Low-temperature renormalization group for ferromagnets with long-range interactions

Ferromagnets with the number of spin components $n>\mathrm{}2\mathrm{}$ and with power-law interactions $r^{-d-\sigma }$ are studied in dimensions $d=2+\epsilon$ for small $\epsilon$. The following theorem is proved: Let ${\eta }_{\mathrm{SR}}$ be the value of the critical exponent $\eta$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2-\sigma <{\eta }_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\sigma$. In the opposite case, the value of $\eta$ is ${\eta }_{\mathrm{LR}}=2-\sigma$ and the critical exponent $\nu$ depends on $\sigma$ in addition to $d$ and $n$.