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Acta Physica Polonica A

Effective Field Methods for Magnetic Systems

G. Kamieniarz, Raf Dekeyser, Grzegorz Musiał

Published Feb 1, 1994• 3 citations
Publisher

Abstract

The self-consistency conditions inherent in the mean-field approximation are extended to take into account some pair correlations.Ii is shown that this extension is systematically improvable and applicable to the spin systems with short-range interactions in the entire temperature region.With two correlations included and the renormalization-group ideas implemented, the critical properties of the Ising model are satisfactorily revealed.Moreover, the improved estimates of the thermodynamical quantities for the quantum linear XY model from the finite-size calculations are found down to very low temperatures.

DOI
10.12693/aphyspola.85.413

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The self-consistency conditions inherent in the mean-field approximation are extended to take into account some pair correlations.Ii is shown that this extension is systematically improvable and applicable to the spin systems with short-range interactions in the entire temperature region.With two correlations included and the renormalization-group ideas implemented, the critical properties of the Ising model are satisfactorily revealed.Moreover, the improved estimates of the thermodynamical quantities for the quantum linear XY model from the finite-size calculations are found down to very low temperatures.
The selfconsistency conditions inherent in the mean field approximation are generalized in several ways according to the symmetry of the Hamiltonian. A substantial improvement on the estimates of thermodynamic properties and critical parameters is obtained through rather simple calculations. Non-classical values of critical exponents are found, implementing the mean-field renormalization group method. These techniques can be applied both to classical and to quantum-mechanical systems, especially in low dimensions.
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We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blöte who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .
The recently proposed multi-interaction mean-field renormalization technique [C. N. Likos and A. Maritan, Phys. Rev. E 53, 3303 (1996)] is applied to the Ising model on a triangular lattice with equal couplings in two directions and a different coupling in the third, a model equivalent to a square Ising model with additional second-neighbor interactions along a single diagonal direction. Three different clusters are considered and the possible mappings between them are discussed. The estimates for critical couplings and exponents are in satisfactory agreement with exact results. An explanation of the way in which the finite-size features of the method bring about a systematic overestimation of the critical temperature is also given.
Applying the master equation to an Ising model, equations of motion are obtained for the average of a single spin and for the average of a pair of spins. These equations relax to equilibrium values that predict a self-consistency relationship among the internal field, temperature, and exchange energy. The critical temperature obtained is that of the Bethe approximation, and is extended to include a static external field in the magnetization direction. With this, the relaxation of an Ising model near the Curie temperature has been investigated in a self-consistent manner. The results are similar to those of Suzuki and Kubo, but in the Bethe approximation instead of the Bragg-Williams approximation, and hence predict the relaxation of the correlation as well. In particular we studied a nearly Ising spin system that is a system with relatively small isotropic transverse coupling. An oscillating, linear, and external magnetic field applied transverse to the strongly coupled direction of the spins leads to lowering of the critical temperature.