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

• URL: http://arxiv.org/abs/2211.06765v1
• Date: Sat, 12 Nov 2022 23:29:23 GMT
• Title: Spectral properties of the 2D magnetic Weyl-Dirac operator with a short-range potential
• Authors: M.B. Alves, O.M. Del Cima, D.H.T. Franco, E.A. Pereira
• Abstract summary: We study the spectral properties of the Weyl-Dirac or massless Dirac operators in dimension 2 in a homogeneous magnetic field. As a by-product of this, we have the stability of bipolarons in graphene in the presence of magnetic fields.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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