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

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

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

Well defined for a closed path Stokes theorem Berry Curvature

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

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

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

Electron dynamics in Dirac bands

Wave-packet in upper bands

Wave packet size Minimum size:

Mechanical observables

Zeeman energy Magnetic moment from self-rotation

Spin is a spin after all !

Wave packet dynamics

Pauli equation Effective quantum mechanic for non-relativistic electrons

Inconsistency between Pauli and Dirac

What is wrong with Pauli ?

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?

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

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