Geometric Phase and Orbital Moment in Quantization Rules for Magnetic Breakdown

The modern semiclassical theory of a Bloch electron in a magnetic field encompasses the orbital magnetization and geometric phase. Beyond this semiclassical theory lies the quantum description of field-induced tunneling between semiclassical orbits, known as magnetic breakdown. Here, we synthesize the modern semiclassical notions with quantum tunneling-into a single Bohr-Sommerfeld quantization rule that is predictive of magnetic energy levels. This rule is applicable to a host of topological solids with unremovable geometric phase, that also unavoidably undergo breakdown. A notion of topological invariants is formulated that nonperturbatively encode tunneling, and is measurable in the de Haas-van Alphen effect. Case studies are discussed for topological metals near a metal-insulator transition and overtilted Weyl fermions.