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Physical review. B, Solid state

Scaling function for two-point correlations. II. Expansion to order 1n

Amnon Aharony A

Published Oct 1, 1974• 20 citations
Publisher

Abstract

The zero-field two-point correlation function of an $n$-vector system in $d$ dimensions is calculated to order $\frac{1}{n}$ for $T\ensuremath{\ge}{T}_{c}$ and $2<d<4$. The critical-point scaling function is obtained as a closed cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector $q$, the variation of the Fourier transform of the correlation function is ${\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}$, where $\ensuremath{\alpha}$ is the specific-heat exponent and ${\stackrel{^}{E}}_{2}$, ${\stackrel{^}{E}}_{3}$ are of order $\frac{1}{n}$. At $d=3$ the term ${t}^{2}$ is modified by logarithmic corrections. As $q\ensuremath{\rightarrow}0$ at fixed nonzero $t$, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}$ are of order $\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\ensuremath{\epsilon}$-expansion results of previous work.

DOI
10.1103/physrevb.10.2834

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The zero-field two-point correlation function of an nnn-vector system in ddd dimensions is calculated to order 1n\frac{1}{n}n1​ for T\ensuremath≥TcT\ensuremath{\ge}{T}_{c}T\ensuremath≥Tc​ and 2<d<42<d<42<d<4. The critical-point scaling function is obtained as a closed cutoff-independent integral. As t=(T\ensuremath−Tc)Tc\ensuremath→0t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0t=Tc​(T\ensuremath−Tc​)​\ensuremath→0 at fixed wave vector qqq, the variation of the Fourier transform of the correlation function is {\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}, where \ensuremathα\ensuremath{\alpha}\ensuremathα is the specific-heat exponent and {\stackrel{^}{E}}_{2}, {\stackrel{^}{E}}_{3} are of order 1n\frac{1}{n}n1​. At d=3d=3d=3 the term t2{t}^{2}t2 is modified by logarithmic corrections. As q\ensuremath→0q\ensuremath{\rightarrow}0q\ensuremath→0 at fixed nonzero ttt, deviations of the scaling function from the Ornstein-Zernike form (1+\ensuremathξ2q2)\ensuremath−1{(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}(1+\ensuremathξ2q2)\ensuremath−1 are of order 10\ensuremath−3\ensuremathξ4q4n\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}n10\ensuremath−3\ensuremathξ4q4​. Results are compared with high-temperature series-expansion estimates and with the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion results of previous work.
The zero-field, two-point correlation function of an nnn-vector system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions is calculated to order ${\ensuremath{\epsilon}}^{2}$ for T> TcT>~{T}_{c}T> Tc​. The scaling function is obtained as a closed, cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector qqq, the leading variation is ${{\stackrel{^}{E}}_{1}}^{n,d}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{{\stackrel{^}{E}}_{2}}^{n,d}(q)t$, where $\ensuremath{\alpha}$ is the specific-heat exponent; thence the maximum in the scattering above Tc{T}_{c}Tc​ is located, in good agreement with high-TTT series-expansion estimates.
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A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below Tc{T}_{c}Tc​ than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.
The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the ddd-dimensional, spin-\textonehalf{} Ising model in general field HHH. For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as H2{H}^{2}H2 and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.
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.
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