历连续的体相转变，并且可以用(O(n))对称哈密顿量来描述。在平板上施加周期性边界条件。我们研究长程对相互作用的影响，其势能在(x\to\infty)时按(bx^{-(d + \sigma)})衰减，其中(2 < \sigma < 4)且(2 < d + \sigma \leq 6)，对于(2 < d < 4)，在体临界温度(T_{c,\infty})及附近的卡西米尔效应。这些相互作用衰减得足够快，以使体临界指数和其他通用体数量保持不变——即，它们在重整化群（RG）意义上是无关的。然而，它们对过量自由能和卡西米尔力(F_c)的标准标度行为进行了重要修改。我们通过纳入这些长程相互作用，对这些量的唯象标度假设进行了推广。对于每单位横截面积的标度化约化卡西米尔力，我们得到形式(L^dF_c/k_BT \approx \Xi_0(L/\xi_{\infty}) + g_{\omega}L^{-\omega}\Xi_{\omega}(L/\Xi_{\infty}) + g_{\sigma}L^{-\omega_{\sigma}}\Xi_{\sigma}(L/\Xi_{\infty}))。这里(\Xi_0)、(\Xi_{\omega})和(\Xi_{\sigma})是通用标度函数；(g_{\omega})和(g_{\sigma})分别是与标度的主导修正以及长程相互作用的修正相关的标度场；(\omega)和(\omega_{\sigma} = \sigma + \eta - 2)是相关的标度修正指数，其中(\eta)表示没有长程相互作用的系统的标准体关联指数；(\xi_{\infty})是（二阶矩）体关联长度（其本身涉及标度修正）。与(g_{\sigma})成比例的贡献在(T\neq T_{c,\infty})时在(L)中代数衰减而不是指数衰减，因此在适当的温度和(L)范围内变得占主导。我们推导了球形模型和高斯模型的精确结果，证实了这些发现。在(d + \sigma = 6)的情况下，其中包括三维中无延迟的范德瓦尔斯相互作用的情况，发现球形模型中标度修正与(b)成比例的幂律会被对数修改。使用一般的RG思想，我们表明这些对数奇点源于当(d + \sigma = 6)时球形模型出现的简并(\omega = \omega_{\sigma} = 4 - d)，以及(g_{\omega})对(b)的依赖性。
