The problem of a system of classical spins of arbitrary dimensionality interacting through an arbitrary, isotropic nearest-neighbor-exchange Hamiltonian is solved exactly for the closed linear chain. The
2m-uple spin-correlation function for spins a distance
k apart is shown to be the
kth power of the ratio of the
mth eigenvalue to the largest one. It is proposed that experimental data be used to find the general functional form of the exchange interaction.
The partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next-nearest-neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next-nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian ℋ=−ΣNi=1[J1(cosϑ2i−1,2i +cosϑ2i,2i+1)+J2cosϑ2i−1,2i+1]. When the nnn interaction is antiferromagnetic (J2<0) it competes with the nn interaction J1, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J2/‖J1‖ ≳−1/2=rc, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at rD=−1/21/2, at which point the ground state energy is a maximum. The result that rD differs from rc is thought to be peculiar to one-dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.
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
Sj;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the
S=21 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.
Thermodynamic properties of one-dimensional Ising chains with random nearest neighbor exchange interaction J are studied exactly in the case of distributions of the type P(J) = A|J|ν for Jmin ≤J ≤Jmax and P(J) = 0, otherwise. The results are compared with those for a periodic chain. We find that the specific heat C ∼Tν+1 for Jmin = 0 and kT ≪Jmax. The dependence of spin-spin correlation function on distance and temperature is discussed. The absence of the peak in the zero field χ⊥ as a function of temperature when ν = 0 is noted in contrast to the case of the periodic chain. The effect of randomness on the low temperature behaviour of these thermodynamic quantities is different. For instance whereas C becomes analytic at T = 0, χ|| either remains singular (ferromagnetic) or becomes analytic (antiferromagnetic). The effect of random xy interaction (∑iJi⊥(SixSi + 1x + SiySi + 1y)) on the high temperature behaviour of χ|| and χ⊥ is discussed.