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Acta Physica Sinica

A GENERALIZED MIXED SPIN MODEL: A RENORMALISATION GROUP APPROACH

TANG KUN-FA, HU JIA-ZHEN, (1)上海交通大学凝聚态物理研究所; (2)上海交通大学应用物理系

Published Jan 1, 1986• 3 citations
Publisher

Abstract

We propose a generalized Ising spin model, which contains the pure Ising model, the mixed spin model proposed by Schofield and Bowers and the particular site-diluted Ising spin model as special oases. A real space renormalisation group is applied to this model, the phase diagram is found to have three nontrivial fixed points (a Ising, a tric-ritical and a first order phase transition point). And it is demonstrated that mixed spin model and pure Ising spin model belong to the same universality, which is in agreement with the conclusion made by Schofield and Bowers who calculated the critical exponents of mixed model directly.

DOI
10.7498/aps.35.1048

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We propose a generalized Ising spin model, which contains the pure Ising model, the mixed spin model proposed by Schofield and Bowers and the particular site-diluted Ising spin model as special oases. A real space renormalisation group is applied to this model, the phase diagram is found to have three nontrivial fixed points (a Ising, a tric-ritical and a first order phase transition point). And it is demonstrated that mixed spin model and pure Ising spin model belong to the same universality, which is in agreement with the conclusion made by Schofield and Bowers who calculated the critical exponents of mixed model directly.
A simple modification of the real-space renormalisation group method is used to discuss critical behaviour of site diluted spin- 1 / 2 Ising and Heisenberg ferromagnets in two and three dimensions. The critical parameters and critical exponents are evaluated, and are shown to be in good agreement with the exact results and series calculations for site-diluted Ising ferromagnets on the square and triangular lattices. Good values of the critical parameters have also been obtained for the site-diluted Heisenberg and Ising models in two and three dimensions using the Migdal-Kadanoff approximation (1976).
The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.
The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.
Mixed spin Ising models have less translations symmetry than their 'single spin' counterparts and are well adapted for the study of a certain type of ferromagnetism. In this articles the high-temperature series expansion work of Schofield and Bowers (1981)-who studied mixed spin- 1 / 2 -spin-1 Ising models-is generalised by replacing the spin-1 objects by arbitrary spin-S ones. The new series are used to investigate the spin dependence or independence of critical parameters in a novel setting. There is no evidence to suggest that the exponents studied are spin dependent.
The mixed spin-7/2 and spin-1/2 Ising model with a crystal field in the absence and presence of an external longitudinal magnetic field is investigated within the framework both of the renormalization group (RG) method and Monte Carlo simulation (MCS). First, we have determined the ground-state phase diagram. Then, using RG technique, the finite temperature phase diagrams of this model are plotted for the two- and three-dimensional cases by following the flow in the parameters space of the Hamiltonian and evaluating the free energy and its isotherm derivative at low temperatures. Our results show that the model exhibits first- and second-order phase transitions as well as critical end-points. In addition, by linearizing the RG transformation at the vicinity of the fixed points of the second-order transition, we have determined the critical exponents. The MCS is also employed to validate the results obtained previously by renormalization, and to examine the influence of an external magnetic field on the magnetic properties and hysteresis loops of the system.
The spin-3/2 Ising model, with nearest-neighbor interactions only, is the prototypical system with two different ordering species, with concentrations regulated by a chemical potential. Its global phase diagram, obtained in d=3 by renormalization-group theory in the Migdal-Kadanoff approximation or equivalently as an exact solution of a d=3 hierarchical lattice, with flows subtended by 40 different fixed points, presents a very rich structure containing eight different ordered and disordered phases, with more than 14 different types of phase diagrams in temperature and chemical potential. It exhibits phases with orientational and/or positional order. It also exhibits quintuple phase transition reentrances. Universality of critical exponents is conserved across different renormalization-group flow basins via redundant fixed points. One of the phase diagrams contains a plastic crystal sequence, with positional and orientational ordering encountered consecutively as temperature is lowered. The global phase diagram also contains double critical points, first-order and critical lines between two ordered phases, critical end points, usual and unusual (inverted) bicritical points, tricritical points, multiple tetracritical points, and zero-temperature criticality and bicriticality. The four-state Potts permutation-symmetric subspace is contained in this model.