在本論文裡，我們探討了電子系統中拓樸與關聯效應的交互影響；
並檢視了三種電子關聯系統中的拓樸效應: 從拓樸安德森晶格模型中的
近藤-狄拉克費米子、薄拓樸絕緣體中由不同表面態電子配對產生的p-波
超導、到受張力石墨烯中超導態的庫柏對密度波。
對於一個在導帶與局域化電子間具有自旋-軌道耦合形式混成的廣義
安德森晶格模型，我們的研究表明在廣泛的溫度與其他參數空間內，狄
拉克費米子可以出現於強與弱拓樸絕緣相之間。而當考慮導帶電子間的
庫倫作用後，臨界點附近產生的
狄拉克費米子將表現出狄拉克液體的行為。
在拓樸絕緣體系統中，我們發現由於幾何和超導不穩性導致拓樸超導
的可能性，即藉由調控薄拓樸絕緣體的厚度，來控制拓樸絕緣體不同表面
態間的電子配對。此外我們也發現由於曲率效應，球面的表面態導致的超
導可以自發產生渦旋以及在其中心的馬約拉那費米子。
最後，當石墨烯的皺褶足夠大時將會產生拓樸平能帶。而當
強赫巴德作用存在時，我們發現對於一給定週期的皺褶，手性d-波超導可以
穩定於輕微參雜的石墨烯。更重要的是，研究表明在有限溫度時可以產生
兩倍皺褶波長的庫柏對密度波超導態。

In this thesis, we investigate the interplay of topological and correlation in electronic
systems. Topological effects in three correlated electronic systems are examined: from
Kondo-Dirac fermions in a topological Anderson lattice, inter-surface p-wave pairing
in thin topological insulators, to strain induced superconducting pair density waves in
graphene. It is shown that in a generalized Anderson lattice with spin-orbit type hy-
bridization between conduction electrons and localized electrons, Dirac fermions emerge
over large temperature and parameter regime between strong and weak topological in-
sulating phases. The massless Dirac fermions form a critical point with nearby regime
characterized by the Dirac liquids when Coulomb interaction for conduction electrons is in-
cluded. In the system of topological insulators, we find that the interplay of geometry and
superconducting instability leads to the possibility of forming topological superconductiv-
ity due to inter-surface pairing by tuning the thickness of the thin topological insulators.
Furthermore, it is shown that superconductivity on spherical surfaces can spontaneously
generate vortices with a Majorana fermion at the center due to the curvature effect. Fi-
nally, it is shown that ripples in graphene, if their amplitudes are large enough, generate
topological flat bands. In the presence of strong Hubbard U interaction, we find that for
a given wavelength of ripple, chiral d-wave superconductivity can be stablized even in
slightly doped graphene. Most importantly, it is shown that superconducting pair density
wave state emerges at a finite temperature regime with doubled wavelength.

Contents IV
List of Figures VIII
1 Introduction 1
2 Kondo effect with the presence of spin-orbit type hybridization 4
2.1 Anderson model with the presence of spin-orbit type hybridization . . . . . 4
2.1.1 Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Anderson lattice model . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Generalized Kondo model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Single impurity Kondo model . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Kondo lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Poor man’s RG analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Analysis of single impurity Kondo model . . . . . . . . . . . . . . . 6
2.3.2 Analysis of Kondo lattice model . . . . . . . . . . . . . . . . . . . . 8
3 Numerical renormalization group(NRG) analysis 11
3.1 Introduction of Wilson’s NRG . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Kondo model in continuous energy basis . . . . . . . . . . . . . . . 11
3.1.2 Logarithmic discrete Hamiltonian . . . . . . . . . . . . . . . . . . . 12
3.1.3 Wilson’s chain model . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.4 Iteration and truncation of Wilson’s NRG . . . . . . . . . . . . . . 14
3.1.5 Symmetry of Wilson’s chain model . . . . . . . . . . . . . . . . . . 17
3.1.6 Completed basis and Hamiltonian matrix element . . . . . . . . . . 17
3.1.7 RG flow and fixed point . . . . . . . . . . . . . . . . . . . . . . . . 18
II
3.1.8 Calculation of thermodynamic and static properties . . . . . . . . . 18
3.2 NRG in the generalized Kondo model . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 2D generalized Kondo model in angular momentum space . . . . . . 20
3.2.2 Two channel Wilson’s chain model . . . . . . . . . . . . . . . . . . 22
3.2.3 Symmetry of the two channel Kondo model(2CK) . . . . . . . . . . 23
3.2.4 Detail of 2CK calculation . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.5 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Dirac fermion in Kondo lattice at finite temperature 26
4.1 Slave boson mean-field approach . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Topological classification and phase diagram . . . . . . . . . . . . . . . . . 27
4.3 Dirac fermion at finite temperature . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Fermionic finite-temperature critical point and corresponded physical prop-
erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Self energy by including boson correction . . . . . . . . . . . . . . . 32
4.4.2 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.3 Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.4 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Superconductive states with triplet pairing induced by geometry 42
5.1 P-wave pairing induced in thin film geometry . . . . . . . . . . . . . . . . 42
5.1.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1.2 The mean-field Hamiltonian and the p-wave gap equation . . . . . . 44
5.1.3 Thickness-dependent T c and SC phase transition . . . . . . . . . . . 46
5.2 Triplet pairing on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.1 The Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.2 The mean-field Hamiltonian and gap equation . . . . . . . . . . . . 50
5.2.3 Vortex on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Strain field induced pair density wave in graphene 54
6.1 Flat band under strain field and the t-J model in honeycomb lattice . . . . 55
6.2 Mean-field Hamiltonian and Equations . . . . . . . . . . . . . . . . . . . . 57
6.3 Exotic superconducting pair density with momentum Q/2 at zero temper-
ature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
III
6.4 Finite temperature superconducting phase and superfluid density . . . . . 61
7 Conclusion and outlook 65
A Calculation details of Wilson’s chain model 67
A.1 U S (1) × U Q (1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2 SU S (2) × U Q (1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.3 SU S (2) × SU Q (2) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B Calculation details of 2CK model 79
B.1 SU j (2) × SU Q (2) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 79
C Superconductive properties of the TI nanoflakes 95
C.1 Longitudinal electron-phonon interaction . . . . . . . . . . . . . . . . . . . 95
C.2 The eigenfunctions of Dirac Hamiltonian at finite size . . . . . . . . . . . . 95
C.3 Interaction projecting into surface fermion basis . . . . . . . . . . . . . . . 98
C.4 The p-wave SC gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . 100
D Superconductive properties of Spherical TI 103
D.1 Exact eigenfunctions of H D in global frame . . . . . . . . . . . . . . . . . . 103
D.1.1 Surface state energy in large R . . . . . . . . . . . . . . . . . . . . 106
D.1.2 Rotating eigenfunctions into local coordinate . . . . . . . . . . . . . 108
D.2 Derivation of effective surface Hamiltonian . . . . . . . . . . . . . . . . . . 109
D.3 Mean-field equations of ∆ ˜ s
˜
s 0
jj 0 , ¯ m ¯ m 0
. . . . . . . . . . . . . . . . . . . . . . . . 112
E Singlet superconducting Hamiltonian on the
strain graphene 114
E.1 Matrix elements of the effective 1D chain model . . . . . . . . . . . . . . . 114
E.2 The effective 1D chain model with translational symmetry . . . . . . . . . 117
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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