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

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 classificati...

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Bibliographic Details
Online Access:	Full text (MCPHS users only)
Main Author:	Pires, A. (Antonio) (Author)
Format:	Electronic eBook
Language:	English
Published:	Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, 2021
Edition:	Second edition.
Subjects:	Mathematical physics. Condensed matter > Mathematics. Topology. Geometry, Differential.
Local Note:	ProQuest Ebook Central

Table of Contents:

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-Poincaré 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.

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

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