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

Abstract

The isotropic Hamiltonian ${\mathcal{H}}^{\left(\nu \right)}=-J\Sigma {j=1\mathrm{}}^{}{}_{}{}^{N-1\mathrm{}}{}_{}{}^{}\mathrm{S}_j^{}{}_{}{}^{}·{\mathrm{S}}_{j+1\mathrm{}}$ is considered for an open linear chain of $N$ $\nu$-dimensional vector spins ${\mathrm{S}}_j$;${\mathcal{H}}^{\left(\nu \right)}$ reduces to the $S=\frac{1}{2}$ Ising, planar, and Heisenberg models for $\nu =1,2,\mathrm{and} 3$. The thermodynamic properties (including the susceptibility) of ${\mathcal{H}}^{\left(\nu \right)}$ are found for ferromagnetic ($J>\mathrm{}0\mathrm{}$) and antiferromagnetic ($J<\mathrm{}0\mathrm{}$) exchange interactions for all temperatures $T$ and all spin dimensionalities $\nu$. The manner in which the various properties depend upon $T$ and $\nu$ is studied; in particular we find (a) that although the chain of spins does not display long-range order except at $T=0\mathrm{}$ for any value of $\nu$ most of the properties vary monotonically with $\nu$ (in such a way that, e.g., the degree of "short-range order" decreases with increasing $\nu$; 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.