NO9900079 Field Theory of Anyons and the Fractional Quantum Hall Effect Susanne F. Viefers Thesis submitted for the Degree of Doctor Scientiarum Department of Physics University of Oslo November 1997 30- 50 Abstract This thesis is devoted to a theoretical study of anyons, i.e. particles with fractional statistics moving in two space dimensions, and the quantum Hall effect. The latter constitutes the only known experimental realization of anyons in that the quasiparticle excitations in the fractional quantum Hall system are believed to obey fractional statistics. First, the properties of ideal quantum gases in two dimensions and in particular the equation of state of the free anyon gas are discussed. Then, a field theory formulation of anyons in a strong magnetic field is presented and later extended to a system with several species of anyons. The relation of this model to fractional exclusion statistics, i.e. intermediate statistics introduced by a generalization of the Pauli principle, and to the low-energy excitations at the edge of the quantum Hall system is discussed. Finally, the Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect is studied, mainly focusing on edge effects; both the ground state and the low- energy edge excitations are examined in the simple one-component model and in an extended model which includes spin effets. Contents I General background 5 1 Introduction 7 2 Identical particles and quantum statistics 10 3 Introduction to anyons 13 3.1 Many-particle description 14 3.2 Field theory description 18 3.3 Anyons in a (strong) magnetic field 19 3.4 The anyon gas 21 4 The Quantum Hall Effect 25 4.1 The integer effect 27 4.2 The fractional effect 29 4.3 Edge excitations 30 II Background for the papers 33 5 Field theory of anyons 35 5.1 Motivation 35 5.2 Full field theory 36 5.3 Field theory in the lowest Landau level 37 5.4 Generalization to several components 42 6 Fractional exclusion statistics 43 6.1 Definition of fractional exclusion statistics 44 6.2 State counting and the distribution function 44 6.3 Thermodynamics 46 6.4 Realizations of FES 47 6.5 Connection to the model on the circle 47 7 Chiral Luttinger liquids and QHE edge states 51 7.1 Luttinger theory and bosonization 52 7.2 Theory of QHE edge excitations 55 7.2.1 Hydrodynamical approach 55 7.2.2 Edge states from bulk effective theory 56 7.3 Connection to lowest Landau level anyons 57 7.4 Multicomponent systems 58 8 The Chern-Simons Ginzburg-Landau theory of the FQHE 60 8.1 The model 61 8.2 Bulk properties 62 8.3 Edge properties 64 8.4 Two-component model with one Chern-Simons field 65 8.5 Several Chern-Simons fields 66 Bibliography 68 III Papers 73 Summary of the papers 75 Accompanying papers • Paper I: S. Viefers, F. Ravndal and T. Haugset, "Ideal Quantum Gases in Two Dimensions" Am. J. Phys. 63 (1995) 369-376. • Paper II: T.H. Hansson, J.M. Leinaas and S. Viefers, "Field Theory of Anyons in the Lowest Landau Level", Nucl. Phys. B 470 (1996) 291-316. • Paper III: Serguei Isakov and Susanne Viefers, "Model of Statistically Coupled Chiral Fields on the Circle", Int. J. Mod. Phys. A 12 (1997) 1895-1914. • Paper IV: J.M. Leinaas and S. Viefers, "Bulk and edge properties of the Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect", (Preprint) Acknowledgements Many people have contributed to the successful completion of this work, and this is the time to thank them. First of all, I want to thank "The boss" Jon Magne Leinaas for being an excellent supervisor, always patiently answering my questions and explaining to me the many things I didn't understand. His enormous knowledge of physics and enjoyable company have made it a great pleasure to be his student. Jon Magne also introduced me to Hans Hansson whose enthusiasm and physical intuition are only matched by his sense of humour. It has been very inspiring to collaborate with Hans on one of the papers in this thesis and to profit from his knowledge in many discussions. Further, I want to thank him for inviting me to Fysikum in Stockholm for two weeks in the spring of 1995. I also enjoyed very much the collaboration with Serguei Isakov on one of the papers, and having him around as a postdoc has greatly improved the working envi- ronment at the theory group. The paper with Finn Ravndal and Tor Haugset was my first publication ever; thanks to both of them and especially to Finn, without whom this article would have never seen the light of day. The most exciting time during this project was the year we spent at the Center for Advanced Study at the Norwegian Academy of Science. I feel very grateful to Jon Magne for letting me participate there and giving me the opportunity to meet physicists, philosophers and tibetologists from all over the world. In addition to the professional benefit, I made several good friends during that year, and in particular I enjoyed very much sharing the office with Heidi Kj0nsberg. My friends and fellow students at the theory group deserve special thanks for making this a nice environment to be in. It has been fun to share the office with Tor Haugset and Elias Bergan and to have Jens Andersen around even at times when nobody else was crazy