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Journal of Physics C Solid State Physics

On phase transitions in Ising systems with many-spin interactions

Moshe Gitterman, M.A. Mikulinsky

Published Oct 28, 1977• 23 citations
Publisher

Abstract

The behaviour of thermodynamical quantities near the critical points of Ising Hamiltonians with many-spin interactions is investigated. A critical analysis of the Griffiths and Wood (1974) numerical calculation is performed demonstrating that the critical behaviour of Ising systems is determined by pair rather than many-spin interactions.

DOI
10.1088/0022-3719/10/20/016

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The behaviour of thermodynamical quantities near the critical points of Ising Hamiltonians with many-spin interactions is investigated. A critical analysis of the Griffiths and Wood (1974) numerical calculation is performed demonstrating that the critical behaviour of Ising systems is determined by pair rather than many-spin interactions.
The Hamiltonian H=-J 2 Sigma (ij) sigma i sigma j -(J 4 /N) Sigma (ij)(kl) sigma i sigma j sigma k sigma l where the first term is a nearest-neighbour pair interaction and the second term is an infinite-range pair-pair interaction, is proposed as a model of an Ising magnet with lattice coupling. The free energy is reduced to that of a rigid Ising model on the same lattice but with a modified pair interaction. In two and three dimensions, the phase transition is found to be first order for all J 4 . The relationship between this result and recent renormalization group calculations is discussed.
The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.
Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.
Abstract This study investigates the critical properties of the mean-field Ising model incorporating five-spin interactions and external magnetic field. By using a variational approach to minimize free energy, the behaviour of the magnetic order parameter is systematically analysed across different phases. The research establishes the conditions for phase transitions, identifies critical points, and calculates the associated critical exponents. The model demonstrates unique phase behaviours, including the coexistence of thermodynamic equilibria and smooth transitions in magnetization. These results offer valuable insights into the impact of higher-order spin interactions in magnetic systems, with implications for understanding complex multi-spin dynamics in physical and biological networks.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.
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.