Quasiparticles in the Quantum Hall Effect Janik Kailasvuori Stockholm University Department of Physics 2006 Thesis for the degree of Doctor of Philosophy in theoretical physics Department of Physics Stockholm University Sweden c Janik Kailasvuori 2006 c American Physical Society (papers) ISBN 91-7155-320-7 Printed by Universitetsservice US AB, Stockholm Cover: Ballet des quasiparticules, courtesy of Mirabelle Ricard. i Abstract The fractional quantum Hall effect (FQHE), discovered in 1982 in a two- dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncom- mutativity imposed on the classical space variables captures the incompress- ibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n−1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n − k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n−1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a ver- sion of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n−1 for 2n vortices. ii Acknowledgements I owe my deepest gratitude to my supervisor Hans Hansson, who gave me the opportunity to make this thesis and the great privilege of working with him. He gave me full freedom to choose subject but, well aware of my tastes in physics, suggested many interesting projects. When I have gone through tough times he was a good person to talk to also when it came to other issues than physics. I am about equally indebted to his colleague, and in a way my second supervisor, Anders Karlhede. His clarity and carefulness has nicely com- plemented Hans’ wilder temperament in physics, and a kinder and more down-to-earth professor is hard to find. Together they make a great team and, as their collaborator, they made me feel like an equal though their insight into physics is of such a completely different caliber than mine. The good care they take of their students contributes to making their little subgroup with my fellow students and roommates Emil Johansson- Bergholtz, Maria Hermanns and our newcomer Emma Wikberg such a re- laxed, generous and stimulating environment, that I unfortunately will leave just as we all start to converge in our projects and see hints of a fascinating bigger picture emerging. The absolute lack of threshold in our interactions was a rewarding experience. In spite of her youth Maria has already taught me loads of things, and I am much indebted to Emil for pushing the idea of a joint project, for all his valuable input to our physics discussion, for all the constructive proof readings and for looking out after me. My former roommate Dr. Joachim Sj¨ostrand, a kind-hearted natural- born party devil, worked on completely different problems, but mainly out- side physics we have probably discussed everything under the sun and I will miss all the great fun we had. The theory corridor in general has been a very enjoyable place. In par- ticular Fawad Hassan and Ingemar Bengtsson have been a great source of knowledge. I have enjoyed very stimulating lunch discussions with S¨oren Holst, Michael Gustafsson, Asa˚ Ericsson, Torsten Bringmann, Martin Eriks- son, Malcolm Fairbarn (whom I also thank for proof readings of the thesis), Rahman Amanullah, Joakim Edsj¨o,Stefan Hofmann and others, and many corridor chats with Lidia Pieri, Narit Pidokrajt, Mohammmed Bourennane and in particular Alexander Zheltukhin, who as a fan of mine has pointed out different career options (like succeeding Hans Blix) and has told me a lot of interesting things about the Soviet Union/Russian language/Ukraina. I would also like to thank Edwin Langmann, Jack Lidmar, my men- tor Christoph Bargholtz and my molecular physics friends Moses Salci and iii Svetlana Verbitskaja; my student buddy Moses for all the physics and career discussions and Svetlana for her delightful physics monologues and all our dancing. Thanks to the friendly Vetenskapens Hus crew, and in particular the endlessly enthusiastic Christer Nilsson, for having offered me interesting and creative teaching opportunities. Thanks also to S¨orenHolst for all the fun teaching collaborations. Among physicists out in the world I would like