One-dimensional Coulomb problem in Dirac materials

Abstract

We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated Coulomb problems, with the wavefunctions expressed in terms of special functions (namely Whittaker functions), whilst the energy spectrum must be determined via solutions to transcendental equations. Most notably, there are critical bandgaps below which certain low-lying quantum states are missing in a manifestation of atomic collapse.

7 pages, 5 figures

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• Magnetic quantum dots and rings in two dimensions
• Localization of massless Dirac particles via spatial modulations of the Fermi velocity
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• Construction of zero energy states in graphene through the supersymmetry formalism
• General Green's function formalism for layered systems: Wave function approach
• Pair states in one-dimensional Dirac systems
• Zero-energy vortices in Dirac materials
• Supercriticality of novel type induced by electric dipole in gapped graphene
• Guided modes and terahertz transitions for two-dimensional Dirac fermions in a smooth double-well potential
• Confinement and edge effects on atomic collapse in graphene nanoribbons
• mKdV equation approach to zero energy states of graphene
• Mapping borophene onto graphene: Quasi-exact solutions for guiding potentials in tilted Dirac cones
• Trapping charge carriers in low-dimensional Dirac materials
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• Bound states in the continuum (BIC) protected by self-sustained potential barriers in a flat band system
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• Critical Charge in Gapped Graphene: the role of the screening of the interaction potential by $\sigma$-orbitals
• Excitonic physics in a Dirac quantum dot
• Electron-electron interactions in partially mixed helical states
• Lorentzian quantum wells in graphene: the role of shape invariance in zero-energy states trapping
• Consequences of the Inherent Density Dependence in Dirac Materials