Topological States on Interfaces Protected by Symmetry (eBook)

Dr. Ryuji Takahashi Department of Applied Physics, The University of Tokyo takahashi@appi.t.u-tokyo.ac.jp

Supervisor’s Foreword 7
Acknowledgments 8
Contents 9
1 Introduction 11
1.1 Topological Phase of Matters 11
1.2 Organization of the Dissertation 13
References 14
2 Topological Invariant and Topological Phases 15
2.1 Integer Quantum Hall Effect 15
2.1.1 Integer Quantum Hall Effect as a Topological Insulator 17
2.1.2 Adiabatic Charge Polarization by the Berry Phase 19
2.1.3 Laughlin's Gedanen Experiment 21
2.1.4 Physical Picture of the Edge State 24
2.1.5 Berry Curvature in Systems with Broken Time-Reversal Symmetry 26
2.2 Topological Insulator 26
2.2.1 Topological Invariant Under Time-Reversal Symmetry 28
2.2.2 Surface Effective Hamiltonian 34
2.3 Topological Flat-Band States in Honeycomb Lattice 35
2.3.1 Dispersion of the Honeycomb Lattice Model 35
2.3.2 Topological Explanation for Existence of the Flat-Band States 39
2.3.3 Completely Localized Edge States 40
References 43
3 Gapless Interface States Between Two Topological Insulators 45
3.1 Refractive Phenomena of the Surface States ƒ 45
3.2 Gapless Interface States Between Two TIs 49
3.2.1 Paradox in the Refractive Phenomena 49
3.2.2 Gapless States from the Surface Effective Model 50
3.2.3 Proof for the Existence of the Gapless States by the Mirror Chern Number 53
3.3 Gapless Interface States in a Lattice Model 56
3.3.1 Model 56
3.3.2 Calculation of the Mirror Chern Number for the Model 57
3.3.3 Gapless Interface States 63
3.4 Interfacial Fermi Loops and Interfacial Symmetries 65
3.4.1 Interfacial Fermi Loops in the FKM Model 65
3.4.2 Interfacial Fermi Loops from Interfacial Symmetries 69
3.4.3 Interfacial Fermi Loops in the ?-junction Interface 71
References 72
4 Weyl Semimetal in a Thin Topological Insulator 73
4.1 Gapless States Protected by Mirror Symmetry in a Topological Insulator 73
4.1.1 Weyl Semimetal Under Broken Inversion Symmetry 77
4.2 Weyl Semimetal Phase in a Lattice Model 78
4.3 Estimation of the Magnetization at the Phase Transition 80
References 81
5 Summary and Outlook 82
6 Properties of the Chern Numbers 86
6.1 Chern Number 86
6.1.1 Berry Curvature with Time-Reversal Symmetry 87
6.2 Kramers Theorem 88
6.3 Berry Curvature and Chern Number 89
6.3.1 Change of the Chern Number 90
Reference 92
7 Calculation for the Interface Fermi Loops 93
7.1 Classification of Interfacial Systems 93
7.2 Fermi Loop for the IPHS 94
Reference 96
Curriculum Vitae 97