Andreev-Bashkin effect and knot solitons in interacting mixture of a charged and a neutral superfuids with possible releveance for neutron stars

Abstract

We discuss a mixture of interacting a neutral and a charged Bose condensates, which is supposed being realized in interior of neutron stars in form of coexistent neutron superfluid and protonic superconductor. We show that in this system, besides ordinary vortices of $S^1\to S^1$ map, the neutron condensate also allows for (meta)stable finite-length knotted solitons, which are characterized by a nontrivial Hopf invariant and in some circumsatnces are stablized by Faddeev-Skyrme term induced by drag effect. We also consider a helical protonic fluxtube in this system and show that, in contrast, it does not induce a Faddeev-Skyrme term.