Critical Exponents for Long-Range Interactions

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