Renormalization-group calculation of the critical-point exponentηfor a critical point of arbitrary order

Abstract

The critical-point exponent $\eta$ for a critical point of order $\mathcal{O}$ in dimensions less than $d_{\mathcal{O}}\equiv \frac{2\mathcal{O}}{(\mathcal{O}-1)}$, is calculated to leading nonvanishing order in the parameter ${\epsilon }_{\mathcal{O}}\equiv d_{\mathcal{O}}-d$. The result is given for $n$-component isotropically interacting magnetic systems. For Ising systems, $n=1\mathrm{}$, the result is ${\eta }_{\mathcal{O}}=\frac{{\epsilon }_{\mathcal{O}}^{2}4{(\mathcal{O}-1)}^2}{{\left(\left\{2\mathcal{O}\right\}\left\{\mathcal{O}\right\}\right)}^3}$. As $\mathcal{O}$ increases, the coefficient of ${\epsilon }_{\mathcal{O}}^2$ rapidly becomes very small, varying as $2^{-6\mathrm{}\mathcal{O}}$ for $\mathcal{O}$ large. In the limit of large $n$, ${\eta }_{\mathcal{O}}$ for odd order points approaches a constant and, for even order points, is proportional to $\frac{1}{n}$.