Exact Solution for a Linear Chain of Isotropically Interacting Classical Spins of Arbitrary Dimensionality

The isotropic Hamiltonian H^(ν)=-JΣj=1^^N-1Sⱼ·Sⱼ₊₁ is considered for an open linear chain of N ν-dimensional vector spins Sⱼ;H^(ν) reduces to the S=1/2 Ising, planar, and Heisenberg models for ν=1,2,and 3. The thermodynamic properties (including the susceptibility) of H^(ν) are found for ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) exchange interactions for all temperatures T and all spin dimensionalities ν. The manner in which the various properties depend upon T and ν is studied; in particular we find (a) that although the chain of spins does not display long-range order except at $T=0$ for any value of ν most of the properties vary monotonically with ν (in such a way that, e.g., the degree of "short-range order" decreases with increasing ν; and (b) that as the spin dimensionality increases without limit, all of the calculated properties approach precisely those predicted by the Berlin-Kac spherical model.