The Aharonov-Casher Theorem and the Axial Anomaly in the Aharonov-Bohm Potential

TL;DR

This paper analyzes the spectral properties of the Dirac Hamiltonian in the Aharonov-Bohm potential, revealing how bound states, phase shifts, and anomalies are affected by singular magnetic fields, with implications for physical quantities like energy and fermion number.

Contribution

It provides a corrected form of the Aharonov-Casher theorem for singular fields and explores the spectral and topological effects in the Dirac Hamiltonian with Aharonov-Bohm potential.

Findings

Bound states are always accompanied by (anti)resonances.

No zero (threshold) modes exist in the Aharonov-Bohm potential.

Phase-shift flip occurs at positive energies, affecting the density of states.

Abstract

The spectral properties of the Dirac Hamiltonian in the the Aharonov-Bohm potential are discussed. By using the Krein-Friedel formula, the density of states (DOS) for different self-adjoint extensions is calculated. As in the nonrelativistic case, whenever a bound state is present in the spectrum it is always accompanied by a (anti)resonance at the energy. The Aharonov-Casher theorem must be corrected for singular field configurations. There are no zero (threshold) modes in the Aharonov-Bohm potential. For our choice of the 2d Dirac Hamiltonian, the phase-shift flip is shown to occur at only positive energies. This flip gives rise to a surplus of the DOS at the lower threshold coming entirely from the continuous part of the spectrum. The results are applied to several physical quantities: the total energy, induced fermion-number, and the axial anomaly.