A Brief Introduction to Topology and Differential Geometry in Condensed Matter Physics

Contents Preface Author biography
1 Path integral approach 1.1 Path integral 1.2 Spin 1.3 Path integral and statistical mechanics 1.4 Fermion path integral References and further reading
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 References and further reading
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 fields References and further reading
4 Metric and curvature 4.1 Metric in a vector space 4.2 Metric in a manifold 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 in a manifold 4.8 Integration on manifolds 4.9 Stokes’s theorem 4.10 Homology 4.11 Cohomology 4.12 Degree of a map 4.13 Hopf-Poincare theorem 4.14 Connection 4.15 Covariant derivative 4.16 Curvature 4.17 The Gauss-Bonnet theorem 4.18 Surfaces References and further reading
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 References and further reading
6 Berry connection and the Aharonov-Bohm effect 6.1 Introduction 6.2 Berry phase 6.3 The Aharonov-Bohm effect 6.4 Non-Abelian Berry connections References and further reading
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- Simmons action 7.9 The fractional quantum Hall effect References and further reading
8 Topological insulators 8.1 Two bands insulator 8.2 Nielsen Ninomya theorem 8.3 Haldane model 8.4 Stats at the edge 8.5 Z2 topological invariants
9 magnetic models 9.1 One- dimensional magnetic models 9.2 Two-dimensional nonlinear sigma model 9.3 XY model 9.4 Theta terms
Appendix A A.1 Integral curve A.2 Lie derivative A.3 Interior product
Appendix B B.1 complex manifolds B.2 Complex projective space B.4 Hopf bundle References and further reading
Appendix C C.1 Fubini-Study metric C.2 Quaternions References and further reading
Appendix D D.1 Rings D.2 Equivalence relations D.3 K - theory