Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: general considerations and exact spherical-model results.
Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: general considerations and exact spherical-model results.
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具有范德华型长程相互作用的系统中的过量自由能和卡西米尔力：一般考虑因素和精确的球形模型结果。
DOI：
10.1103/physreve.73.016131
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Daniel Grüneberg
中科院分区：
文献类型：
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作者：
D. Dantchev;H. Diehl;Daniel Grüneberg
我们考虑系统局限于宏观横向延伸和有限厚度L的d维板，在极限L ->无穷大经历连续的体相转变，并可由O（n）对称哈密顿量描述。周期性的边界条件应用于整个板。我们研究了长程对相互作用对Casimir效应的影响，长程对相互作用的势能随x ->无穷大而衰减，其中2<sigma<4和2<d+sigma<或=6，在体临界温度Tc，无穷大和附近，2<d<4。这些相互作用衰减得足够快，使得体临界指数和其他普适体量不变--即，它们在重整化群（RG）意义上是不相关的。然而，他们需要的过量自由能和卡西米尔力Fc的标准标度行为的重要修改。我们推广这些量的唯象标度Ansätze纳入这些长程相互作用。对于每单位横截面积的换算Casimir力，我们得到LdFc/kBt的形式，近似为Xi 0（L/Xiinfinity）+ g ω L-ω Xiomega（L/Xiinfinity）+ g sigmaL-ω sigmaXisigma（L/Xiinfinity）。其中Xi 0、Xiomega和Xisigma是通用标度函数; g omega和g sigma分别是与标度的主要校正和长程相互作用的主要校正相关联的标度场; omega和omega sigma = sigma + eta - 2是相关联的标度校正指数，其中eta表示没有长程相互作用的系统的标准体相关指数; xi infinity是（二阶矩）整体相关长度（其本身涉及对缩放的校正）。对于T不= Tc，贡献比例变量g σ在L中以代数方式而不是指数方式衰减，因此在适当的温度和L范围内占主导地位。我们得到确切的结果，球面和高斯模型，证实了这些发现。在d + sigma = 6的情况下，其中包括d = 3维的非滞后货车德瓦耳斯相互作用，发现球模型中与B成比例的标度修正的幂律得到修正.使用一般RG的想法，我们表明，这些对数奇点起源于退化Ω = Ω Σ = 4 - d发生的球形模型时，d + Σ = 6，在与B依赖的g Ω。
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.