Choose the orientation of spherical coordinates as
$$
\widetilde{\mathbf n}
=(\sin\theta\cos\phi,-\sin\theta\sin\phi,\cos\theta),
$$
which differs from the opposite azimuth convention only by $\phi\mapsto-\phi$. With $\mathbf B=B\widehat{\mathbf x}$, $\theta=\pi/2-\delta\theta$, and $\phi=\delta\phi$,
$$
\widetilde{\mathbf n}
=(1,-\delta\phi,\delta\theta)+O(\delta^2).
$$
Substitution into the effective action gives
$$
\boxed{
\mathcal L_{eff}
=-\alpha^2\left[(\nabla\delta\theta)^2
+(\nabla\delta\phi)^2\right]
+\beta^2\left[(\partial_t\delta\theta+B\delta\phi)^2
+(\partial_t\delta\phi-B\delta\theta)^2\right],}
$$
where
$$
\boxed{\alpha^2=\frac{JS^2}{2},
\qquad
\beta^2=\frac1{16Ja^2}.}
$$

Let $\psi=\delta\theta+i\delta\phi$ and define the zero-field <spin-wave velocity>
$$
c=\frac\alpha\beta=2\sqrt2\,JSa.
$$
The linearized equation is
$$
(\partial_t-iB)^2\psi-c^2\nabla^2\psi=0.
$$
For a plane wave, the two circular polarizations therefore obey
$$
\boxed{(\omega\pm B)^2=c^2k^2,}
$$
or, 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 splitting>[Zeeman-splits] the two opposite circular polarizations by shifting their frequencies in opposite directions.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 2 / g / Solution

{title2=$c$}

The spin-wave velocity is the slope $c=\lim_{k\to0}\omega(k)/k$ of a linearly dispersing spin wave. For the leading square-lattice Heisenberg antiferromagnet continuum theory, $c=2\sqrt2JSa$.


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