Spectral flow of the Dirac spectrum in intersecting vortices

TL;DR

This paper investigates how the spectrum of the Dirac Hamiltonian changes in the presence of crossing vortices on a four-torus, revealing the behavior of zero modes and spectral flow during vortex intersections.

Contribution

It provides a detailed analysis of the spectral flow of the Dirac operator in intersecting vortex backgrounds, including idealized fat and thin vortex cases, using the index theorem.

Findings

Zero modes can be expressed in terms of eigenspinors crossing zero energy.

Spectral flow is steepest at the crossing time for thin vortices.

Zero modes relate to the crossing points of vortices.

Abstract

The spectrum of the Dirac Hamiltonian in the background of crossing vortices is studied. To exploit the index theorem, and in analogy to the lattice the space-time manifold is chosen to be the four-torus $\T^4$. For sake of simplicity we consider two idealized cases: infinitely fat and thin transversally intersecting vortices. The time-dependent spectrum of the Dirac Hamiltonian is calculated and in particular the influence of the vortex crossing on the quark spectrum is investigated. For the infinitely fat intersecting vortices it is found that zero modes of the four-dimensional Dirac operator can be expressed in terms of those eigenspinors of the Euclidean time-dependent Dirac Hamiltonian, which cross zero energy. For thin intersecting vortices the time gradient of the spectral flow of the Dirac Hamiltonian is steepest at the time at which the vortices cross each other.