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

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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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Pires, Antonio Sergio Teixeira (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: San Rafael [California] (40 Oak Drive, San Rafael, CA, 94903, USA) : Morgan & Claypool Publishers, [2019]
Colección:IOP (Series). Release 5.
IOP concise physics.
Temas:
Mathematical physics.
Condensed matter > Mathematics.
Topology.
Geometry, Differential.
SCIENCE / Physics / Mathematical & Computational.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1. Path integral approach
  • 1.1. Path integral
  • 1.2. Spin
  • 1.3. Path integral and statistical mechanics
  • 1.4. Fermion path integral
  • 2. 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. Pfaffian
  • 3. 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 field
  • 4. 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-Poincaré theorem
  • 4.14. Connection
  • 4.15. Covariant derivative
  • 4.16. Curvature
  • 4.17. The Gauss-Bonnet theorem
  • 4.18. Surfaces
  • 5. 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 forms
  • 6. 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 connections
  • 7. 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 effect
  • 8. 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 invariants
  • 9. Magnetic models
  • 9.1. One-dimensional antiferromagnetic model
  • 9.2. Two-dimensional non-linear sigma model
  • 9.3. XY model
  • 9.4. Theta terms
  • Appendices.
  • A. Lie derivative
  • B. Complex vector spaces
  • C. Fubini-Study metric and quaternions
  • D. K-theory.

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