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A connected isotropic spin-$S$ ferromagnet with positive exchanges has a fully polarized ground state and a maximal-total-spin multiplet. For $N$ sites the total spin is $NS$, with $2NS+1$ magnetic states generated by total-spin lowering. Global rotations span the same degenerate space. Slowly varying reorientations produce gapless ferromagnetic magnons; finite-volume degeneracy must be distinguished from a thermodynamic statement about spontaneous order.

Use dimensionless spin angular momentum operators, so their commutators have no additional $\hslash$. A fully polarized state $|F⟩={⨂}_n|S,S{⟩}_n$ is an exact ground state of the Heisenberg ferromagnet, with $E_0=-NJ{S}^2$. On each bond the largest eigenvalue of ${\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$ is $S^2$, obtained in the two-spin total-spin-$2S$ sector. Hence the bondwise lower bound on the Hamiltonian is attained by this state.

The ferromagnetic ground-state multiplet has total spin $NS$. Applying the global lowering operator $(S_{\mathrm{tot}}^{-}{)}^r$, $r=0,\dots ,2NS$, generates its $2NS+1$ magnetic states; their linear combinations are also ground states. Global spin rotations generate polarized ground states pointing in arbitrary directions and span the same multiplet. Uniform reorientation therefore costs no energy. Slowly varying orientations give gapless ferromagnetic magnons, with quadratic rather than linear long-wavelength dispersion. In the ordered thermodynamic ground state, the two broken spin generators are canonically paired because $⟨[S_{\mathrm{tot}}^x,S_{\mathrm{tot}}^y]⟩=i⟨S_{\mathrm{tot}}^z⟩\ne 0$, producing one type-B Goldstone boson. For a finite chain the zero-wavevector motion remains inside the exact ground multiplet; nonzero wavevectors have a finite-size spacing that vanishes as $N^{-2}$. This is a ground-state statement, not a claim of finite-temperature long-range order in one dimension.

Put $q=\delta \theta$, $p=\delta \varphi$, so $\theta =\pi /2-q$ and $\varphi =p$. The quadratic Lagrangian is obtained by takingThe two real fluctuations form a coordinate and its canonical momentum rather than two independent modes. Their Euler-Lagrange equations combine to giveand similarly for $p$. The ferromagnetic magnon therefore has the quadratic dispersion relation

Write $\theta =\pi /2-\vartheta$ and $\varphi =\phi$. To quadratic order,After dropping the total derivative $S{\phi }_t$, the quadratic Lagrangian density isIts Euler--Lagrange equations areFor modes proportional to $e^{i(kx-\omega t)}$, nontrivial amplitudes requireThusAt short wavelength the anisotropy is negligible and the quadratic dispersion is that of a conventional ferromagnetic magnon. At long wavelength, easy-plane anisotropy makes the out-of-plane fluctuation the conjugate density of the in-plane phase; eliminating it produces the linear Goldstone sound mode characteristic of a superfluid.