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Journal of Physics C Solid State Physics

A renormalisation group approach for the mixed Ising model

Julia M. Yeomans, Robin B. Stinchcombe

Published Feb 28, 1979• 25 citations
Publisher

Abstract

A real-space renormalisation group method is applied to the mixed, spin- 1 / 2 Ising ferromagnet in which each exchange interaction may take values J A , J B with probabilities p, 1-p respectively. The T c versus p curve for the square lattice is obtained for chosen values of alpha . For all alpha the slopes of the critical curves at p=0 and 1 are in very close agreement with exact results and the value for T c (p= 1 / 2 ) agrees with an exact result to within a few per cent.

DOI
10.1088/0022-3719/12/4/008

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We propose a generalized Ising spin model, which contains the pure Ising model, the mixed spin model proposed by Schofield and Bowers and the particular site-diluted Ising spin model as special oases. A real space renormalisation group is applied to this model, the phase diagram is found to have three nontrivial fixed points (a Ising, a tric-ritical and a first order phase transition point). And it is demonstrated that mixed spin model and pure Ising spin model belong to the same universality, which is in agreement with the conclusion made by Schofield and Bowers who calculated the critical exponents of mixed model directly.

Acta Physica Sinica•Jan 1, 1986•3 citations
A real-space renormalisation group method is applied to the mixed, spin- 1 / 2 Ising ferromagnet in which each exchange interaction may take values J A , J B with probabilities p, 1-p respectively. The T c versus p curve for the square lattice is obtained for chosen values of alpha . For all alpha the slopes of the critical curves at p=0 and 1 are in very close agreement with exact results and the value for T c (p= 1 / 2 ) agrees with an exact result to within a few per cent.
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 renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.
We propose a generalized Ising spin model, which contains the pure Ising model, the mixed spin model proposed by Schofield and Bowers and the particular site-diluted Ising spin model as special oases. A real space renormalisation group is applied to this model, the phase diagram is found to have three nontrivial fixed points (a Ising, a tric-ritical and a first order phase transition point). And it is demonstrated that mixed spin model and pure Ising spin model belong to the same universality, which is in agreement with the conclusion made by Schofield and Bowers who calculated the critical exponents of mixed model directly.