Novel Aspects of Localization, Delocalization, and Anomalous Transport in One-dimensional Systems QIFANG ZHAO BSc (Hons), NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2014 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Qifang Zhao 21 August 2014 i ii Dedicated to my parents and wife iii iv Acknowledgements First of all, I would like to thank my supervisor, A/Prof Gong Jiang Bin Throughout my four years of PhD study, he provided me with guidance, inspirations and support He also shared with me his personal experiences in doing independent research, teaching me efficient ways to achieve fruitful outcomes through meaningful processes During the early stage, his insightful suggestions laid the paths to successful projects In the later stages of the course, his emphasis on independent research has successfully forged a young research mind It is a great honour and blessing to be his student His knowledge, wisdom and kindness have always been and will always be what I look up to Next, I would like to thank my co-supervisor, A/Prof Cord A Müller We collaborated for the past three years, during which I picked up useful skills such as presentation, poster-designing and many more from him I would like to express my deepest gratitude to him for his guidance in my first project which really means a lot to me in the early stage of PhD study I am also very grateful to my fellow group members Derek, Hailong, Long Wen, Adam, Gao Yang, Yon Shin and Da Yang for all the beneficial and delightful discussions I would like to offer my special thanks to Derek who helped me clarify some doubts through rigorous derivations, Hailong for introducing me to the research of periodically driven systems, and Long Wen who elaborated some physics concepts to me, sharing his profound knowledge I also thank fellow PhD students in block S16, Taolin, Qinglin, Feng Ling, Liu Sha, Lina, Qin Chu, Qiao Zhi and so on, for their companionship v Last but not least, I would like to thank my family members, my wife Chen Zhixiu, my father Wang Zaiqin, mother Zhao Zonghua and little brother Wang Guangwei for their continuous support and encouragement throughout my PhD study vi Contents Introduction 1.1 Physics of Disorder in Quantum Systems 1.2 Physics of Disorder in Current Research Frontiers 1.3 The Structure of Research Topics and the Outline of Thesis Physics Background and Mathematical Preliminaries 2.1 The Transfer Matrix Formalism in 1D 9 2.1.1 Properties of the Reflection and Transmission Coefficients 10 2.1.2 The Transfer Matrix 11 2.2 The Application of the Transfer Matrix Formalism in Kronig-Penney Model 13 2.3 The Formalisms for Periodically Driven Systems in Classical and Quantum Mechanics 15 2.3.1 The Kicked Rotor in Classical Mechanics 16 2.3.2 The Kicked Rotor in Quantum Mechanics 17 Localization Behavior of Dirac Particles in Disordered Graphene Superlattices 20 3.1 Introduction 21 3.1.1 Graphene Superlattices 21 3.1.2 Disordered Graphene Superlattices 22 Localization Length in Disordered Graphene Superlattices 24 3.2 vii Contents 3.2.1 Modeling Disordered Graphene Superlattices 24 3.2.2 Transfer Matrix Formalism 26 3.2.3 Clean Graphene Superlattices 28 3.2.4 Disordered Graphene Superlattices 29 3.2.5 Randomly Spaced, Identical Barriers 30 3.2.6 Weak-disorder Expansion 30 Scalar Potential 32 3.3.1 Amplitude-disordered Delta Scalar Potential 33 3.3.2 Disordered Square Scalar Potential 37 Vector Potential 40 3.4.1 Amplitude-disordered Delta Vector Potential 40 3.4.2 Disordered Square Vector Potential 43 3.5 Wave Packet Dynamics: Disorder-Induced Filtering 45 3.6 Concluding Remarks 48 3.A Details of Weak-disorder Expansion 49 3.A.1 Absence of Mixed-fluctuation Terms 50 3.A.2 Diagonalization Procedure 51 3.3 3.4 1D Dirac Model with Random Mass 4.1 53 Introduction 53 4.1.1 1D Dirac Model with Square Mass Barriers 53 4.1.2 Disordered Vector Potential GSL 55 4.1.3 The Connection between the Dirac Model and the Quantum Walk, and the Implications 56 Lyapunov Exponents in 1D Dirac Model 57 4.A 2D Dirac Models with Random Mass 61 4.A.1 Graphene with 2D Random Mass 61 4.A.2 Graphene with 1D Random Mass 62 4.2 viii Contents From Disordered Quantum Walk to Physics of Off-diagonal Disorder 5.1 64 65 Off-diagonal Disorder 67 Set-up of Our QW Model 73 5.2.1 Solving the Spectrum of U 75 Exploring the DOS and Localization Length near Special Energies 77 5.3.1 Case of Zero Quasi-energy 78 5.3.2 Analyzing Quasi-energy Values 79 5.3.3 Re-interpretation of Chained Transfer Matrices 80 5.3.4 Integrated Density of States (DOS) 84 5.3.5 Derivation of the DOS 86 5.3.6 Derivation of the Localization Length 88 5.3.7 Numerical Analysis of 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