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Instructors: Univ.-Prof. Dr. Yuriy Mokrousov
Event type:
Lecture
Displayed in timetable as:
08.128.7001
Hours per week:
2
Language of instruction:
Englisch
Min. | Max. participants:
- | -
Requirements / organisational issues:
The physics and required mathematical background of geometrical and topological phases in non-relativistic quantum physics are discussed in depth. The course seeks to maintain a balance between presenting mathematical background uncommon to most physicists with profound physical applications. The course aims to be self-contained and requires basic knowledge of mathematics and band theory of solids.
Mathematical foundations (this semester)
- Topological and differential manifolds
- Tensor fields
- Fiber-bundles and connections
- Homotopy, holonomy and cohomology theory
- Characteristic classes and Chern-Simons forms
Partly based on the expertise in the understanding of mathematical background, in the next semester
the following physical effects will be considered in detail.
Applications in solids (next semester):
- Berry phases in solids: Sundaram-Niu equations, orbital magnetization
- Berry phases in solids: Chern numbers and invariants of band manifolds
- Berry phases in solids: Quantum, spin, quantum spin, anomalous and quantum anomalous Hall effects
- Berry phases in solids: Magnetization dynamics
- Berry phases in solids: Chiral magnetism, skyrmions
- Berry phases in solids: computational aspects in the band theory of solids
Contents:
- Topological and differential manifolds
- Tensor fields
- Fiber-bundles and connections
- Homotopy, holonomy and cohomology theory
- Characteristic classes and Chern-Simons forms
Recommended reading list:
M. Nakahara, "Geometry, Topology and Physics", CRC Press (2003)
A. Bohm et al. "The Geometric Phase in Quantum Systems", Springer (2003)
Xiao, Chang, Niu, "Berry phase effects on electronic properties", Reviews of Modern Physics (2010)
Digital teaching:
The course takes place online, with Q&A sessions on-site, if possible.
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