Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 75 / 1 / c / ii / Solution

Expanding $r_k$ at large positive $g$ gives $r_k=g-\cos k+{\sin }^{2}k/(2g)+O(g^{-2})$, soThe 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 $E_0/N=-J[g+1/(4g)+O(g^{-3})]$.