Finite-ring spin-wave threshold with antisymmetric exchange ($u_c(L)={\max }_{k\in (2\pi /L)\mathbb{Z}}[\cos k+(D/J)\sin k]-1$)

For a polarized spin chain with dispersion relation $ϵ(k)=2S[J(1+u)-J\cos k-D\sin k]$, 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.