Modern theory of magnetic breakdown

TL;DR

This paper develops a unified quantum and semiclassical framework for magnetic breakdown in topological materials, enabling prediction of magnetic energy levels and measurement of topological invariants through quantum tunneling effects.

Contribution

It introduces a combined Bohr-Sommerfeld quantization rule that incorporates magnetic breakdown and topological invariants, applicable to topological solids with geometric phases.

Findings

Predicts magnetic energy levels in topological materials.

Formulates measurable topological invariants via de-Haas-van-Alphen effect.

Analyzes case studies of topological metals and Weyl fermions.

Abstract

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 \emph{unremovable} geometric phase, that also \emph{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 over-tilted Weyl fermions.