Fermion Zero Modes on Vortices

AbstractThis chapter considers fermion zero modes on vortex lines and cosmic strings. The quantum numbers, which characterize the energy spectrum of these fermionic excitations, are the linear (pz) and angular (Q) momenta along the string. There are two types of fermion zero modes: true and approximate. The approximate fermion zero modes have spectrum which crosses zero as function of Q. Due to discrete nature of angular momentum these modes have a small gap called minigap. This occurs in the core of vortices in conventional s-wave superconductors. The number of the anomalous branches is determined by the winding number of a vortex. This analog of index theorem is obtained using topology in combined (p,r) space. The true fermion zero modes have spectrum which crosses zero as function of pz. These modes take place inside vortices in triplet superconductors and 3He-B. The singly quantized vortex in 3He-A contains branch with exactly zero energy for all pz, while half-quantum vortex contains Majorana fermions. Both true and approximate zero modes are obtained using the semi-classical approach which is valid because the core size is much larger than the inverse wavelength of fermions. Analysis is extended to fermions on asymmetric vortices.