博碩士論文 108222032 詳細資訊




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姓名 連士權(Shih-Chuan Lien)  查詢紙本館藏   畢業系所 物理學系
論文名稱
(Modification of Distributional Exact Diagonalization Approach for Single Impurity Anderson Model)
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摘要(中) 近藤效應(Kondo effect)在1934年首先在實驗中發現,在非磁性金屬中參雜帶磁性的原子在電阻溫度線圖上會發現一最小值。近藤淳以sd exchange 成功解釋此最小值的出現,因而稱為近藤效應。這與Anderson Impurity model中雜質項加入的交互作用能量息息相關。
此篇論文我們利用Distributional Exact Diagonalization近似法計算出Anderson Single Impurity model的譜函數(spectral function)。雖然有許多不同的方法可以用來計算,但是他們存在某些缺點,而不能廣泛應用在所有情況。Distributional Exact Diagonalization近似法藉由隨機選取交互作用前譜函數(spectral function)的能量點,將連續的Anderson模型分割成數組有限模型。接著計算各組self-energy做平均後求得連續函數對應的self-energy。Granath and Strand提到若接受所有隨機選取的點,Kondo resonance peak 不會出現,除非只選擇交互作用前後基態電子數一致的組。此限制可以用Friedel sum rule理解。一個連續Anderson模型在費米能量附近的態符合費米液體(Fermi liquid)的假設,因此可藉由Friedel sum rule將費米能量電子數與金屬中巡游電子波函數的相位偏移作連結。不論交互作用大小Friedel sum rule都適用,因此在無交互作用情況下費米能量附近的Kondo peak被保留。
若是交互作用前後基態電子數一致代表Friedel sum rule同樣可用在有限Anderson模型。我們以各組兩個態的情況舉例,說明在能量不對稱情況下Friedel sum rule該如何應用,並提出此電子數一致的限制應被調整以得到更高精度的解。
摘要(英) In 1934 Kondo effect was first observed in experiments when magnetic impurities were embedded in metallic materials. A resistance minimum appears when lowering the temperature of the system. Jun Kondo explained this phenomenon with a sd-exchange energy. This is connected to the interaction term in Anderson single impurity model.
In this thesis we calculate the energy spectral function by using Distributional Exact Diagonalization approach. Although there are several methods to numerically calculate the Anderson single impurity model, there are flaws that could not be apply to general conditions. The Distributional Exact Diagonalization approach divides the continued spectrum in Anderson impurity model into several groups of finite Anderson impurity model. Then the interacting energy spectrum can be obtained by averaging all the self-energy of each individual group. Granath and Strand mentioned that the Kondo resonance peak can only appear if requiring that the electron number of ground state of the finite Anderson impurity model to be the same in both interacting and non-interacting case. This constrain can be understood with Friedel sum rule. Since the Anderson impurity model meets the Fermi liquid assumption in the vicinity of Fermi energy, we can connect the number of electron and phase shift at the Fermi energy. This assumption is valid for different strength of interaction, and that is how Kondo peak appear.
If the constrain in Granath and Strand is valid this impies that Friedel sum rule is also valid for finite Anderson impurity model. We take a two-pole group of finite Anderson impurity model for example, and explain how Friedel sum rule can be apply for non-symmetry energy case. We conclude that the constrain could be modified to find a better solution in low number of group cases.
關鍵字(中) ★ 近藤效應
★ 安德森模型
★ 精確對角化
關鍵字(英) ★ Kondo Effect
★ Anderson Impurity Model
★ Distributional Exact Diagonalization
論文目次 1 Introduction 1
2 Quantum Dot 3
2.1 Darwin-Fock Spectrum 3
2.1.1 Dimensionless Units 3
2.1.2 Hamiltonian 4
3 Kondo Effect 7
3.1 Single Impurity Anderson Model 8
3.1.1 Impurity Green’s function 9
3.1.2 Coulomb interaction 12
3.1.3 Self-consistence procedure 14
3.2 Schrieffer-Wolff Transformation 17
3.3 Kondo Hamiltonian 23
4 Distributional Exact Diagonalization Simulation 30
4.1 Exact Diagonalization 30
4.2 Lehmann representation 33
4.3 The Distributional Diagonalization Algorithm 35
5 Results and Discussion 39
5.1 Impurity Spectrum 39
5.2 Modification of Constraint 42
5.2.1 Ground state conditions 42
5.2.2 Friedel sum rule 50
5.2.3 Kondo resonance peak 52
5.2.4 possible solutions 57
6 Conclusion 58
Appendix A Spectral Representation 59
A.1 Real Time Green’s Function 59
A.2 Imaginary Time Green’s Function 60
Appendix B Equation of Motion 63
References 67
參考文獻 [1] de Haas, de Boer, and van den Berg, The electrical resistance of gold, copper, and lead at low temperatures, Physica 1, 1115 (1934).
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[3] P. W. Anderson, Localized magnetic states in metals, Phys. Rev. 124, 41–53 (1961).
[4] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).
[5] D. G.-G. et al., Science 391, 156 (1998).
[6] M. Kastner, Physics Today 46, 24 (1993).
[7] Quaas N, Wenderoth M, Weismann A, Ulbrich R G and Schönhammer K, Phys. Rev. B 69, 201103 (2004).
[8] Antoine Georges, The beauty of impurities: Two revivals of Friedel′s virtual bound-state concept, Comptes Rendus Physique, Volume 17, Issues 3–4, 430-446 (2016).
[9] M. P. Sarachik, E. Corenzwit, L. D. Longinotti, Resistivity of Mo-Nb and MoRe Alloys Containing 1% Fe. Phys. Rev A 135, A1041 (1964).
[10] Motahari, S. et al. “Kondo physics of the Anderson impurity model by distributional exact diagonalization.” Physical Review B 94: 235133 (2016).
[11] M. Granath and H. U. R. Strand. Distributional exact diagonalization formalism for quantum impurity models. Phys. Rev B, 86,115111 (2012).
指導教授 唐毓慧(Yu-Hui Tang) 審核日期 2022-1-24
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