Ground State Stability for the spin-1/2 ferromagnetic Heisenberg spin ring

For the quantum Heisenberg spin-j model on a bipartite, balanced graph, the ground state of the antiferromagnet is a spin singlet by the Lieb-Mattis theorem, "Ordering of energy levels." Moreover, the minimum energy in the spin S sector E(S) is monotonically increasing as S moves from 0 to j|V|. The Lieb-Mattis theorem also implies that for the ferromagnet the absolute ground state is E₀^FM(S_max). However, their theorem does not necessarily imply that E(j|V|) is monotonically decreasing as a function of S for the ferromagnet, as one might expect. We use the tool of first-order perturbative linear spin wave theory, and show that in that context E(0) < E(1). This is modelled on the Lieb-Schultz-Mattis theorem about gapless spin systems. This tool is applicable for sufficiently large systems, such as long spin rings, in the presence of a spectral gap. For spin j = 1/2, all this was already proved by Sutherland, using the Bethe ansatz. But we present numerical evidence for spin rings for j > 1/2, as well.