Spectrum of bound fermion states on vortices in $^3$He-B

TL;DR

This paper analyzes the subgap fermion spectra within vortex cores in superfluid $^3$He-B, revealing how different vortex types influence the presence of zero modes and spectral branches crossing the Fermi level.

Contribution

It provides an analytical framework for understanding fermion spectra in various vortex types in $^3$He-B, highlighting the relationship between vortex core size and zero modes.

Findings

Singly quantized vortices have two anomalous branches crossing the Fermi level.

Singular vortices exhibit spectra similar to Caroli-de Gennes-Matricon states.

Nonsingular vortices have zero modes proportional to vortex core size.

Abstract

We study subgap spectra of fermions localized within vortex cores in $^3$He-B. We develop an analytical treatment of the low-energy states and consider the characteristic properties of fermion spectra for different types of vortices. Due to the removed spin degeneracy the spectra of all singly quantized vortices consist of two different anomalous branches crossing the Fermi level. For singular $o$ and $u$ vortices the anomalous branches are similar to the standard Caroli-de Gennes -Matricon ones and intersect the Fermi level at zero angular momentum yet with different slopes corresponding to different spin states. On the contrary the spectral branches of nonsingular vortices intersect the Fermi level at finite angular momenta which leads to the appearance of a large number of zero modes, i.e. energy states at the Fermi level. Considering the $v$, $w$ and $uvw$ vortices with superfluid cores we show that the number of zero modes is proportional to the size of the vortex core.