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理论物理通讯：英文版

A Direct Calculation of Critical Exponents of Two-Dimensional Anisotropic Ising Model

Xiong Xiong, Gäng, Wang, Xiangrong

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

用一个维的量的一个准确答案横向地的 Isingmodel，我们计算二维的各向异性的古典 Ising 模型(IM ) 的批评代表。我们证实代表与各向同性的古典 IM 的那些一样。我们的途径提供获得并且验证这些著名结果的一个其他的工具。

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The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.
We study the critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the Ising model (IM) with random bonds, the dilute N-color Ashkin–Teller model (ATM) and some its generalizations. It is shown that all these models exhibit the same critical behavior as that of the 2D-IM apart from some logarithmic corrections. The minimal conformal field theory (CFT) models with randomness are found to be described by critical exponents which are numerically very close to those of the pure 2D-IM.
The form of the universal scaling law of the critical dynamic exponent, , is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents and . So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, , on the site models of the regular fractals with . Meanwhile, we discuss Daniel Kandal's correction to the formula of the, critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.