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Full Description
This book introduces aspects of topology and applications to problems in condensed matter physics. Basic topics in mathematics have been introduced in a form accessible to physicists, and the use of topology in quantum, statistical and solid state physics has been developed with an emphasis on pedagogy. The aim is to bridge the language barrier between physics and mathematics, as well as the different specializations in physics. Pitched at the level of a graduate student of physics, this book does not assume any additional knowledge of mathematics or physics. It is therefore suited for advanced postgraduate students as well. A collection of selected problems will help the reader learn the topics on one's own, and the broad range of topics covered will make the text a valuable resource for practising researchers in the field.
The book consists of two parts: one corresponds to developing the necessary mathematics and the other discusses applications to physical problems. The section on mathematics is a quick, but more-or-less complete, review of topology. The focus is on explaining fundamental concepts rather than dwelling on details of proofs while retaining the mathematical flavour. There is an overview chapter at the beginning and a recapitulation chapter on group theory. The physics section starts with an introduction and then goes on to topics in quantum mechanics, statistical mechanics of polymers, knots, and vertex models, solid state physics, exotic excitations such as Dirac quasiparticles, Majorana modes, Abelian and non-Abelian anyons. Quantum spin liquids and quantum information-processing are also covered in some detail.
Contents
Chapter 1. Overview of Topological Ideas in Condensed Matter Physics.- Chapter 2. Set Topology.- Chapter 3. Homotopy theory.- Chapter 4. Homology.- Chapter 5. Differential Topology and Differential Geometry.- Chapter 6. Vector Bundles.- Chapter 7. Special Topics: A Crash Course on Knots.- Chapter 8. Special Topics: A Short Course on Group Theory.- Chapter 9. Use of Topology in physical problems.- Chapter 10. What is dimension?.- Chapter 11. Quantum Geometry and Topology.- Chapter 12. Topology, geometry and quantum interference in condensed matter physics.- Chapter 13. Dirac quasiparticles and Majorana modes in condensed matter systems.- Chapter 14. Vertex Models and Knot invariants.- Chapter 15. Concepts of polymer statistical topology.- Chapter 16. Introduction to abelian and non-abelian anyons.-Chapter 17. An introduction to Quantum Spin Liquids.- Chapter 18. Topological aspects of quantum information processing.
