Topological Quantum

Steven H. Simon

Steven H. Simon is a theoretical physics professor at Oxford University and a professorial fellow of Somerville College, Oxford. Before coming to Oxford he was director of theoretical physics research at Bell Laboratories. He has served on the UK EPSRC Physical Sciences Strategic Advisory Board. He is known for his work on Topological Phases of Matter, Topological Quantum Computation, and Fractional Quantum Hall Effect. He is the author of a popular introductory book on solid state physics, entitled The Oxford Solid State Basics which has been used at over a hundred universities worldwide.

1. Introduction and Ancient History
2. Kauffman Bracket Invariant and Relation to Physics
I Anyons and Topological Quantum Field Theories
3. Particle Quantum Statistics
4. Aharonov-Bohm Effect and Charge-Flux Composites
5. Chern-Simons Theory Basics
6. Short Digression on Quantum Gravity
7. Defining Topological Quantum Field Theory
II Anyon Basics
8. Fusion and Structure of Hilbert Space
9. Change of Basis and F-Matrices
10. Exchanging Identical Particles
11. Computing with Anyons
III Anyon Diagrammatics
12. Planar Diagrams
13. Braiding Diagrams
14. Seeking Isotopy
15. Twists
16. Nice Theories with Planar or Three-Dimensional Isotopy
17. Further Structure
IV Some Examples: Planar Diagrams and Anyon Theories
18. Some Simple Examples
19. Anyons From Discrete Groups Elements
20. Bosons and Fermions from Group Representati
21. Quantum Groups (In Brief)
22. Temperly-Lieb Algebra and Jones-Kauffman Anyons
V Applications of TQFT Diagrammatics
23. State Sum TQFTs
24. Formal Construction of “Chern-Simons” TQFT: Surgery and More Complicated Manifolds
25. Anyon Condensation
VI Toric Code Basics
26. Introducing Quantum Error Correction
27. Introducing the Toric Code
28. The Toric Code as a Phase of Matter and a TQFT
29. Robustness of Topologically-Ordered Matter
30. Abstracting the Toric Code: Introduction to Tube Algebra
VII More General Loop-Gas and String-Net Models
31. Kitaev Quantum Double Model
32. Doubled-Semion Model
33. Levin-Wen String-Net
VIII Entanglement and Symmetries
34. Topological Entanglement
35. SPT Phases of Matter
36. Anyon Permuting Symmetry
IX Further Thoughts
37. 37 Experiments (In Brief)
38. Final Comments
39. Appendix: Kac and Other Resources for TQFTs
40. Appendix: Some Mathematical Basics
Index