Winding Hopfions on R2×S1R2×S1

We study Hopfions in the Faddeev–Skyrme model with potential terms on R2×S1R2×S1. 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)U(1) modulus twisted P   times along S1S1, 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+1d=2+1.