Lorentz-Covariant Landau Levels of Tilted Dirac Fermions in Nonuniform Fields

Abstract

We develop an analytical theory of Landau quantization for tilted anisotropic Dirac fermions in an exponentially decaying magnetic field. Using anisotropy scaling and a Lorentz transformation, we recast the laboratory-frame problem into the isotropic Dirac equation in the boosted frame, where the exponentially decaying magnetic-field problem admits an exact solution. Transforming the boosted-frame spectrum back to the laboratory frame yields an implicit quantization condition with an intrinsically energy-dependent guiding-center parameter. We identify the conditions for physically admissible states. We further show that the formalism recovers the known uniform-field spectrum of tilted anisotropic Dirac fermions in the appropriate limit. Our results extend the Lorentz-covariant treatment of tilted anisotropic Dirac fermions to an exponentially decaying magnetic field and reveal the energy-dependent guiding-center structure generated by the inverse transformation to the laboratory frame.