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Physical Review

Influence of the Pair Correlations on the Phase Transition in an Ising Lattice

Raf Dekeyser, Francis L. Halzen

Published May 10, 1969• 6 citations
Publisher

Abstract

We propose a method to incorporate pair correlations of an Ising lattice system in molecular field theory to higher orders. The theory is applied to ferromagnetism, where a system of equations is obtained for the response function (here the susceptibility). We obtain important corrections on the Weiss model and the spherical model. This is illustrated by an explicit calculation of the critical temperature and the spurious phase, appearing for $T>{T}_{c}$, near the critical point in the case of a simple cubic lattice.

DOI
10.1103/physrev.181.949

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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 2000•30 citations
We propose a method to incorporate pair correlations of an Ising lattice system in molecular field theory to higher orders. The theory is applied to ferromagnetism, where a system of equations is obtained for the response function (here the susceptibility). We obtain important corrections on the Weiss model and the spherical model. This is illustrated by an explicit calculation of the critical temperature and the spurious phase, appearing for T>TcT>{T}_{c}T>Tc​, near the critical point in the case of a simple cubic lattice.
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.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.
In the Weiss molecular-field approximation, the S=1S=1S=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〉〈{S}_{i}〉〈Si​〉, 〈Si2〉〈S_{i}^{}{}_{}{}^{2}〉〈Si​​2〉, $〈{S}_{i}{S}_{{i}^{\ensuremath{'}}}〉$, $〈{S}_{i}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$, $〈S_{i}^{}{}_{}{}^{2}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$ (iii, ${i}^{\ensuremath{'}}$ 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 model of ferromagnetism proposed by Ising is treated by synthesizing a cluster treatment with the spherical model. The spherical model is used to determine approximately the molecular field acting on the cluster and the effective interactions between dipoles of the cluster.The method is applied to the square net and to the cubic lattice and the critical temperatures are obtained. The result for the square net is that EkTc=0.831\frac{E}{k{T}_{c}}=0.831kTc​E​=0.831 and for the cubic lattice that EkTc=0.442\frac{E}{k{T}_{c}}=0.442kTc​E​=0.442. In these results EEE is the energy required to change an adjacent pair of dipoles from parallel to antiparallel relative orientation.