enough to be at work. Harek Haugerud has been doing a great job running the computer network and helping to organize several "theoretical" Christmas parties. Then, of course, I want to thank my friends and family for making life outside the physics building worth living and for cheering me up on bad days. In particular, my mother deserves to be mentioned for all the moral as well as practical support she has been giving me all these years. Finally, I would like to thank the Norwegian Research Council (NFR), the Norwe- gian Academy of Science and Letters, the Department of Physics (Oslo) and Akershus Fylkeskommune for financial support, and NORDITA both for financial support and kind hospitality during several conferences and a one-week visit in the autumn of 1996. Part I General background Chapter 1 Introduction The world we experience in everyday life is three-dimensional. So at first sight it may seem strange that someone should be interested in studying the physics of two- dimensional - or even worse: one-dimensional - systems. Nevertheless, this is exactly what this thesis is about. Several people, both physicists and non-physicists, have asked me about the motivation for studying abstract systems which do not seem to have much to do with reality and which do not have any "useful" practical applica- tions. So, before giving a rough description of what this field of physics is all about, let me start by trying to answer this question. Actually, there are two answers. First of all, it is not the main purpose of theo- retical physics to develop new technology. The main goal is to obtain a fundamental description and understanding of Nature; demanding practical usefulness of every project would strongly limit the radius of scientific activity. However, a new theoreti- cal understanding of some phenonemon may often lead to new and useful applications even though one was not looking for them in the first place. A good example of this is quantum mechanics which was developed in the first part of this century because peo- ple had discovered phenonema which classical mechanics failed to predict correctly. But in addition to providing a theoretical description of a wide range of phenonema in Nature, quantum mechanical principles are the basis of many modern technolog- ical devices such as lasers, semiconductors or SQUIDS (very sensitive devices for measuring weak magnetic fields, used in medicine). Even worse: It is not even crucial if new theories can be tested in any experiment at the present stage. Sometimes, testing a theory simply demands technology which does not yet exist. ] One of the greatest achievements in modern physics is Einstein's theory of general relativity describing gravity. Einstein developed this theory because he thought it was a logical continuation of special relativity, even though there were, at the time, no indications that classical Newtonian gravity might not be exact. After Einstein had presented his theory, it took several years until the first experimental confirmation: In 1919, English scientists verified that the gravitational field indeed deflects light, as predicted by general relativity. What this teaches us is that the lack of immediate applications or experimental 1 For example, in order to examine the existence of some particles predicted by elementary particle physics one first has to build new and larger accelerators. Introduction verifications does not necessarily make a theory less interesting. In particular, a theoretical study of the physics of lower-dimensional systems is interesting by itself, since it involves trying to understand very fundamental concepts. Secondly - and perhaps more convincingly to some - we shall see that there do exist systems in Nature which are effectively two-dimensional. For example, there are crystals built up of layers consisting of different kinds of atoms in such a way that the electrical resistance experienced by electrons moving within some of these layers is much smaller than that for motion perpendicular to the planes. This means that, at least at low enough temperatures, the electrons do not have enough energy to leave the layer they are in, and their motion is confined to two dimensions. Actually, one has even constructed systems where electrons effectively move in only one dimension: they are called quantum wires. So, in this sense, the models discussed in this thesis do have something to do with the real world. Furthermore, a theoretical understanding of these systems may well prove important in the development of new and smaller semiconductor microelectronic devices. So, what is so special about lower-dimensional systems? Until 20 years ago, it was believed that all particles in Nature could be classified as either bosons or fermions. The physical properties of a particle strongly depend on its statistics, i.e. which of these two categories it belongs to. Electrons, protons and neutrons are examples of fermions, whereas photons, the particles that make up light, are bosons. Indeed it is true that in three dimensions, only Bose- and Fermi statistics are allowed. In 1977, however, the two Norwegian physicists Jon Magne Leinaas