Winding Hopfions on R^2 x S^1

Understand

We study Hopfions in the Faddeev-Skyrme model with potential terms on R^2 x S^1.

• Apart from the conventional Hopfions, there exist winding Hopfions, that is, the lump (baby Skyrmion) strings with the lump charge Q with the U(1) modulus twisted P times along S^1, having the Hopf charge PQ.
• We consider two kinds of potential terms, that is, the potential linear in the field and the ferromagnetic potential with two easy axes, and present stable solutions numerically.
• We also point out that a Q-lump carries the unit Hopf charge per the period in d=2+1.