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Journal of the Physical Society of Japan

The Effect of Lattice Vibration on the Phase Transition of the Ising Model

Noboru Matsudaira

Published Nov 1, 1968• 30 citations
Publisher

Abstract

The effect of lattice vibration on the second-order phase transition for the Ising model is investigated. The total Hamiltonian of the system is separated into the phonon part and the spin part, if the displacement of the atoms from their equilibrium position is assumed to be small. The spin part of the free energy is then calculated (i) in the molecular-field approximation for the square and the simple cubic lattices, and (ii) by the method due to Kadanoff, a modification of the Onsager's exact solution, for the square lattice. The results show that (i) the phase transition remains to be of the second order, (ii) the critical temperature increases, and (iii) the magnetic part of the heat capacity decreases, both for ferro- and antiferromagnetic Ising models.

DOI
10.1143/jpsj.25.1225

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The effect of lattice vibration on the second-order phase transition for the Ising model is investigated. The total Hamiltonian of the system is separated into the phonon part and the spin part, if the displacement of the atoms from their equilibrium position is assumed to be small. The spin part of the free energy is then calculated (i) in the molecular-field approximation for the square and the simple cubic lattices, and (ii) by the method due to Kadanoff, a modification of the Onsager's exact solution, for the square lattice. The results show that (i) the phase transition remains to be of the second order, (ii) the critical temperature increases, and (iii) the magnetic part of the heat capacity decreases, both for ferro- and antiferromagnetic Ising models.
The phase transitions in two-dimensional antiferromagnetic Ising model are studied on a square lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transition of second order and the frustration points depending on relative interaction values and magnetic field. A effect, heat capacity splitting near the frustration points, is found.
Dynamics of a spin-3/2 Ising system Hamiltonian with bilinear and biquadratic nearest-neighbour exchange interactions is studied by a simple method in which the statistical equilibrium theory is combined with the Onsager's theory of irreversible thermodynamics. First, the equilibrium behaviour of the model in the molecular-field approximation is given briefly in order to obtain the phase transition temperatures, i.e. the first- and second-order and the tricritical points. Then, the Onsager theory is applied to the model and the kinetic or rate equations are obtained. By solving these equations three relaxation times are calculated and their behaviours are examined for temperatures near the phase transition points. Moreover, the z dynamic critical exponent is calculated and compared with the z values obtained for different systems experimentally and theoretically, and they are found to be in good agrement.
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Using the Ising Model, the phase transition between ferromagnetic and paramagnetic phases of a 2-D square spin lattice are explored. A phase transition into the ferromagnetic phase occurs when the temperature of the system falls below some critical value. The ferromagnetic phase is characterized by a non-zero magnetization on the macroscopic level. Using the Mote Carlo method to obtain system's configurations (i.e., sets of microstates) in thermal equilibrium, we will explore system's macroscopic properties such as magnetization, magnetic susceptibility, and specific heat capacity as functions of temperature. We will evaluate the transition temperature, and discuss how the lattice size affects the results.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
Abstract The effect of an external magnetic field on the phase transitions and the thermodynamic and magnetic properties of the three-dimensional antiferromagnetic Ising model on a body-centered cubic lattice has been studied using a replica algorithm of the Monte Carlo method, taking into account the interactions of next nearest neighbors. A phase diagram of the dependence of the critical temperature on an external magnetic field has been built. It is shown that a second-order phase transition is observed in the magnetic field range under consideration.