Physics 250 (public): Geometry and topology in many-particle physics
Physics 250 (public): Geometry and topology in many-particle physics
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Meetings: TuTh 2-3:30 pm, 397 Le Conte (except 1/29, in 395).
Outline:
I. Preliminaries, including some simple mathematical examples
Landau theory of broken-symmetry phases
Integrals over classical or quantum fluctuations
II.Topological defects in broken-symmetry phases via homotopy theory
III. Integer quantum Hall effect and Chern number
IV. Polarization in solids as an example of a Chern-Simons form
V.Berry-phase contribution to anomalous Hall effect
VI.Topological insulators and quantum spin Hall effect
VII.Fractional quantum Hall effect
VIII. Chern-Simons theory and topological degeneracy
IX.Braiding and two-dimensional statistics
X.(as time permits) Geometric and topological properties of critical theories
III-VI involve “Thouless-type” topological phases: the response of a system is determined by a topological invariant.
VII-IX involve “Wen-type” topological phases: the low-energy theory of the phase is a topological field theory.
Mathematical techniques:
Diverse ideas from homotopy and cohomology, fiber bundles, differential geometry, etc., presented at a rather low level of rigor.
Lecture notes:
(At this time, I am planning to provide lecture notes for topics for which I feel no suitable tutorial is available.)
preliminaries.pdf (includes topological defects)
thoulessphases.pdf (static; updated 3/5)
thoulesspart2.pdf (static; updated 3/19)
wenphases.pdf (static; updated 4/21)
Reading assignments and problems:
assignments1.pdf (due 2/05/09) solutions1.pdf
assignments2.pdf (due 2/19/09) solutions2.pdf (refers to this file from Stone)
assignments3.pdf (due 3/12/09) solutions3.pdf
assignments4.pdf (due 4/14/09) solutions4.pdf
assignments5.pdf (due 5/07/09)
Public-domain readings:
Book-in-progress on mathematical methods by Michael Stone, UIUC, including considerably more detail on many of the methods we discuss
Notes on Berry-phase theory of polarization by R. Resta
Notes on Berry-phase theory of anomalous Hall effect by N. P. Ong
Notes on Kitaev models by A. Kitaev and C. Laumann
Reference for mathematical details:
Nakahara, Geometry, topology, and physics, IOP Publishing.
References for condensed matter background:
Chaikin and Lubensky, Principles of condensed matter physics, Cambridge.
Thouless, Topological quantum numbers in nonrelativistic physics, World Sci.