Vortex Fractional Fermion Number through Heat Kernel methods and Edge States

TL;DR

This paper applies heat kernel methods to compute the fermion number in vortex backgrounds, demonstrating boundary condition independence and revealing fractional charges carried by edge states in a Higgs phase.

Contribution

It tests a new heat kernel-based method for calculating fermion number on solitons, confirming its validity and uncovering fractional edge charges in vortex systems.

Findings

Fermion number invariant under certain boundary conditions.

Edge states in vortex backgrounds carry fractional charge.

Spectrum exhibits nontrivial features in the Higgs phase.

Abstract

Computing the vacuum expectation of fermion number operator on a soliton background is often challenging. A recent proposal in arXiv:2305.13606 simplifies this task by considering the soliton in a bounded region and relating the $\eta$ invariant, and thus the fermion number, to a specific heat kernel coefficient and to contributions from the edge states. We test this method in a system of charged fermions living on an Abrikosov-Nielsen-Olesen (ANO) vortex background. We show that the resulting $\eta$ invariant does not depend on boundary conditions (within a certain class), thereby supporting the validity of the method. Our analysis reveals a nontrivial feature for the fermionic spectrum in the vortex-induced Higgs phase. As a by-product, we also find that for a vortex living on a disk, the edge states carry fractional charge.