and Jan Myrheim discovered that in two or lower dimensions it is possible to imagine infinitely many "intermediate" categories of particles [1]. Since these particles can have any statistics, Wilczek later named them anyons [2]; fermions and bosons are then merely special cases of anyons. Anyons and lower dimensional physics were a purely theoretical concept until the experimental discovery of the quantum Hall effect in 1980 [3]. In the quantum Hall experiment, electrons are trapped at the interface between a semiconductor and an insulator or between two semiconductors; this means that they "live" in two dimen- sions. This system is exposed to a strong magnetic field directed perpendicular to the plane the electrons move in. One then examines what happens if the strength of this magnetic field is varied. It turns out that at special values of the field strength, the electrons are distributed smoothly, so they have the same density everywhere. If the field is changed slightly, small "lumps" of higher or lower density will form and in a sense behave like particles. These so-called quasi-particles turn out to be anyons! In this thesis both anyons in general and certain aspects of the quantum Hall effect are studied. In order to make the thesis readable for non-experts, such as graduate students in other areas of physics, I will start at an elementary level: Chapters 2 through 4 give an introduction to some of the basic concepts in this field, namely quantum statistics (explaining the possibility of the existence of anyons), anyons and the quantum Hall effect. This general introduction is supplemented by paper I which, in addition to some original results, gives a pedagogical introduction to anyons and 1.0 Introduction quantum statistical mechanics in two dimensions. Chapters 5 through 8 are more directly related to the areas of physics I have been working in; they give the theoretical background for the papers and a summary of our own results. In chapter 5 the field theoretical description of anyons, in particular in the presence of a strong magnetic field (papers II and III), is discussed. In chapter 6, the concept of fractional exclusion statistics, a way of introducing intermediate statis- tics as a generalization of Pauli's exclusion principle, used in paper III, is explained. Chapter 7 contains a review of the theory of Luttinger liquids (strongly correlated electrons in one dimension) and how they are used to describe the low-energy exci- tations at the edge of a quantum Hall system; these concepts are used in papers II and III. Finally, in chapter 8, the Chern-Simons Ginzburg-Landau theory, an effective model of the quantum Hall system, is introduced, providing the background for paper IV. A short summary of each paper is included along with the papers themselves. Chapter 2 Identical particles and quantum statistics We introduce the basic ideas of identical particles and particle statistics and explain what is so special about two dimensions. By definition, two particles are said to be identical if they cannot be distinguished by any experiment. This means that no intrinsic difference between them, such as charge or mass, can be detected. This definition is valid both in classical physics and in quantum mechanics. The important difference is that classically, the particles, even though they are identical are not indistinguishable; in principle, one can put "labels" on the particles and distinguish different configurations by following their trajecto- ries (histories). In quantum mechanics, on the other hand, there is no such thing as a continuous trajectory which allows us to track a particle's history. Therefore, identical particles are truly indistinguishable in quantum mechanics. This implies that configurations which only differ by a permutation of identical particles are to be considered as one and the same physical state. Now, two physically equivalent states can (at most) differ by a phase factor a, e.g. \x1X2) = a\x2X\) where Xi denotes the position of particle i. There is a traditional argument (see, e.g. [4]) saying that since interchanging two particles twice brings us back to the original state, a2 must be equal to 1, thus a — ±1. These two cases correspond to the particles having different statistics: Those particles that pick up a (—1) under particle exchange are called fermions and are characterized by totally antisymmetric wavefunctions. This symmetry property leads directly to Pauli's ex- clusion principle stating that two identical fermions cannot be in the same quantum state. The other case, a = 1, gives us bosons; their wavefunctions are completely symmetrical, and there is no restriction on the number of bosons allowed to occupy the same quantum state. At first sight, the above argument is very convincing, and it was believed for a long time that all particles occurring in Nature had to be either bosons or fermions. Indeed, all experimentally observed fundamental particles did fit into one of these categories. However, it turns out that things are fundamentally different in two (or one) space dimensions, as was first pointed out by Leinaas and Myrheim [1]. In the following 10