"Version: 202111"--Title page verso.

Includes bibliographical references.

1. Path integral approach -- 1.1. Path integral -- 1.2. Path integral in quantum field theory -- 1.3. Spin -- 1.4. Path integral and statistical mechanics -- 1.5. 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 groups -- 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 * 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. Surfaces -- 4.19. Geodesics -- 4.20. Fundamental theorem of the Riemann geometry

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 -- 6.5. The Aharonov-Casher effect

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- and three-band insulators -- 8.2. Nielsen-Ninomiya theorem -- 8.3. Haldane model -- 8.4. Checkerboard lattice -- 8.5. States at the edge -- 8.6. The Z2 topological invariants -- 8.7. The Kane-Mele model -- 8.8. Three-dimensional topological insulators -- 8.9. Calculation of edge modes

9. Topological phases in one dimension -- 9.1. The Su-Schrieffer-Heeger model -- 9.2. Winding number and Zak phase -- 9.3. Finite chain -- 9.4. Alternative form of the SSH Hamiltonian -- 9.5. Localized states at a domain wall -- 9.6. The Ising chain in a transverse field -- 9.7. The Kitaev chain -- 9.8. Majorana fermion operators -- 9.9. Rashba spin-orbit superconductor in one dimension

10. Topological superconductors -- 10.1. Basics of superconductivity -- 10.2. Two-dimensional chiral p-wave superconductors -- 10.3. Two-dimensional chiral p-wave superconductor on a lattice -- 10.4. Continuum limit -- 10.5. Non-Abelian statistics -- 10.6. d-Wave pairing symmetry

11. Higher-order topological insulators -- 11.1. Crystalline symmetries -- 11.2. Second-order topological insulator in two dimensions -- 11.3. Gapless corner states -- 11.4. A three-dimensional chiral HOTI

12. Classification of topological states with symmetries -- 12.1. Symmetries -- 12.2. Time-reversal symmetry -- 12.3. Particle-hole symmetry -- 12.4. Chiral symmetry -- 12.5. Periodic table -- 12.6. Complex classes -- 12.7. Real classes -- 12.8. Classification for zero dimensions -- 12.9. Dirac Hamiltonians -- 12.10. Dimension reduction -- 12.11. Topological defects

13. Weyl semimetals -- 13.1. The Weyl equation -- 13.2. Linear Weyl modes -- 13.3. Chern numbers -- 13.4. An example -- 13.5. Fermi arcs -- 13.6. Weyl semimetal in an external magnetic field -- 13.7. Type II Weyl semimetals -- 13.8. Weyl semimetals with spins higher than 1/2 -- 13.9. Chiral anomaly -- 13.10. Dirac semimetals

14. Kubo theory and transport -- 14.1. Linear response theory -- 14.2. Electron transport -- 14.3. Anomalous Hall effect -- 14.4. Orbital magnetization -- 14.5. Spin transport -- 14.6. Interacting topological insulators

15. Magnetic models -- 15.1. One-dimensional antiferromagnetic model -- 15.2. Sine-Gordon soliton -- 15.3. Two-dimensional non-linear sigma model -- 15.4. XY model -- 15.5. Theta terms

16. Topological magnon insulators -- 16.1. Magnon Hall effect -- 16.2. The ferromagnetic honeycomb lattice -- 16.3. Generalized Bogoliubov transformation -- 16.4. Antiferromagnetic honeycomb lattice -- 16.5. Thermal Hall conductivity

17. K-theory -- 17.1. Rings -- 17.2. Equivalence relations -- 17.3. Grothendieck group -- 17.4. Sum of vector bundles -- 17.5. K-theory -- 17.6. K-theory and topological insulators -- 17.7. The 2Z invariant -- 17.8. The Atiyah-Singer index theorem.

This book provides a self-consistent introduction to the mathematical ideas and methods from these fields that will enable the student of condensed matter physics to begin applying these concepts with confidence. This expanded second edition adds eight new chapters, including one on the classification of topological states of topological insulators and superconductors and another on Weyl semimetals, as well as elaborated discussions of the Aharonov-Casher effect, topological magnon insulators, topological superconductors and K-theory.

Also available in print.

Mode of access: World Wide Web.

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Antonio S.T. Pires graduated from the University of California in Santa Barbara in 1976. He is a Professor of Physics at the Universidade Federal de Minas Gerais, Brazil researching quantum field theory applied to condensed matter. He is a member of the Brazilian Academy of Science, was the Editor of the Brazilian Journal of Physics, and a member of the Advisory Board of the Journal of Physics: Condensed Matter. He has published the books ADS/CFT correspondence in condensed matter and theoretical tools for spin models in magnetic systems.

Title from PDF title page (viewed on December 6, 2021).