Midgap spectrum of the fermion-vortex system

TL;DR

This paper analyzes the midgap energy spectrum of fermions interacting with vortices in two dimensions, revealing bound states beyond zero modes and providing analytical solutions for different vortex core sizes.

Contribution

It presents the first complete analytical solutions for the bound states in the fermion-vortex system, extending previous zero mode results to include a spectrum of bound states.

Findings

Existence of infinite bound states in the fermion-vortex system.

Analytical solutions in terms of Laguerre polynomials for different vortex core sizes.

Spectrum squared corresponds to Coulomb potential and harmonic oscillator.

Abstract

I study the midgap spectrum of the fermion-vortex system in two spatial dimensions. The existence of bound states, in addition to the zero modes found by Jackiw and Rossi, is established. For a singly quantized vortex, I present complete analytical solutions in terms of generalized Laguerre polynomials in the opposite limits of vanishing and large vortex core size. There is an infinite number of such bound states, with a spectrum that is, when squared, given by, respectively, the Coulomb potential and the isotropic harmonic oscillator. Possible experimental signatures of this spectrum in condensed-matter realizations of the system are pointed out.