Fermion zero modes in symmetric vortices in superfluid He

Fermions localized within vortex cores are considered for the simplest most symmetric vortices in the superfluid phases of ³He. Axisymmetric vortices in the A, A₁ and planar phases contain the fermionic condensate with a completely flat band and zero energy. The o-vortex in the B-phase contains 1D Fermi liquids, formed by fermions occupying the branches which cross the zero level. The number of such Fermi liquids increases with increasing external magnetic field. The fermionic spectrum in this vortex is described by the “orbital” and spin momenta interacting with the effective internal magnetic field produced by the vortex and with the external field. Most of the information is obtained using the vortex symmetry which determines symmetry properties of the fermionic spectrum.