The Role of Topology in Materials

Foreword

Preface

Contents

Contributors

Introductory

1 Importance of Topology in Materials Science

1.1 Introduction

1.2 Essentials of Topology

1.2.1 Genus and Euler Characteristics

1.2.2 Network Topology

1.2.3 Geometry-Topology Interrelationship

1.3 Topological Taxonomy of Functional Materials

1.3.1 Nanocarbons

1.3.2 Soft and Polymeric Materials

1.3.3 Minimal Periodic Surfaces

1.4 Topological Phases in Condensed Matter

1.4.1 Real-Space Topological Materials

1.4.2 Dirac Materials

1.4.3 Topological Insulators and Topological Superconductors

1.4.4 Weyl Semimetals

1.4.5 Other Topological Materials

1.5 Metrology and Techniques

1.5.1 High-Resolution Electron Microscopy

1.5.2 Nonlinear Optical Imaging

1.5.3 X-Ray Tomography and Electron Holography

1.5.4 X-Ray and Neutron Scattering

1.5.5 Elasticity and Deformation Energy Characterization

1.5.6 Topological Correlators and Other Metrics

1.6 Computational Topology of Materials

1.6.1 Topological Databases and Visualizing Topology

1.6.2 Miscellaneous Topics

1.7 Conclusion

References

2 Topology and Geometry in Condensed Matter

2.1 Topology

2.1.1 Introduction

2.1.2 Classification of Vector Fields with Homogeneous Boundary Conditions

2.1.3 Classification of Defects in Vector Fields (Mainly Spin Fields)

2.1.4 Defects and Homogeneous Boundary Conditions

2.2 Geometry

2.2.1 Energy

2.2.2 Geometry with Intrinsic Length: The Cylinder

2.2.3 Geometry with Intrinsic Length: Plane with a Disc Missing

2.2.4 Interaction Between Geometry and Physical Field

2.2.5 Chirality of 1d Spin Configurations

2.3 Quantum Potential, Thin Tubes, Knots

2.4 Conclusions

References

Condensed Matter Materials Physics

3 Topology-Induced Geometry and Properties of Carbon Nanomaterials

3.1 Introduction

3.1.1 Carbon as a Building Block

3.1.2 Defect in sp2 Nanocarbon

3.2 Topology-Induced Geometry in sp2 Nanocarbon

3.2.1 Surface Curvature Generation in Graphene Sheets

3.2.2 Plastic Deformation of Carbon Nanotubes

3.3 Stone-Wales Defect

3.3.1 Symmetry Breaking by C–C Bond Rotation

3.3.2 Formation Energy

3.3.3 Out-of-Plane Displacement

3.3.4 Microscopic Observation

3.4 Defect of 5–7 Paired Type

3.4.1 Dissociation of a SW Defect

3.4.2 As a Seed of Surface Curvature

3.5 Peanut-Shaped C60 Polymers

3.5.1 Fusion of C60 Molecules

3.5.2 TLL State in C60 Polymers

3.5.3 Topology-Based Understanding

3.5.4 Curvature-Based Understanding

3.5.5 Electron-Phonon Coupling in C60 Polymers

3.6 Carbon Nanocoil

3.6.1 Benefit from Coiled Structure

3.6.2 Atomistic Modeling

3.6.3 Experimental Realization

3.6.4 Theoretical Prediction

3.7 State-of-the-Art Curved sp2 Nanocarbons

3.7.1 Nano-“Pringles”

3.7.2 Nano- “Tetrapod”

3.7.3 Nano-“Schwarzite”

3.8 Perspective

References

4 Topology by Design in Magnetic Nano-materials: Artificial Spin Ice

4.1 Introduction

4.2 Frustration, Topology, Ice, and Spin Ice

4.3 Simple Artificial Spin Ices

4.3.1 Kagome Spin Ice

4.3.2 Square Ice

4.4 Exotic States Through Vertex-Frustration

4.5 Emergent Ice Rule, Charge Screening, and Topological Protection: Shakti Ice

4.6 Dimensionality Reduction: Tetris Ice

4.7 Polymers of Topologically Protected Excitations: Santa Fe Ice

4.8 Conclusions

References

5 Topologically Non-trivial Magnetic Skyrmions in Confined Geometries

5.1 Introduction

5.2 Topological Effect in Magnetic Skyrmions

5.2.1 Topology in Magnetic Materials

5.2.2 Topological Stability of Magnetic Skyrmions and Emergent Magnetic Monopoles

5.2.3 Topological and Skyrmion Hall Effect

