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Journal of Physics C Solid State Physics

Phase diagrams for correlated dilution problems

N. S. Branco, S. L. A. de Queiroz, Raimundo R. dos Santos

Published May 10, 1988• 12 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/21/13/011

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Journal of Physics C Solid State Physics•Jun 10, 1983•6 citations
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 authors study correlated dilution of two-dimensional Ising spins. In this 'bootstrap dilution' model, a (randomly occupied) site only develops a magnetic moment if surrounded by at least m magnetic neighbours. By generalising an ansatz proposed by Coniglio (1981, 1982) for ordinary dilution, they find that for some values of m, temperature is irrelevant (in a renormalisation-group sense) near the percolation fixed point, contrarily to the uncorrelated case. Real-space renormalisation-group calculations illustrate their findings.
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.
Exact analytic results are presented for the critical properties of the dilute Ising model on a Bethe lattice, where a fraction p of the sites are occupied at random by spins. It is found that the critical behaviour is the same as that of the pure system (p=1) except for transitions which occur by varying p at zero temperature, where the critical behaviour is different and corresponds to the percolation problem. For T to 0 and p to p c , the percolation concentration, the zero-field magnetization shows crossover behaviour of the form expected from scaling theories. Outside the critical region, a rapidly convergent numerical method is developed and used to calculate the spontaneous magnetization.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
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).