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Zeitschrift für Naturforschung A

Correlation Function Approach to Short-Range Order in the Ising-Model

Rainer J. Jelitto

Published Feb 1, 1970• 0 citations
Publisher

Abstract

Abstract By pushing forward the decoupling from the three-to the four-point correlation function short-range order is systematically introduced into the description of the statistical behaviour of the Ising-model. Application of a procedure which is a natural generalization of that invented by Bogoljubov and Tjablikov for the Heisenberg-model, leads to an overall-approximation for the magnetization and the nearest-neighbour correlation which may be compared with the Ising-model variant of Oguchi's two-spin-cluster molecular field theory. The results of both approximations are very similar both for low and high temperatures, but for the transition point the new approach yields values which lie considerably lower and therefore are more reliable than those following from Oguchi's theory. Moreover, the comparison with a slight modification of the theory which is also presented in this paper, illuminates the physical mechanism which is responsible for the formation of correlations within the order of approximation, considered.

DOI
10.1515/zna-1970-0206

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Abstract By pushing forward the decoupling from the three-to the four-point correlation function short-range order is systematically introduced into the description of the statistical behaviour of the Ising-model. Application of a procedure which is a natural generalization of that invented by Bogoljubov and Tjablikov for the Heisenberg-model, leads to an overall-approximation for the magnetization and the nearest-neighbour correlation which may be compared with the Ising-model variant of Oguchi's two-spin-cluster molecular field theory. The results of both approximations are very similar both for low and high temperatures, but for the transition point the new approach yields values which lie considerably lower and therefore are more reliable than those following from Oguchi's theory. Moreover, the comparison with a slight modification of the theory which is also presented in this paper, illuminates the physical mechanism which is responsible for the formation of correlations within the order of approximation, considered.
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In this paper we present Green function technique applied to calculations of spin-spin correlations in systems governed by Ising Hamiltonian. This offers approximate yet reasonably accurate analytical results, as an alternative approach to direct computer simulation. Local spin operators are represented in terms of particle operators for fermions. Chain of Green functions for local magnetization on given site involves nearest neighbors and it is decoupled at the next-nearest-neighbors level. Therefore, still accounts at the lowest level for magnetic correlations between nearest neighbors. This technique was applied to binary alloy A x B 1-x with given atomic short range order. Calculations for stoichiometric x=1/2 composition show how temperature dependence of magnetization and spin-spin correlations are modified when atomic order parameter changes. We scanned this parameter from the case of ordered binary alloy, via random alloy, to clusterlike structure. In model calculations we (a) confirm characteristic λ -like shape of the magnetic specific heat near Curie temperature and (b) recover essential results of exact, two-dimensional Ising model. This supports our claim of usefulness of such analytical approach, which is easy and flexible to describe different geometries (clusters, thin films) and conditions imposed on spins located on specific atoms (spins partly pinned on the surface atoms).
Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent Ising models in two and three dimensions. This constitutes the first-order approximation which improves the mean-field theory (the zeroth-order approximation) by incorporating specific short-range correlations into the distribution function. We first calculate the single-spin expectation value. Although the dynamic equation may need to be solved by using a numerical method, its approximate solution in the pretransitional region can be studied in detail. We then calculate the two-spin correlation function. This requires yet another extension of Bethe's method. Again approximate solutions can be obtained in the pretransitional region. Our results for the static pair correlation functions are equivalent to that of Elliott and Marshall, but our derivation is simpler. We also calculate the time-dependent correlation function for the magnetization, and reproduce the fluctuation-dissipation relation.
To calculate thermodynamic functions in statistical mechanics the authors propose a new variational approach which has as particular cases the approaches of Bogoliubov, Oguchi and Tsallis-da Silva (1982), but introduces some improvements, mainly in the short-range order effects above the critical temperature. It requires a choice of a trial Hamiltonian that defines an approximate ensemble through which the thermodynamic functions may be calculated. To check the efficiency of their approach they apply it to the ferromagnetic Ising model by using the simplest possible trial Hamiltonian.
The aim of this series of papers is to help bridge the gap in our knowledge of the properties of the Ising and Heisenberg models between very short-range forces corresponding to nearest-neighbour interactions, and very long-range forces corresponding to the mean-field approximation. The present paper is concerned with the equivalent neighbour model in which the interactions between a spin and a certain finite number of its neighbours are equal, and the remaining interactions are all zero. Methods derived previously for obtaining series expansions at high temperatures are generalized to include second- and third-neighbour shells for standard two- and three-dimensional lattices. It is found that as the co-ordination number q increases the properties of the model approach an asymptotic value which depends on dimension, and on q in a given dimension, but not significantly on the type of lattice structure. An estimate is made of this asymptotic behaviour as a function of q .