Physical review. B, Solid state

Critical Behavior of the Four-Dimensional Ising Ferromagnet and the Breakdown of Scaling

M A Moore

Published Mar 1, 1970• 35 citations

Abstract

Experience obtained from an analysis of the high-temperature series expansion of the spin-spin correlation function of the four-dimensional Ising model shows that the apparent failure of scaling in the three-dimensional Ising model cannot easily be attributed to logarithmic factors modifying the simple power-law singularities at the critical point.

DOI: 10.1103/physrevb.1.2238

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Solution and Critical Behavior of Some "Three-Dimensional" Ising Models with a Four-Spin Interaction

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An exact solution is obtained for some "three-dimensional" Ising models with a four-spin interaction. The spontaneous magnetization is vanishing for all temperatures. The temperature derivative of the susceptibility shows a divergence of the form ${({T}_{c}\ensuremath{-}T)}^{\frac{\ensuremath{-}7}{8}}$ below the critical point. The specific heat shows a logarithmic singularity. Several other related models are introduced with new types of singularities.

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Critical behavior of the three-dimensional Ising model

Henk W. J. Blöte, Robert H. Swendsen

The simple-cubic Ising model has been investigated with the Monte Carlo renormalization-group method. The results are found to be consistent with the critical exponents obtained from series and $\ensuremath{\epsilon}$ expansions.

Physical review. B, Condensed matterSep 1, 197944 citations

Physical review. B, Solid state

Critical Behavior of the Four-Dimensional Ising Ferromagnet and the Breakdown of Scaling

M A Moore

Published Mar 1, 1970• 35 citations

Publisher

Abstract

DOI: 10.1103/physrevb.1.2238

Related papers

Solution and Critical Behavior of Some "Three-Dimensional" Ising Models with a Four-Spin Interaction

Masuo Suzuki

Physical Review LettersFeb 21, 197273 citations

University of Critical Correlations in the Three-Dimensional Ising Ferromagnet

Martin V. Ferer, M A Moore, Michael Wortis

Physical review. B, Solid stateJun 1, 197122 citations

One-dimensional random Ising ferromagnets and antiferromagnets

M. Ya. Azbel, Michael Rubinstein

The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.

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Critical behavior of the random-field Ising model

Misha Gofman, Joan Adler, Amnon Aharony A +2 more

We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$ ${\mathrm{〉}}_{\mathit{T}}$ -〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ ${\mathrm{〉}}_{\mathit{T}}$ 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$ ${\mathrm{〉}}_{\mathit{T}}$ ${]}_{\mathit{h}}$ and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$ ${\mathrm{〉}}_{\mathit{T}}$ ${]}_{\mathit{h}}$ , where 〈 ${\mathrm{〉}}_{\mathit{T}}$ indicates a canonical average at temperature T for an arbitrary configuration of random fields and [ ${]}_{\mathit{h}}$ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each ${\mathit{h}}_{\mathit{i}}$ =\ifmmode\pm\else\textpm\fi{} ${\mathit{h}}_{0}$ and a Gaussian distribution in which each ${\mathit{h}}_{\mathit{i}}$ has variance ${\mathit{h}}_{0}^{2}$ . We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$ ${\mathit{a}}_{\mathit{n}}$ (g,d)(J/T ${)}^{\mathit{n}}$ , where J is the exchange constant and the coefficients ${\mathit{a}}_{\mathit{n}}$ (g,d) are polynomials in g\ensuremath{\equiv} ${\mathit{h}}_{0}^{2}$ / ${\mathit{J}}^{2}$ and in d. We assume that as T approaches its critical value, ${\mathit{T}}_{\mathit{c}}$ , one has \ensuremath{\chi}\ensuremath{\sim}(T- ${\mathit{T}}_{\mathit{c}}$ ${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T- ${\mathit{T}}_{\mathit{c}}$ ${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)( ${\mathit{T}}^{2}$ )/( ${\mathit{gJ}}^{2}$ ). We find that A approaches a constant value as T\ensuremath{\rightarrow} ${\mathit{T}}_{\mathit{c}}$ (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{〉}}_{\mathit{T}}$〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$${]}_{\mathit{h}}$ and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$${]}_{\mathit{h}}$, where 〈${\mathrm{〉}}_{\mathit{T}}$ indicates a canonical average at temperature T for an arbitrary configuration of random fields and [${]}_{\mathit{h}}$ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each ${\mathit{h}}_{\mathit{i}}$=\ifmmode\pm\else\textpm\fi{}${\mathit{h}}_{0}$ and a Gaussian distribution in which each ${\mathit{h}}_{\mathit{i}}$ has variance ${\mathit{h}}_{0}^{2}$. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$${\mathit{a}}_{\mathit{n}}$(g,d)(J/T${)}^{\mathit{n}}$, where J is the exchange constant and the coefficients ${\mathit{a}}_{\mathit{n}}$(g,d) are polynomials in g\ensuremath{\equiv}${\mathit{h}}_{0}^{2}$/${\mathit{J}}^{2}$ and in d. We assume that as T approaches its critical value, ${\mathit{T}}_{\mathit{c}}$, one has \ensuremath{\chi}\ensuremath{\sim}(T-${\mathit{T}}_{\mathit{c}}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-${\mathit{T}}_{\mathit{c}}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(${\mathit{T}}^{2}$)/(${\mathit{gJ}}^{2}$). We find that A approaches a constant value as T\ensuremath{\rightarrow}${\mathit{T}}_{\mathit{c}}$ (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterMar 1, 199659 citations

Low-Temperature Critical Exponents from High-Temperature Series: The Ising Model

David S. Gaunt, George Allen Baker Jr.

First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.

Physical review. B, Solid stateFeb 1, 1970113 citations

Critical behavior of the three-dimensional Ising model

Henk W. J. Blöte, Robert H. Swendsen

Physical review. B, Condensed matterSep 1, 197944 citations

{\mathrm{〉}}_{\mathit{T}}

-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$

{\mathrm{〉}}_{\mathit{T}}

〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$

{\mathrm{〉}}_{\mathit{T}}

{]}_{\mathit{h}}

and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$

{\mathrm{〉}}_{\mathit{T}}

{]}_{\mathit{h}}

, where 〈

{\mathrm{〉}}_{\mathit{T}}

indicates a canonical average at temperature T for an arbitrary configuration of random fields and [

{]}_{\mathit{h}}

indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each

{\mathit{h}}_{\mathit{i}}

=\ifmmode\pm\else\textpm\fi{}

{\mathit{h}}_{0}

and a Gaussian distribution in which each

{\mathit{h}}_{\mathit{i}}

has variance

{\mathit{h}}_{0}^{2}

. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$

{\mathit{a}}_{\mathit{n}}

(g,d)(J/T

{)}^{\mathit{n}}

, where J is the exchange constant and the coefficients

{\mathit{a}}_{\mathit{n}}

(g,d) are polynomials in g\ensuremath{\equiv}

{\mathit{h}}_{0}^{2}

{\mathit{J}}^{2}

and in d. We assume that as T approaches its critical value,

{\mathit{T}}_{\mathit{c}}

, one has \ensuremath{\chi}\ensuremath{\sim}(T-

{\mathit{T}}_{\mathit{c}}

${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-

{\mathit{T}}_{\mathit{c}}

${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(

{\mathit{T}}^{2}

)/(

{\mathit{gJ}}^{2}

). We find that A approaches a constant value as T\ensuremath{\rightarrow}

{\mathit{T}}_{\mathit{c}}

(consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.