Vortex Fractional Fermion Number through Heat Kernel methods and Edge States

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.