O(n)Heisenberg model with long-range interactions

The $O\mathrm{}\left(n\right)\mathrm{}$ Heisenberg model in $d$ dimensions with long-range interactions $r^{-d-\sigma }$ is studied near $d=n=\sigma =1\mathrm{}$ and $d=n=\sigma =2\mathrm{}$. Renormalization-group (RG) equations are constructed by making minimal assumptions of analyticity and requiring that known results be reproduced for particular limiting cases. The fixed points of these RG equations are determined, and the existence of boundaries in the $n-d-\sigma$ space, across which the fixed points collide and exchange stabilities, is demonstrated. The physical characteristics of the various regions are discussed.