由簇系数求临界指数
Critical exponents from cluster coefficients
摘要：
对于一大类排斥相互作用模型，Mayer集团积分可以转化为一个三对角实对称矩阵$R_{mn}$，其元素收敛于两个常数.这使得一个有效的外推这些模型的状态方程。由于在负实z轴附近的（非物理的）奇点，标准方法（例如基于簇积分展开的Padè近似）无法捕捉这些模型在有序转变附近的行为，特别是不能检测临界点。最近的一项工作（Eisenberg和Baram，PNAS { \bf 104}，5755（2007））表明，如果$R$矩阵元素衰减到其渐近常数遵循$1/n^2$定律，则可以精确计算临界指数$σ$和$σ'$，它们表征密度奇异性作为活动的函数。在这里，我们采用重正化参数扩展这一结果和分析的情况下，渐近的方法对它们的极限值的$R$矩阵元素是一个更一般的形式。相关的渐近校正项（RG意义上）的确定，然后我们提供了一个校正的临界指数的精确公式。我们确定的公式的使用限制，并证明了一个物理模型，这是超出其有效性范围。对新公式进行了数值验证，并将其应用于多个混凝土物理模型的分析。
英文摘要：
For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix $R_{mn}$, whose elements converge to two constants. This allows for an effective extrapolation of the equation of state for these models. Due to a nearby (nonphysical) singularity on the negative real z axis, standard methods (e.g. Padè approximants based on the cluster integrals expansion) fail to capture the behavior of these models near the ordering transition, and, in particular, do not detect the critical point. A recent work (Eisenberg and Baram, PNAS {\bf 104}, 5755 (2007)) has shown that the critical exponents $σ$ and $σ'$, characterizing the singularity of the density as a function of the activity, can be exactly calculated if the decay of the $R$ matrix elements to their asymptotic constant follows a $1/n^2$ law. Here we employ renormalization arguments to extend this result and analyze cases for which the asymptotic approach of the $R$ matrix elements towards their limiting value is of a more general form. The relevant asymptotic correction terms (in RG sense) are identified and we then provide a corrected exact formula for the critical exponents. We identify the limits of usage of the formula, and demonstrate one physical model which is beyond its range of validity. The new formula is validated numerically and then applied to analyze a number of concrete physical models.
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