Topological Quantum - Steven H Simon

. .

Résumé :
At the intersection of physics, mathematics, and computer science, an exciting new field of study has formed, known as Topological Quantum. This research field examines the deep connections between the theory of knots, special types of subatomic particles known as anyons, certain phases of matter, and quantum computation. This book elucidates this nexus, drawing in topics ranging from quantum gravity to topology to experimental condensed matter physics. Topological quantum has increasingly been a focus point in the fields of condensed matter physics and quantum information over the last few decades, and the forefront of research now builds on the basic ideas presented in this book. The material is presented in a down-to-earth and entertaining way that is far less abstract than most of what is in the literature. While introducing the crucial concepts and placing them in context, the subject is presented without resort to the highly mathematical category theory that underlies the field. Requiring only an elementary background in quantum mechanics, this book is appropriate for all readers, from advanced undergraduates to the professional practitioner. This book will be of interest to mathematicians and computer scientists as well as physicists working on a wide range of topics. Those interested in working in these field will find this book to be an invaluable introduction as well as a crucial reference....

Biographie:
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....

Sommaire:

• 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
• Free