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

- Antonio Sergio Teixeira Pires

- March 2019

### Description

In recent 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 other textbooks.

### About Editors

Antonio Sergio Teixeira Pires is a professor of physics at the Universidade Federal de Minas Gerais, in Brazil. He works in quantum field theory applied to condensed matter, is a member of the Brazilian Academy of Science, was an editor of the*Brazilian Journal of Physics*and a member of the Advisory Board of the

*Journal of Condensed Matter Physics*. Pires is also the author of

*AdS/CFT Correspondence in Condensed Matter*, which is also published as part of the Concise Physics series.

### Table of Contents

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

### Bibliographic

Paperback ISBN: 9780750329736

Ebook ISBN: 9781643273730

DOI: 10.1088/2053-2571/aaec8f

Publisher: Morgan & Claypool Publishers