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Physical Review Letters

Skeleton Graph Expansion for Critical Exponents

T. Tsuneto, Elihu Abrahams

Published Feb 5, 1973• 62 citations
Publisher

Abstract

Critical exponents are calculated using conventional many-body perturbation theory by means of a self-consistent skeleton graph expansion in $4\ensuremath{-}\ensuremath{\epsilon}$ deminsions, where $\ensuremath{\epsilon}$ is small. The self-consistency is expressed through Ward identities. The results, the same as those of Wilson, are obtained without recourse to renormalization group arguments.

DOI
10.1103/physrevlett.30.217

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Critical exponents are calculated using conventional many-body perturbation theory by means of a self-consistent skeleton graph expansion in 4\ensuremath−\ensuremathϵ4\ensuremath{-}\ensuremath{\epsilon}4\ensuremath−\ensuremathϵ deminsions, where \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ is small. The self-consistency is expressed through Ward identities. The results, the same as those of Wilson, are obtained without recourse to renormalization group arguments.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.
The "subcritical" exponent ${\ensuremath{\Delta}}_{2s}$ for corrections to asymptotic scaling in the renormalization-group theory of critical phenomena, has been calculated for the continuous-spin versions of the classical, d=3d=3d=3 Ising, $X\ensuremath{-}Y$, and Heisenberg models by numerical integrations of Wilson's approximate recursion formula. The results are ${\ensuremath{\Delta}}_{2s}(\mathrm{Ising})=\ensuremath{-}0.640$, ${\ensuremath{\Delta}}_{2s}(X\ensuremath{-}Y)=\ensuremath{-}0.644$, and ${\ensuremath{\Delta}}_{2s}(\mathrm{Heisenberg})=\ensuremath{-}0.647$. Comparison is made with results from perturbation expansions in $\ensuremath{\epsilon}=4\ensuremath{-}d$ (ddd being the dimensionality), high-temperature series expansions, and experimental measurements.
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An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit $n=\ensuremath{\infty}$ of the nnn-vector model are calculated. We obtain agreement with the results of Wilson and Fisher, and with the spherical model.