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Physical Review Letters

Critical Exponents in 3.99 Dimensions

Kenneth Geddes Wilson, Michael Ellis Fisher

Published Jan 24, 1972• 1,651 citations
Publisher

Abstract

Critical exponents are calculated for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}$ small, using renormalization-group techniques. To order $\ensuremath{\epsilon}$ the exponent $\ensuremath{\gamma}$ is $1+\frac{1}{6}\ensuremath{\epsilon}$ for an Ising-like model and $1+\frac{1}{5}\ensuremath{\epsilon}$ for an $\mathrm{XY}$ model.

DOI
10.1103/physrevlett.28.240

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Critical exponents are calculated for dimension d=4\ensuremath−\ensuremathϵd=4\ensuremath{-}\ensuremath{\epsilon}d=4\ensuremath−\ensuremathϵ with \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ small, using renormalization-group techniques. To order \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ the exponent \ensuremathγ\ensuremath{\gamma}\ensuremathγ is 1+16\ensuremathϵ1+\frac{1}{6}\ensuremath{\epsilon}1+61​\ensuremathϵ for an Ising-like model and 1+15\ensuremathϵ1+\frac{1}{5}\ensuremath{\epsilon}1+51​\ensuremathϵ for an XY\mathrm{XY}XY model.
The critical-point exponent $\ensuremath{\eta}$ for a critical point of order O\mathcal{O}O in dimensions less than ${d}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, is calculated to leading nonvanishing order in the parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}{d}_{\mathcal{O}}\ensuremath{-}d$. The result is given for nnn-component isotropically interacting magnetic systems. For Ising systems, n=1n=1n=1, the result is ${\ensuremath{\eta}}_{\mathcal{O}}=\frac{{\ensuremath{\epsilon}}_{\mathcal{O}}^{2}4{(\mathcal{O}\ensuremath{-}1)}^{2}}{{({2\mathcal{O}}{\mathcal{O}})}^{3}}$. As O\mathcal{O}O increases, the coefficient of ${\ensuremath{\epsilon}}_{\mathcal{O}}^{2}$ rapidly becomes very small, varying as ${2}^{\ensuremath{-}6\mathcal{O}}$ for O\mathcal{O}O large. In the limit of large nnn, ${\ensuremath{\eta}}_{\mathcal{O}}$ for odd order points approaches a constant and, for even order points, is proportional to 1n\frac{1}{n}n1​.
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.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).