Berry phase effects on Electrons Qian Niu University

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 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 nonrelativistic 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+e. A) • 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