Landau Levels and Electronic States for Pseudospin-1 Lattices with a Bandgap: Application to a Lieb Lattice

We have carried out detailed theoretical and numerical calculations and developed a physics-based model for quantitatively describing the Landau levels of several pseudospin-1 structures with a flat band and a finite bandgap in their electronic-energy spectrum under a strong and uniform magnetic field. We have investigated the Landau-level-based dynamics, as well as the corresponding eigenstates, for gapped graphene, a dice lattice with both a zero and finite bandgap and, eventually, for the Lieb lattice, which represents a separate type of square lattice with a very special non-symmetric (elevated) location of the flat band which intersects the conduction band at its lowest point. Exact analytical consideration of Landau-level states has been performed and explained when dealing with all types of considered lattices. Our model could be further generalized for treating cases with an arbitrary position for the flat band between the valence and conduction bands. Our current results have direct implications for a deep-level investigation of the quantum Hall effect, as well as other magnetic and topological properties of these novel materials.