Linear Magnetic Chains with Anisotropic Coupling

Abstract

Linear chains (and rings) of $S=\frac{1}{2}$ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=-2J\Sigma \stackrel{N}{i=1\mathrm{}}\{{S_i}^z{S_{i+1\mathrm{}}}^z+\gamma ({S_i}^x{S_{i+1\mathrm{}}}^x+{S_i}^y{S_{i+1\mathrm{}}}^y\left)\right\}-g\beta \Sigma \stackrel{N}{i=1\mathrm{}}\mathrm{H}·{\mathrm{S}}_i$ have been studied by exact machine calculations for $N=2\mathrm{} \mathrm{to} 11$, $\gamma =0\mathrm{} \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\to \infty$ behavior is accurately indicated, for all $\gamma$, in the region $\frac{\mathrm{kT}}{\mathrm{}\left|J\right|\mathrm{}}>~0.5$ which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at $T=0\mathrm{}$ are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\gamma =1\mathrm{}$ and confirm the accuracy of Walker's perturbation series in $\gamma$.