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Berry phase in quantum mechanics

Univ.-Prof. Dr. Yuriy Mokrousov

Kurzname: 08.128.611
Kursnummer: 08.128.611

Voraussetzungen / Organisatorisches

The course is dedicated to selected aspects of geometry and topology in solid state 
physics, trying to keep it as concise and as self-contained as possible. The course
can be followed from the beginning to the end with minimal reference to other
sources.

We start by discussing the mathematical foundations of the Berry, or, geometric
phase, formulate the adiabatic approximation for quantum dynamics and discuss
such fundamental concepts as Berry connection, Berry curvature, gauge freedom,
parallel transport and first Chern number. We also discuss the mathematical fibre
bundle structure of the quantum mechanical setup and some other selected issues.
The physical examples we choose to apply the introduced concepts to are the
Aharonov-Bohn effect and spin-1/2 in a magnetic field.

We further discuss the geometrical and topological nature of electric polarization in
insulators, bringing to attention its relation to the Chern number, the emergence of
the Chern insulators in two-dimensional reciprocal space, and the interplay between
various versions of quantized Hall effects. We derive the expression for the velocity
of a state due to time-evolution of the quantum system and express it in geometrical
terms, and use it to arrive at the equations of motion which govern the dynamics of
electrons in a solid in response to electro-magnetic fields and general perturbations.
We discuss the consequences of these equations for such properties as Hall
conductance and orbital magnetization, as well as dynamics in skyrmions.

Empfohlene Literatur

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) 

Inhalt

- Berry phase in quantum mechanics
- Abelian and non-abelian Berry phase
- Introduction to underlying mathematical structure of fibre bundles
- Kato theorem and Aharonov-Anandan viewpoint
- Non-adiabatic Berry phase
- Berry phase in the band theory of solids
- Berry phase and semiclassical dynamics of electrons
- Flavors of quantized Hall effects
- Electric polarization, orbital magnetization and skyrmions

Termine:

Datum (Wochentag)UhrzeitOrt
23.04.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
30.04.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
07.05.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
14.05.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
28.05.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
04.06.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
18.06.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
25.06.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
02.07.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik
09.07.2020 (Donnerstag)14.15 - 17.00 Uhr01 231 Seminarraum E
2413 - Neubau Physik/Mathematik

Semester: WiSe 2020/21