Cluster Variation Method, Effective Field Method, and the Method of Integral Equation on the Regular and Random Ising Models

  • Shigetoshi Katsura


The effective field method for the Ising model is closely related to the cluster variation method. In the first part we consider an effective field theory of the simple cubic lattice. Density matrices of a vertex, ρ (1), a pair, ρ (2), a square, ρ (4), and a cube, ρ (8), are written in terms of a single bond effective field λ, an effective field due to a square, λ′, that due to a cube, λ″, an effective interaction due to a square, ν, and that due to a cube ν′. The reducibilities of ρ (2) to ρ (1), ρ (4) to ρ (2), and ρ (8) to ρ (4) lead to relations between λ, λ′, λ″, ν, and v′. These relations give the critical temperature of the Ising model on the simple cubic lattice as κT/J = 4.60402.

In the second part the ± J random Ising model is considered. The distribution of the single bond effective field h is denoted by g(h). The consistency among the effective fields leads to an integral equation. We can derive phase diagrams of the system on the square and the simple cubic lattices, the triangular and the face-centered cubic lattices. The shape of the phase diagram of the former is symmetric with respect to +J and − J sides, while that of the latter is asymmetric in reflection of the crystal structure. Our method also gives a qualitatively well behaved phase diagram of Eu p Sr1−p S. The integral equation can be exactly solved at T = 0 and discrete solutions and continuous solutions are obtained. The latter give the phase boundary between spin glass, MSG, and ferromagnetic phases.


Ising Model Spin Glass Effective Interaction Effective Field Random Ising Model 
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Copyright information

© Plenum Press, New York 1996

Authors and Affiliations

  • Shigetoshi Katsura
    • 1
  1. 1.Tohoku College of Engineering and Information SciencesAobaku, SendaiJapan

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