Ferromagnetic magnon

A ferromagnetic magnon is a quantum of a collective spin wave about ferromagnetic order. In a uniform isotropic exchange ferromagnet, slowly varying misalignment costs less energy than rapid variation. Its long-wavelength energy is typically quadratic, $E(\mathbf{k})\simeq D{k}^2$, and its angular frequency is $E/\hslash$.

A one-magnon state in a fully polarized magnet reduces total spin projection by one quantum while spreading that change coherently across the sample. It is therefore different from assigning each site an independent classical tilt. The ferromagnetic spin chain gives an explicit cosine dispersion relation whose small-wavenumber limit is quadratic. An applied magnetic field or symmetry-breaking anisotropy can introduce a nonzero excitation gap, while interactions between magnons matter when their population is no longer dilute. The mode picture is most useful when the ordered reference and low-excitation approximation are justified.

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 $k^2$.

For a periodic spin-one-half Heisenberg model $H=-(J/{\hslash }^2){\sum }_{n=1}^N{\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$, $J>0$, the fully aligned state has $E_0=-NJ/4$. In its single-excitation subspace, let $|n⟩$ denote a down spin at site $n$ in an otherwise up background. Direct action of each bond givesThe normalized translation eigenstates are $|q⟩=N^{-1/2}{\sum }_n{e}^{iqna}|n⟩$, with $q=2\pi \ell /(Na)$ modulo $2\pi /a$. Their energy eigenvalues are $E_0+J[1-\cos (qa)]$. This is an exact one-magnon calculation, not a many-magnon harmonic approximation. Near $q=0$ the energy above $E_0$ is $J{a}^2{q}^2/2$.

The wavenumber restriction comes from requiring a periodic plane-wave coefficient pattern, equivalently requiring this vector to be an eigenvector of a one-site translation with eigenvalue $e^{-iqa}$. The identity $T^N=I$ alone imposes no restriction on a general state: it already holds throughout the ring Hilbert space. An arbitrary finite sum with coefficients $e^{iqna}$ still returns after $N$ translations, but need not be a one-site translation eigenstate unless $e^{iqNa}=1$.

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 Quantum 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.