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Journal of Physics C Solid State Physics

The dilute spin-I Ising model with crystal-field interactions

Noureddine Benayad, A. Benyoussef, N. Boccara

Published Mar 30, 1985• 54 citations
Publisher

Abstract

2D spin-1 Ising models with nearest-neighbour and crystal-field interactions exhibit tricritical behaviour. A new method that applies to a wide class of random systems is used to study the influence of site and bond dilution on this behaviour. The authors have calculated temperature-crystal-field-concentration phase diagrams and determined, in particular, the influence of dilution on the zero-temperature critical field and on the tricritical temperature. The results are in agreement with those obtained recently for bond dilution by a real-space renormalisation-group technique. The method appears more tractable.

DOI
10.1088/0022-3719/18/9/023

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2D spin-1 Ising models with nearest-neighbour and crystal-field interactions exhibit tricritical behaviour. A new method that applies to a wide class of random systems is used to study the influence of site and bond dilution on this behaviour. The authors have calculated temperature-crystal-field-concentration phase diagrams and determined, in particular, the influence of dilution on the zero-temperature critical field and on the tricritical temperature. The results are in agreement with those obtained recently for bond dilution by a real-space renormalisation-group technique. The method appears more tractable.
The authors use a Migdal-Kadanoff renormalisation group to study the behaviour of the Blume-Emery-Griffiths model when quenched, random-bond dilution is introduced. Phase diagrams are calculated for varying dilution. The behaviour of the multicritical points of the pure system as dilution is introduced is of particular interest. They find that the tricritical temperature tends to zero as the percolation threshold is approached but that the tricritical field remains finite in agreement with experimental results on the dilute metamagnet Fe p Mg (1-p) Cl 2 .
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.
Real space renormalisation group techniques and the replica method of Edwards and Anderson (1975), combined with successive variational approximations going beyond mean field, are used to calculate phase diagrams of various Ising systems. For a bond dilute 2D Ising model with nearest-neighbour interactions, real-space renormalisation group techniques provide very precise results which are used to test the reliability of the replica method. Combined with variational approximations going beyond mean field, this last method is applied to 3D Ising models with competing nearest-neighbour and next-nearest-neighbour interactions.
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).