Topology in condensed matter systems
Lecture in summer term 2026.
Lecturer: Thore Posske
Script & cheat sheet
Script and coursebook
Handwritten script
Cheat sheet
(write me additionally suggested content for the cheat sheet)
Organization
Schedule:
Lecture: Wed, 10:00 a.m. - 11:30 a.m., Notkestr. 9/11
Exercises: Wed, 11:45 p.m. - 1:15 p.m., Notkestr. 9/11
No lecture: Start of May (holiday), May 27 (but exercise group!)
Last lecture: Wed, July 8, 2026.
Topics, recommendations, and literature
Consent for video recording
Zoom-link
Questions and answers session
There is a questions and answers session on
Friday, July 24, 11 a.m. CEST
at the lecture's seminar room and online.
Send questions until two days in advance.
Exam
Written exam, 90 minutes on Wednesday, July 29, 10 a.m. CEST
Registration at Stine additionally obligatory.
You are allowed to take a printout of the TICMS cheat sheet with you.
Please suggest additional equations for this sheet.
Content
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Lecture 13 (July 15)
The particle on a ring - example for quantum theory of non-simply connected configuration spaces
Video 13 |
Blackboard 13 |
Notes 13
|
Exercise Sheet 13 |
Solutions 13
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Lecture 12 (July 8)
Theory of classical indistinguishable particles and its path integral quantization
Video 12
|
Blackboard 12 |
Notes 12
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Exercise Sheet 12
|
Solutions 12
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Lecture 11 (July 1)
Geometric interpretation of the adiabiatc theorem, Majorana braiding, anyons
Video 11
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Blackboard 11 |
Notes 11
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Exercise Sheet 11
|
Solutions 11
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Lecture 10 (June 24)
The adiabatic theorem of quantum mechanics and topological quantum computing
Video 10
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Blackboard 10 |
Notes 10
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Exercise Sheet 10
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Solutions 10
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Lecture 9 (June 17)
The Kitaev Majorana chain II - topological classification
Video 9
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Blackboard 9
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Notes 9
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Exercise Sheet 9
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Solutions 9
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Lecture 8 (June 10)
The Kitaev Majorana chain I - pedestrian's approach
Video 8
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Blackboard 8
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Notes 8
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Exercise Sheet 8
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Solutions 8
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Lecture 7 (June 3)
Classification of topological phases and the tenfold way
Video 7
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Blackboard 7
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Notes 7
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Exercise Sheet 7
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Solutions 7
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Lecture 6 (May 20)
Properties of the fundamental symmetries T, P, C, and the tenfold way
Video 6
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Blackboard 6
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Notes 6
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Exercise Sheet 6
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Solutions 6
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Lecture 5 (May 6)
Classification of topological phases, (anti)unitary operators
Video 5
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Blackboard 5
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Notes 5
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Exercise Sheet 5
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Solutions 5
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Lecture 4 (Aptil 27)
Fibre bundles, topological invariants and the first homotopy group (fundamental group)
Video 4
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Blackboard 4
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Notes 4
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Exercise Sheet 4
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Solutions 4
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Lecture 3 (April 22)
Constructing topological spaces, sums, products, quotients, initial and final topology)
Video 3
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Blackboard 3
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Notes 3
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Exercise Sheet 3
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Solutions 3
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Lecture 2 (April 15)
Continuous functions, homeomorphisms, metric spaces
Video 2
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Blackboard 2
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Notes 2
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Exercise Sheet 2
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Solutions 2
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Lecture 1 (April 8)
Administration, inspiration, and basic topology
Video 1
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Blackboard 1
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Notes 1
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Exercise Sheet 1
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Solutions 1
toggler:
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Additional material
Video lecture by John Milnor (Fields medalist, "Milnor spheres") about differential topology
Video lectures by Norman Wildberger about algebraic topology explained very intuitively
Video that the torus can be folded to a Moebius strip and
corresponding Mathematica notebook
Notebook for numerically experimenting with the Kitaev chain's spectrum and topological phases
Recommended reading/references
Topology In Condensed Matter: An Introduction - Miguel A N Araujo and Pedro Sacramento
Quantum Computation and Quantum Information - Michael Nielsen and Isaac Chuang
Altland's
and Simons' book on Condensed Matter theory
Course on topology in condensed matter physics by Akhmerov et al.
Topology and physics - a historical essay, C. Nash
Topological Quantum Computation-From Basic Concepts to First Experiments, Stern, Ady and Lindner, Netanel H.
Seminal paper of Leinaas and Myrheim '77 describing anyons
Philosophical background on the "principle of indiscernibles"
Kitaev's periodic table of gapped, non-interacting, topological phases
Ryu, Schnyder, Furusaki, Ludwig, classification of gapped, non-interacting, topological phases
Katos seminal paper about adiabatic time evolution
