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We developed a new Bohr-Sommerfeld quantization rule that unifies semiclassical and quantum theories for Bloch electrons in magnetic fields. This rule predicts magnetic energy levels in topological solids, incorporating orbital magnetization and geometric phase.

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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.