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 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.
In the present paper we show how nonclassical, quite accurate, critical exponents can be extracted in a very simple way from the Pad\'e analysis of the results obtained by mean-field-like approximation schemes, and in particular by the cluster variation method. We study the critical behavior of the Ising model on several lattices (quadratic, triangular, simple cubic and face centered cubic) and two problems of surface critical behavior. Both unbiased and biased approximants are used, and results are in very good agreement with the exact or numerical ones.
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.
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.
I show that the cluster variation method, long used as a powerful hierarchy of approximations for discrete (Ising-like) two-dimensional lattice models, yields exact results on the disorder varieties which appear when competitive interactions are put into these models. I consider, as an example, the plaquette approximation of the cluster variation method for the square lattice Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, and, after rederiving known results, report simple closed-form expressions for the pair and plaquette correlation functions.
Renormalization group recursions based on mean-field approximations [J. O. Indekeu, A. Maritan, and A. L. Stella, J. Phys.A15, L291 (1982)], commonly referred to as mean-field renormalization group methods (MFRG), have proven to be efficient and easily applicable for computing non-classical critical properties of lattice models. We give a fairly complete bibliography of applications to date, and extend previous test calculations of bulk, surface, and corner critical exponents in the two-dimensional Ising model to larger cluster sizes on triangular, square (including crossing bonds), and honeycomb lattices. Without much effort the exact value of the critical exponent ratioyH/yT is reproduced systematically with a precision of 2%. This ratio turns out to be the most accurate probe of non-classical critical behaviour that is available in the MFRG method.
A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of ``correlations'' between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.