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Journal of the Physical Society of Japan

Some Dynamical Properties of the Ising Ferromagnet. II. Cubic Lattices

Noboru Matsudaira

Published Aug 1, 1967• 15 citations
Publisher

Abstract

A method to calculate the dynamical as well as the static properties of the Ising ferromagnet, developed in the previous paper for the two-dimensional square lattice, is extended to cubic lattices. The temperature dependence of the frequency-dependent magnetic susceptibility, approach to equilibrium of the magnetization of the system, the fluctuation-dissipation theorem, and some static properties are calculated. It is proved that all these properties are essentially the same for cubic lattices and for two-dimensional model. It is further demonstrated by explicit calculation that the non-linear effect is very important for the critical slowing-down near the Curie point, especially in the ferromagnetic region.

DOI
10.1143/jpsj.23.232

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A method to calculate the dynamical as well as the static properties of the Ising ferromagnet, developed in the previous paper for the two-dimensional square lattice, is extended to cubic lattices. The temperature dependence of the frequency-dependent magnetic susceptibility, approach to equilibrium of the magnetization of the system, the fluctuation-dissipation theorem, and some static properties are calculated. It is proved that all these properties are essentially the same for cubic lattices and for two-dimensional model. It is further demonstrated by explicit calculation that the non-linear effect is very important for the critical slowing-down near the Curie point, especially in the ferromagnetic region.
Time-dependent statistics of the Ising model proposed by Glauber for the one-dimensional chain are extended to the example of a two-dimensional square lattice. Each spin is assumed to change its state through the interaction with a heat bath. The equations of motion for both the single spin and the spin correlation functions are solved approximately by using a decoupling procedure where the many-body correlation functions are taken as sums of products of pair correlation functions. As a special case, our theory allows the approximate calculation of the equilibrium properties of the system and it turns out that, in this case, our result is an improvement over the Bethe approximation. Both the frequency-dependent magnetic susceptibility and the decay of the magnetic moment to the equilibrium state are calculated above and below the Curie temperature. The fluctuation–dissipation theorem developed by Glauber for the linear chain is shown to hold in the two-dimensional case also.
The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.
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.
The Ising model on a triangular lattice with antiferromagnetic nearest neighbor and ferromagnetic next nearest neighbor interactions is investigated by treating the master equation. To terminate hierarchy structure of the equation of motion, the cluster decoupling approximation (CDA) is applied in a similar way as in the previous paper. But here, we retain not only single cumulants but also second ones. The fluctuation effect in the occurrence of the successive phase transition and “turn round” phenomena in a relaxation process are discussed. The formula of the frequency dependent susceptibility is derived within the present approximation, and the response of the system to the external oscillating field is studied. Finally, the fluctuation-dissipation theorem is derived for this spin system.
We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.
Two properties of the Ising model are considered. Firstly, by extrapolating appropriate series expansions, the spontaneous magnetization of a ferromagnet for three-dimensional lattices is obtained. From the results accurate values for the critical temperature are found. Secondly, the high temperature susceptibility of an antiferromagnet is derived from the approximation of Kikuchi. The resulting susceptibility curves show that for loose-packed lattices a maximum value occurs at a temperature slightly above the critical point, while for the close-packed lattices no such effect is obtained.