Intensive Course on Topology in Physics

Organization

Lecture at the CUI/SFB graduate school, January 2021
ZOOM-LINK, 10:00-12:00 a.m.
kahoot.it
Schedule: Mo, Tue, Wed, Thu, Fr, Jan 11 2021 to Jan 15 2021, 10:00 - 12:00 a.m..

Content

I) Topology and why it is important Jan 11 - Jan 12 (two lectures)

Notes 1 (topology basics) | Notes 2 (special topological spaces) | Notes 3 (Topological invariants, homotopy)
Additional material: metric spaces | Additional material: constructing top. spaces
Exercise sheet 1 (basic topology) | Exercise sheet 2 (basic topology II) | Exercise sheet 3 (homotopies) | Exercise sheet 4 (braid group)
II) Classification of topological matter Jan 13 - Jan 15 (three lectures)
Notes 1 (What is gapped topological matter) | Notes 2 (fundamental symmetries, periodic table) | Notes 3 (The Kitaev Majorana chain I) | Notes 3 (The Kitaev Majorana chain II)
Exercise sheet 5 (symmetry reduction) | Exercise sheet 6 (understanding time reversal symmetry) | Exercise sheet 7 (Kitaev Majorana chain)
III) Topological/adiabatic quantum computing Better luck next time (no lecture)
Notes 1 (introduction)
Additional literature:
Topological Quantum Computation-From Basic Concepts to First Experiments, Stern, Ady and Lindner, Netanel H.
Preskill's lecture on Topological Quantum Computing

All scripts in svg format (8x zoom needed)

Topology - Introduction
Contructing topological spaces
Topological classifications - Homotopy
What is gapped topological matter?
Classification of gapped topological matter (and fundamental symmetries)
The Kitaev Majorana chain
Z2 invariant of the Kitaev chain

Additional material

Fun with topology: Wind and Mr. Ug
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

Three lectures by Edward Witten on topology in condensed matter physics
Topology and physics - a historical essay, C. Nash
Topological Quantum Computation-From Basic Concepts to First Experiments, Stern, Ady and Lindner, Netanel H.
Preskill's lecture on Topological Quantum Computing
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 Time reversal symmetric Kitaev chains, Dumitrescu and Tewari 2013