Two-Dimensional Heisenberg Model with Nonlinear Interactions

We investigate a two-dimensional classical $N$-vector model with a nonlinear interaction $(1+{\text{bsigma}}_i\cdot {\text{bsigma}}_j{)}^p$ in the large-N limit. As observed for N=3 by Bl"ote {\em et al.} [Phys. Rev. Lett. {\bf 88}, 047203 (2002)], we find a first-order transition for $p>p_c$ and no finite-temperature phase transitions for $p<p_c$. For $p>p_c$, both phases have short-range order, the correlation length showing a finite discontinuity at the transition. For $p=p_c$, there is a peculiar transition, where the spin-spin correlation length is finite while the energy-energy correlation length diverges.