Geometric theory of magnetic breakdown

We complete the theory of magnetic breakdown in solids to include the geometric Berry phase. Our theory describes the spectrum of Bloch electrons where the semiclassical approximation breaks down -- specifically, where bands approach each other near a topological intersection, as well as at saddle-point dispersions where the semiclassical velocity of a Bloch wave packet vanishes. In these situations, the energy levels of Bloch electrons are determined by generalized Bohr-Sommerfeld quantization conditions which incorporate both quantum tunneling and Berry phase. Specific case studies are discussed for inter-band breakdown that arises in tilted Dirac fermions, and for intra-band breakdown on the surface of topological crystalline insulators.