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

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

13. Electron dynamics in Dirac bands

14. Wave-packet in upper bands

15. Wave packet size
Minimum size:

16. Mechanical observables

17. Zeeman energy
Magnetic moment from self-rotation

18. Spin is a spin after all !

19. Wave packet dynamics

20. Pauli equation
Effective quantum mechanic for non-relativistic electrons

21. Inconsistency between Pauli and Dirac

22. What is wrong with Pauli ?

23. 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?

24. 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

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