Research group Hermanns

Introduction to strange and topological quantum matter (summer term 2016)

This lecture discusses modern concepts of condensed matter theory, such as strange metals and topological phases of matter. The lectures are shared with Philipp Strack, who will give the first part concerning strange metals. More information can be found on his course webpage, as well as in the course syllabus. More information about the second part of the course, as well as lecture notes will come soon.

Third exercise sheet

The third exercise sheet is now online and can be downloaded here: ExerciseSheet3. In this exercise, you will revisit the Toric Code (nice lecture notes by Alexei Kitaev and Chris Laumann can be found here ), albeit in a new shape, where it becomes clear that the e and m particles are identical types of anyons. The alternative description is also very suitable to study lattice dislocations, which have the amazing feature that they can turn an abelian anyon theory into an effectively nonabelian one. This exercise sheet is heavily based on the Bombins very nice paper on Topological Order with a Twist: Ising Anyons from an Abelian Model.

Further reading

A short introduction to Fibonacci anyon models by Simon Trebst, Matthias Troyer, Zhenghan Wang, and Andreas W.W. Ludwig gives a pedagogical introduction to Fibonacci anyons, and also give a lot of the technical details. For instance, they explicitly derive the two possible F-matrices that are consistent with the pentagon equation.

If you had not yet enough of mathematics, then read Parsa Bondersons PhD Thesis on Non-Abelian Anyons and Interferometry. In particular, it gives all the details that I swept under the rug in the lectures and tells you how to use it to compute something useful.

In my course, the field theory side of topological phases was basically skipped. The standard reference to read up on this is Eduardo Fradkins Book on Field Theories of Condensed Matter Physics. There are also very nice lecture notes by Hans Hansson discussing Effective Field Theories of topological Phases of Matter, from a 2015 summer school at the IIP.

A nice introduction to anyon theories and how you would use them for quantum computation is the review article on Non-Abelian anyons and topological quantum computation by Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. This review is, however, strongly biased towards quantum Hall effect.

Alexei Kitaev is famous for inventing simple models that give profound insight. One example is of course the Toric Code, another example is the Kitaev honeycomb model. This paper has also an extensive section on anyons, and explicitly derives the solutions to the pentagon and hexagon equation for Ising anyons.