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Physica Scripta

A generalization of the two-dimensional Ising model and the Bragg–Williams mean-field theory

Navinder Singh

Published Aug 6, 2014• 1 citation
Publisher

Abstract

We develop a generalization of the two-dimensional (2D) Ising model in which a 'spin variable' lies in the 2D plane and can have four internal degrees of freedom (four possible pointing directions: up, down, right and left). By considering various forms of near-neighbour interactions, we are able to solve the model with a generalization of the Bragg–William approximation. Within this mean-field theory we examined the behaviour of heat capacity and an 'order parameter' which captures the short-range correlations between the spins. Below a transition temperature these short-range correlations are enhanced and lead to a long-range order. This is examined (for the first time) through the breakdown of statistical independence.

DOI
10.1088/0031-8949/89/9/095204

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We develop a generalization of the two-dimensional (2D) Ising model in which a 'spin variable' lies in the 2D plane and can have four internal degrees of freedom (four possible pointing directions: up, down, right and left). By considering various forms of near-neighbour interactions, we are able to solve the model with a generalization of the Bragg–William approximation. Within this mean-field theory we examined the behaviour of heat capacity and an 'order parameter' which captures the short-range correlations between the spins. Below a transition temperature these short-range correlations are enhanced and lead to a long-range order. This is examined (for the first time) through the breakdown of statistical independence.
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.
[Synopsis] In this paper, the Bragg-Williamsa pproximation of the Ising model is interpreted. The number N_＋/N is a measure of the "long-range order" in the lattice, and it is a fraction of up spins of all spins. The number N_<＋＋>/(_γN/2) is a measure of the "short-range order" in the lattice, and if it is definite that a given spin is up, the number is the fraction of its nearest neighbor with spin up. the parameter of long-range oeder L and that of short-range order σ are defined as N_+/N≡(1/2)(L+1) (-1L≦L≦＋1) and as N_<＋＋>/(γN/2)≡(1/2)(σ＋1)(-1≦σ≦＋1) respectively. The Bragg-Williams approximation consists of putting N_<＋＋>/(γN/2)≒N_<＋＋>/N. It states that there is no short-range order. The energy per spin is (1/N)E(L)＝-(1/2)εγL^2-βHL. The partition function in the Bragg-Williams approximation is Z(H, T)＝Σ__{S_i}e^<1/ N(1/2εγL^2＋βHL)>. We calculated the thermodynamic functions id' est' Helmholtz's free energy F, the spontaneous magnetization per spin , the internal energy U, the specific heat at constant volume C_V. The equation of state for the lattice gas is as follows : P_G＝βH-(1/8)ε_0γ(1＋ ^^^-)-(1/2)k_BTlog_e((1- ^^^-)/4) and (1/ν)＝(1/2)(1＋(h/2)L^^-) in the Bragg-Williams approximation.
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 .
Boehm and Bak [11] have analysed, within the mean field approximation, a model which leads to a phase diagram including multiple phase transitions between commensurate phases. It happens that an exactly soluble two-dimensional Ising model exhibiting similar mean field features, is available. This model is built on a brickwork lattice, and its exact partition function is obtained within the following assumptions : i) In the vertical direction, nearest neighbours are coupled by ferromagnetic interactions 2 J' (J' > 0). ii) In the horizontal direction, nearest neighbours are coupled by ferromagnetic interactions J (J > 0) while nextnearest neighbours are coupled by arbitrary interactions J2. Results are discussed with respect to the value of the ratio J2/J. It is found in particular that : 1) For J2/J > — 1/2, the system orders ferromagnetically at a critical temperature Tc = 1/kB βc defined by tanh 2 βc J'. sinh 2 βc J. exp 4 βc J 2 = 1. 2) For J2/J ≤ - 1/2, no ordering occurs even at zero temperature. Ground states and their degeneracy are examined. These results are compared with those obtained for the same system in the mean field approximation, which predicts the existence of a devil's staircase behaviour for the periodicity versus temperature curve, and with those given for the two-dimensional ANNNI model by a recent Monte-Carlo study [13].