Journal of Physics C Solid State Physics

Critical properties of the mixed Ising ferromagnet

Julia M. Yeomans, Robin B. Stinchcombe

Published Jan 10, 1980• 15 citations

Abstract

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.

DOI: 10.1088/0022-3719/13/1/012

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Journal of Physics C Solid State Physics

Critical properties of the mixed Ising ferromagnet

Julia M. Yeomans, Robin B. Stinchcombe

Published Jan 10, 1980• 15 citations

Publisher

Abstract

DOI: 10.1088/0022-3719/13/1/012

Related papers

A renormalisation group approach for the mixed Ising model

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Journal of Physics C Solid State PhysicsFeb 28, 197925 citations

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Julia M. Yeomans, Robin B. Stinchcombe

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.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.

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Approximate critical surface of the bond-mixed square-lattice Ising model

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The critical surface of the quenched bond-mixed square-lattice spin- $\frac{1}{2}$ first-neighbor-interaction ferromagnetic Ising model (with exchange interactions ${J}_{1}$ and ${J}_{2}$ ) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ in the variables ${t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})$] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely ${p}_{c}=\frac{1}{2}$ , ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for $p=\frac{1}{2}$ . Furthermore an analytic approximation [namely, $(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2$] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than ${10}^{\ensuremath{-}3}$ (in the $t$ variables) for any point in the surface.The critical surface of the quenched bond-mixed square-lattice spin-$\frac{1}{2}$ first-neighbor-interaction ferromagnetic Ising model (with exchange interactions ${J}_{1}$ and ${J}_{2}$) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ in the variables ${t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})$] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely ${p}_{c}=\frac{1}{2}$, ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for $p=\frac{1}{2}$. Furthermore an analytic approximation [namely, $(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2$] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than ${10}^{\ensuremath{-}3}$ (in the $t$ variables) for any point in the surface.

Physical review. B, Condensed matterApr 1, 198035 citations

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Enaldo F. Sarmento, R. B. Muniz, Solange Bessa Cavalcanti

The ferromagnetic-phase stability limit and magnetization of a quenched bond-mixed spin-1/2 transverse Ising model in an anisotropic simple square lattice are studied within the framework of an effective-field theory considering both competing and noncompeting interactions. In the bond-disordered case, where bonds ${J}_{1}$ and ${J}_{2}$ occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv} ${J}_{1}$ / ${J}_{2}$ ), and the critical bond concentrations ${p}_{c}$ at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.The ferromagnetic-phase stability limit and magnetization of a quenched bond-mixed spin-1/2 transverse Ising model in an anisotropic simple square lattice are studied within the framework of an effective-field theory considering both competing and noncompeting interactions. In the bond-disordered case, where bonds ${J}_{1}$ and ${J}_{2}$ occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv}${J}_{1}$/${J}_{2}$), and the critical bond concentrations ${p}_{c}$ at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.

Physical review. B, Condensed matterJul 1, 198725 citations

The critical surface of the quenched bond-mixed square-lattice spin-

\frac{1}{2}

first-neighbor-interaction ferromagnetic Ising model (with exchange interactions

{J}_{1}

and

{J}_{2}

) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ in the variables ${t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})$] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely

{p}_{c}=\frac{1}{2}

, ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for

p=\frac{1}{2}

. Furthermore an analytic approximation [namely, $(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2$] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than ${10}^{\ensuremath{-}3}$ (in the

t

variables) for any point in the surface.

{J}_{1}

and

{J}_{2}

occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv}

{J}_{1}

{J}_{2}

), and the critical bond concentrations

{p}_{c}

at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.