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Physical review. B, Solid state

Examination of the phenomenological scaling functions for critical scattering

Craig A. Tracy, Barry M. McCoy

Published Jul 1, 1975• 120 citations
Publisher

Abstract

In the scaling limit $k\ensuremath{\rightarrow}0$, $\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $y=k\ensuremath{\xi}$ is fixed, the $k$-dependent susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$ can, according to the scaling hypotheses of Kadanoff and Fisher, be written as $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)={\ensuremath{\xi}}^{\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}}X_{\ifmmode\pm\else\textpm\fi{}}(y)+o({\ensuremath{\xi}}^{\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}})$. We exactly compute the scale functions $X_{\ifmmode\pm\else\textpm\fi{}}(y)$ for the two-dimensional Ising model in zero magnetic field. We then compare the various phenomenological scale functions (Ornstein-Zernike pole approximate, Fisher approximate, Fisher-Burford approximate, Tarko-Fisher approximates, etc.) with the exact $X_{\ifmmode\pm\else\textpm\fi{}}(y)$ for the two-dimensional Ising model. This comparison provides insight into those regions of $y=k\ensuremath{\xi}$ where these phenomenological scale functions are applicable. Such insight is important since the region of experimentally accessible $y$ is rather limited. We then use these results to examine the method of data analysis used in critical scattering experiments. We conclude that no experiment to date unambiguously and directly establishes that the critical exponent $\ensuremath{\eta}$ is greater than zero.

DOI
10.1103/physrevb.12.368

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Theory of Critical-Point Scattering and Correlations. II. Heisenberg Models

Douglas S. Ritchie, Michael Ellis Fisher

The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin, $S=\frac{1}{2}$ to $\ensuremath{\infty}$, on the sc, bcc, and fcc lattices are studied on the basis of high-temperature series expansions along the lines developed in Paper I [M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967)]. Subject to increased uncertainties for low spin, it is concluded that the exponents $\ensuremath{\gamma}={1.375}_{\ensuremath{-}0.01}^{+0.02}$, $2\ensuremath{\nu}={1.405}_{\ensuremath{-}0.01}^{+0.02}$, and $\ensuremath{\eta}=0.043\ifmmode\pm\else\textpm\fi{}0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\ensuremath{\chi}}_{0}(T)$, for the inverse correlation length ${\ensuremath{\kappa}}_{1}(T)$, for the effective interaction range ${r}_{1}(T)$, and using the Fisher-Burford approximant, for the total scattering $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$. The shape parameter ${\ensuremath{\varphi}}_{c}$ attains the "universal" value ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 11 for large spin but shows signs of spin dependence (and lattice dependence) for low spin. At fixed k the scattering is predicted to display a maximum above ${T}_{c}$ determined by $\frac{{\ensuremath{\kappa}}_{1}({T}_{max})}{k}\ensuremath{\simeq}0.10 (\mathrm{for} S\ensuremath{\gtrsim}2)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈{S}_{0}^{z}{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉}_{c}$ for various models. This leads to the revised, universal estimate ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 15 for all three cubic lattice, spin-\textonehalf{} Ising models. The results are compared d briefly with various experiments which support $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.

