Linear Antiferromagnetic Chain with Anisotropic Coupling

The exact solution is given for a linear chain of $N$ atoms of spin ½ coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\Sigma \stackrel{N}{i=1\mathrm{}}[{S_i}^z{S_{i+1\mathrm{}}}^z+(1-\alpha \left)\right({S_i}^x{S_{i+1\mathrm{}}}^x+{S_i}^y{S_{i+1\mathrm{}}}^y\left)\right].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\alpha$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\Sigma i^{}{S_i}^z{S_{i+1\mathrm{}}}^z$, is calculated exactly from the variation of the ground-state energy with $\alpha$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\partial }^{2}E}{\partial {\alpha }^2}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.