Topological Phases of Matter

MSc Physik / MSc Physics

SS 21


Please sign up for this course on C@mpus (LV-Nr. 046465000).

Until further notice, this course is provided online via ILIAS.


Time and Date

There will be a new lecture available on ILIAS every week on Wednesday.

First lecture: Wednesday, 21 April 2021

General Information

  • The lecture and the tutorials will be given in English.
  • All exercises must be handed in via ILIAS and are graded/corrected by the tutor. 
  • Admission to the final exam requires 80% of the written scores and active participation in the tutorials.
  • Please register for the exercise groups online. To do so, you need the Lecture Key given in the first lecture.
  • If you assign a password at the registration, you can request your current scores here.


There will be an oral examination at the end of the course. Details will be given in the lecture.


A list of references is given in the script (see below); this includes both original research and textbooks.


For this course, we assume that students are familiar with the following concepts:

  • Non-relativistic quantum mechanics and second quantization (fermions, bosons, spins, ...)
  • Basics of condensed matter theory (band theory, quasi particles, ...)
  • Basics of quantum information (qubits, quantum gates, ...)
  • Basics of group theory ((non-)abelian groups, linear representations, ...)

If you think that you lack knowledge in one of these fields, you may still participate in the course if you close the gaps as we go along.


The goal of this course is to explore the realm of “topological quantum phases”—quantum phases that cannot be characterized by the paradigm of spontaneous symmetry breaking. This field is a very active domain of research with potential applications in quantum information processing. I plan to cover topics such as ...

  • Integer- [and fractional] quantum Hall effects
  • Symmetry protected topological phases
  • Classification of non-interacting fermionic topological phases
  • Topological band structures, Berry phases and Chern numbers
  • Classification of interacting bosonic topological phases
  • Projective representations, group cohomology and symmetry fractionalization
  • [Topological order and topological quantum field theories]
  • [Anyonic statistics and fusion categories]
  • [Topological quantum memories and quantum computation]

As an in-depth review of the these topics would require a dedicated course for each bullet point alone, our goal shall be a bit more modest: At the end of this course you should know the various key concepts that allow for topological characterizations of phases of matter, and you should be aware of their relations and differences (i.e. we aim more for the "big picture" than the details).


Download PDF

This script is updated continuously and is mostly identical to the notes that I use for the lecture. For more details on the covered topics, I refer the reader to the references given in the script.


No. Date Notes Topics
0 19.04.2021 PDF Preliminaries
1 21.04.2021 PDF Landau paradigm, Topological order, SPT phases, SET phases
2 28.04.2021 PDF Integer quantum Hall effect, Landau levels
3 05.05.2021 PDF Berry phase, Chern number
4 12.05.2021 PDF Kubo formula, TKNN formula
5 19.05.2021 PDF Disorder, Edge states, Classification
6 02.06.2021 PDF 2-band models, Skyrmions, Time-reversal symmetry
7 09.06.2021 PDF Dirac fermions, Qi-Wu-Zhang Model, Haldane Model
8 16.06.2021 PDF Kane-Mele model, Pfaffian, Z2 topological invariant
9 30.06.2021 PDF SSH chain, Sublattice symmetry, Zak phase
10 07.07.2021 PDF Majorana chain, Particle-hole symmetry
11 14.07.2021 PDF 10 symmetry classes, Classification of top. insulators & superconductors
12 21.07.2021 PDF Topological edge modes in classical systems

Problem Sets

No. Problem Set Date Comments
1 Problem Set 01 30.04.2021  
2 Problem Set 02 12.05.2021  
3 Problem Set 03 02.06.2021  
4 Problem Set 04 16.06.2021  
5 Problem Set 05 30.06.2021  
6 Problem Set 06 14.07.2021  


Tutor Room Day Time
Dr. Nicolai Lang Online Wednesday 13:00
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