Linear Antiferromagnetic Chain with Anisotropic Coupling

The exact solution is given for a linear chain of N atoms of spin {} coupled together by the anisotropic Hamiltonian H=2JΣNi=1[SᵢᶻSᵢ₊₁ᶻ+(1-α)(SᵢˣSᵢ₊₁ˣ+SᵢʸSᵢ₊₁ʸ)]. The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter α is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, Σi^SᵢᶻSᵢ₊₁ᶻ, is calculated exactly from the variation of the ground-state energy with α. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in ∂²E∂α² is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.