Oliveira T P, Branco N S
Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Santa Catarina, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jan;85(1 Pt 1):011113. doi: 10.1103/PhysRevE.85.011113. Epub 2012 Jan 6.
We employ a mean-field approximation to study the Ising model with aperiodic modulation of its interactions in one spatial direction. Two different values for the exchange constant, J(A) and J(B), are present, according to the Fibonacci sequence. We calculate the pseudocritical temperatures for finite systems and extrapolate them to the thermodynamic limit. We explicitly obtain the exponents β, δ, and γ and, from the usual scaling relations for anisotropic models at the upper critical dimension (assumed to be 4 for the model we treat), we calculate α, ν, ν(∥), η, and η(∥). Within the framework of a renormalization-group approach, the Fibonacci sequence is a marginal one and we obtain exponents that depend on the ratio r≡J(B)/J(A), as expected; however, the scaling relation γ=β(δ-1) is obeyed for all values of r we studied. We characterize some thermodynamic functions as log-periodic functions of their arguments, as expected for aperiodic-modulated models, and obtain precise values for the exponents from this characterization.
我们采用平均场近似来研究在一个空间方向上相互作用具有非周期调制的伊辛模型。根据斐波那契数列，存在两种不同的交换常数J(A)和J(B)值。我们计算有限系统的伪临界温度并将其外推到热力学极限。我们明确得到了指数β、δ和γ，并且根据上临界维度（对于我们所研究的模型假定为4）下各向异性模型的常用标度关系，我们计算出了α、ν、ν(∥)、η和η(∥)。在重整化群方法的框架内，斐波那契数列是一个边缘数列，正如预期的那样，我们得到了依赖于比率r≡J(B)/J(A)的指数；然而，对于我们研究的所有r值，都满足标度关系γ=β(δ - 1)。正如非周期调制模型所预期的那样，我们将一些热力学函数表征为其自变量的对数周期函数，并从这一表征中得到指数的精确值。
