Informal course on
Topological terms in condensed matter physics
Spring 2007, EPFL, Lausanne, Switzerland
Lecture 1: May 21, Monday, room PH31 (EPFL) 11h15 - 13h00.
Instructor: Dr. Alexander (Sasha) Abanov,
Lecture 2: May 30, Wednesday, room PH31 (EPFL) 11h15 - 13h00.
Lecture 3: June 4, Monday, room PH33 (EPFL) 15h15-17h00
Lecture 4: June 11, Monday, room PH33 (EPFL) 15h15-17h00
Office: PH H2 487
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 non-trivial 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
To lecture 1
To lecture 2
To lecture 3
To lecture 4
Geometry and topology
B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov,
Modern Geometry-Methods 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 0-12-514080-0.
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, ISBN-10: 0521845084, ISBN-13: 978-0521845083.
Field Theories of Condensed Matter Systems (Advanced Books Classics), Westview Press; Reissue edition, 2001, ISBN-10: 0201328593, ISBN-13: 978-0201328592
A. M. Tsvelik,
Quantum Field Theory in Condensed Matter Physics, Cambridge University Press; 2 edition, 2003, ISBN-10: 052182284X,
Topology in physics
N. D. Mermin,
The topological theory of defects in ordered media, Rev. Mod.
Phys., 51, 591 (1979).
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, Theta-terms in nonlinear sigma-models.
Nucl. Phys. B 570 , 685-698 (2000). (PDF, subscription required )
This is the article on which my lectures are based.
Some of the homotopy groups used in physics
Last updated June 11, 2007