Zero modes on cosmic strings in an external magnetic field

TL;DR

This paper investigates how an external magnetic field affects charge carriers on cosmic strings, revealing that zero modes are shifted into the bulk but remain below the expulsion energy threshold, with a modified dispersion relation.

Contribution

The study provides a quantum analysis of zero modes on cosmic strings in magnetic fields, showing a significant change in dispersion relations and charge carrier behavior.

Findings

Zero modes are shifted into the bulk in magnetic fields.

Dispersion relation changes from linear to a hyperbolic tangent form.

Charge carriers remain below the expulsion energy threshold.

Abstract

A classical analysis suggests that an external magnetic field can cause trajectories of charge carriers on a superconducting domain wall or cosmic string to bend, thus expelling charge carriers with energy above the mass threshold into the bulk. We study this process by solving the Dirac equation for a fermion of mass $m_f$ and charge $e$, in the background of a domain wall and a magnetic field of strength $B$. We find that the modes of the charge carriers get shifted into the bulk, in agreement with classical expectations. However the dispersion relation for the zero modes changes dramatically -- instead of the usual linear dispersion relation, $\omega_k =k$, the new dispersion relation is well fit by $\omega \approx m_f tanh(k/k_*)$ where $k_*=m_f$ for a thin wall in the weak field limit, and $k_*=eBw$ for a thick wall of width $w$. This result shows that the energy of the charge carriers on the domain wall remains below the threshold for expulsion even in the presence of an external magnetic field. If charge carriers are expelled due to an additional perturbation, they are most likely to be ejected at the threshold energy $\sim m_f$.