= Solution
In this part $0<\Delta=1+\alpha<1$, so the ordered reference lies in the $xy$ plane. Choosing one direction breaks the continuous rotation <symmetry> within that plane. \b[A gapless, linearly dispersing phase mode is therefore expected] in the spin-wave approximation, while tilting the Néel axis out of the plane costs anisotropy energy. It is incorrect to continue the easy-axis expansion about an unstable $z$-ordered reference and interpret its imaginary frequencies as physical modes.
For the <easy-plane XXZ antiferromagnetic spin-wave spectrum>, take the Néel direction along old $x$, then rotate alternating local <spins> so that all longitudinal components point up. One such local frame gives the full-zone quadratic coefficients
$$
A_k=1+\frac{\Delta-1}{2}\cos k,\qquad B_k=\frac{\Delta+1}{2}\cos k,
$$
and therefore
$$
\boxed{\epsilon_k=2JS\sqrt{(1-\cos k)(1+\Delta\cos k)}.}
$$
Near $k=0$, $\epsilon_k\simeq JS\sqrt{2(1+\Delta)}|k|$. The zone-edge mode has gap $\epsilon_\pi=2JS\sqrt{2(1-\Delta)}$. A different sublattice phase convention shifts the zero by $\pi$; folding into the magnetic zone displays an acoustic and a gapped polarization. The figure uses this explicitly chosen frame. One-dimensional long-wavelength fluctuations again limit the ordered-reference interpretation.
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