Solution
= Solution
Expanding $r_k$ at large positive $g$ gives $r_k=g-\cos k+\sin^2k/(2g)+O(g^{-2})$, so
$$
\boxed{\epsilon_k=2Jg-2J\cos k+O(J/g).}
$$
The branch is strongly gapped and relatively flat, with width approaching $4J$ against a mean energy of order $2Jg$. These are field-polarized spin-flip <quasiparticles> with a small relative exchange modulation. Averaging the expansion over <momentum> also gives $E_0/N=-J[g+1/(4g)+O(g^{-3})]$.