Physical review. B, Solid state•Apr 1, 1972•196 citations

Dispersion-theory approach to the correlation function for critical scattering

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A dispersion-theory approach provides an exact representation of the two-point correlation function $G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},t)=C{t}^{\ensuremath{-}\ensuremath{\gamma}}g({q}^{2}{\ensuremath{\xi}}^{2})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and $\ensuremath{\xi}$ is the correlation length, which describes the critical scattering near a second-order phase transition. The threshold property of the spectral weight function $F(\frac{x}{3})\ensuremath{\propto}\mathrm{Im}{g}^{\ensuremath{-}1}(\ensuremath{-}{x}^{2})$ is used to motivate an approximation to the scaling function $g({x}^{2})$ based on the asymptotically exact Fisher-Langer approximant ${g}_{\mathrm{FL}}({x}^{2})=(\frac{{C}_{1}}{{x}^{2\ensuremath{-}\ensuremath{\eta}}})(1+\frac{{C}_{2}}{{x}^{\frac{(1\ensuremath{-}\ensuremath{\alpha})}{\ensuremath{\nu}}}}+\frac{{C}_{3}}{{x}^{\frac{1}{\ensuremath{\nu}}}})$, where $\ensuremath{\alpha}$, $\ensuremath{\eta}$, $\ensuremath{\nu}$ are the usual critical exponents. The approximation consists of truncating the spectral function associated with ${g}_{\mathrm{FL}}^{\ensuremath{-}1}({x}^{2})$ in a manner designed to simulate the known threshold property of the exact spectral function, namely $F(\frac{x}{3})=0$ for $x\ensuremath{\le}3$. The new approximant is checked on the two-dimensional Ising model and shown to agree with the exact result to better than 0.03% for all $x$. Near four dimensions, the agreement with exact results is also excellent. A phenomenological approach to the intermediate three-dimensional case is presented, and shown to be in good agreement with high-temperature series and $\ensuremath{\epsilon}$-expansion results. The dispersion-theory approach is also shown to provide a convenient framework for calculations within the $\ensuremath{\epsilon}$ expansion and is used to compute $g({x}^{2})$ to $O({\ensuremath{\epsilon}}^{3})$, a new result.

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Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

Tai Tsun Wu, Barry M. McCoy, Craig A. Tracy +1 more

We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ and $T<{T}_{c}$. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where $t$ is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-$t$ behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

Physical review. B, Solid state•Jan 1, 1976•694 citations

Theory of critical point scattering and correlations. III. The Ising model below Tc and in a field

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Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field $H$ and temperature $T$. In three dimensions P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general $H$ and $T$ are found to be considerably greater than for $H=0$, $T>{T}_{c}$. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for $T={T}_{c}$ and for $T\ensuremath{\gtrless}{T}_{c}$ with $H=0$.

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Scaling laws and correlation functions near magnetic critical points

Robert Hecht

"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."

