Spectral properties of the 2D magnetic Weyl-Dirac operator with a short-range potential

TL;DR

This paper investigates the spectral properties of 2D magnetic Weyl-Dirac operators with short-range potentials, establishing the existence of discrete spectrum states and bounds on negative eigenvalues, with implications for stability of bipolarons in graphene.

Contribution

It provides new results on the spectral analysis of Weyl-Dirac operators with magnetic and electric perturbations in 2D, including existence of discrete states and eigenvalue bounds.

Findings

Existence of discrete spectrum states between Landau levels.

Lower bounds for sums of negative eigenvalues with electric potentials.

Stability results for bipolarons in graphene under magnetic fields.

Abstract

This paper is devoted to the study of the spectral properties of the Weyl-Dirac or massless Dirac operators, describing the behavior of quantum quasi-particles in dimension 2 in a homogeneous magnetic field, $B^{\rm ext}$, perturbed by a chiral-magnetic field, $b^{\rm ind}$, with decay at infinity and a short-range scalar electric potential, $V$, of the Bessel-Macdonald type. These operators emerge from the action of a pristine graphene-like QED$_3$ model recently proposed in Eur. Phys. J. B93} (2020) 187. First, we establish the existence of states in the discrete spectrum of the Weyl-Dirac operators between the zeroth and the first (degenerate) Landau level assuming that $V=0$. In sequence, with $V_s \not= 0$, where $V_s$ is an attractive potential associated with the $s$-wave, which emerges when analyzing the $s$- and $p$-wave M{\o}ller scattering potentials among the charge carriers in the pristine graphene-like QED$_3$ model, we provide lower bounds for the sum of the negative eigenvalues of the operators $|\boldsymbol{\sigma} \cdot \boldsymbol{p}_{\boldsymbol{A}_\pm}|+ V_s$. Here, $\boldsymbol{\sigma}$ is the vector of Pauli matrices, $\boldsymbol{p}_{\boldsymbol{A}_\pm}=\boldsymbol{p}-\boldsymbol{A}_\pm$, with $\boldsymbol{p}=-i\boldsymbol{\nabla}$ the two-dimensional momentum operator and $\boldsymbol{A}_\pm$ certain magnetic vector potentials. As a by-product of this, we have the stability of bipolarons in graphene in the presence of magnetic fields.