Ferromagnetic spin chain

A ferromagnetic spin chain is a one-dimensional array whose exchange interactions favor aligned neighbouring spins. For $J>0$, the nearest-neighbour Heisenberg model conventionhas fully aligned ground states. A collective small deviation from alignment produces a ferromagnetic magnon rather than a set of unrelated local spin flips.

For spin length $S$, lattice spacing $a$ and this exchange convention, the one-magnon energy is $E(k)=2JS[1-\cos (ka)]$, giving $E(k)\simeq JS{a}^2{k}^2$ at long wavelength. The gapless point reflects the freedom to rotate a uniform aligned state. This ground-state and excitation statement does not establish finite-temperature spontaneous order in an infinite one-dimensional short-range model. Finite length, anisotropy and coupling between chains can change what an experiment observes.

Easy-plane anisotropy makes spin orientations within one plane energetically preferable to orientations normal to it. For example, a classical single-site term $D(S^z{)}^2$ with $D>0$ penalizes out-of-plane components and leaves the $xy$ plane degenerate. At sufficiently low energies, an in-plane angle can then be the main orientational degree of freedom.

The resulting approximate angular description connects a magnet to the XY model, but the microscopic form matters. For a quantum spin one-half, $(S^z{)}^2$ is proportional to the identity, so that single-site term alone does not generate orientational anisotropy. Anisotropic exchange interactions provide another route. Stating the spin size and Quantum Hamiltonian prevents transferring a classical mechanism to a quantum model where it is ineffective. Residual in-plane anisotropy can further select a finite set of directions rather than a continuous circle.

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.