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Journal of Physics C Solid State Physics

Low-temperature behaviour at the percolation threshold for Ising spins with correlated dilution

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

Published Mar 30, 1987• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/20/9/001

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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 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.
We consider a diluted Ising model in which the absence of a spin affects the exchange coupling of a nearest-neighbor pair along the line joining the three spins; that is, it acquires the value $\ensuremath{\alpha}J$, where $\ensuremath{\alpha}$ is a phenomenological parameter $\ensuremath{\epsilon}[0,1]$. This model has been proposed to explain the experimental phase diagram for $\mathrm{KN}{\mathrm{i}}_{x}{\mathrm{Mg}}_{1\ensuremath{-}x}{\mathrm{F}}_{3}$. A position-space renormalization-group analysis clearly distinguishes two percolation thresholds depending on whether $\ensuremath{\alpha}=0$ or $\ensuremath{\alpha}>0$, though both cases seem to be in the same universality class. Further, thermal fluctuations dominate over the geometrical ones as in the uncorrelated case, and the critical curve (critical temperature versus concentration of magnetic sites) displays an upward curvature for intermediate degrees of correlation $0<\ensuremath{\alpha}<1$, as experimentally observed.
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 bond-diluted classical Heisenberg ferromagnet is treated by a position space renormalisation group method valid when the valid behaviour is at low temperatures. The method is outlined and used to obtain exact (one-dimensional) and approximate (two- and three-dimensional) results for the critical behaviour of the correlation length near the percolation limit. The procedure agrees in the pure limit with exact results of Polyakov and Migdal for the two-dimensional case. At general concentrations an exact correspondence with related treatments of random resistor networks is exhibited. This implies that the percolation-to-thermal crossover exponents of the Ising and Heisenberg systems are the same. The generalisations required for a proper comparison with neutron scattering experiments are briefly discussed.