Anisotropic Linear Magnetic Chain

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Abstract
The ground‐state and the spin‐wave states of the Hamiltonian, H= ∑ i (S i x S i+1 x +S i y S i+1 y +ρS i z S i+1 z ) ,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 gapG(ρ) appears.