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Physical Review Letters

Predicted Critical Correlation Function for Three-Dimensional Phase Transitions

Alan J. Bray

Published Feb 9, 1976• 15 citations
Publisher

Abstract

A new approximant is proposed for the critical correlation function $g(y)$ of $y=k\ensuremath{\xi}$, where $\ensuremath{\xi}$ is the correlation length and $k$ the wave number of an order-parameter fluctuation. The approach involves truncating the spectral function associated with the three-term Fisher-Langer approximant in a manner suggested by the requirement of a three-particle threshold. The resulting $g(y)$ is accurate to better than 0.03% everywhere for the two-dimensional Ising model. Results for the three-dimensional case are consistent with $\ensuremath{\epsilon}$-expansion and high-temperature series results.

DOI
10.1103/physrevlett.36.285

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A new approximant is proposed for the critical correlation function g(y)g(y)g(y) of y=k\ensuremathξy=k\ensuremath{\xi}y=k\ensuremathξ, where \ensuremathξ\ensuremath{\xi}\ensuremathξ is the correlation length and kkk the wave number of an order-parameter fluctuation. The approach involves truncating the spectral function associated with the three-term Fisher-Langer approximant in a manner suggested by the requirement of a three-particle threshold. The resulting g(y)g(y)g(y) is accurate to better than 0.03% everywhere for the two-dimensional Ising model. Results for the three-dimensional case are consistent with \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion and high-temperature series results.
A one-parameter scaling function s(x)s(x)s(x) of $x=k\ensuremath{\xi}$, where $\ensuremath{\xi}$ is the correlation length and kkk the wave number of a density or concentration fluctuations in a fluid, is derived from a phenomenological field theoretic approach. On the basis of a positive-definite spectral function, we exploit the requirements of a three-particle threshold and the correct asymptotic approach as $x\ensuremath{\rightarrow}\ensuremath{\infty}$. The resulting s(x)s(x)s(x) is compared with the Fisher-Burford approximant obtained from numerical studies of the three-dimensional Ising model.
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.
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We compute the 2n-point renormalized coupling constants in the symmetric phase of the three-dimensional (3D) Ising model on the simple cubic lattice in terms of the high temperature expansions O(${\mathrm{\ensuremath{\mathrm{B}}}}^{17}$) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3D scalar field at the infrared fixed point or, equivalently, in the critical equation of state of the 3D Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations, as well as previous high temperature expansions.
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