Lehrende/r: Univ.-Prof. Dr. Yuriy Mokrousov
Veranstaltungsart: online: Vorlesung
Anzeige im Stundenplan: 08.128.7003
Semesterwochenstunden: 2
Unterrichtssprache: Englisch
Min. | Max. Teilnehmerzahl: - | -
Voraussetzungen / Organisatorisches: Registration via Jogustine "Registration -> Auditor registration" / Anmeldung über Jogustine "Anmeldung -> Höreranmeldung" 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.
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
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)
