Nearest-neighbour ferromagnetic magnon dispersion
= Nearest-neighbour ferromagnetic magnon dispersion
{title2=$E_k=4JS\sin^2(k/2)$}
For $H=-J\sum_n\mathbf S_n\cdot\mathbf S_{n+1}$ and unit spacing, the one-magnon energy is $2JS(1-\cos k)=4JS\sin^2(k/2)$. The exact one-magnon hopping matrix has diagonal $2JS$ and neighbour entries $-JS$. The same result follows from the <linear spin-wave approximation>. Its long-wavelength stiffness is $JS$, with energy proportional to $k^2$.