Vortex fractional fermion number through heat kernel methods and edge states

Computing the vacuum expectation of a fermion number operator on a soliton background is often challenging. A recent proposal in [1] simplifies this task by considering the soliton in a bounded region and relating the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>η</a:mi></a:math> 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 vortex background. We show that the resulting <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>η</c:mi></c:math> 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 a fractional charge.