Disordered Landau levels of single-cone massless Dirac fermions with broken particle-hole symmetry

Recently, a single two-dimensional gapless Dirac cone has been experimentally observed on the surface of a semimagnetic topological insulator, providing a platform to study the physics of a single Dirac fermion. Here, we focus on Landau levels (LLs) in the presence of Anderson disorder. We find that the particle-hole (PH) asymmetric term is essential for determining the evolution of LLs with magnetic flux and disorder. With particle-hole symmetry (PHS), the zeroth LL ${E}_{0}$ is pinned at the charge-neutrality point and independent of magnetic field and weak disorder. However, with PHS broken, ${E}_{0}$ either ascends or descends depending on the sign of the PH asymmetric term. Furthermore, neighboring LLs ${E}_{n\ensuremath{\ne}0}$ adhere to the antilevitation scenario for a vanishing or small PH asymmetric term. In contrast, for a sufficiently large PH asymmetric term, LLs above and below ${E}_{0}$ exhibit different behaviors. We also propose a theoretical method to understand these evolutionary behaviors of disordered LLs based on self-consistent Born approximation. Finally, we find that a rare-region-like effect occurs for a small disorder concentration, leading to rich scenarios. In particular, the zeroth LL ${E}_{0}$ becomes dependent on magnetic flux, and its dependence on disorder strength becomes nonmonotonic. Our findings offer valuable insights for understanding disordered LLs of Dirac fermions and provide guidance for experiments involving topological insulators in strong magnetic fields.