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For a nearest-neighbour Heisenberg antiferromagnet on a hypercubic lattice, the linear spin-wave approximation gives ${\gamma }_{\mathbf{k}}=d^{-1}{\sum }_j\cos (k_{j}a)$ and the displayed frequency. Near a gapless point its spin-wave velocity is $2JSa\sqrt{d}$. Its ground-state boson occupation is $[1/\sqrt{1-{\gamma }_{\mathbf{k}}^2}-1]/2$, producing quantum depletion of Néel order. Exact zero modes need an infrared regulator or separate global-rotation treatment.

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 $a$.

Choose the orientation of spherical coordinates aswhich differs from the opposite azimuth convention only by $\varphi \mapsto -\varphi$. With $\mathbf{B}=B\hat{\mathbf{x}}$, $\theta =\pi /2-\delta \theta$, and $\varphi =\delta \varphi$,Substitution into the effective action giveswhere

Let $\psi =\delta \theta +i\delta \varphi$ and define the zero-field spin-wave velocityThe linearized equation isFor a plane wave, the two circular polarizations therefore obeyor, with signed-frequency branches, $\omega =ck\pm B$ and their negative-frequency partners. At $B=0$ these are the two degenerate, linearly dispersing antiferromagnetic spin waves. The field Zeeman-splits the two opposite circular polarizations by shifting their frequencies in opposite directions.