Topology in Condensed Matter

The idea behind topological systems is simple: if there exists a quantity, which cannot change in an insulating system where all the particles are localized, then the system must become conducting and obtain propagating particles when the quantity (called a "topological invariant") finally changes.

The practical applications of this principle are quite profound, and already within the last eight years they have lead to prediction and discovery of a vast range of new materials with exotic properties that were considered to be impossible before.

What will you gain from this course?

Learn about the variety of subtopics in topological materials, their relation to each other and to the general principles.

Learn to follow active research on topology, and critically understand it on your own.

Acquire skills required to engage in research on your own, and to minimize confusion that often arises even among experienced researchers.

What is the focus of this course?

Applications of topology in condensed matter based on bulk-edge correspondence.

Special attention to the most active research topics in topological condensed matter: theory of topological insulators and Majorana fermions, topological classification of "grand ten” symmetry classes, and topological quantum computation

Extensions of topology to further areas of condensed matter, such as photonic and mechanical systems, topological quantum walks, topology in fractionalized systems, driven or dissipative systems.

What tools does this course use?

Simple thought experiments that rely on considerations of symmetry or continuity under adiabatic deformations

Computer simulations similar to those used in actual research will give a more detailed and visual understanding of the involved concepts

Dissecting research papers that teaches you to simply understand the idea even in the rather involved ones.

This course is a joint effort of Delft University of Technology, QuTech, NanoFront, University of Maryland, and Joint Quantum Institute.

FAQs

Are there any books that are required for the course?

No, the course will only rely on materials and software freely available online.

Is it possible to get credit for this course at my university?

Not by default, but we invite anyone to use the course materials as a basis for a graduate course, with course materials studied as preparation and followed by a classroom discussion. Such courses are planned at universities of Copenhagen, Delft, Leiden, and University of Maryland. Following such a course will obviously give you credit points.

Would it not be better to take a more formal approach, and to describe the math in a more rigorous and systematic way?

While advanced math is certainly relevant for some researchers, in our experience it is the simple things that are the most confusing. We aim for the course to stay accessible and relevant to advanced undergraduate/beginner graduate students, both the theorists and experimentalists.

I do not know enough about condensed matter physics, but I have attended an exciting talk/read a cool article, and I'd like to learn more. Would the course be useful for me?

We are not sure. On the one hand, we will aim the course at people familiar with basic condensed matter physics and the necessary math, hence we will always assume that we don't need to explain e.g. band structures from scratch. However, a good share of the course materials are just discussions which would give you some sort of overview and understanding what this is all about.

Why didn't you discuss my favorite topic, which is certainly relevant and exciting?

Hey, that's a great idea! We aim to start from covering the basic questions, and then let the course evolve together with the field. So if you want, please help us by preparing the materials that would be helpful for the course, and they will become a bonus topic. By the way, same holds if you spot an error, or know how to improve the course: everything about this course is open, so don't hesitate to contribute.

What you'll learn

• Understand the variety of subtopics in topological materials, their relation to each other and to the general principles.
• Follow active research on topology, and critically understand it on your own.
• Acquire skills required to engage in research on your own, and to minimize confusion that often arises even among experienced researchers.