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Journal of Physics C Solid State Physics

Theory of the phase boundary for cubic Ising lattices with pair-quartet interactions

B. Frank, Ole G. Mouritsen

Published May 10, 1983• 6 citations
Publisher

Abstract

A theory is presented for the determination of the transition temperature T c for spin- 1 / 2 Ising models on cubic lattices with pair (J 2 ) and quartet (J 4 ) interactions. A linear relationship between T c and J 4 /J 2 is found to have a base wider in scope than that of previous theories. The calculation of the linear coefficient is performed through the derivation of a new equation for special four-spin correlation functions which is solved in a novel way by the use of a ratio approximation. The results are compared with those derived from previous theories, series analysis and Monte Carlo calculations which are also presented here. The linear coefficient is shown to depend on a parameter which turns out to have approximately the same value for all cubic lattices, thus suggesting a set of scaled variables in terms of which the phase boundary for all cubic lattices may be fitted by a common curve. From the authors' work a coherent picture emerges for the phase behaviour of cubic Ising lattices with pair and quartet interactions, involving regions of continuous and first-order transitions separated by a tricritical point which occurs at the same value of the scaled variables for all cubic lattices.

DOI
10.1088/0022-3719/16/13/011

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Physica Scripta•Oct 1, 1988•2 citations
A theory is presented for the determination of the transition temperature T c for spin- 1 / 2 Ising models on cubic lattices with pair (J 2 ) and quartet (J 4 ) interactions. A linear relationship between T c and J 4 /J 2 is found to have a base wider in scope than that of previous theories. The calculation of the linear coefficient is performed through the derivation of a new equation for special four-spin correlation functions which is solved in a novel way by the use of a ratio approximation. The results are compared with those derived from previous theories, series analysis and Monte Carlo calculations which are also presented here. The linear coefficient is shown to depend on a parameter which turns out to have approximately the same value for all cubic lattices, thus suggesting a set of scaled variables in terms of which the phase boundary for all cubic lattices may be fitted by a common curve. From the authors' work a coherent picture emerges for the phase behaviour of cubic Ising lattices with pair and quartet interactions, involving regions of continuous and first-order transitions separated by a tricritical point which occurs at the same value of the scaled variables for all cubic lattices.
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.
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- (J2{J}_{2}J2​) and four - (J4{J}_{4}J4​) spin interactions. In the limit J2=0{J}_{2}=0J2​=0 the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.
The behaviour of a three-dimensional Ising-like system at temperatures above the critical value of T c is studied in the approximation of the sextic distribution for modes of spin density oscillations (ρ 6 model). The original collective variables method is developed in this higher non-Gaussian approximation for calculating the thermodynamic characteristics of the system near T c taking into account the first confluent correction. The calculations are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The main distinctive feature of the method is the separate inclusion of the contributions to the thermodynamic functions from the short-wave and long-wave spin density oscillation modes. The dependences of the phase transition temperature, leading critical amplitudes, and correction-to-scaling amplitudes for the specific heat and susceptibility on the microscopic parameters of the system are investigated.
An analytic method for finding the highly accurate values of the critical point for cubic Ising models with arbitrary spin is presented. The thermodynamic quantities of interest are expressed through some higher-order correlations with a reference spin excluded. A reduction scheme for these higher-order correlations is constructed. It leads to two types of resulting equations for correlation functions. The coincidence of two different representations of critical correlations is ensured by the introduced parameter of the reduction scheme. The estimation of critical points for all cubic lattices is compatible with series works to within 1.69%, 0.70% for the limiting values of spin, ½ and ∞, respectively.