Renormalization-group calculations of exponents for critical points of higher order

Abstract

Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order $\mathcal{O}$, $\mathcal{O}$ phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points ($\mathcal{O}=2$) and Gaussian tricritical points ($\mathcal{O}=3$) to critical points of arbitrary order $\mathcal{O}$. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for $\mathcal{O}=3,4$ are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for $\mathcal{O}=3$, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.