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Physica Scripta

Critical Properties of the Ferromagnetic Dyson Chains With Discrete Spins

S. A. Bulgadaev

Published Aug 1, 1984• 0 citations
Publisher

Abstract

The critical properties of the one-dimensional ferromagnets with classical discrete spins interacting ∼ r -(1+σ) (1 - σ ≪ 1) in external magnetic field are investigated. Using low T expansion, relating these models with classical gases with generalized charges, the thermal and magnetic critical properties are found as for σ ≠ 1 (in first order on 1 - σ), as for σ = 1.

DOI
10.1088/0031-8949/30/2/010

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The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.
Critical properties of Dyson's hierarchical Ising model in one dimension with 'potential' falling off like r -(1+ sigma ) are examined in the range 0< sigma <1 where a phase transition is known to occur. A new exact renormalisation group recursion relation is derived for a 'dual' spin probability density function. Together with a scaling-type assumption the authors obtain eta =2- sigma and delta =(1+ sigma )/(1- sigma ) for all 0< sigma <1. Independent evidence, however, suggests that delta =3 for 0< sigma <or=1/2, along with other classical critical exponents in this region. In the non-classical region 1/2< sigma <1 accurate numerical values are obtained for the critical exponents nu and gamma and various Delta sigma = sigma -1/2 expansions to third order in Delta sigma .
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