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Journal of Physics C Solid State Physics

Critical properties of the site-diluted Ising ferromagnet

Julia M. Yeomans, Robin B. Stinchcombe

Published Jul 14, 1978• 28 citations
Publisher

Abstract

The site-diluted Ising ferromagnet is treated by a position space renormalisation group method. The method is briefly outlined and results obtained by considering three different two-dimensional clusters are presented. The results include the dependence of transition temperature T c on concentration p and the thermal exponents for the percolation and Ising fixed points. Corresponding results are also presented for the bond-diluted Ising ferromagnet.

DOI
10.1088/0022-3719/11/13/006

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The site-diluted Ising ferromagnet is treated by a position space renormalisation group method. The method is briefly outlined and results obtained by considering three different two-dimensional clusters are presented. The results include the dependence of transition temperature T c on concentration p and the thermal exponents for the percolation and Ising fixed points. Corresponding results are also presented for the bond-diluted Ising ferromagnet.
A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.
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.
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 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.