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Physics Letters

The decay of thermodynamical fluctuations in magnetisation in the constant coupling approximation for a Heisenberg ferromagnet

J. Kociński

Published Apr 1, 1963• 1 citation
Publisher
DOI
10.1016/0031-9163(63)90355-x

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A statistical theory of the localized model of ferromagnetism is developed by considering the effective density matrices for a single spin, and two nearest-neighbor spins, respectively. The Ising Model and the Heisenberg model, and even the case of the anisotropic exchange interaction, are treated within a unified mathematical scheme. The result is reduced to that of the Bethe approximation in the case of the Ising model, and the constant coupling approximation in the case of the Heisenberg model. The method is generalized to calculate the static spin-correlation function. It is obtained in Ornstein-Zernike type. In the nonequilibrium state, the expectation value of the spin decays by the homogeneous damping as well as the spin-diffusion process. This relaxation equation is different from that by Van Hove because of the existence of the homogeneous damping. The diffusion constant does not vanish even at the Curie point TC, contrary to Van Hove's result, although the damping constant vanishes at TC. The latter fact causes the critical slowing down of the regression of fluctuation. The solution of this relaxation equation gives the time-dependent spin-correlation function. The method can be applied to the antiferromagnetic case, too.
Results of numerical studies on the equilibrium scaling stiffness energy for 2D nearest-neighbour Heisenberg systems at T=0 are presented. The authors first consider the percolation problem and estimate the critical exponent for the spin wave stiffness to be close to 1. They then consider systems in which -J-bonds appear with probability x and +J-bonds with probability (1-x). The scaling stiffness energy, a measure of sensitivity to changes in the boundary conditions, disappears exponentially with x at x c approximately=0.13 (an estimate for large systems). This value of concentration is identified as an upper boundary of the usual ferromagnetic phase. Nevertheless the magnetisation persists above x c and disappears at X c approximately=0.30 when the system enters into the spin glass phase. The region between x c and X c is identified as corresponding to the Gabay-Toulouse mixed phase in which the system looks like a spin glass but has some non-zero magnetisation. This interpretation receives further support from studies of the two- and four-spin correlation functions in local energy minima. The correlations decay exponentially, at least on small distances. The correlation lengths of the two-spin correlations start to drop at x c . The spin components parallel to the magnetisation are actually less correlated than the transverse ones.
Using a recently introduced renormalisation group method, the authors study the behaviour of the spontaneous surface and bulk magnetisations as functions of the temperature for an Ising ferromagnet in a semi-infinite cubic lattice for various J S /J B ratios (J S and J B respectively are the surface and bulk coupling constants). In particular, they study the extraordinary transition where the surface maintains its magnetisation as the bulk disorders: the authors find a discontinuity in the first derivative of the surface magnetisation at the bulk transition temperature. The criticality of the system (universality classes, critical exponents and amplitudes) is discussed as well.