PHY 680-04: Special Topics in Theoretical Physics
Topological terms in condensed matter physics
Spring 2009, Stony Brook
Tue, Thu: 11:20-12:40, in B-131

Instructor: Dr. Alexander (Sasha) Abanov, Associate Professor
Office: B-102
Phone: 2-8174
Web page:


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.

Syllabus of the course


The main prerequisite for the course is the knowledge of basics of quantum field theory. We will use path integral approach to quantum mechanics and quatnum field theory. All necessary methods of topology will be introduced in the course.

The final grade for this course will be based on grades for homeworks (60%), which will be given every two weeks, and the grade for the presentation (40%) which should be made at the end of the semester.

Topics to be touched


Suggested presentation topics

You can find some suggested presentation topics here . You are welcome to choose (and discuss with instructor) your own topics related to topological aspects either in condensed matter systems or in quantum field theory. The length of the presentation is about 30 minutes. It should be pedagogical. The main goal is that we all should learn something from the presentation.


Chapter Title Format Updated
1 Introduction PDF 02/05/09
2 Particle on a ring PDF 01/29/09
3 Topology and topological spaces PDF 02/05/09
4 Homotopy theory primer PDF 02/19/09
5 Topological defects and textures in ordered media PDF 03/02/09
6 Theta terms
7 Path integral for a single spin. WZW term in zero dimensions PDF 03/17/09
8 Brief introduction to differential forms PDF 03/24/09
9 Spin chains
10 Quantum magnets in higher dimensions. Hopf term etc
11 Integer Quantum Hall Effect
12 Fractional Quantum Hall Effect
References References
Appendix 1 Homotopy groups used in physics PDF 02/05/09

Recommended Books

Geometry and topology

Quantum field theory in condensed matter physics

Topology in physics

Syllabus of the course

Last updated May 7, 2009