A Brief Introduction to Topology and Differential Geometry in Condensed Matter Physics / Antonio Sergio Teixeira Pires.

Format:

Book

Author/Creator:

Pires, Antonio Sergio Teixeira, author.

Contributor:

Institute of Physics (Great Britain), publisher.

Series:

IOP expanding physics.

IOP Ebooks Series

Language:

English

Subjects (All):

Condensed matter--Mathematics.

Condensed matter.

Geometry, Differential.

Mathematical physics.

Topology.

Physical Description:

1 online resource (various pagings) : illustrations (some color).

Edition:

Second edition.

Place of Publication:

Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, [2021]

System Details:

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.

Biography/History:

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.

Summary:

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.

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.

Notes:

"Version: 202111"--Title page verso.

Includes bibliographical references.

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

Description based on print version record.

ISBN:

9780750339544

0750339543

9780750339551

0750339551

OCLC:

1288247152