Linear Antiferromagnetic Chain with Anisotropic Coupling

Abstract

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