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

TL;DR

This paper provides explicit breather soliton solutions in a 1D ferromagnetic chain, revealing topological sectors distinguished by total twist and illustrating the belt trick's role in the chain's configuration space.

Contribution

It introduces explicit breather solutions and identifies topological sectors in the 1D Heisenberg ferromagnetic chain, highlighting the belt trick's geometric significance.

Findings

Identification of two topological sectors distinguished by total twist

Explicit breather soliton solutions in the chain

Finite energy gap between sectors inversely proportional to chain size

Abstract

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.