= Ferromagnetic spin chain
{wiki}
A <ferromagnetic spin chain> is a one-dimensional array whose exchange interactions favor aligned neighbouring <spins>. For $J>0$, the nearest-neighbour <Heisenberg model> convention
$$
H=-J\sum_i\mathbf S_i\cdot\mathbf S_{i+1}
$$
has 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 $E(k)\simeq JSa^2k^2$ 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.
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