Physical review. B, Solid state

University of Critical Correlations in the Three-Dimensional Ising Ferromagnet

Martin V. Ferer, M A Moore, Michael Wortis

Published Jun 1, 1971• 22 citations

Abstract

Evidence from exact enumeration series is presented to support the hypothesis that the functional form of spin-spin correlations for three-dimensional zero-field Ising ferromagnets in their high-temperature critical region is independent of lattice and spin magnitude.

DOI: 10.1103/physrevb.3.3911

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Physical review. B, Solid state

University of Critical Correlations in the Three-Dimensional Ising Ferromagnet

Martin V. Ferer, M A Moore, Michael Wortis

Published Jun 1, 1971• 22 citations

Publisher

Abstract

DOI: 10.1103/physrevb.3.3911

Related papers

High-Temperature Series and Critical Amplitudes for the Spin-Spin Correlations of the Three-Dimensional Ising Ferromagnet

Martin V. Ferer, Michael Wortis

High-temperature series for the inverse spin-spin correlation function of the $s=\frac{1}{2}$ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at ${T}_{c}$ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.High-temperature series for the inverse spin-spin correlation function of the $s=\frac{1}{2}$ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at ${T}_{c}$ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.

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Martin V. Ferer, M A Moore, Michael Wortis

The spin-spin correlation function for the $S=\frac{1}{2}$ Ising ferromagnet is evaluated in zero magnetic field and for $T>~{T}_{c}$ by the method of series expansions. Evidence is presented showing that the scaling of correlations is valid in the (weak) limit $r\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\kappa}r$ fixed, but is not valid in the (strong) limit $\ensuremath{\kappa}a\ensuremath{\rightarrow}0$, $r\ensuremath{\gg}a$, with $\ensuremath{\kappa}r$ arbitrary.The spin-spin correlation function for the $S=\frac{1}{2}$ Ising ferromagnet is evaluated in zero magnetic field and for $T>~{T}_{c}$ by the method of series expansions. Evidence is presented showing that the scaling of correlations is valid in the (weak) limit $r\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\kappa}r$ fixed, but is not valid in the (strong) limit $\ensuremath{\kappa}a\ensuremath{\rightarrow}0$, $r\ensuremath{\gg}a$, with $\ensuremath{\kappa}r$ arbitrary.

Physical Review LettersJun 23, 196959 citations

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M A Moore

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Updated tests of scaling and universality for spin-spin correlations in the two- and three- dimensional spin- S Ising models using high-temperature expansions

P. Butera, M. Comi

We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin- $S$ Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-$S$ Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.

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P F Fox, David S. Gaunt

Journal of Physics C Solid State PhysicsOct 31, 197226 citations

Spin-Spin Correlation Function of the Three-Dimensional Ising Ferromagnet Above the Curie Temperature

M A Moore, David Jasnow, Michael Wortis

High-temperature series expansions for the Ising model's spin-spin correlation function are found to order ${(\frac{J}{\mathrm{kT}})}^{12}$ on the simple cubic, bcc, and fcc latices. Analysis of moment series indicates that the scaling form of the correlation function is correct, at least in the limit $\ensuremath{\kappa}r$ fixed, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$. Values are obtained for the correlation indices $\ensuremath{\nu}$ and $\ensuremath{\eta}$-the latter by two methods, thereby providing a direct check on the scaling law.High-temperature series expansions for the Ising model's spin-spin correlation function are found to order ${(\frac{J}{\mathrm{kT}})}^{12}$ on the simple cubic, bcc, and fcc latices. Analysis of moment series indicates that the scaling form of the correlation function is correct, at least in the limit $\ensuremath{\kappa}r$ fixed, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$. Values are obtained for the correlation indices $\ensuremath{\nu}$ and $\ensuremath{\eta}$-the latter by two methods, thereby providing a direct check on the scaling law.

Physical Review LettersMay 5, 1969128 citations

High-temperature series for the inverse spin-spin correlation function of the

s=\frac{1}{2}

Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at

{T}_{c}

and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.

The spin-spin correlation function for the

S=\frac{1}{2}

Ising ferromagnet is evaluated in zero magnetic field and for

T>~{T}_{c}

by the method of series expansions. Evidence is presented showing that the scaling of correlations is valid in the (weak) limit $r\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\kappa}r$ fixed, but is not valid in the (strong) limit $\ensuremath{\kappa}a\ensuremath{\rightarrow}0$, $r\ensuremath{\gg}a$, with $\ensuremath{\kappa}r$ arbitrary.

We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-

S

Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.

High-temperature series expansions for the Ising model's spin-spin correlation function are found to order

{(\frac{J}{\mathrm{kT}})}^{12}

on the simple cubic, bcc, and fcc latices. Analysis of moment series indicates that the scaling form of the correlation function is correct, at least in the limit $\ensuremath{\kappa}r$ fixed, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$. Values are obtained for the correlation indices $\ensuremath{\nu}$ and $\ensuremath{\eta}$-the latter by two methods, thereby providing a direct check on the scaling law.