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

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, $B^{\mathrm{ext}}$, perturbed by a chiral-magnetic field, $b^{\mathrm{ind}}$, 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 $V_s\ne 0$, where $V_s$ is an attractive potential associated with the $s$-wave, which emerges when analyzing the $s$- and $p$-wave Møller 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 $|\mathbf{\sigma }\cdot {\mathbf{p}}_{{\mathbf{A}}_{\pm }}|+V_s$. Here, $\mathbf{\sigma }$ is the vector of Pauli matrices, ${\mathbf{p}}_{{\mathbf{A}}_{\pm }}=\mathbf{p}-{\mathbf{A}}_{\pm }$, with $\mathbf{p}=-i\nabla$ the two-dimensional momentum operator and ${\mathbf{A}}_{\pm }$ certain magnetic vector potentials. As a by-product of this, we have the stability of bipolarons in graphene in the presence of magnetic fields.

23 pages, 2 figures