Spectrum of the Vortex Bound States of the Dirac and Schrodinger Hamiltonian in the presence of Superconducting Gaps

TL;DR

This paper analyzes vortex bound states in superconductors using Schrödinger and Dirac Hamiltonians, revealing differences in their energy responses to magnetic fields that could indicate the presence of Majorana fermions.

Contribution

It provides a numerical and quasi-classical analysis of vortex bound states for both Hamiltonians, highlighting qualitative differences relevant to Majorana fermion detection.

Findings

Bound state energies proportional to vortex angular momentum at high chemical potential

Dirac Hamiltonian bound states are insensitive to magnetic fields

Schrödinger Hamiltonian bound states shift proportionally with magnetic field

Abstract

We investigate the vortex bound states both Schrodinger and Dirac Hamiltonian with the s-wave superconducting pairing gap by solving the mean-field Bogoliubov-de-Gennes equations. The exact vortex bound states spectrum is numerically determined by the integration method, and also accompanied by the quasi-classical analysis. It is found that the bound state energies is proportional to the vortex angular momentum when the chemical potential is large enough. By applying the external magnetic field, the vortex bound state energies of the Dirac Hamiltonian are almost unchanged; whereas the energy shift of the Schrodinger Hamiltonian is proportional to the magnetic field. These qualitative differences may serve as an indirect evidence of the existence of Majorana fermions in which the zero mode exists in the case of the Dirac Hamiltonian only.