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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

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  • Katsura, Shigetoshi
  • Inawashiro, Sakari
  • Fujiki, Sumiyoshi

Abstract

The long ranged Gaussian random Ising bond model and short ranged binary random Ising bond model are discussed by the method of the pair approximation of the cluster variation and by the method of the integral equation for the distribution function of the effective fields. For the long ranged model, Sherrington and Kirkpatrick's result is generalized and rederived without use of the replica method. For the short ranged model, i.e. a binary mixture of JA = -JB, the integral equation is solved exactly and the energy of the spin glass state is obtained at T = 0.

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  • 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.
    • Handle: RePEc:eee:phsmap:v:99:y:1979:i:1:p:193-216
    DOI: 10.1016/0378-4371(79)90130-4
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    References listed on IDEAS

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    1. Storing, Herbert J. & Strauss, Leo & Berns, Walter & Storing, Herbert J. & Weinstein, Leo & Horwitz, Robert, 1963. "Replies to Schaar and Wolin: I," American Political Science Review, Cambridge University Press, vol. 57(1), pages 151-160, March.
    2. Kirkpatrick, Jeane, 1975. "Representation in the American National Conventions: the Case of 1972," British Journal of Political Science, Cambridge University Press, vol. 5(3), pages 265-322, July.
    3. Morita, T. & Horiguchi, T., 1975. "Classical one-dimensional Heisenberg model with an interaction of finite range," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 83(3), pages 519-532.
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    1. Katsura, Shigetoshi & Nagahara, Izuru, 1980. "Effect of the frustration to the ground state energy and entropy of the spin-glass in the random bond Ising model on the square lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 104(3), pages 397-416.
    2. Morita, T., 1984. "Spin-glass and ferromagnetic phases of the random-bond Ising model on the Bethe lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 125(2), pages 321-343.
    3. Katsura, Shigetoshi & Miyamoto, Nahomi, 1982. "Random Ising model in the magnetic field at T = 0," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 112(3), pages 393-404.
    4. Morita, T., 1983. "Free energy of the random Ising model in terms of the magnetizations of sites," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 143-152.
    5. Horiguchi, Tsuyoshi, 1981. "On the bethe approximation for the random bond Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 107(2), pages 360-370.
    6. Shirakura, Takayuki & Katsura, Shigetoski, 1983. "Spin glass in the infinitely long-ranged bond-random anisotropic classical planar model-longitudinal and transverse spin glasses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 117(2), pages 281-299.
    7. Katsura, Shigetoshi & Kikuchi, Ryoichi, 1984. "The cluster variation method and the method of the reducibility of density matrices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 123(2), pages 595-604.
    8. 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.
    9. Akagi, Eiichi & Seino, Mitsuhiro & Katsura, Shigetoshi, 1991. "The free energy at T = 0 of the spin glass in the case of the continuous distribution of the effective field on the Bethe lattice for Z = 4, 5, 6," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 178(3), pages 406-414.
    10. de Oliveira, Mário J., 1988. "Ising spin glass in a field zero temperature in the Bethe approximation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 150(3), pages 614-626.
    11. Katsura, Shigetoshi & Suenaga, Toshiyuki & Imaizumi, Takahiro, 1982. "The boundary between the spin-glass and ferromagnetic states of the bond-random Ising model in the square lattice at T = 0," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(3), pages 368-379.
    12. Katsura, Shigetoshi, 1980. "Entropy of the spin-glass state in the binary mixture of the ferro- and antiferromagnetic random Ising model at T = 0," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 104(1), pages 333-338.
    13. de Oliveira, M.J., 1988. "Ising spin glass in the Bethe approximation at zero temperature," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 148(3), pages 567-574.

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