5.2.4 Skyrmion-Based Racetrack Memory (RM)

5.3 Origin of Magnetic Skyrmion

5.3.1 Magnetic Phase Diagram in Chiral Magnets

5.3.2 Mechanism of DM Interaction

5.4 Magnetic Skyrmions in Confined Geometries

5.4.1 Sample Fabrication Techniques

5.4.2 Lorentz TEM

5.4.3 Off-Axis Electron Holography for Imaging Magnetic Contrast

5.4.4 Edge-Mediated Skyrmion Phase and Field-Driven Cascade Phase Diagram

5.4.5 High Flexibility of Geometrically-Confined Skyrmions

5.5 Conclusions

References

6 Topological Phases of Quantum Matter

6.1 Introduction

6.2 Topology in Condensed Matter Physics

6.3 HgTe/CdTe Quantum Wells and Quantum Spin-Hall Insulators

6.4 Z2 Topological Insulators in Three Dimensions

6.5 Topological Crystalline Insulators

6.6 Topological Semi-metals

6.7 Topological Superconductivity

6.8 Strongly Correlated Topological Materials

6.9 Outlook and Conclusions

References

Biology and Mathematics

7 Theoretical Properties of Materials Formed as Wire Network Graphs from Triply Periodic CMC Surfaces, Especially the Gyroid

7.1 Introduction

7.1.1 Classical Geometry of the Gyroid and Graph Approximation for the Channels

7.1.2 P and D Surfaces

7.2 Theory

7.2.1 Overview

7.2.2 Summary of the Methods

7.2.3 The Gyroid Without Magnetic Field

7.2.4 Enhanced Symmetries from a Re-gauging Groupoid

7.2.5 Slicing, Chern Classes and Stability Under Perturbations

7.2.6 Possible Experimental Verification

7.2.7 The P Wire Network Without Magnetic Field

7.2.8 The D Wire Network and the Honeycomb Lattice Without Magnetic Field

7.3 Noncommutative Approach in the Presence of a Magnetic Field

7.3.1 Gyroid in the Presence of a Magnetic Field

7.3.2 P Wire Network in a Magnetic Field

7.3.3 D Wire Network in a Magnetic Field

7.3.4 Honeycomb in a Magentic Field

7.3.5 Possible 3d Quantum Hall Effect

7.4 General Theory and Possible Material Design

7.5 Discussion and Conclusion

References

8 Entangled Proteins: Knots, Slipknots, Links, and Lassos

8.1 Introduction

8.2 Entanglement in Proteins

8.3 Proteins with Knots and Slipknots

8.3.1 Classification and Description of Knots

8.3.2 Proteins with Knots and Slipknots – KnotProt Server and Database

8.3.3 Knotting Fingerprint for Knots and Slipknots

8.3.4 Folding of Knotted Proteins

8.3.5 Function of Knotted Proteins

8.4 Links in Proteins

8.5 Proteins with Lassos

8.6 Conclusions

References

Soft Matter and Biophotonics

9 Topology in Liquid Crystal Phases

9.1 Introduction

9.2 Schlieren Textures and Two-Dimensional Nematics

9.3 The Homotopy Theory of Defects

9.3.1 Point Defects: Hedgehogs

9.3.2 Disclination Loops

9.3.3 The Pontryagin–Thom Construction

9.4 Illustrations in Liquid Crystals

9.4.1 Skyrmions

9.4.2 Colloids

9.4.3 Torons and Hopf Textures

9.5 Smectics

9.6 Geometry of Line Fields

9.6.1 Umbilics

9.6.2 Chirality Pseudotensor

9.7 Cholesterics

9.7.1 ? Lines: Defects in the Pitch

9.8 Knotted Fields

9.8.1 Homotopy Classification

9.8.2 Construction of Knots in Nematics

References

10 Topologically Complex Morphologies in Block Copolymer Melts

10.1 Introduction

10.2 AB Block Copolymers

10.3 ABC Block Copolymers

10.4 Blending Molecular Architectures

10.5 Concluding Remarks

References

11 Topology of Minimal Surface Biophotonic Nanostructures in Arthropods

11.1 Introduction

11.2 Topology of Arthropod Biophotonic Nanostructures

11.3 Self-assembly of Minimal Surface Biophotonic Nanostructures

11.4 Biomimetic Potential of Minimal Surface Biophotonic Nanostructures

11.5 Conclusion

References

Index