•Jan 1, 1967•0 citations
In the scaling limit k\ensuremath→0k\ensuremath{\rightarrow}0k\ensuremath→0, \ensuremathξ\ensuremath→\ensuremath∞\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathξ\ensuremath→\ensuremath∞ such that y=k\ensuremathξy=k\ensuremath{\xi}y=k\ensuremathξ is fixed, the kkk-dependent susceptibility \ensuremathχ(k\ensuremath→,T)\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)\ensuremathχ(k\ensuremath→​,T) can, according to the scaling hypotheses of Kadanoff and Fisher, be written as \ensuremathχ(k\ensuremath→,T)=\ensuremathξ\ensuremathγ\ensuremathνX\ifmmode±\else\textpm\fi(y)+o(\ensuremathξ\ensuremathγ\ensuremathν)\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)={\ensuremath{\xi}}^{\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}}X_{\ifmmode\pm\else\textpm\fi{}}(y)+o({\ensuremath{\xi}}^{\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}})\ensuremathχ(k\ensuremath→​,T)=\ensuremathξ\ensuremathν\ensuremathγ​X\ifmmode±\else\textpm\fi​(y)+o(\ensuremathξ\ensuremathν\ensuremathγ​). We exactly compute the scale functions X\ifmmode±\else\textpm\fi(y)X_{\ifmmode\pm\else\textpm\fi{}}(y)X\ifmmode±\else\textpm\fi​(y) for the two-dimensional Ising model in zero magnetic field. We then compare the various phenomenological scale functions (Ornstein-Zernike pole approximate, Fisher approximate, Fisher-Burford approximate, Tarko-Fisher approximates, etc.) with the exact X\ifmmode±\else\textpm\fi(y)X_{\ifmmode\pm\else\textpm\fi{}}(y)X\ifmmode±\else\textpm\fi​(y) for the two-dimensional Ising model. This comparison provides insight into those regions of y=k\ensuremathξy=k\ensuremath{\xi}y=k\ensuremathξ where these phenomenological scale functions are applicable. Such insight is important since the region of experimentally accessible yyy is rather limited. We then use these results to examine the method of data analysis used in critical scattering experiments. We conclude that no experiment to date unambiguously and directly establishes that the critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη is greater than zero.
The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin, S=12S=\frac{1}{2}S=21​ to $\ensuremath{\infty}$, on the sc, bcc, and fcc lattices are studied on the basis of high-temperature series expansions along the lines developed in Paper I [M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967)]. Subject to increased uncertainties for low spin, it is concluded that the exponents $\ensuremath{\gamma}={1.375}_{\ensuremath{-}0.01}^{+0.02}$, $2\ensuremath{\nu}={1.405}_{\ensuremath{-}0.01}^{+0.02}$, and $\ensuremath{\eta}=0.043\ifmmode\pm\else\textpm\fi{}0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\ensuremath{\chi}}_{0}(T)$, for the inverse correlation length ${\ensuremath{\kappa}}_{1}(T)$, for the effective interaction range r1(T){r}_{1}(T)r1​(T), and using the Fisher-Burford approximant, for the total scattering $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$. The shape parameter ${\ensuremath{\varphi}}_{c}$ attains the "universal" value ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 11 for large spin but shows signs of spin dependence (and lattice dependence) for low spin. At fixed k the scattering is predicted to display a maximum above Tc{T}_{c}Tc​ determined by $\frac{{\ensuremath{\kappa}}_{1}({T}_{max})}{k}\ensuremath{\simeq}0.10 (\mathrm{for} S\ensuremath{\gtrsim}2)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈{S}_{0}^{z}{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉}_{c}$ for various models. This leads to the revised, universal estimate ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 15 for all three cubic lattice, spin-\textonehalf{} Ising models. The results are compared d briefly with various experiments which support $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.
A dispersion-theory approach provides an exact representation of the two-point correlation function $G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},t)=C{t}^{\ensuremath{-}\ensuremath{\gamma}}g({q}^{2}{\ensuremath{\xi}}^{2})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and $\ensuremath{\xi}$ is the correlation length, which describes the critical scattering near a second-order phase transition. The threshold property of the spectral weight function $F(\frac{x}{3})\ensuremath{\propto}\mathrm{Im}{g}^{\ensuremath{-}1}(\ensuremath{-}{x}^{2})$ is used to motivate an approximation to the scaling function g(x2)g({x}^{2})g(x2) based on the asymptotically exact Fisher-Langer approximant ${g}_{\mathrm{FL}}({x}^{2})=(\frac{{C}_{1}}{{x}^{2\ensuremath{-}\ensuremath{\eta}}})(1+\frac{{C}_{2}}{{x}^{\frac{(1\ensuremath{-}\ensuremath{\alpha})}{\ensuremath{\nu}}}}+\frac{{C}_{3}}{{x}^{\frac{1}{\ensuremath{\nu}}}})$, where $\ensuremath{\alpha}$, $\ensuremath{\eta}$, $\ensuremath{\nu}$ are the usual critical exponents. The approximation consists of truncating the spectral function associated with ${g}_{\mathrm{FL}}^{\ensuremath{-}1}({x}^{2})$ in a manner designed to simulate the known threshold property of the exact spectral function, namely F(x3)=0F(\frac{x}{3})=0F(3x​)=0 for $x\ensuremath{\le}3$. The new approximant is checked on the two-dimensional Ising model and shown to agree with the exact result to better than 0.03% for all xxx. Near four dimensions, the agreement with exact results is also excellent. A phenomenological approach to the intermediate three-dimensional case is presented, and shown to be in good agreement with high-temperature series and $\ensuremath{\epsilon}$-expansion results. The dispersion-theory approach is also shown to provide a convenient framework for calculations within the $\ensuremath{\epsilon}$ expansion and is used to compute g(x2)g({x}^{2})g(x2) to $O({\ensuremath{\epsilon}}^{3})$, a new result.
Critical point scaling in a field HHH applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function f(R)f(R)f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically f(R)f(R)f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3d=3d=3, we have determined the exponents and the critical scaling function f(R)f(R)f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f(R)f(R)f(R) in d=3d=3d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of RRR for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function f(R)f(R)f(R) at some negative value of RRR, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.
We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where ttt is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-ttt behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field HHH and temperature TTT. In three dimensions P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general HHH and TTT are found to be considerably greater than for H=0H=0H=0, T>TcT>{T}_{c}T>Tc​. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for T=TcT={T}_{c}T=Tc​ and for $T\ensuremath{\gtrless}{T}_{c}$ with H=0H=0H=0.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."