On the Dirac bag model in strong magnetic fields

TL;DR

This paper analyzes the spectral behavior of two-dimensional Dirac operators with magnetic fields and MIT bag boundary conditions, revealing asymptotic energy behavior and spectral gaps in strong magnetic fields.

Contribution

It provides new asymptotic results for Dirac operators under strong magnetic fields and boundary conditions, including spectral gap characterization and universal constants.

Findings

Asymptotic behavior of low-lying energies in strong magnetic fields.

Existence of a spectral gap of size proportional to √B on the half-plane.

Universal constant a₀ characterizes energies in bounded domains.

Abstract

In this work we study Dirac operators on two-dimensional domains coupled to a magnetic field perpendicular to the plane. We focus on the infinite-mass boundary condition (also called MIT bag condition). In the case of bounded domains, we establish the asymptotic behavior of the low-lying (positive and negative) energies in the limit of strong magnetic field. Moreover, for a constant magnetic field $B$, we study the problem on the half-plane and find that the Dirac operator has continuous spectrum except for a gap of size $a\_0\sqrt{B}$, where $a\_0\in (0,\sqrt{2})$ is a universal constant. Remarkably, this constant characterizes certain energies of the system in a bounded domain as well. We discuss how these findings, together with our previous work, give a fairly complete description of the eigenvalue asymptotics of magnetic two-dimensional Dirac operators under general boundary conditions.