The cluster-variation method was used to calculate the critical temperature for the fcc Ising ferromagnet in three different cluster approximations. A scheme for determing a set of independent cluster variables is presented, which considerably simplifies the minimization of the free energy. The system of equations arising from such minimization is solved, in the disordered state, by means of a simple iteration technique. The highest level of approximation treated in this work yields a critical temperature which is only 1.5% above the one estimated by the exact high-temperature expansion of the zero-field susceptibility. Furthermore, the high-temperature expansion for the specific heat gives four exact coefficients and the fifth is determined to within 0.4%.
In the Weiss molecular-field approximation, the
S=1 Ising-model ferromagnet possessing additional single-ion-type uniaxial anisotropy is known to exhibit (depending upon the ratio of anisotropy and interaction constants) both first-order (discontinuous) and second-order (continuous) magnetic phase transitions which join at a tricritical point. We further investigate the above model system in the spin-pair approximation using the cluster-variation method. Within this improved approximation, results are found for the temperature and anisotropy dependences of the thermal-average quantities
〈Si〉,
〈Si2〉, $〈{S}_{i}{S}_{{i}^{\ensuremath{'}}}〉$, $〈{S}_{i}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$, $〈S_{i}^{}{}_{}{}^{2}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$ (
i, ${i}^{\ensuremath{'}}$ are nearest-neighbor lattice sites), a better estimate for the location of the tricritical point is achieved, and the specific heat and initial paramagnetic susceptibility are calculated. To illustrate these results, the simple cubic lattice is used as an example of application.
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.
Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.
The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is
z1 where
z is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}$ where
vij is the exchange potential. This is the molecular field value for the Curie point
Tc. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment
M in the same approximation,
M being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N$. Here ${\ensuremath{\mu}}_{i}$ is the "spin" per particle and
N is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field,
M is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in
O(z1) in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.
We use an effective field model which explicitly incorporates correlations to determine the critical temperatures for Ising ferromagnets on the common lattices. We specifically retain only the first nontrivial odd correlation function, i.e., the triplet correlation function, while neglecting higher order correlations. Within this approximation, assuming the equivalence of triplet correlations involving both central and noncentral sites in the nearest neighbor cluster yields values for the critical temperature below the Bethe approximations for z⩾4. Likewise, assuming the inequivalence of triplet correlations involving the central and noncentral sites yields values of the critical temperature only slightly higher than the Bethe approximation for z⩾4. For lattices with four- and six-fold coordination this equivalence and inequivalence of triplet correlation functions is sufficient to cause the critical temperatures to approach their two- and three-dimensional values. For instance, for quadratic lattices we obtain values for the critical temperature of 2.5284 and 2.9796 as compared to the exact and series results of 2.2692 and 2.7044 for the simple quadratic and diamond lattices, respectively, and the Bethe approximation of 2.8854.