Toroidal domain walls as Hopfions

We construct stable domain walls in a shape of a torus in the Faddeev-Skyrme model with a quadratic potential term admitting two discrete vacua. The phase modulus of the domain wall is twisted P and Q times along the toroidal and poloidal cycles of the torus, respectively, where Q represents the constituent lump charge. We find that these solutions carry the Hopf charge PQ and can be regarded as Hopfions. The toroidal domain walls are characterized by the two topological charges (P,Q), unlike the conventional Hopfions characterized by only the Hopf charge. We explicitly construct stable solutions of the toroidal domain walls with P, Q = 1,2,3, yielding the Hopf charges 1,2,3,4,6,9.

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