Topology and Condensed Matter Physics

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. Front Matter ....Pages i-xxviii Overview of Topological Ideas in Condensed Matter Physics (Siddhartha Sen)....Pages 1-21 Front Matter ....Pages 23-23 Set Topology (Somnath Basu, Atreyee Bhattacharya)....Pages 25-43 Homotopy theory (Samik Basu, Soma Maity)....Pages 45-63 Homology (Dheeraj Kulkarni)....Pages 65-77 Differential Topology and Differential Geometry (Kingshook Biswas, Soma Maity)....Pages 79-108 Vector Bundles (Utsav Choudhury, Atreyee Bhattacharya)....Pages 109-141 Special Topics: A Crash Course on Knots (Mahan Mj)....Pages 143-151 Special Topics: A Short Course on Group Theory (Bobby Ezhuthachan)....Pages 153-168 Front Matter ....Pages 169-169 Use of Topology in physical problems (Somendra M. Bhattacharjee)....Pages 171-216 What is dimension? (Somendra M. Bhattacharjee)....Pages 217-251 Quantum Geometry and Topology (R. Shankar)....Pages 253-279 Topology, geometry and quantum interference in condensed matter physics (Alexander Abanov)....Pages 281-331 Dirac quasiparticles and Majorana modes in condensed matter systems (K. Sengupta)....Pages 333-342 Vertex Models and Knot invariants (P. Ramadevi)....Pages 343-357 Concepts of polymer statistical topology (Sergei Nechaev)....Pages 359-398 Introduction to abelian and non-abelian anyons (Sumathi Rao)....Pages 399-437 An introduction to Quantum Spin Liquids (Subhro Bhattacharjee)....Pages 439-470 Topological aspects of quantum information processing (Ville Lahtinen, Jiannis K. Pachos)....Pages 471-500 Back Matter ....Pages 501-507