to acknowledge Georgios Kavoulakis, Alexios Polychronakos, Bogdan Morairu and Richard von Unge for interactions and collaborations, and Ioanna Kyriakou for valuable inter- actions. I would also like to thank the Nordforsk Nordic Network and in particular Susanne Viefers for the many stimulating network meetings. Former school and high school teachers that influenced the start of my path towards this PhD include most notably Walle Steinmo, Stig Larsson and Olof Lev´en. For catalysing my evolution and my thinking in a bigger perspective that includes physics but goes far beyond that, I am deeply grateful to Mairo Burdette, Bitte Sl¨attebol, Satoshi Kudo, Tania Sopilnichenko and my ”guru” Carl-Philippe Carr. For the cover picture and for her lovely friendship I thank Mirabelle Ricard. Not directly related to this PhD work but of immeasurable importance for my life in general are my wonderful friends—Andreas, Kaisa, Laurent and Sofija just to mention a very few not already mentioned—and my family. Here I must refrain from completeness lest the acknowledgement would risk out-sizing the rest of the thesis (which is already quite long). However, my sisters Heidi and Mikaela deserve a very special thanks for all their love and care. My late mother Ritva was not there to influence my student years but gave me her special sensitivity (whereas most of her warm temperament went to my dear brother Petri) and hence she essentially defined my core. Finally, my warmest thanks go to my father Folke for his loving support and for his endurance that took the family through the heavy seas throughout the years. iv Contents 1 Introduction and overview 1 1.1 The thesis put into a general context . 1 1.2 The quantum Hall system—a new state of two-dimensional matter . 3 1.3 Collective phenomena and quasiparticles . 6 1.4 Incompressible liquids . 7 1.5 Fractional charge . 7 1.6 Effective Chern-Simons theories . 7 1.7 Fractional statistics . 9 1.8 Nonabelian statistics and connections to p-wave superconduc- tors . 10 1.9 Thin torus studies of the Moore-Read state . 10 1.10 Outline of the thesis . 11 1.11 Suggested literature . 11 2 The noncommutative Chern- Simons theory of QH liquids 13 2.1 Chern-Simons theories as effective theories of the FQHE . 13 2.1.1 Chern-Simons terms in the FQHE . 14 2.1.2 An effective theory of the FQHE . 15 2.2 Noncommutative Chern-Simons theory and matrix models . 16 2.2.1 The noncommutative plane . 17 2.2.2 Functions, integrals, transforms, kernels, derivatives . 18 2.2.3 Noncommutative Chern-Simons lagrangians and ma- trix models . 19 2.2.4 Gauge fixed and gauge covariant formulations . 21 2.2.5 Quantization gives the Laughlin fractions . 22 2.2.6 The finite matrix model for the FQHE . 23 2.3 Charge and current operators in the QH matrix model . 25 vi CONTENTS 2.3.1 Coupling the QH matrix model to an electromagnetic field . 25 2.3.2 Current and density operators . 26 2.3.3 Quantization and the magnetic algebra . 27 2.3.4 Gaussian LLL particles in a classical model . 27 2.3.5 The droplet state . 29 2.3.6 Comparison of Weyl ordering and anti-ordering . 30 2.4 Quasiparticles in the QH matrix model . 32 2.4.1 Quasiholes in the matrix model . 33 2.4.2 Quasielectrons and their φ4-description . 33 2.5 Conclusions and discussion . 36 3 Thin torus studies of nonabelian quantum Hall states 39 3.1 Background . 39 3.2 The general thin limit picture . 41 3.2.1 The one-dimensional lattice and 1D spin chains . 41 3.2.2 Fock states on the torus . 42 3.2.3 The thin limit . 42 3.2.4 Ground states, gap and degeneracy . 43 3.2.5 Quasiparticles as fractionally charged domain walls . 44 3.2.6 Degeneracy for quasiparticles . 45 3.2.7 Interpolation to bulk physics . 46 3.3 The Moore-Read state in the plane . 47 3.3.1 The three-body interaction . 48 3.3.2 Quasiholes in the Moore-Read state . 49 3.4 The Moore-Read state in the thin torus limit . 50 3.4.1 Ground state in the thin limit . 50 3.4.2 Half-quasiparticles in the thin limit . 50 3.4.3 Counting argument that gives 2n−1 . 52 3.4.4 The MR state as ground state of the Coulomb inter- action .