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The asymmetric continuous distribution function of the effective field of the Ising model in the spin glass and the ferromagnetic states on the Bethe lattice

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  • Sasaki, Mitsuhiro
  • Katsura, Shigetoshi

Abstract

The distribution of effective fields of the Ising spin glass on the Bethe lattice at T=0 is obtained for a general concentration p of ±J bonds, by solving the integral equation. The solutions of the integral equations are classified into (i) paramagnetic solution, (ii) spin glass with symmetric discrete distribution with more than three delta functions, (ii′) spin glass with symmetric distribution with more than three delta functions, (iii) discrete asymmetric ferromagnetic distribution, (iv) spin glass with symmetric continuous distribution, (v) asymmetric continuous distribution. The values of the concentration p of the +J bonds at which the solutions (ii) and (iii), (iv) and (v), (v) and (iii) are connected, are obtained to be p1=78, p2=0.869427 and p3=1112 respectively. It is expected that the symmetric spin glass, asymmetric continuous distribution, and discrete asymmetric distribution are realized in 12 ⩽p⩽p2, p2⩽p⩽p3 and p3⩽p⩽1, respectively. The distribution functions in each region are shown explicitly.

Suggested Citation

  • Sasaki, Mitsuhiro & Katsura, Shigetoshi, 1989. "The asymmetric continuous distribution function of the effective field of the Ising model in the spin glass and the ferromagnetic states on the Bethe lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 157(3), pages 1195-1202.
    • Handle: RePEc:eee:phsmap:v:157:y:1989:i:3:p:1195-1202
    DOI: 10.1016/0378-4371(89)90039-3
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    1. Gebauer, Rolf H, 1987. "Socio-economic Classification of Farm Households--Conceptual, Methodological and Empirical Considerations," European Review of Agricultural Economics, Oxford University Press and the European Agricultural and Applied Economics Publications Foundation, vol. 14(3), pages 261-283.
    2. Katsura, Shigetoshi & Inawashiro, Sakari & Fujiki, Sumiyoshi, 1979. "Spin glasses for the infinitely long ranged bond Ising model and for the short ranged binary bond Ising model without use of the replica method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 99(1), pages 193-216.
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