電子系統中拓樸與關聯效應的交互影響｜國立清華大學博碩士論文庫

簡易檢索 / 詳目顯示

回結果列表

研究生：	周柏豪 Chou, Po-Hao
論文名稱：	電子系統中拓樸與關聯效應的交互影響 Interplay of topology and correlation effects in electronic systems
指導教授：	牟中瑜 Mou, Chung-Yu
口試委員:	王道維 Wang, Daw-Wei 陳柏中 Chen, Po-Chung 仲崇厚 Chung, Chung-Hou 張明哲 Chang, Ming-Che
學位類別：	博士 Doctor
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2018
畢業學年度：	106
語文別：	英文
論文頁數：	122
中文關鍵詞：	近藤絕緣體、拓樸、超導、數值重整化群
外文關鍵詞：	Kondo insulator, topology, superconductivity, numerical renormalization group
相關次數：	點閱：2 下載：0
分享至:	分享至facebook 分享至twitter

查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

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
                                

[1] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95 146802 (2005).
[2] M. Konig, S. Wiedmann, C. Brune, A Roth, H. Buhmann, L. W. Molenkamp, X. L.
Qi, and S. C. Zhang, Sience 318 776 (2007).
[3] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314 1757 (2006).
[4] T. Yoshida, S. Fujimoto, and N. Kawakami, Phys. Rev. B 85, 125113 (2012).
[5] Y. Tada, R. Peters, M. Oshikawa, A. Koga, N. Kawakami, and S. Fujimoto, Phys.
Rev. B 85, 165138 (2012).
[6] S. Raghu, Xl. L. Qi, C. Henerkampp, and S. C. Zhang, Phys. Rev. Lett. 100, 156401
(2008).
[7] Y. Zhang, Y. Rau, and A. Vishwanath, Phys. Rev. B 79, 245331 (2009).
[8] B. Yan, L. Muchler, X.-L. Qi, S.-C. Zhang, and C. Felser, Phys. Rev. B 85, 165125
(2012).
[9] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Rev. Lett. 104 106408 (2010).
[10] D. J. Kim, J. Xia, and Z. Fisk, Nat. Mater. 13, 466 (2014).
[11] T. Yoshida, R. Peters, S. Fujimoto and N. Kawakami, Phys. Rev. B 87, 085134
(2013)
[12] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).
[13] P.W. Anderson, J. Phys. C: Solid St. Phys. 3, 2436 (1970).
[14] K. G. Wilson Rev. Mod. Phys. 47, 773 (1975).
[15] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, K.G., Phys. Rev. B 21,
1003 (1980).
[16] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, K.G., Phys. Rev. B 21,
1044 (1980).
[17] A. C. Hewson, The Kondo problem to heavy fermions (Cambridge Univ. Press, Cam-
bridge New York, 1997).
[18] R. Bulla, T. A. Costi, T. Pruschke, Rev. Mod. Phys. 80, 395 (2008).
[19] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
[20] A. A. Burkov, M. D. Hook, and L. Balents: Phys. Rev. B 84, 235126 (2011).
[21] D. E. Sheehy and J. Schmalian, Phys. Rev. Lett. 99, 226803 (2007).
[22] A. Auerbach and K. Levin, Phys. Rev. Lett. 57, 877 (1986).
[23] A.J. Millis and P.A. Lee, Phys. Rev. B 35, 2294 (1987).
[24] N. Read and D. M. Newns, J. Phys. C: Solid State Phys. 16, 3273 (1983).
[25] L. Zhao, H. Deng, I. Korzhovska, M. Begliarbekov, Z. Chen, E. Andrade, E. Rosen-
thal, A. Pasupathy, V. Oganesyan, and L. Krusin-Elbaum, Nat. Commun. 6, 8279
(2015).
[26] A.-P. Jauho and H. Smith, Phys. Rev. B 47, 4420 (1993).
[27] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. B 82,
045122 (2010).
[28] McMillan, W. L. Phys. Rev. 167, 331 (1968).
[29] Bessas, D; Sergueev, I; Wille, H.-C.; Peron, P; Ebling, D; Hermann R.P. Phys. Rev.
B 86, 224301 (2012).
[30] A. A. Abrikosov, Int. J. Mod. Phys. A 17, 885 (2002).
[31] K.-I. Imura, Y. Yoshimura, Y. Takane, and T. Fukui, Phys. Rev. B 86, 235119 (2012).
[32] Zhao, L; Deng, H; Korzhovska, I; Begliarbekov, M; Chen, Zh; Andrade, E; Rosenthal,
E; Pasupathy, A; Oganesyan, V; Krusin-Elbaum, L. Nat .Commun, 6, 8279 (2015).
[33] Y. E. Kraus, A. Auerbach, H. A. Fertig, and S. H. Simon, Phys. Rev. Lett. 101,
267002 (2008).
[34] Y. E. Kraus, A. Auerbach, H. A. Fertig, and S. H. Simon, Phys. Rev. B 79, 134515
(2009).
[35] Dunster TM (2010) Legendre and related functions. In: Olver Frank W J, Lozier
Daniel M, Boisvert Ronald F, Clark Charles W (eds) NIST handbook of mathematical
functions.
[36] T. Dray, J. Math. Phys. 26, 1030 (1985).
[37] T. T. Wu and C. N. Yang, Nucl. Phys. B 107, 365 (1976).
[38] T. T. Wu and C. N. Yang, Phys. Rev. D 16, 1018 (1977).
[39] N. B. Kopnin, T. T. Heikkil, and G. E. Volovik, Phys. Rev.B 83, 220503 (2011).
[40] V. A. Khodel and V. R. Shaginyan, JETP Lett. 51, 553 (1990).
[41] V. J. Kauppila, F. Aikebaier, and T. T. Heikkil, Phys. Rev. B 93, 214505 (2016).
[42] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-
Herrero, Nature 556, 43 (2018).
[43] J. E. Hirsh, Phys. Rev. Lett. 54, 1317 (1985).
[44] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).
[45] F. C. Zhang, C. Gros, T. M. Rice, H. Shiba, Supercond. Sci. Tech. 1, 36 (1988).
[46] M. U. Ubben and P. A. Lee, Phys. Rev. B 46, 8434 (1992)
[47] A. M. Black-Schaffer and C. Honerkamp, J. Phys.: Condens. Matter 26, 423201
(2014).
[48] D. F. Agterberg and H. Tsunetsugu, Nat. Phys. 4, 639 (2008).
[49] D. J. Scalapino, S. R. White, and S. C. Zhang, Phys. Rev. B 47, 7995 (1993).
[50] P.-H. Chou, L.-J. Zhai, C.-H. Chung, C.-Y. Mou, and T.-K. Lee, Phys. Rev. Lett.
116, 177002 (2016).
[51] S. Chadov,X-L. Qi, J. Kubler, G. H. Fecher,C. Felser, and S-C Zhang, Nature Mate-
rials 9, 541 (2010).
[52] S. Dzsaber, L. Prochaska, A. Sidorenko, G. Eguchi, R. Svagera, M. Waas, A.
Prokofiev, Q. Si, and S. Paschen, Phys. Rev. Lett. 118, 246601 (2017).

簡易檢索 / 詳目顯示

相關論文