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An antiferromagnetic spin wave is a small collective variation of the ordered spin pattern in an antiferromagnet. In a bipartite Néel state, neighbouring spins point oppositely; the slowly varying Néel order parameter describes their staggered orientation. A long-wavelength disturbance changes that orientation smoothly rather than independently flipping each site.

For an isotropic ordered antiferromagnet in the usual long-wavelength regime, the low-energy dispersion relation is linear, $\omega \simeq c|\mathbf{k}|$, with spin-wave velocity $c$. This differs from the quadratic low-wavenumber ferromagnetic magnon in a conventional isotropic ferromagnet. Anisotropy or an applied magnetic field can alter the branches or introduce gaps. The starting ordered state also needs justification: strong quantum fluctuations, especially in low dimension, can invalidate a naive linear spin-wave approximation. A computed divergence is then a warning about the approximation, not a negative physical magnetization.

For $H=J{\sum }_{⟨i,j⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, with $J>0$, dimensionless spin length $S$ and lattice spacing $a$ on a $d$-dimensional hypercubic lattice, the linear spin-wave approximation givesThe angular frequency is ${\omega }_{\mathbf{k}}=E_{\mathbf{k}}/\hslash$. Near a gapless point, $E_{\mathbf{k}}\simeq 2JSa\sqrt{d}|\mathbf{k}|$, so the physical spin-wave velocity is $2JSa\sqrt{d}/\hslash$. Writing the energy formula directly as an angular frequency assumes $\hslash =1$.

The zero-temperature spin-boson occupation is $[1/\sqrt{1-{\gamma }_{\mathbf{k}}^2}-1]/2$, producing quantum depletion of Néel order. Exact zero modes need an infrared cutoff or separate global-rotation treatment. The energy, angular frequency and spin normalization must remain consistent when comparing different conventions.

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.