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Transport in Quantum Hall Systems : Probing Anyons and Edge Physics

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Transport in Quantum Hall Systems : Probing Anyons and Edge Physics Transport in Quantum Hall Systems : Probing Anyons and Edge Physics by Chenjie Wang B.Sc., University of Science and Technology of China, Hefei, China, 2007 M.Sc., Brown University, Providence, RI, 2010 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Department of Physics at Brown University PROVIDENCE, RHODE ISLAND May 2012 c Copyright 2012 by Chenjie Wang This dissertation by Chenjie Wang is accepted in its present form by Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Prof. Dmitri E. Feldman, Advisor Recommended to the Graduate Council Date Prof. J. Bradley Marston, Reader Date Prof. Vesna F. Mitrovi´c, Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii Curriculum Vitae Personal Information Name: Chenjie Wang Date of Birth: Feb 02, 1984 Place of Birth: Haining, China Education Brown University, Providence, RI, 2007-2012 • - Ph.D. in physics, Department of Physics (May 2012) - M.Sc. in physics, Department of Physics (May 2010) University of Science and Technology of China, Hefei, China, 2003- • 2007 - B.Sc. in physics, Department of Modern Physics (July 2007) Publications 1. Chenjie Wang, Guang-Can Guo and Lixin He, Ferroelectricity driven by the iv noncentrosymmetric magnetic ordering in multiferroic TbMn2O5: a first-principles study, Phys. Rev. Lett. 99, 177202 (2007) 2. Chenjie Wang, Guang-Can Guo, and Lixin He, First-principles study of the lattice and electronic structures of TbMn2O5, Phys. Rev B 77, 134113 (2008) 3. Chenjie Wang and D. E. Feldman, Transport in line junctions of ν = 5/2 quantum Hall liquids, Phys. Rev. B 81, 035318 (2010) 4. Chenjie Wang and D. E. Feldman, Identification of 331 quantum Hall states with Mach-Zehnder interferometry, Phys. Rev. B 82, 165314 (2010) 5. Chenjie Wang and D. E. Feldman, Rectification in Y-junctions of Luttinger liquid wires, Phys. Rev. B 83, 045302 (2011) 6. Chenjie Wang and D. E. Feldman, Fluctuation-dissipation theorem for chiral systems in non-equilibrium steady states, Phys. Rev. B 84, 235315 (2011) 7. Chenjie Wang and D. E. Feldman, in preparation. v Acknowledgements Well, it is time to say goodbye to Brown. How time flies! Five years ago when I arrived in Providence, a beautiful and quiet small town, I was curious and uncertain about everything. Now after five years of struggling, looking back to my graduate life, I find it is more than satisfying. I would like to thank several people without whom my PhD would be impossible and my life would be harder. Without a doubt, my advisor Professor Dima Feldman is the first one to whom I would like to express my thanks for his guidance, encouragement and most im- portantly his unique personality influence. Dima’s intelligence and deep insight in science influence me a lot. “You are not thinking logically.” I still remember these words he said to me when I was learning quantum many-body physics. These words were the point where I actually started to think scientifically, and I am still benefiting from them. Thanks, Dima! I would like to thank several condensed matter professors, Professor Michael Kosterlitz, Professor Xinsheng Ling, Professor Brad Marston, Professor Vesna Mitro- vic and Professor See-Chen Ying for their teaching, helpful discussions and/or being on my prelim and defense committee. In particular, I thank Professor Marston with whom I had many fruitful discussions on numerical methodology. Professor Koster- litz also deserves my special acknowledgements. I joined two study groups that he vi supervised, from which I learned much knowledge of the renormalization group and quantum phase transition. I also thank them for being my recommendation letter writers. I am grateful to my current and former officemates Hao Tu, Guang Yang, Wan- ming Qi, Florian Sabou, Pengyu Liu and Lei Wang, Feifei Li and Dina Obeid. We all have or had been suffering from the windowless office for many years, with me the luckiest – my desk is close to the door. There are many other classmates and friends that I need to thank, without whom my life would have been harder and of less fun : Feifei Li, Dima’s former student, who helped a lot at the beginning of my graduate research; Jun He, my former roommate, who offered me many helps when I was still learning to survive in this country; Congkao Wen for introducing me to the view of high-energy physics; Chao Li, a nice life companion; Xuqing Huang, a very good friend who always offered free hospitality and meals when I went to Boston; Hong Pan, a very good friend with whom I had so many useful discussions on physics experiments; Bosheng Zhang, a nice friend with whom I had many useful discussions on life philosophy; and