Geometric Phase and Orbital Moment in Quantization Rules for Magnetic Breakdown

Area of Science:

• Condensed matter physics
• Quantum mechanics
• Solid-state physics

Background:

• Modern semiclassical theory describes Bloch electrons using orbital magnetization and geometric phase.
• Quantum tunneling between semiclassical orbits, termed magnetic breakdown, extends beyond this theory.

Purpose of the Study:

• To synthesize modern semiclassical concepts with quantum tunneling into a unified Bohr-Sommerfeld quantization rule.
• To predict magnetic energy levels in topological solids with inherent geometric phase and magnetic breakdown.

Main Methods:

• Developed a novel Bohr-Sommerfeld quantization rule.
• Formulated a concept of topological invariants that nonperturbatively encode tunneling.
• Investigated case studies including topological metals and overtilted Weyl fermions.

Main Results:

• The synthesized rule accurately predicts magnetic energy levels.
• The rule is applicable to topological solids exhibiting unavoidable magnetic breakdown.
• Topological invariants measurable via the de Haas-van Alphen effect were formulated.

Conclusions:

• The unified Bohr-Sommerfeld rule provides a predictive framework for magnetic energy levels in complex topological materials.
• The concept of topological invariants offers a new method for characterizing tunneling phenomena.