Spectrum of the Dirac Hamiltonian with the mass-hedgehog in arbitrary dimension

TL;DR

This paper derives the complete spectrum of the Dirac Hamiltonian with a mass-hedgehog potential in any dimension, revealing its relation to bosonic and fermionic number operators and uncovering underlying symmetries.

Contribution

It provides a general method to determine the spectrum and degeneracies of the Dirac Hamiltonian with hedgehog mass configurations across arbitrary dimensions.

Findings

Spectrum expressed via fictitious bosons and fermions

Degeneracies and symmetries identified in various dimensions

Connection to physical systems like vortices in graphene and topological insulators

Abstract

It is shown that the square of the Dirac Hamiltonian with the isotropic mass-hedgehog potential in d dimensions is the number operator of fictitious bosons and fermions over d quantum states. This result allows one to obtain the complete spectrum and degeneracies of the Dirac Hamiltonian with the hedgehog mass configuration in any dimension. The result pertains to low-energy states in the core of a general superconducting or insulating vortex in graphene in two dimensions, and in the superconducting vortex at the topological - trivial insulator interface in three dimensions, for example. The spectrum in d=2 is also understood in terms of the underlying accidental SU(2) symmetry and the supersymmetry of the Hamiltonian.