Semiclassical theory of Landau levels and magnetic breakdown in topological metals

The Bohr-Sommerfeld quantization rule lies at the heart of the semiclassical theory of a Bloch electron in a magnetic field. This rule is predictive of Landau levels and quantum oscillations for conventional metals as well as for a host of topological metals, which have emerged in the recent synergy between band theory, crystalline symmetries, and topology. Here, the authors systematically formulate quantization rules that apply to energy-degenerate bands and to orbits that intersect at saddle points and type-II Dirac points where quantum tunneling is unavoidable. The formulation extends previous work by incorporating the orbital magnetic moment and the geometric Berry phase. The latter is an indispensable property of topological metals.