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

  1. II.Topological defects in broken-symmetry phases via homotopy theory

  2. III. Integer quantum Hall effect and Chern number

  3. IV. Polarization in solids as an example of a Chern-Simons form

  4. V.Berry-phase contribution to anomalous Hall effect

  5. VI.Topological insulators and quantum spin Hall effect

  6. VII.Fractional quantum Hall effect

  7. VIII. Chern-Simons theory and topological degeneracy

  8. IX.Braiding and two-dimensional statistics

  9. 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.