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Exploring Berry Phase Effects on Electrons: A Comprehensive Overview

Dive into the fascinating world of Berry phase effects on electrons, covering topics such as Bloch electron dynamics, anomalous velocity, and the orbital nature of spin. Discover applications in AHE and orbital magnetism, unraveling the complexities of geometric phase and Berry curvature.

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Exploring Berry Phase Effects on Electrons: A Comprehensive Overview

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  1. Berry phase effects on Electrons Qian Niu University of Texas at Austin Supported by DOE-NSET NSF-Focused Research Group NSF-PHY Welch Foundation International Center of Quantum Structures

  2. Outline • Berry phase—an introduction • Bloch electron in weak fields • Anomalous velocity • Correction to phase space measure (DOS) • Apllications: AHE, orbital magnetism, etc. • Dirac electron --- degenerate bands • Orbital nature of spin • Anomalous velocity: spin orbit coupling • Incompleteness of Pauli and Luttinger Hamiltonians • Summary

  3. Berry Phase Parameter dependent system: Adiabatic theorem: Geometric phase:

  4. Well defined for a closed path Stokes theorem Berry Curvature

  5. Analogies Berry curvature Magnetic field Berry connection Vector potential Geometric phase Aharonov-Bohm phase Chern number Dirac monopole

  6. Applications • Berry phase interference, energy levels, polarization in crystals • Berry curvature spin dynamics, electron dynamics in Bloch bands • Chern number quantum Hall effect, quantum charge pump

  7. Other Physical Effects Density of states and specific heat: Magnetoconductivity:

  8. Electron dynamics in Dirac bands

  9. Wave-packet in upper bands

  10. Minimum size: Wave packet size

  11. Mechanical observables

  12. Magnetic moment from self-rotation Zeeman energy

  13. Spin is a spin after all !

  14. Wave packet dynamics

  15. Pauli equation • Effective quantum mechanic for non-relativistic electrons

  16. Inconsistency between Pauli and Dirac

  17. What is wrong with Pauli ?

  18. Caution on effective Hamiltonians • Peierles substitution for non-degerate bands: en(k) en(p+eA) • Luttinger Hamiltonians: • Two-band model for conduction electrons (Rashba) • Four-band model for heavy and light holes • Six-band model: including spin/orbit split off • Eight-band model (Kane): Zincblend semiconductors • Pauli Hamiltonian: for non-relativistic electrons • Dirac Hamiltonian: complete, or is it?

  19. Summary Berry phase A unifying concept with many applications Bloch electron dynamics in weak fields Berry curvature: a ‘magnetic field’ in the k space. Anomalous velocity: AHE A fundamental modification of density of states Dirac electron dynamics in weak fields Orbital nature of spin Anomalous velocity: spin-orbit coupling Incompleteness of effective Hamiltonians

  20. Ming-Che Chang Chih-Piao Chuu Dimitrie Culcer Ganesh Sundaram Jun-Ren Shi Di Xiao Yu-Gui Yao Chuan-Wei Zhang Ping Zhang Acknowledgements

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