Calculation of the Critical Exponentηvia Renormalization-Group Recursion Formulas

This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\eta$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\eta$. They are intended to demonstrate, qualitatively, how nonzero values for $\eta$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\nu$ and $\gamma$. The equations are solved both by $\epsilon$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\epsilon }^2$ we obtain $\eta =0.05\mathrm{}{\epsilon }^2$. Numerical results in three dimension are $\eta =0.058\mathrm{}$, $\nu =0.588\mathrm{}$, and $\gamma =1.14\mathrm{}$. The relation $\gamma =(2-\eta )\nu$ is confirmed.