XXX one-magnon dispersion E(k) = 2J(1 - cos k); ratio E(π)/E(π/3) = 4

Exact one-magnon spectrum of the 1D XXX ferromagnet. Acting with H = -J·Σ S_i·S_{i+1} on the single-spin-flip state |k⟩ = (1/√N)·Σ_x exp(ikx)·S^-_x|↑↑…↑⟩ yields H|k⟩ = E(k)|k⟩ with E(k) = 2J(1 - cos k). Dispersion: gapless (E(0) = 0,…