= Finite-ring spin-wave threshold with antisymmetric exchange
{title2=$u_c(L)=\max_{k\in(2\pi/L)\mathbb Z}[\cos k+(D/J)\sin k]-1$}
For a polarized <spin> chain with <dispersion relation> $\epsilon(k)=2S[J(1+u)-J\cos k-D\sin k]$, the first zero mode appears at the maximum over the allowed ring <momenta>. In the <thermodynamic limit> this threshold is $\sqrt{1+(D/J)^2}-1$, since the maximum of $J\cos k+D\sin k$ is $\sqrt{J^2+D^2}$. At finite size the minimizing <wave vector> need not be available, so a positive continuum threshold need not give a positive finite-ring threshold. For $L=4$ and $0<D<J$, the latter is zero.
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