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Journal of Physics C Solid State Physics

Critical properties of site-diluted Heisenberg and Ising magnets

Andrzej Komoda, Andrzej Pękalski

Published Jun 10, 1983• 6 citations
Publisher

Abstract

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).

DOI
10.1088/0022-3719/16/16/018

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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).
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 authors discuss the critical frontier and critical exponents associated with the quenched bond-dilute quantum anisotropic spin- 1 / 2 Heisenberg ferromagnet in square lattice. To perform the calculations, they extend (and analyse in some detail) an approximate real-space renormalisation group framework recently developed by some of the authors for the pure model. Whenever comparison with available exact results is possible, the agreement is either perfect or quite satisfactory. Some effort has been dedicated to the extraction of the main asymptotic behaviour of the critical frontier. Also several interesting quantum effects appearing in the composition laws of (Heisenberg) bond arrays are exhibited.
A real-space rescaling method is applied to the mixed, spin- 1 / 2 , Ising ferromagnet. Both bond and site disorder are considered where, respectively, bonds J A , J B or sites A, B occur with probabilities p, 1-p. For the bond-disordered case, critical curves are obtained for the square lattice for a range of alpha identical to J B /J A . Slopes at p=0,1 are in very good agreement with exact results previously obtained and T c (p- 1 / 2 ) agrees to within a few per cent with an exact result obtained by Fisch. For the more realistic site-disordered problem only the case where bond strengths are related by tanh J AA tanh J BB =tanh 2 J AB is considered. Critical curves are plotted for various alpha for the square and triangular lattices. The results agree well with exact limiting slopes available for the square lattice. Thermal, percolation and crossover exponents are also calculated for both the site- and bond-disordered cases.
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.