In the last 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. Physicists have been creative in producing models for actual physical phenomena which realize mathematically exotic concepts and new phases have been discovered in condensed matter in which topology plays a leading role. An important classification paradigm is the concept of topological order, where the state characterizing a system does not break any symmetry, but it defines a topological phase in the sense that certain fundamental properties change only when the system passes through a quantum phase transition.
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. It conveys to physicists the basis for many mathematical concepts, avoiding the detailed formality of most textbooks.
Author(s): Antonio Sergio Teixeira Pires
Series: IOP Concise Physics
Publisher: IOP Publishing
Year: 2019
Language: English
Pages: 172
City: Bristol
PRELIMS.pdf
Preface
Acknowledgements
Author biography
Antonio Sergio Teixeira Pires
CH001.pdf
Chapter 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
CH002.pdf
Chapter 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
CH003.pdf
Chapter 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
References and further reading
CH004.pdf
Chapter 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
References and further reading
CH005.pdf
Chapter 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
CH006.pdf
Chapter 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
References and further reading
CH007.pdf
Chapter 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
References and further reading
CH008.pdf
Chapter 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
References and further reading
CH009.pdf
Chapter 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
References and further reading
APP1.pdf
Chapter
A.1 Integral curve
A.2 The Lie derivative
A.3 Interior product
References and further reading
APP2.pdf
Chapter
B.1 Complex manifolds
B.2 Complex projective space
B.3 Hopf bundle
References and further reading
APP3.pdf
Chapter
C.1 Fubini–Study metric
C.2 Quaternions
References and further reading
APP4.pdf
Chapter
D.1 Rings
D.2 Equivalence relations
D.3 Sum of vector bundles
D.4 K-theory
References and further reading
