Ferromagnetic spin chain

A ferromagnetic spin chain favors aligned neighbouring spins. Its conventional long-wavelength magnons have quadratic dispersion.

Easy-plane anisotropy penalizes spin components normal to a preferred plane, leaving a low-energy angular degree of freedom within that plane.

A ferromagnetic magnon is a quantized spin wave. Without a linear restoring term its long-wavelength frequency is proportional to $k^2$.

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 $k^2$.

Positive nearest- and next-nearest-neighbour exchanges give one-magnon energy $2S[J_1(1-\cos k)+J_2(1-\cos 2k)]$. At small momentum this is $S(J_1+4J_2)k^2$. The quadratic-boson result describes the harmonic many-magnon spectrum; higher-order spin-wave interactions remain in the full Hamiltonian.

A damped ferromagnetic spin wave has a complex pole $\omega =\pm {\omega }_k-i{\Gamma }_k$. Both poles of its retarded Green function must lie in the lower half of the complex frequency plane so that the response respects causality.