A two-layered Ising model is presented which maps to the Ashkin-Teller (AT) model (1943) for certain ratios of the two-spin couplings. Thus the model is able to exhibit the same critical exponents as the AT model, which are in general different from the usual Ising exponents; even continuous dependence of the critical exponents on the energy parameters is to be expected.

Journal of Physics C Solid State PhysicsMar 28, 19763 citations

Correlations in inhomogeneous Ising models

W. F. Wolff, J. Zittartz

The European Physical Journal BDec 1, 198228 citations

Lecture notes in mathematics

Continuous exponents of spin correlation functions of inhomogeneous layered ising models

Barry M. McCoy, Jacques H. H. Perk

Published Jan 1, 1982• 1 citation

Publisher

DOI: 10.1007/bfb0093496

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The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength $J$ , and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where $d=2$ is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength $J$, and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where $d=2$ is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.

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Correlations in inhomogeneous Ising models

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The European Physical Journal BJun 1, 198316 citations

Two-layered Ising model with short-range two-spin interactions showing the critical behaviour of the Ashkin-Teller model

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Journal of Physics C Solid State PhysicsMar 28, 19763 citations

Correlations in inhomogeneous Ising models

W. F. Wolff, J. Zittartz

The European Physical Journal BDec 1, 198228 citations

The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength

J

, and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where

d=2

is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.