Critical Properties of Isotropically Interacting Classical Spins Constrained to a Plane

High-temperature expansions of the susceptibility and internal energy (specific heat) are presented for general lattice structure for a system of isotropically interacting unit vectors (or "classical spins") which are constrained to lie in a plane. A phase transition ($T_c>0\mathrm{}$) is indicated for two-dimensional lattices; the expected result $T_c=0\mathrm{}$ is found in one dimension, but only upon choosing a more suitable expansion parameter than $\frac{J}{\mathrm{kT}}$. Similarities with the corresponding expansions of the $S=\frac{1}{2}$ Ising and classical Heisenberg models are pointed out; in particular, it is found that certain critical properties of this planar model appear to be bounded on one side by the Ising model and on the other side by the Heisenberg model.