Topological insulators and superfluids – basic theory and recent progress (2018)

ECTS credits: 3


Course parameters:

Language: English

Level of course: PhD

Time of year: Spring 2018

No. of contact hours/hours in total incl. preparation, assignment(s) or the like:

16 hours of lectures

20 hours of study for lectures and problem solving

35 hours of self-study for oral presentation  

10 hours listening to oral presentations

Capacity limits: 15 participants


Objectives of the course:

This course intends to cover the underlying mathematical and physical concepts behind topological insulators. It will in addition introduce the students to cutting edge topics including Floquet engineering and the realization of topological phases with cold atom systems, as well as the topological phases in the presence of strong interactions.  


Learning outcomes and competences:

At the end of the course, the student should be able to:

-Outline the connection between fiber bundles and topological states in physics

-Give an account of topological insulators and their classification
-Explain the concept of symmetry protected topological phases and their classification

-Design Hamiltonians with interesting topological properties using Floquet engineering

-Give an account of recent progress concerning topological phases with cold atom systems

-Characterize interacting topological states with Green’s functions


Course contents:

An introduction to fiber bundles and topological states

The classification of topological phases

Time dependent Hamiltonians and Floquet engineering

Symmetry protected topological phases

The use of Green’s functions to characterize interacting topological phases

Recent experimental progress with cold atom atoms



Three years of study including quantum mechanics and statistical physics.


Name of lecturers:

Victor Gurarie, University of Colorado, Boulder, USA

Nathan Goldman, Université libre de Bruxelles, Belgium


Type of course/teaching methods:

Lectures, student talks, and exercises



Lecture notes and original papers.


Background literature:

- “Topological insulators and Topological superconductors”, Bernevig.

- Geometry, Topology and Physics (book) by Nakahara

- Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields, N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014).

- Topological Quantum Matter with Ultracold Gases in Optical Lattices, N. Goldman, J. C. Budich, P. Zoller, Nature Physics 12, 639-645 (2016).


Course homepage:

Will be set up


Course assessment:

The students will be required to give an oral presentation of a chosen subject, relevant to the course. The students will be graded based on their presentation on a pass/non-pass basis. Grading on the usual scale will be available on request in advance. Students are expected to attend all presentations of their fellow students.

Georg Bruun will be responsible for the student evaluation.



Department of Physics



To be announced



To be announced



Deadline for registration is 1 April 2018.

If you have any questions, please contact Georg M. Bruun, e-mail: