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Accurate critical exponents for Ising like systems in non-integer dimensions

J. C. Le Guillou, Jean Zinn‐Justin

Current physics sources and commentsJan 1, 199028 citations

Monte Carlo Study of Ising Critical Isotherms

J. G. García, Manuel I. Marqués, Julio A. Gonzalo

Monte Carlo investigations of magnetization versus field, Mc(H), at the critical temperature provide direct accurate results on the critical exponent δ−1 for one, two, three and four-dimensional lattices: δ−1 1D =0, δ−1 2D =0.0666(2)≈1/15, δ−1 3D =0.1997(4)≈1/5, δ−1 4D =0.332(5)≈1/3. This type of Monte Carlo data on δ, which is not easily found in studies of Ising lattices in the current literature, as far as we know, defines extremely well the numerical value of this exponent within very stringent limits.Monte Carlo investigations of magnetization versus field, Mc(H), at the critical temperature provide direct accurate results on the critical exponent δ−1 for one, two, three and four-dimensional lattices: δ−1 1D =0, δ−1 2D =0.0666(2)≈1/15, δ−1 3D =0.1997(4)≈1/5, δ−1 4D =0.332(5)≈1/3. This type of Monte Carlo data on δ, which is not easily found in studies of Ising lattices in the current literature, as far as we know, defines extremely well the numerical value of this exponent within very stringent limits.

FerroelectricsJan 1, 20120 citations

Monte Carlo simulations of Ising models

Walter Selke

Computer Physics CommunicationsAug 1, 19961 citation

Universality of the Ising and the S=1 model on Archimedean lattices: A Monte Carlo determination

Anastasios Malakis, Gül Gülpınar, Yenal Karaaslan +2 more

The Ising models S=1/2 and S=1 are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well-known 2D Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S=1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2D Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S=1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.The Ising models S=1/2 and S=1 are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well-known 2D Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S=1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2D Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S=1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.

arXiv:1203.4507Physical Review EMar 29, 201210 citations

Monte Carlo Simulations of Dilute Ising Models

Walter Selke

Springer proceedings in physicsJan 1, 19938 citations

Accurate Monte Carlo critical exponents for Ising lattices

J. G. García, Julio A. Gonzalo, Manuel I. Marqués

A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.

arXiv:cond-mat/0211270Nov 14, 20020 citations

Accurate Monte Carlo critical exponents for Ising lattices

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Accurate Monte Carlo critical exponents for Ising lattices

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