Rogue breather modes: Topological sectors, and the `belt-trick', in a one-dimensional ferromagnetic spin chain

We present explicit solutions for breather soliton modes of excitation in the one-dimensional Heisenberg ferromagnetic spin chain. We identify a characteristic geometrical feature of these breather modes wherein a helicoidal configuration of spins is continuously transformed to one which differs from the initial helicoid by a total twist of `2'. This is a curious manoeuvre popularly known as the `belt trick', an illustration of the simple connectedness of the $\rm{SU}(2)$ group manifold, and its rotation period $4\pi$. We show that this effectively splits the configuration space of the ferromagnetic chain in one-dimension into two topological sectors, distinguished by their total twist -- either `0', or `1'. Further, the energy lower bound of the two sectors is separated by a finite gap varying inversely with the size of the lattice.