Critical Exponents for Long-Range Interactions

Abstract

Critical exponents for a $d$-dimensional system with an isotropic $n$-component order parameter and long-range attractive interactions decaying as $\frac{1}{{r}^{d+\ensuremath{\sigma}}} (\ensuremath{\sigma}>0)$ are derived, using the renormalization group approach, as power series in $\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d>0$ ($\ensuremath{\sigma}\ensuremath{\ne}2$, fixed) or $\ensuremath{\Delta}\ensuremath{\sigma}=\ensuremath{\sigma}\ensuremath{-}\frac{1}{2}d>0$ ($d$ fixed) and, separately, to order $\frac{1}{n}$ for all $d$ and $\ensuremath{\sigma}\ensuremath{\ne}2$. For $\ensuremath{\epsilon}<0$ the exponents have fixed ("classical") values; when $\ensuremath{\epsilon}=\ensuremath{\Delta}\ensuremath{\sigma}=0$ fractional powers of $\mathrm{ln}(\frac{\ensuremath{\Delta}T}{{T}_{c}})$ appear; when $\ensuremath{\sigma}>2$ the exponents assume their short-range values.