ABSTRACT
This chapter presents the delicately theoretical frameworks about the diverse quantization phenomena, especially for those due to a uniform perpendicular magnetic field and lattice symmetries in layered condensed-matter systems. The generalized tight-binding model, the dynamic Kubo formula, the static one, and the sublattice- and layer-dependent random-phase approximation are developed/modified to thoroughly explore the electronic properties, optical absorption spectra, quantum transports, and Coulomb excitations under the magnetic quantization, respectively. The analytic derivations and numerical evaluations are available in fully exploring the diverse quantization phenomena. The developed theoretical framework is suitable for the various geometric structures, the uniform/nonuniform external fields, the multi-orbital hybridizations in chemical bonds, and the environment-induced spin-orbital couplings. The electronic energy spectra, van Hove singularities, optical transition spectra, and magnetic subenvelop functions strongly depend on the multi-combined effects arising from the stacking symmetries/twisted angles, gate voltages, and magnetic fields.
