Topological Quantum Many-Body Physics

Dr. Nicolai Lang

Thursday, 10. April 2025

• Thursday, 08:00 - 09:30, V57.05, Pfaffenwaldring 57
• Friday, 08:00 - 09:30, V57.05, Pfaffenwaldring 57

• The lecture and the tutorials will be given in English.
• There are two types of exercises:
  "Written" exercises are handed in in class and graded/corrected by the tutor.
  "Oral" exercises are discussed and presented in the tutorials by students.
• Admission to the final exam requires 66% of the written scores, 66% of the oral scores, and presenting a problem on the blackboard twice.
• Please register for the exercise groups online.
  To do so, you need the Lecture Key given in the first lecture.
• You can request your current scores here.

Recordings of the lectures can be accessed via ILIAS.

This course can be used for the following modules:

There will be an oral examination at the end of the course.
Details will be given in the lecture.

This course follows no particular textbook and topics are compiled from various sources.
Have a look at the script for references and literature recommendations.

The goal is to gain a thorough understanding of topological concepts in modern quantum many-body physics. You aquire the mathematical tools needed to describe topological quantum phases, understand the physical features that characterize these systems, and learn about potential applications. Planned topics are (this list is not final!):

Topological phases of non-interacting fermions:

• Integer quantum Hall effect
• Berry connection, Berry holonomy, Chern number
• Anomalous quantum Hall effect (Haldane model)
• Quantum spin Hall effect, topological insulators (Kane-Mele model)
• Pfaffian topological invariant
• Winding numbers, sublattice symmetry, edge modes (SSH model)
• Topological superconductivity (Majorana chain)
• Tenfold way and periodic table of topological insulators/superconductors
• Effects of interactions
• Topological bands in classical systems (topological metamaterials ...)

Symmetry-protected topological phases of interacting bosons:

• Tensor network states, matrix product states, PEPS
• Projective representations and (twisted) cohomology groups
• Classification of bosonic topological phases in one dimension
• Haldane chain and AKLT model

Intrinsic topological order and long-range entanglement:

• Statistics of indistinguishable particles in 2+1 dimensions (Braid group)
• Toric code (anyonic excitations, top. entanglement entropy ...)
• Topological quantum memories
• Fibonacci anyons
• String-net condensates
• Topological quantum computation (non-abelian anyons, braiding, fusion ...)
• Mathematical framework:
  Modular tensor categories, pentagon & hexagon relations,
  quantum dimension, topological spin ....
• Application to foundational questions of high-energy physics (fermions ...)

The concept of second quantization is a prerequisite to follow this course. If you did not learn about second quantization in your advanced quantum mechanics course, I suggest that you catch up by self-study (any textbook on advanced quantum mechanics covers this topic).

Beyond that, you should be familiar with the following concepts:

• Non-relativistic quantum mechanics and second quantization
  (fermions, bosons, spins, ...)
• Basics of condensed matter theory
  (band theory, quasi particles, ...)
• Basics of quantum information
  (qubits, quantum gates, ...)
• Basics of group theory
  ((non-)abelian groups, linear representations, ...) 

Knowledge of relativistic quantum mechanics it not required.

The script will be continuously expanded and published in parallel to the lectures.

The script for upcoming lectures will be published (at least) one day before the lecture. We recommend that you print these notes (or download them to your tablet) so that you can concentrate on the lecture.

Warning: We are well aware of the capabilities and limitations of state of the art transformer models. In case you feel tempted: Contemplate why you decided to study physics in the first place, and why you signed up for this particular course. And, last but not least, remember the final exam.