= Solution
Rotate alternate sites by $\pi$ about a transverse axis so all local reference spins point up. Choosing the transverse axes to give positive anomalous coupling, the <linear spin-wave approximation> yields
$$
H=-JNS^2+JS\sum_m[2a_m^\dagger a_m+a_ma_{m+1}+a_m^\dagger a_{m+1}^\dagger]+O(S^0).
$$
Use lattice spacing one and $a_m=N^{-1/2}\sum_ke^{ikm}a_k$, with \b[all $N$ momenta] $k=2\pi j/N$, $j=-N/2,\ldots,N/2-1$, in $[-\pi,\pi)$. Then $\gamma_k=\cos k$ and
$$
H=-JNS^2+JS\sum_k[2a_k^\dagger a_k+\gamma_k(a_ka_{-k}+a_k^\dagger a_{-k}^\dagger)].
$$
Reordering $a_{-k}a_{-k}^\dagger=a_{-k}^\dagger a_{-k}+1$ gives the requested Nambu form with constant $-JNS(S+1)$. Its extra $JSN$ restores $-JNS^2$ before diagonalization. This full-zone convention has no additional sublattice or half-zone factor.
Back to article page
