A Exact Solution of the Anisotropic Heisenberg Model with Long Range 'kac' Interactions.

Kenneth Millard, Harvey S. Leff

Paul A. Pearce, Colin J. Thompson

Tom Kennedy

Jose Antonio Rogan, Miguel Kiwi

Vahid Karimipour, Mohammad Khorrami

Eugene Stanley

The isotropic Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}=\ensuremath{-}J\ensuremath{\Sigma}{j=1}^{}^{N\ensuremath{-}1}\mathrm{S}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ is considered for an open linear chain of $N$ $\ensuremath{\nu}$-dimensional vector spins ${\mathrm{S}}_{j}$;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the $S=\frac{1}{2}$ Ising, planar, and Heisenberg models for $\ensuremath{\nu}=1,2,\mathrm{and} 3$. The thermodynamic properties (including the susceptibility) of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found for ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) exchange interactions for all temperatures $T$ and all spin dimensionalities $\ensuremath{\nu}$. The manner in which the various properties depend upon $T$ and $\ensuremath{\nu}$ 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 $\ensuremath{\nu}$ most of the properties vary monotonically with $\ensuremath{\nu}$ (in such a way that, e.g., the degree of "short-range order" decreases with increasing $\ensuremath{\nu}$; 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.