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A spin-S model on a Bethe lattice

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  • Tamashiro, M.N.
  • Salinas, S.R.A.

Abstract

We present an exact formulation of an Ising spin-S model on a Cayley tree as a 2S-dimensional nonlinear discrete map. The attractors of the map are associated with the thermodynamic solutions on a Bethe lattice. We analyse in detail the typical half-integer case S = 32, with bilinear interactions and the inclusion of a crystal field. There is a stable paramagnetic fixed point at high temperatures. There is also a low-temperature region of stability of two distinct ferromagnetic fixed points. In addition to these usual attractors, we detect the presence of several unstable fixed points, which are however irrelevant to the thermodynamic behavior. In the limit of infinite coordination of the tree, the problem is simplified and we regain the standard results of a mean-field approximation. For S = 32, with biquadratic interactions but no crystal field, we confirm the existence of a ferrimagnetic phase for finite coordinations of the tree and analyse the occurrence of re-entrant boundaries inside the ordered region. For the typical integer spin case S = 2, without biquadratic interactions, we show the splitting of the first-order paramagnetic boundary, in agreement with mean-field calculations. Also, we confirm the existence of a ferrimagnetic phase in the S = 1 case for negative biquadratic interactions.

Suggested Citation

  • Tamashiro, M.N. & Salinas, S.R.A., 1994. "A spin-S model on a Bethe lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 124-146.
    • Handle: RePEc:eee:phsmap:v:211:y:1994:i:1:p:124-146
    DOI: 10.1016/0378-4371(94)90073-6
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    References listed on IDEAS

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    1. Nihat Berker, A., 1993. "Critical behavior induced by quenched disorder," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 194(1), pages 72-76.
    2. Carneiro, C.E.I. & Henriques, V.B. & Salinas, S.R., 1989. "On the equivalence of different Landau expansions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 162(1), pages 88-98.
    3. Coutinho, S. & SáBarreto, F.C. & Vasconcelos dos Santos, R.J., 1993. "Ising model randomly decorated with general spin angular momentum," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 196(3), pages 461-475.
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