本論文以理論計算及模型研究兩個具有新穎電子特性之狄拉克半金屬。其一為KHgX系列拓樸絕緣體，透過壓力，我們將KHgX轉變為具有鏡像陳數為負三之狄拉克半金屬。再近而透過對稱性破壞，此半金屬向更進一步轉變為另一拓樸絕緣體。其二為PbO2，我們發現PbO2是一個具有二次方曲率的狄拉克半金屬。我們也提出一個分類法去辨別該狄拉克金屬的拓樸類。此論文由四個章節架構而成。第一章簡介拓樸數的定義。第二章簡介侷域近似法。第三及四章介紹KHgX及PbO2。

Dirac semimetals (DSMs) are characterized by having four-fold degenerate band crossing points in their band structures, where the low-energy behavior is described by the relativistic Dirac equation. In this work, based on the density functional theory calculation and theoretical modeling, we investigate two Dirac semimetals having novel electronic properties. For the first one, we found that the topological non-symmorphic insulator (TNCI) KHgX where X=As,Sb,Bi can be transformed into a DSM by stress. The DSM phase of KHgX has mirror Chern number Cm=-3 on the kz=0 plane in the Brillouin zone, leading to unique pattern of the surface band structure. With the increasing stress and with symmetry-breaking, we observed the evolution of the surface band topology in accordance with the bulk topological phase transition, i.e., TNCI(X=2)>DSM>TNCI(X=3), where X denotes the Z4 invariant defined for the TNCI. For the second, we found that the Dirac node in the material PbO2 has quadratic energy dispersion on the kx-ky plane, dubbed quadratic Dirac node (QDN). We show that the minimal model describing the QDN in PbO2 can be block diagonalized into two weakly-interacting double Weyl nodes with opposite chiral charges. To further investigate the topology of the QDN, we propose a classification scheme for DNs protected by four-fold rotation which is based on the Wilson loop. According to our classification scheme, we argue that the quadratic behavior manifested by the QDN is not robust, in the sense that perturbations can “linearize” these quadratic energy bands on the plane perpendicular to the fold-fold axis. This thesis contains four chapters which are written independently, briefly described as follows: Chapter 1 introduces the topological invariants in condense matter systems, paving the way for the study of topological materials. The introduction was written in detail, including many proofs that are usually omitted in the literature. Chapter 2 demonstrates the implementation of the tight-binding approximation to crystal problems, written in the language of ket and bra, and with the the notation similar to technical articles of the software Wannier90. Due to the small dimensionality, tight-binding Hamiltonian is very useful for numerically calculating the topological invariants. Chapter 3 and 4 respectively discuss the materials KHgX and PbO2 just mentioned.

1 Introduction to Topological Invariants in Condense Matter Systems 4
1.1 Berry Phase, Connection and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 The Abelian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 The non-Abelian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Chern Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Chern Number as the Winding Number of the Berry Phase . . . . . . . . . . . . . . 15
1.3 Mirror Chern Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Mirror Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Mirror Gauge, Mirror Subspace and Mirror Chern Number . . . . . . . . . . . . . . 18
1.4 Z2 Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Time-Reversal Operator and the Sewing Matrix . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Time-Reversal Gauge and Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.3 Z2 Invariant as the Time-Reversal Polarization and Spin Chern Number . . . . . . . 29
1.4.4 Z2 Invariant in terms of the Sewing Matrix . . . . . . . . . . . . . . . . . . . . . . 32
1.4.5 Appendix-About the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Introduction to the Tight-Binding Approximation 41
2.1 The Tight-Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.1 Step I-Setting up the Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.2 Step II-Writing down the exact Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 42
2.1.3 Step III-Approximating the Exact Hamiltonian using Atomic Orbitals . . . . . . . . 43
2.1.4 Lemmas for the Basis States in Real and Reciprocal space . . . . . . . . . . . . . . 45
2.1.5 Step IV-Transforming to the Reciprocal Space . . . . . . . . . . . . . . . . . . . . 47
2.2 Wannier Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Position Operator and Electric Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.1 Position Operator and its Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2 One Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.3 Three Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 Dirac Semimetal Phase in the Topological Non-symmorphic Insulator KHgX (X=As,Sb,Bi) 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 The Crystal and Bulk Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 The Z4 Invariant and Mirror Chern Number . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Surface States Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Quadratic Dirac Node and the Topological Classification 70
4.1 Crystal Structure and Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Effective Hamiltonian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Appendix-k.p Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 k.p Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Quasi-Degenerate Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Symmetry Constraints (Method of Invariants) . . . . . . . . . . . . . . . . . . . . . 75
4.4 Appendix-k.p Effective Hamiltonian for PbO2 . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.1 Step I-Obtaining the Matrix Representation of the Symmetry Operators . . . . . . . 76
4.4.2 Step II-Classifying the Irreducibe Representation of the Dirac Matrices . . . . . . . 79
4.4.3 Step III-Constructing the Effective Hamiltonian to Desired Order . . . . . . . . . . 81
4.5 Appendix-Classification of DNs protected by four-fold Rotation . . . . . . . . . . . . . . . 84
4.5.1 The generalized rotation matrix and the quantum number q=0,1,2,3 . . . . . . 84
4.5.2 The quantum number q changes when the Wilson band goes across +-pi. . . . . . . 86
4.5.3 Constraints by inversion and time-reversal symmetry . . . . . . . . . . . . . . . . . 87
4.5.4 Classification of DNs by the Wilson band structures . . . . . . . . . . . . . . . . . 88

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