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Use bosonic Bogoliubov diagonalization with $A_k=2JS$, $B_k=2JS{\gamma }_k$. Away from the zero modes, letHere $u_k^2-v_k^2=1$ preserves the canonical commutation relations, while the sign of $v_k$ cancels anomalous pairing. In hyperbolic notation $u_k=\cosh {\theta }_k$, $v_k=\sinh {\theta }_k$, this condition is $\tanh (2{\theta }_k)=-B_k/A_k$. Combining the normal ordering constants givesThe dispersion vanishes linearly near $k=0$ and $\pi$, with spin-wave velocity $2JS$ in unit lattice spacing. The exact zero modes make the Bogoliubov coefficients singular; use a small infrared regulator or a symmetry-selected reference and treat the global rotations separately. A finite transformation at ${\omega }_k=0$ is not asserted.

On a hypercubic lattice of coordination $z=2d$, the same hypercubic antiferromagnetic spin-wave dispersion usesThe corresponding constant is $-dNJS(S+1)$, and ${\omega }_{\mathbf{k}}\sim 2JS\sqrt{d}|\mathbf{k}|$ near a Goldstone point. Restoring lattice spacing $a$ multiplies this velocity by $a$.