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Journal of Physics C Solid State Physics

Tricritical behaviour in a bond-dilute spin model

H Dickinson, Julia M. Yeomans

Published Apr 30, 1983• 31 citations
Publisher

Abstract

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 .

DOI
10.1088/0022-3719/16/12/002

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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 .
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 randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold p∗{p}^{\mathrm{*}}p∗ at which the critical temperature goes to zero depends upon q.
The authors study spins with correlated ('bootstrap') dilution in which a (randomly occupied) site only develops a magnetic moment if surrounded by at least m magnetic neighbours. A generalisation of the Harris criterion indicates that the tendency of correlated disorder is to favour a new critical behaviour upon dilution. They also present real-space renormalisation group calculations for phase diagrams and discuss the different approximations involved. The structure of the infinite cluster at the percolation threshold of bootstrap dilution is compared with those of fractal geometries. Other correlated percolation models are compared with the present model.
The effect of bond randomness on tricritical and critical end-point phenomena is studied by renormalization-group theory. In three dimensions, the pure-system tricritical point is replaced by a line segment of second-order transitions dominated by randomness and bounded by a multicritical point and a random-bond tricritical point, which reaches zero temperature at threshold randomness. This topology indicates a violation of the empirical universality principle. The random-bond tricritical point renormalizes onto the fixed distribution of random-field Ising criticality.
A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu T and nu p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.
In the frame of Ising model, we investigate the critical properties of bond dilution ferromagnetic spin system with random crystal field and obtain the magnetization expressions by using effective field theory. Special emphasis is placed on tricritical properties of the square lattice. At the range of concern random concentration of crystal field and bond dilution,the present system shows some new critical properties,such as fluctuation of tricritical point. We discuss in detail the influence of two disorder factors on the critical properties and give phase diagram.