Topological Phases in Condensed Matter Physics

✍ Scribed by Saurabh Basu

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 226
• Category: Library

✦ Synopsis

The book is mainly designed for post-graduate students to learn modern-day condensed matter physics. While emphasizing an experiment called the ‘Quantum Hall effect’, it introduces the subject of 'Topology' and how the topological invariants are related to the quantization of the Hall plateaus. Thus, the content tries to deliver an account of the topological aspects of materials that have shaped the study of condensed matter physics in recent times. The subject is often quite involved for a student to grasp the fundamentals and relate them to physical phenomena. Further, these topics are mostly left out of the undergraduate curriculum, although they often require a simplistic view of the concepts involved to be presented pedagogically. The book contains examples, worked-out concepts, important derivations, diagrams for illustration, etc. to aid the understanding of the students. The book also emphasizes the experimental discoveries that put the subject in its perspective and elaborate on the applications which are likely to be of interest to scientists and engineers.

✦ Table of Contents

Foreword
Preface
Acknowledgements
Contents
About the Author
1 Symmetry and Topology
1.1 Introduction
1.2 Gauss-Bonnet Theorem
1.3 Berry Phase
1.4 Discrete Symmetries
1.4.1 Inversion Symmetry
1.4.2 Time Reversal Symmetry
1.4.3 Particle-Hole Symmetry
1.4.4 Chiral Symmetry
References
2 Topology in One-Dimensional (1D) Tight Binding Models
2.1 Su-Schrieffer-Heeger (SSH) Model
2.1.1 Introduction
2.1.2 The SSH Hamiltonian
2.1.3 Topological Properties
2.1.4 Chiral Symmetry of the SSH Model
2.2 Kitaev Model: Topological Superconductivity
2.2.1 Introduction
2.2.2 Two-Site Kitaev Chain
2.2.3 Particle-Hole Symmetry of the Kitaev Model
2.2.4 Winding Number
2.2.5 Majorana Fermions in the Kitaev Model
2.2.6 Energy Spectrum of ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper N) /StPNE pdfmark [/StBMC pdfmarkto.ps: [/EMC pdfmark [/Artifact <> /BDC pdfmark Nps: [/EMC pdfmark [/StBMC pdfmark ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-site Kitaev Model
2.2.7 Topological Properties of the Majorana Modes
2.2.8 Experimental Realization of the Kitaev Chain
2.3 Detecting Majoranas: 4π Periodic Josephson Junctions
2.4 s-Wave Kitaev Chain
2.4.1 Topological Properties
2.4.2 Winding Number and the Phase Diagram
2.4.3 Real Space Analysis: Edge Properties
References
3 Quantum Hall Effect
3.1 Introduction
3.2 General Perspectives
3.3 Quantum Hall Effect and SI Unit of Resistance
3.4 Why Is 2D Important?
3.5 Why Are the Conductivity and the Resistivity Tensors Antisymmetric?
3.6 Translationally Invariant System: Classical Limit of QHE
3.7 Charge Particles in a Magnetic Field: Landau Levels
3.8 Degeneracy of the Landau Levels
3.9 Conductivity of the Landau Levels
3.10 Spin and the Electric Field
3.11 Laughlin's Argument: Corbino Ring
3.12 Edge Modes and Conductivity of the Single Landau Level
3.13 Incompressibility of the Quantum Hall States
3.14 Derivation of the Hall Resistance
3.15 Hall Conductivity and the Chern Number
References
4 Graphene: A Hobby Horse for Studying Topology
4.1 Introduction
4.2 Tight Binding Hamiltonian
4.2.1 Electronic Properties of Graphene
4.3 Dirac Cones: Experimental Confirmation
4.4 Graphene Nanoribbon
4.4.1 Hofstadter Butterfly
4.4.2 Landau Levels in Graphene
4.5 Hall Conductivity of a Graphene Nanoribbon
4.6 Experimental Observation of the Landau Levels in Graphene
4.7 Bilayer Graphene
4.8 Quantum Hall Effect in Bilayer Graphene
References
5 Anomalous Quantum Hall and Spin Hall Effects in Graphene
5.1 Introduction
5.1.1 Berry Phase in Graphene
5.1.2 Symmetries of Graphene
5.2 Semenoff Insulator
5.3 Haldane (Chern) Insulator
5.4 Quantum Anomalous Hall Effect
5.5 Quantum Spin Hall Insulator
5.6 Kane-Mele Model
5.7 Bulk-Boundary Correspondence
5.8 Spin Hall Conductivity
5.9 Rashba Spin-Orbit Coupling
5.9.1 Rashba Spin-Orbit Coupling in Graphene
5.10 Topological Properties: The Z2 Invariant
5.11 Spin Hall Effect
5.11.1 Spin Current
References
6 Fractional Quantum Hall Effect: Role of Coulomb Interactions
6.1 Introduction
6.2 Electrons in the Symmetric Gauge
6.3 The Lowest Landau Level
6.4 The Filling Fraction Revisited
6.5 Fractional Charge and the Hall Conductivity
6.6 Fractional Hall Fluid and the Plasma
6.7 Composite Fermions
6.8 Hierarchy Approach to FQHE
6.9 Fractional Statistics
6.10 Non-abelian Anyons
6.11 The Braid Group
6.12 Fractional Quantum Hall Effect in Graphene
References
7 Topology Beyond Fermionic Systems
7.1 Introduction
7.2 Circuit Equations: Kirchhoff's Law
7.3 Topoelectronic Circuits
7.4 Non-Hermitian Systems
7.4.1 Non-reciprocity in Circuits: Probing Non-Hermitian Topology
7.5 Non-reciprocal SSH Model: PT Broken System
7.5.1 Circuits
7.6 Optomechanical Systems
7.7 Topological Acoustics
7.8 Topological Phenomena in Bosonic Systems
References
Appendix
A.1 Kubo Formula and the Hall Conductivity
A.2 Periodic Table of Topological Materials: Ten-fold Classification
A.2.1 Mathematical Representation of the Symmetries
A.3 Chern Number
A.3.1 Chern Number Using Fukui's Method
A.4 Z2 Invariant
A.4.1 Z2 Invariant Using Fu and Kane Method
Postscript
Bibliography