Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions

Recent calculations on ferromagnets with $n$-component spins and with long-range forces (the interaction between spin decays as $\frac{1}{r^{d+\sigma }}$, where $d$ is the dimensionality of the system and $\sigma >0\mathrm{}$) have given exponents which are apparently discontinuous at $\sigma =2\mathrm{}$, i.e., when the transition to short-range interactions is made. By solving Wilson's exact recursion relations to order ${\epsilon }^2(\epsilon =2\mathrm{}\sigma -d)$ we show that the exponents $\eta$, $\gamma$, and $\varphi$ are continuous functions of $\sigma$ and that there exists a region of long-range potentials defined by $2>\mathrm{}\sigma >2-{\eta }_{\mathrm{SR}}$ where the exponents assume their short-range values.