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On some mean-field like approximations for the triangular Ising antiferromagnet

Alessandro Pelizzola, Marco Pretti

Published Feb 3, 1999• 0 citations
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Abstract

Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B {\bf 56}, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field like approximations for this model, namely the Bethe approximation itself, the cluster variation method and the hard-spin mean-field theory. We show: (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered phase which is stable down to zero temperature; (iv) a few, so far unknown, zero temperature results (entropy and internal energy) of the hard-spin mean-field theory.

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Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B {\bf 56}, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field like approximations for this model, namely the Bethe approximation itself, the cluster variation method and the hard-spin mean-field theory. We show: (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered phase which is stable down to zero temperature; (iv) a few, so far unknown, zero temperature results (entropy and internal energy) of the hard-spin mean-field theory.
Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B 56, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field-like approximations for this model, namely, the Bethe approximation itself, the cluster variation method, and the hard-spin mean-field theory. We show (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero-temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered phase which is stable down to zero temperature; (iv) a few, so far unknown, zero-temperature results (entropy and internal energy) of the hard-spin mean-field theory.
The cluster variation method is a powerful hierarchy of mean-field-like approximations for Ising-like lattice models, which is known to give results that are particularly accurate at high and/or low temperature and converge to the exact ones as the size of the clusters taken into account gets larger. Here we show how these properties can be exploited to obtain non-classical, quite accurate estimates of quantities characterizing the critical behavior of the ordinary Ising model and the ground state of the triangular Ising antiferromagnet.
The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.
Few years ago, application of the mean field Bethe scheme on a given system was shown to produce a systematic change of the system intrinsic symmetry. For instance, once applied on a ferromagnet, individual spins are no more equivalent. Accordingly a new loopwise mean field theory was designed to both go beyond the one site Weiss approach and yet preserve the initial Hamitonian symmetry. This loopwise scheme is applied here to solve the triangular antiferromagnetic Ising model. It is found to yield Wannier's exact result of no ordering at non-zero temperature. No adjustable parameter is used. Simultaneously a non-zero critical temperature is obtained for the triangular Ising ferromagnet. This simple mean field scheme opens a new way to tackle random systems. PACS . 75.25.+z Spin arrangements in magnetically ordered materials (including neutron and spin-polarized electron studies, synchrotron-source X-ray scattering, etc.) -- 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) -- 75.50.-y Studies of specific magnetic materials 1
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
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.