Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

TL;DR

This paper studies the localization properties of massless Dirac fermions in a 2D magnetic quantum dot, showing superexponential and Gaussian-like localization of discrete spectrum states under specific conditions.

Contribution

It demonstrates superexponential and Gaussian-like localization of Dirac fermion states in a magnetic quantum dot with decaying magnetic and electric potentials, under certain symmetry and analyticity assumptions.

Findings

Discrete spectrum states are superexponentially localized.

Existence of states between the zeroth and first Landau levels.

States are Gaussian-like localized under symmetry and analyticity conditions.

Abstract

We consider a two-dimensional massless Dirac operator $H$ in the presence of a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric potential $V$. For $V\in L_{\rm loc}^p(\R^2)$, $p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that states in the discrete spectrum of $H$ are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that V=0. In addition, under the condition that $b$ is rotationally symmetric and that $V$ satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of $H$ are Gaussian-like localized.