About this course - Topology in condensed matter: tying quantum knots

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A word of welcome¶

Through this course, we want to provide an introduction to the topic of topology in condensed matter. We want it to be accessible and useful to people with different backgrounds and motivations.

We want the course to be useful to you if you are a master’s student and want to gain an understanding of what topology is all about.

You could also be a PhD student or a postdoc doing experiments and want to get a better theoretical grasp of what you should expect in your investigation.

You could even be a theorist working in topology and extremely familiar with topological invariants and vector bundles, but would like to get a better overview of how the mathematical ideas apply in physical systems.

Finally, we also want this course to be equally useful if you are, say, a professor working in condensed matter and want to apply the ideas introduced by topology in your domain, so you just need a quick overview of the research activity.

We hope that you find something useful, and we always appreciate your questions and feedback via the course chat or the course repository issue tracker. So whenever you see a typo or you would like to suggest an improvement, you can open an issue (or even make a pull request right away).

Course structure¶

The course is separated into 12 topics, each containing 2–3 lectures on related subjects. Each lecture is introduced by an expert in this subject. We end each topic with suggestions of open-ended questions for self-study: numerical simulations or papers to read and review.

Prerequisites¶

Background knowledge¶

While the math that we use only requires linear algebra and calculus, this course is complex. Topology affects many physical phenomena, and therefore the course touches many different concepts in condensed matter physics. In the next chapter we provide a brief review of band theory—the main physical concept that you will need; however, if you don’t know condensed matter physics yet, you are likely to struggle.

Code¶

We provide source code for all the computer simulations used in the course, as well as suggestions for what you can investigate on your own. To use these, you need to be familiar with Python’s SciPy stack (see e.g. this course), as well as the Kwant quantum transport package. You can run the code right away without installing anything using the Binder project over here: .

To obtain a local version of the code, clone or download the course repository, install the conda environment from environment.yml, and open the source files using the jupytext Jupyter extension.

Literature¶

We are mostly going to focus on the overall structure of the field and study the most basic and general phenomena. We will also skip detailed derivations and some details.

For a more formal and complete source of information on topological insulators and superconductors, we recommend that you look into the reviews below. (Of course we think they will be much easier to follow after you finish the course).

Topological insulator reviews¶

Majorana fermion reviews¶

Advanced topics: Fractional particles and topological quantum computation¶

Extra topics¶

References¶

Koenig, M., Buhmann, H., Molenkamp, L. W., Hughes, T. L., Liu, C.-X., Qi, X.-L., & Zhang, S.-C. (2008). The Quantum Spin Hall Effect: Theory and Experiment. 10.48550/ARXIV.0801.0901
Hasan, M. Z., & Kane, C. L. (2010). Topological Insulators. 10.48550/ARXIV.1002.3895
Qi, X.-L., & Zhang, S.-C. (2010). Topological insulators and superconductors. 10.48550/ARXIV.1008.2026
Beenakker, C. W. J. (2011). Search for Majorana fermions in superconductors. 10.48550/ARXIV.1112.1950
Alicea, J. (2012). New directions in the pursuit of Majorana fermions in solid state systems. 10.48550/ARXIV.1202.1293
Leijnse, M., & Flensberg, K. (2012). Introduction to topological superconductivity and Majorana fermions. 10.48550/ARXIV.1206.1736
Beenakker, C. W. J. (2014). Random-matrix theory of Majorana fermions and topological superconductors. 10.48550/ARXIV.1407.2131
Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Sarma, S. D. (2007). Non-Abelian Anyons and Topological Quantum Computation. 10.48550/ARXIV.0707.1889
Stern, A. (2007). Anyons and the quantum Hall effect - a pedagogical review. 10.48550/ARXIV.0711.4697
Hassler, F. (2014). Majorana Qubits. 10.48550/ARXIV.1404.0897
Cayssol, J., Dóra, B., Simon, F., & Moessner, R. (2012). Floquet topological insulators. 10.48550/ARXIV.1211.5623
Ando, Y., & Fu, L. (2015). Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials. 10.48550/ARXIV.1501.00531