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The dispersion relation of magnons in a ferromagnetic 1d lattice is

\begin{equation} \omega(k)=\omega_0\big[1-\cos(ka)\big] \end{equation}

where $\omega_0$ is a constant and $a$ is the spacing between sites in the lattice. This dispersion relation must be considered in the first brillouin zone; that is, up to $k=\frac{\pi}{a}$.

On the other hand, the dispersion relation of magnons in an antiferromagnetic 1d lattice is

\begin{equation} \omega(k)=\omega_0\big[1-\cos^2(ka)\big]^{\frac{1}{2}} \end{equation}

The problem I'm having when I try to understand this dispersion relatinons is this. While the ferromagnetic one is a monotone growing function up to $k=\frac{\pi}{a}$ (which is a nice property of a dispersion relations), the antiferromagnetic one grows only up to $k=\frac{\pi}{2a}$ and then decays again. I don't understand this. Is this saying that antiferromagnetic magnons must be considered only at low energies because its dispersion relation breaks at higher $k$? Am I doing the math wrong?

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    $\begingroup$ Or maybe it has to do with the fact that in an antiferromagnetic latice (if bipartite) the spacing between sites of each sublattice is $2a$ and not $a$... $\endgroup$ – P. C. Spaniel Nov 7 '17 at 3:52
  • $\begingroup$ If I remember correctly, the zero dispersion at $k=\pi/a$ is due to some symmetry. $\endgroup$ – leongz Nov 9 '17 at 19:25

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