Topology and New Kinds of Order in Condensed Matter Physics

With the discovery of the integer and fractional quantum Hall effect in the 1980s, it was realized that not all phases of matter occurring in nature can be understood using Landau’s theory. The quantum Hall state represents a distinct phase of matter which can occur even when there is no local order parameter or spontaneous breaking of a global symmetry. Phases of this new kind are now usually referred to as topological phases. This module gives an introduction to different theoretical aspects of topological phases and their experimental signatures. The following topics are covered in this module:

• Kosterlitz–Thouless transitions
• Graphene, Dirac Hamiltonian and Chern insulators
• Topological insulators in 2D and 3D
• Weyl semi-metals
• Symmetry protected topological phases
• Topological superconductors and Majorana chains
• Spin liquids and frustrated magnetism
• Axiomatic description of topological order
• Exactly solvable models: toric code and string net models
• Topological quantum computing

After successful completion of the module the students have an overview of the recent developments and open questions related to topological phases of matter in condensed matter physics. In particular, the students are able to

• classify phases of matter according to symmetry and topology
• derive topological invariants from band structures
• understand the topological phenomena of the quantum Hall effects, topological insulators, and topological metals
• understand the axiomatic description of topological order and anyon theories
• solve exactly solvable models that exhibit topological order
• understand the bascis of topological quantum computing

No preconditions in addition to the requirements for the Master’s program in Physics.

The lecture is designed for the presentation of the subject, usually by blackboard presentation. The focus resides on theoretical foundations of the field, presentation of methods and simple, illustrative examples. Command of basic methods is deepened and practised through homework problems, which cover important aspects of the field. The homework problems should develop the analytic skills of the students and their ability to perform calculations. The homework problems are discussed in the exercise classes by the students themselves under supervision in order to develop the skills to explain a physics problem logically.

Blackboard lectures, written notes for download, exercise sheets, course homepage

• A. Bernevig & T. Hughes: Topological Insulators and Topological Superconductors, Princeton University Press, (2013)
• E. Fradkin: Field Theories of Condensed Matter Physics, Cambridge University Press, (2013)
• X.-G. Wen: Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons, Oxford University Press, (2007)
• J. Preskill: Lecture notes, California Institute of Technology, (2004) http://www.theory.caltech.edu/~preskill/ph219/topological.pdf
• C. Nayak et al.: Non-Abelian Anyons and Topological Quantum Computation, arXiv, (2008) http://arxiv.org/pdf/0707.1889v2.pdf

The achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using presentations independently prepared by the students. The exam of 25 minutes consists of the presentation and a subsequent discussion.

For example an assignment in the exam might be:

• Describe the non-local order parameters for symmetry protected topological phases.
• What are the characteristic surface state of a topological insulator?
• How can Majorana modes in semiconductor nanowires be realized?
• What is the "topological entanglement" and how can it be determined?

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.

There will be a bonus (one intermediate stepping of "0,3" to the better grade) on passed module exams (4,3 is not upgraded to 4,0). The bonus is applicable to the exam period directly following the lecture period (not to the exam repetition) and subject to the condition that the student passes the mid-term of active participation in the tutorials and at least 50% of exercise points