A brief introduction to topology and differential geometry in condensed matter physics /

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"Version: 20190301"--Title page verso."A Morgan & Claypool publication as part of IOP Concise Physics"--Title page verso.Includes bibliographical references.1. Path integral approach -- 1.1. Path integral -- 1.2. Spin -- 1.3. Path integral and statistical mechanics -- 1.4. Fermion path integral2. Topology and vector spaces -- 2.1. Topological spaces -- 2.2. Group theory -- 2.3. Cocycle -- 2.4. Vector spaces -- 2.5. Linear maps -- 2.6. Dual space -- 2.7. Scalar product -- 2.8. Metric space -- 2.9. Tensors -- 2.10. p-vectors and p-forms -- 2.11. Edge product -- 2.12. Pfaffian3. Manifolds and fiber bundle -- 3.1. Manifolds -- 3.2. Lie algebra and Lie group -- 3.3. Homotopy -- 3.4. Particle in a ring -- 3.5. Functions on manifolds -- 3.6. Tangent space -- 3.7. Cotangent space -- 3.8. Push-forward -- 3.9. Fiber bundle -- 3.10. Magnetic monopole -- 3.11. Tangent bundle -- 3.12. Vector field4. Metric and curvature -- 4.1. Metric in a vector space -- 4.2. Metric in manifolds -- 4.3. Symplectic manifold -- 4.4. Exterior derivative -- 4.5. The Hodge star operator -- 4.6. The pull-back of a one-form -- 4.7. Orientation of a manifold -- 4.8. Integration on manifolds -- 4.9. Stokes' theorem -- 4.10. Homology -- 4.11. Cohomology -- 4.12. Degree of a map -- 4.13. Hopf-Poincar?e theorem -- 4.14. Connection -- 4.15. Covariant derivative -- 4.16. Curvature -- 4.17. The Gauss-Bonnet theorem -- 4.18. Surfaces5. Dirac equation and gauge fields -- 5.1. The Dirac equation -- 5.2. Two-dimensional Dirac equation -- 5.3. Electrodynamics -- 5.4. Time reversal -- 5.5. Gauge field as a connection -- 5.6. Chern classes -- 5.7. Abelian gauge fields -- 5.8. Non-abelian gauge fields -- 5.9. Chern numbers for non-abelian gauge fields -- 5.10. Maxwell equations using differential forms6. Berry connection and particle moving in a magnetic field -- 6.1. Introduction -- 6.2. Berry phase -- 6.3. The Aharonov-Bohm effect -- 6.4. Non-abelian Berry connections7. Quantum Hall effect -- 7.1. Integer quantum Hall effect -- 7.2. Currents at the edge -- 7.3. Kubo formula -- 7.4. The quantum Hall state on a lattice -- 7.5. Particle on a lattice -- 7.6. The TKNN invariant -- 7.7. Quantum spin Hall effect -- 7.8. Chern-Simons action -- 7.9. The fractional quantum Hall effect8. Topological insulators -- 8.1. Two bands insulator -- 8.2. Nielsen-Ninomya theorem -- 8.3. Haldane model -- 8.4. States at the edge -- 8.5. Z2 topological invariants9. Magnetic models -- 9.1. One-dimensional antiferromagnetic model -- 9.2. Two-dimensional non-linear sigma model -- 9.3. XY model -- 9.4. Theta termsAppendices. -- A. Lie derivative -- B. Complex vector spaces -- C. Fubini-Study metric and quaternions -- D. K-theory.In recent years there have been great advances in the applications of topology and differential geometry to problems in condensed matter physics. Concepts drawn from topology and geometry have become essential to the understanding of several phenomena in the area. The main purpose of this book is to provide a brief, self-contained introduction to some mathematical ideas and methods from differential geometry and topology, and to show a few applications in condensed matter.Also available in print.Mode of access: World Wide Web.System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.Antonio Sergio Teixeira Pires is a Professor of Physics in the Physics Department at the Universidade Federal de Minas Gerais, Belo Horizonte, Brazil. He works in quantum field theory applied to condensed matter. He is a member of the Brazilian Academy of Science, was an Editor of the Brazilian Journal of Physics and a member of the Advisory Board of the Journal of Physics: Condensed Matter. He is the author of AdS/CFT Correspondence in Condensed Matter, which is another book in the Concise Physics series.Title from PDF title page (viewed on April 1, 2019).

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: .,

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1 online resource (various pagings) :illustrations.

Language

English

ISBN/ISSN

9781643273747

Classification

530.15

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Subject(s)

SCIENCE / Physics / Mathematical & Computational.
Mathematical physics.
Geometry, Differential.
Topology.
Condensed matter

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Antonio Sergio Teixeira Pires.

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