Easy-plane XXZ antiferromagnetic spin-wave spectrum (${ϵ}_k=2JS\sqrt{(1-\cos k)(1+\Delta \cos k)}$)

For $0<\Delta <1$, choose a planar Néel reference and rotate local axes to align its longitudinal spins. One phase convention gives the displayed full-zone spectrum, linear near $k=0$ with slope $JS\sqrt{2(1+\Delta )}$, and gapped at $k=\pi$ by $2JS\sqrt{2(1-\Delta )}$. Shifting a sublattice phase can interchange the momentum locations. Folding the magnetic zone separates the gapless in-plane mode from the gapped out-of-plane polarization. An easy-axis reference in this regime is unstable and does not supply this physical expansion.