Informal course on
Topological terms in condensed matter physics
Spring 2007, EPFL, Lausanne, Switzerland
Lecture 1: May 21, Monday, room PH31 (EPFL) 11h15  13h00.
Lecture 2: May 30, Wednesday, room PH31 (EPFL) 11h15  13h00.
Lecture 3: June 4, Monday, room PH33 (EPFL) 15h1517h00
Lecture 4: June 11, Monday, room PH33 (EPFL) 15h1517h00
Instructor: Dr. Alexander (Sasha) Abanov,
Associate Professor
Office: PH H2 487
Phone: 35856
Email:
alexandre.abanov@sunysb.edu
Web page:
http://felix.physics.sunysb.edu/~abanov/
Abstract
The methods of quantum field theory are widely used in condensed matter physics. In particular, the concept of an effective action was proven useful when studying low temperature and long distance behavior of condensed matter systems. Often the degrees of freedom which appear due to spontaneous symmetry breaking or an emergent gauge symmetry, have nontrivial topology. In those cases the terms in the effective action describing low energy degrees of freedom can be metric independent (topological). I will try to give a simple classification of possible topological terms as well as some of their consequences. We will also discuss the origin of these terms and calculate effective actions for several fermionic models. In this approach topological terms appear as phases of fermionic determinants and represent quantum anomalies of fermionic models. In addition to the wide use of topological terms in high energy physics, they appeared to be useful in studies of charge and spin density waves, Quantum Hall Effect, spin chains, frustrated magnets, and some models of high temperature superconductivity.
Topics to be touched
 Effective actions
 Symmetry breaking: explicit, spontaneous, and anomalous
 The order parameter manifold
 Topological terms
 Basics of topology
 Topology and topological spaces
 Homotopy groups
 Differential forms and cohomologies
 Types of topological terms
 Metric independence
 Classification of topological terms in nonlinear sigma models
 Theta terms
 WZW terms
 Topological current and other terms
 Fermionic determinants and the origin of topological terms
 Fermions coupled to nonlinear sigma models
 Fermionic induced effective action
 Calculation of fermionic determinants
 Topological terms as phases of fermionic determinants
 Consequences of topological terms
 Degenerate vacua and gapless modes
 Physics at the boundary
 Spin, charge, and statistics of excitations
 Physical examples
 Quantum spins and spin chains
 Quantum Hall effect
 Topological terms in mesoscopic physics
 Open questions
Exercises

To lecture 1
(PDF )

To lecture 2
(PDF )

To lecture 3
(PDF )

To lecture 4
(PDF )
Recommended Books
Geometry and topology

B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov,
Modern GeometryMethods and Applications : Part II,
the Geometry and Topology of Manifolds,
(Graduate Texts in Mathematics, Vol 104)
Springer Verlag, 1985, ISBN: 0387961623.

C. Nash and S. Sen,
Topology and Geometry for Physicists, Academic Press, 1983, ISBN 0125140800.

Mikio Nakahara,
Geometry, Topology, and Physics,3rd edition,
Cambridge, Massachusetts, MIT press, 1987, ISBN: 0852740956.
Quantum field theory in condensed matter physics

A. Altland and B. Simons,
Condensed Matter Field Theory,
Cambridge University Press, 2006, ISBN10: 0521845084, ISBN13: 9780521845083.

E. Fradkin,
Field Theories of Condensed Matter Systems (Advanced Books Classics), Westview Press; Reissue edition, 2001, ISBN10: 0201328593, ISBN13: 9780201328592

A. M. Tsvelik,
Quantum Field Theory in Condensed Matter Physics, Cambridge University Press; 2 edition, 2003, ISBN10: 052182284X,
ISBN13: 9780521822848
Topology in physics

N. D. Mermin,
The topological theory of defects in ordered media, Rev. Mod.
Phys., 51, 591 (1979).
(PDF,
subscription required )
This is a review article which explains how to use the homotopy theory
for the classification of topological defects in ordered media.

A. G. Abanov and P. B. Wiegmann, Thetaterms in nonlinear sigmamodels.
Nucl. Phys. B 570 , 685698 (2000). (PDF, subscription required )
This is the article on which my lectures are based.
Some of the homotopy groups used in physics
(PDF )
Last updated June 11, 2007