Key points are not available for this paper at this time.
The ground-state and the spin-wave states of the Hamiltonian, H= ∑ i (SixSi+1x+SiySi+1y+ρSizSi+1z),are studied for all values of ρ, and analytical expressions are given for their energies. On the other hand, by using a canonical transformation which changes H(ρ) into -H(- ρ), the states of highest energy can also be obtained. The ground state is ferromagnetic for ρ ≤ − 1 and antiferromagnetic for ρ ≥ −1. For ρ = ±1, the energy has singularities, but it remains continuous. For ρ = 1, all its derivatives are also continuous. In the range − 1 ≤ ρ ≤ 1, the spin-wave states of given momentum are degenerate but for ρ ≥ 1; this degeneracy is removed, and an energy gap G(ρ) appears.
Cloizeaux et al. (Mon,) studied this question.