 Classical ground states of an XXZ spin chain

For an even periodic classical chain with spin length $S$ and $J>0$, each bond attains the negative largest singular value of $\mathrm{diag}(1,1,\Delta )$. For $\Delta >1$ the two minimizing configurations are alternating $z$ spins. For $|\Delta |<1$ they are planar Néel states with arbitrary in-plane angle. For $\Delta <-1$ the two configurations have uniform $z$ spins. At $\Delta =1$ the Néel direction is arbitrary on a sphere; at $\Delta =-1$ a staggered transverse component and uniform longitudinal component parameterize another spherical degeneracy. These are classical configuration manifolds, not a claim about finite quantum-chain degeneracy.