Physical Review Letters

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

M A Moore, David Jasnow, Michael Wortis

Published May 5, 1969• 128 citations

Abstract

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.

DOI: 10.1103/physrevlett.22.940

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Scaling Form of the Spin-Spin Correlation Function of the Three-Dimensional Ising Ferromagnet above the Curie Temperature

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Physical Review Letters

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

M A Moore, David Jasnow, Michael Wortis

Published May 5, 1969• 128 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.22.940

Related papers

Scaling Form of the Spin-Spin Correlation Function of the Three-Dimensional Ising Ferromagnet above the Curie Temperature

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

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

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

Martin V. Ferer, Michael Wortis

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Physical review. B, Solid stateNov 1, 197264 citations

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Howard B. Tarko, Michael Ellis Fisher

Series expansions for spin-spin correlations of simple-cubic, bcc, and square lattice Ising ferromagnets for general field and temperature have been analyzed. To within the attainable precision (of \ifmmode\pm\else\textpm\fi{} 1 to \ifmmode\pm\else\textpm\fi{} 5%) the exponents ${\ensuremath{\nu}}^{\ensuremath{'}}$ and ${\ensuremath{\nu}}^{c}$ are consistent with scaling predictions. Deviations of the scattering from Ornstein-Zernike forms are significantly larger quantitatively than above ${T}_{c}$ ; the second-moment correlation lengths verify $\frac{{{\ensuremath{\xi}}_{1}}^{+}}{{{\ensuremath{\xi}}_{1}}^{\ensuremath{-}}(T)}=1.96\ifmmode\pm\else\textpm\fi{}0.03$ as $T\ensuremath{\rightarrow}{{T}_{c}}^{\ifmmode\pm\else\textpm\fi{}}$.Series expansions for spin-spin correlations of simple-cubic, bcc, and square lattice Ising ferromagnets for general field and temperature have been analyzed. To within the attainable precision (of \ifmmode\pm\else\textpm\fi{} 1 to \ifmmode\pm\else\textpm\fi{} 5%) the exponents ${\ensuremath{\nu}}^{\ensuremath{'}}$ and ${\ensuremath{\nu}}^{c}$ are consistent with scaling predictions. Deviations of the scattering from Ornstein-Zernike forms are significantly larger quantitatively than above ${T}_{c}$; the second-moment correlation lengths verify $\frac{{{\ensuremath{\xi}}_{1}}^{+}}{{{\ensuremath{\xi}}_{1}}^{\ensuremath{-}}(T)}=1.96\ifmmode\pm\else\textpm\fi{}0.03$ as $T\ensuremath{\rightarrow}{{T}_{c}}^{\ifmmode\pm\else\textpm\fi{}}$.

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Energy density correlations in the three-dimensional Ising model

Reginald Rutherford

The energy-density-energy-density correlation function is studied in the three-dimensional Ising model in zero external field, for various spins on the fcc lattice. An extension of the linked-cluster technique is used to obtain high-temperature series expansions for the correlation function and its associated functions, to at least eighth order in the high-temperature interaction variable. Analysis of the series provides values for their critical indices and amplitudes. These values are used to verify the predictions of several theories of critical behavior. The critical index of the coherence length of the energy density correlation function is found to be equal to that of the spin-spin correlation function for all spins within the accuracy obtained here. For spins higher than ~ this critical index appears to obey strong scaling. An analysis of universality indicates that only one scale factor is needed to transform the energy density correlation function from one spin to another. This prediction is checked here and found to be good within our present accuracy. The current-algebra hypothesis predicts definite relations between the leading singularities of various correlation functions at the critical point. This prediction is checked directly and consistent values of the leading coefficients are obtained. The shape of the energy density correlation function at large distances from the origin, both at and above the critical point, is determined. The results are sufficiently accurate to rule out the Ornstein-Zernike form in zero field for T > T , but not sufficient to eliminate the possibility c of anisotropy at T = T .The energy-density-energy-density correlation function is studied in the three-dimensional Ising model in zero external field, for various spins on the fcc lattice. An extension of the linked-cluster technique is used to obtain high-temperature series expansions for the correlation function and its associated functions, to at least eighth order in the high-temperature interaction variable. Analysis of the series provides values for their critical indices and amplitudes. These values are used to verify the predictions of several theories of critical behavior. The critical index of the coherence length of the energy density correlation function is found to be equal to that of the spin-spin correlation function for all spins within the accuracy obtained here. For spins higher than ~ this critical index appears to obey strong scaling. An analysis of universality indicates that only one scale factor is needed to transform the energy density correlation function from one spin to another. This prediction is checked here and found to be good within our present accuracy. The current-algebra hypothesis predicts definite relations between the leading singularities of various correlation functions at the critical point. This prediction is checked directly and consistent values of the leading coefficients are obtained. The shape of the energy density correlation function at large distances from the origin, both at and above the critical point, is determined. The results are sufficiently accurate to rule out the Ornstein-Zernike form in zero field for T > T , but not sufficient to eliminate the possibility c of anisotropy at T = T .

Jan 1, 19730 citations

Extension to order β23 of the high-temperature expansions for the spin- 12 Ising model on simple cubic and body-centered cubic lattices

P. Butera, M. Comi

Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$

arXiv:hep-lat/0006009Physical review. B, Condensed matterDec 1, 200040 citations

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.

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 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.

{T}_{c}

; the second-moment correlation lengths verify $\frac{{{\ensuremath{\xi}}_{1}}^{+}}{{{\ensuremath{\xi}}_{1}}^{\ensuremath{-}}(T)}=1.96\ifmmode\pm\else\textpm\fi{}0.03$ as $T\ensuremath{\rightarrow}{{T}_{c}}^{\ifmmode\pm\else\textpm\fi{}}$.