Yana Cheng, Yuzhen Guan, Xin Jia, Mingming Jiang, Dongfang Li, Wenzhe Zhang, Ilyong Jung and Alex Geringer-Sameth for their help and encouragement. I would like to thank the administrative staff of the Physics Department, Barbara Dailey, Sabina Griffin, and Jane Martin and the Chair of the Physics Department, Prof. James Valles, for their help. It is hard for me to think of a complete list now, because writing the thesis has squeezed all my energy out. I am sorry to those who are important to me but not on this list. Finally my beloved parents. They know nothing about physics, but they have been supporting my career all the time. I cannot imagine getting a PhD without their tremendous support and love. vii Contents Curriculum Vitae iv Acknowledgments vi 1 Introduction 1 1.1 From the Classical Hall Effect to the Quantum Hall effect ....... 4 1.1.1 Energy Scales ........................... 9 1.2 Theories of QHE ............................. 12 1.2.1 IQHE: Landau Quantization ................... 12 1.2.2 Laughlin FQHE States ...................... 16 1.2.3 Quasiparticles, Anyons and Anyonic Statistics ......... 18 1.2.4 FQHE at ν =5/2: Non-Abelian States ............. 22 1.3 Edge theory ................................ 25 1.4 Interferometers .............................. 29 1.4.1 Fabry-Perot Interferometer .................... 30 viii 1.4.2 Mach-Zehnder Interferometer .................. 34 1.5 Transport Theory ............................. 36 1.5.1 Landauer-B¨uttiker Approach ................... 37 1.5.2 Fluctuation-Dissipation Theorem ................ 40 1.5.3 Keldysh Formalism ........................ 41 1.5.4 Example: Tunneling through a Single QPC ........... 43 1.6 Overview .................................. 47 2 Identification of 331 Quantum Hall States with Mach-Zehnder Interferometry 49 2.1 Introduction ................................ 50 2.2 Statistics in the 331 state ........................ 54 2.3 Mach-Zehnder interferometer ...................... 56 2.4 Electric current .............................. 60 2.5 Shot noise ................................. 63 2.6 Summary ................................. 70 3 Transport in Line Junctions of ν =5/2 Quantum Hall Liquids 71 3.1 Introduction ................................ 72 3.2 Proposed 5/2 states ............................ 76 3.3 Qualitative discussion ........................... 79 3.4 Calculation of the current ........................ 89 3.5 The number of singularities ....................... 97 3.6 I-V curves .................................107 ix 3.6.1 Tunneling into integer edge modes. ...............107 3.6.2 K = 8 state ............................108 3.6.3 331 state ..............................112 3.6.4 Pfaffian state ...........................115 3.6.5 Reconstructed Pfaffian state ..................116 3.6.6 Disorder-dominated anti-Pfaffian state .............119 3.6.7 Non-equilibrated anti-Pfaffian state ...............122 3.7 Discussion .................................124 4 Fluctuation-Dissipation Theorem for Chiral Systems in Nonequi- librium Steady States 129 4.1 Introduction ................................130 4.2 Heuristic derivation ............................135 4.3 Proof of the nonequilibrium FDT ....................137 4.3.1 Chiral systems ..........................138 4.3.2 Initial density matrix and Heisenberg current operator . 140 4.3.3 Voltage bias ............................143 4.3.4 Main argument ..........................146 4.4 Discussion .................................150 4.5 Generalization to Multi-Terminal Setup and Heat Transport – Proof by Fluctuation Relations .........................153 4.6 Conclusion .................................163 5 Summary 164 x A Multi-component Halperin States 166 A.1 Quasiparticle statistics and edge modes .................167 A.2 The case of only one flavor allowed to tunnel ..............171 A.2.1 Current ..............................171 A.2.2 Noise ................................173 A.3 General case ................................176 B Integer Edge Reconstruction 181 C FDT in an Ideal Gas Model 184 xi List of Tables 1.1 Energy Scales, estimated for a GaAs-AlGaAs system with the width of the 2D electron gas set to 25 nm and the magnetic field at 5 Tesla (typical for 5/2 FQHE). Data for ν =1/3 and ν =5/2 is from Ref. [34] 12 3.1 The number of conductance singularities for different models in dif- ferent setups. ............................... 98 3.2 Summary of singularities in the voltage dependence of the differential A conductance Gtun for different 5/2 states. The “Modes” column shows the numbers of left- and right-moving modes in the fractional edge, the number in the brackets being the number of Majorana modes. “A” or “N” in the next column means Abelian or non-Abelian statis- tics. The “Singularities” shows the
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