Surface Landau level signatures of a topological transition in antiferromagnetic topological insulators

Layered van der Waals (vdW) antiferromagnetic (AFM) topological insulators (TIs), particularly the recently discovered $\mathrm{MnBi}{}_{2}\mathrm{Te}{}_{4}$-family materials, have attracted significant attention. The gapped surface states in $\mathrm{MnBi}{}_{2}\mathrm{Te}{}_{4}$ are one of its key properties. However, whether the surface states in $\mathrm{MnBi}{}_{2}\mathrm{Te}{}_{4}$ are gapped or gapless remains a persistent and unresolved issue. A recently proposed intrinsic explanation is that a slight expansion of the topmost vdW gap can transform initially gapped surface states into gapless ones, potentially resolving this controversy. Despite this insight, how to experimentally comfirm the signal of the topmost vdW gap expansion in AFM TI is a problem worthy of investigation. In this work, we employ a multi-Dirac model to investigate the evolution of the band structure and Landau levels (LLs) in an AFM TI as the topmost vdW gap expands, revealing a range of intriguing phenomena. Our findings provide interesting perspectives on the tunability of LLs in layered vdW AFM TIs.