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 ($\mathcal{O}=2\mathrm{}$) and Gaussian tricritical points ($\mathcal{O}=3\mathrm{}$) to critical points of arbitrary order $\mathcal{O}$. An expansion scheme in ${\epsilon }_{\mathcal{O}}\equiv \frac{2\mathcal{O}}{(\mathcal{O}-1)-d}$ is proposed. The nontrivial fixed points and the critical exponents for $\mathcal{O}=3,4\mathrm{}$ are calculated to order ${\epsilon }_{\mathcal{O}}$. We present the relevant scaling fields and densities for $\mathcal{O}=3\mathrm{}$, and, in an appendix, justify the validity of the approximate recursion relation to order ${\epsilon }_{\mathcal{O}}$.