1 Introduction
1.1 From the Hall Effect to ntum Spin Hall Effect
1.2 Topological Insulators as Generalization of ntum Spin Hall Effect
1.3 Topological Phases in Superconductors and Superfluids
1.4 Dirac Equation and Topological Insulators
1.5 Summary: The Confirmed Family Members
1.6 Further Reading
References
2 Starting from the Dirac Equation
2.1 Dirac Equation
2.2 Solutions of Bound States
2.2.1 Jackiw-Rebbi Solution in One Dimension
2.2.2 Two Dimensions
2.. Three and Higher Dimensions
. Why Not the Dirac Equation
2.4 dratic Correction to the Dirac Equation
2.5 Bound State Solutions of the Modified Dirac Equation.
2.5.1 One Dimension: End States
2.5.2 Two Dimensions: Helical Edge States
2.5.3 Three Dimensions: Surface States
2.5.4 Generalization to Higher-Dimensional Topological Insulators
2.6 Summary
2.7 Further Reading
References
3 Minimal Lattice Model for Topological Insulator
3.1 Tight Binding Approximation
3.2 From Continuous to Lattice Model
3.3 One-Dimensional Lattice Model
3.4 Two-Dimensional Lattice Model
3.4.1 Integer ntum Hall Effect
3.4.2 ntum Spin Hall Effect
3.5 Three-Dimensional Lattice Model
3.6 Parity at the Time Reversal Invariant Momenta
3.6.1 One-Dimensional Lattice Model
3.6.2 Two-Dimensional Lattice Model
3.6.3 Three-Dimensional Lattice Model
3.7 Summary
References
4 Topological Invariants
4.1 Bloch Theorem and Band Theory
4.2 Berry Phase
4.3 ntum Hall Conductance and Chern Number
4.4 Electric Polarization in a Cyclic Adiabatic Evolution..
4.5 Thouless Charge Pump
4.6 Fu-Kane Spin Pump
4.7 Integer ntum Hall Effect: Laughlin Argument
4.8 Time Reversal Symmetry and the Z2 Index
4.9 Generalization to Two and Three Dimensions
4.10 Phase Diagram of Modified Dirac Equation
4.11 Further Reading
References
5 Topological Phases in One Dimension
5.1 Su-Schrieffer-Heeger Model for Polyacetylene
5.2 Ferromagnet with Spin-Orbit Coupling
5.3 p-Wave Pairing Superconductor
5.4 Ising Model in a Transverse Field
5.5 One-Dimensional Maxwell's Equations in Media
5.6 Summary
References
6 ntum Spin Hall Effect
6.1 Two-Dimensional Dirac Model and the Chern Number
6.2 From Haldane Model to Kane-Mele Model
6.2.1 Haldane Model
6.2.2 Kane-Mele Model
6.3 Transport of Edge States
6.3.1 Landauer-Biittiker Formalism
6.3.2 Transport of Edge States
6.4 Stability of Edge States
6.5 Realization of ntum Spin Hall Effect in HgTe/CdTe ntum Well
6.5.1 Band Structure of HgTe/CdTe ntum Well..
6.5.2 Exact Solution of Edge States
6.5.3 Experimental Measurement
6.6 ntum Hall Effect and ntum Spin Hall Effect:A Case Study
6.7 Coherent Oscillation Due to the Edge States
6.8 Further Reading
References
7 Three-Dimensional Topological Insulators
7.1 Family Members of Three-Dimensional Topological Insulators...
7.1.1 Weak Topological Insulators: PbxSnl-xTe
7.1.2 Strong Topological Insulators: Bil-xSbx
7.1.3 Topological Insulators with a Single Dirac Cone: BiSe and BiTe
7.1.4 Strained HgTe
7.2 Electronic Model for BiSe
7.3 Effective Model for Surface States
7.4 Physical Properties of Topological Insulators
7.4.1 Absence of Backscattering
7.4.2 Weak Antilocalization
7.4.3 Shubnikov-de Haas Oscillation
7.5 Surface ntum Hall Effect
7.6 Surface States in a Strong Magnetic Field
7.7 Topological Insulator Thin Film
7.7.1 Effective Model for Thin Film
7.7.2 Structural Inversion Asymmetry
7.7.3 Experimental Data of ARPES
7.8 HgTe Thin Film
7.9 Further Reading
References
8 Impurities and Defects in Topological Insulators
8.1 One Dimension
8.2 Integral Equation for Bound State Energies
8.2.1 δ-potential
8.3 Bound States in Two Dimensions
8.4 Topological Defects
8.4.1 Magnetic Flux and Zero-Energy Mode
8.4.2 Wormhole Effect
8.4.3 Witten Effect
8.5 Disorder Effect to Transport
8.6 Further Reading
References
9 Topological Superconductors and Superfluids
9.1 Complex (p + ip)-Wave Superconductor of Spinless or Spin-Polarized Fermions
9.2 Spin-Triplet Pairing Superfluidity: 3He-A and 3He-B Phases
9.2.1 3He: Normal Liquid Phase
9.2. He-B Phase
9.. He-A Phase: Equal Spin Pairing
9.3 Spin-Triplet Superconductor: Sr2RuO4
9.4 Superconductivity in Doped Topological Insulators
9.5 Further Reading
References
10 Majorana Fermions in Topological Insulators
10.1 What Is the Majorana Fermion?
10.2 Majorana Fermions in p-Wave Superconductors
10.2.1 Zero-Energy Mode Around a ntum Vortex
10.2.2 Majorana Fermions in Kitaev's Toy Model
10.. si-One-Dimensional Superconductor
10.3 Majorana Fermions in Topological Insulators
10.4 Detection of Majorana Fermions
10.5 Sau-Lutchyn-Tewari-Das Sarma Model for Topological Superconductor
10.6 Non-Abelian Statistics and Topological ntum Computing
10.7 Further Reading
References
11 Topological Anderson Insulator
11.1 Band Structure and Edge States
11.2 ntized Anomalous Hall Effect
11.3 Topological Anderson Insulator
11.4 Effective Medium Theory for Topological Anderson Insulator
11.5 Band Gap or Mobility Gap
11.6 Summary
11.7 Further Reading
References
12 Summary: Symmetry and Topological Classification
12.1 Ten Symmetry Classes for Noninteracting Fermion Systems
12.2 Physical Systems and the Symmetry Classes
12.2.1 Standard (Wigner-Dyson) Classes
12.2.2 Chiral Classes
12.. Bogoliubov-de Gennes (BdG) Classes
1. Characterization in the Bulk
12.4 Five Types in Each Dimension
12.5 Conclusion
12.6 Further Reading
References
A Derivation of Two Formulae
A.1 ntization of the Hall Conductance
A.2 A Simple Formula for the Hall Conductance
B Time Reversal Symmetry
B.1 Classical Cases
B.2 Time Reversal Operator Θ
B.3 Time Reversal for a Spin-1/2 System
Index
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