COMPOSITE FERMIONS When electrons are confined to two dimensions, cooled to near absolute zero temperature, and subjected to a strong magnetic field, they form an exotic new collective state of matter, which rivals superfluidity and superconductivity in both its scope and the elegance of the phenomena associated with it. Investigations into this state began in the 1980s with the observations of integral and fractional quantum Hall effects, which are among the most important discoveries in condensed matter physics. The fractional quantum Hall effect and a stream of other unexpected findings are explained by a new class of particles: composite fermions. A self-contained and pedagogical introduction to the physics and experimental manifestations of composite fermions, this textbook is ideal for graduate students and academic researchers in this rapidly developing field. The topics covered include the integral and fractional quantum Hall effects, the composite fermion Fermi sea, geometric observations of composite fermions, various kinds of excitations, the role of spin, edge state transport, electron solid, and bilayer physics. The author also discusses fractional braiding statistics and fractional local charge. This textbook contains numerous exercises to reinforce the concepts presented in the book. Jainendra Jain is Erwin W. Mueller Professor of Physics at the Pennsylvania State University. He is a fellow of the John Simon Guggenheim Memorial Foundation, the Alfred P. Sloan Foundation, and the American Physical Society. Professor Jain was co-recipient of the Oliver E. Buckley Prize of the American Physical Society in 2002. Pre-publication praise for Composite Fermions: “Everything you always wanted to know about composite fermions by its primary architect and champion. Much gorgeous theory, of course, but also an excellent collection of the relevant experimental data. For the initiated, an illuminating account of the relationship between the composite fermion model and other models on stage. For the novice, a lucid presentation and dozens of valuable exercises.” Horst Stormer, Columbia University, NY and Lucent Technologies. Winner of the Nobel Prize in Physics in 1998 for discovery of a new form of quantum fluid with fractionally charged excitations.

COMPOSITE FERMIONS Jainendra K. Jain The Pennsylvania State University

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge cb2 8ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107404250 © J. K. Jain 2007 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 First paperback edition 2011 A catalogue record for this publication is available from the British Library isbn 978-0-521-86232-5 Hardback isbn 978-1-107-40425-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Manju, Sunil, and Saloni

Contents

Preface List of symbols and abbreviations

page xiii xv

1

Overview 1.1 Integral quantum Hall effect 1.2 Fractional quantum Hall effect 1.3 Strongly correlated state 1.4 Composite fermions 1.5 Origin of the FQHE 1.6 The composite fermion quantum fluid 1.7 An “ideal” theory 1.8 Miscellaneous remarks

1 1 2 4 5 7 7 9 10

2

Quantum Hall effect 2.1 The Hall effect 2.2 Two-dimensional electron system 2.3 The von Klitzing discovery 2.4 The von Klitzing constant 2.5 The Tsui–Stormer–Gossard discovery 2.6 Role of technology Exercises

12 12 14 17 19 21 22 23

3

Landau levels 3.1 Gauge invariance 3.2 Landau gauge 3.3 Symmetric gauge 3.4 Degeneracy 3.5 Filling factor 3.6 Wave functions for filled Landau levels 3.7 Lowest Landau level projection of operators 3.8 Gauge independent treatment

26 27 28 29 32 33 34 36 37 vii

viii

Contents

3.9 3.10 3.11 3.12 3.13 3.14 3.15

Magnetic translation operator Spherical geometry Coulomb matrix elements Disk geometry/parabolic quantum dot Torus geometry Periodic potential: the Hofstadter butterfly Tight binding model Exercises

40 42 52 61 65 67 69 71

4

Theory of the IQHE 4.1 The puzzle 4.2 The effect of disorder 4.3 Edge states 4.4 Origin of quantized Hall plateaus 4.5 IQHE in a periodic potential 4.6 Two-dimensional Anderson localization in a magnetic field 4.7 Density gradient and Rxx 4.8 The role of interaction

77 77 78 81 81 96 97 101 102

5

Foundations of the composite fermion theory 5.1 The great FQHE mystery 5.2 The Hamiltonian 5.3 Why the problem is hard 5.4 Condensed matter theory: solid or squalid? 5.5 Laughlin’s theory 5.6 The analogy 5.7 Particles of condensed matter 5.8 Composite fermion theory 5.9 Wave functions in the spherical geometry 5.10 Uniform density for incompressible states 5.11 Derivation of ν ∗ and B∗ 5.12 Reality of the effective magnetic field 5.13 Reality of the  levels 5.14 Lowest Landau level projection 5.15 Need for other formulations 5.16 Composite fermion Chern–Simons theory 5.17 Other CF based approaches Exercises

105 105 106 109 110 113 115 116 118 138 141 141 145 146 146 153 154 164 173

6

Microscopic verifications 6.1 Computer experiments 6.2 Relevance to laboratory experiments 6.3 A caveat regarding variational approach

174 174 175 176

Contents

6.4 6.5 6.6 6.7 6.8 6.9

Qualitative tests Quantitative tests What computer experiments prove Inter-composite fermion interaction Disk geometry A small parameter and perturbation theory Exercises

ix

176 181 187 188 192 197 199

7

Theory of the FQHE 7.1 Comparing the IQHE and the FQHE 7.2 Explanation of the FQHE 7.3 Absence of FQHE at ν = 1/2 7.4 Interacting composite fermions: new fractions 7.5 FQHE and spin 7.6 FQHE at low fillings 7.7 FQHE in higher Landau levels 7.8 Fractions ad infinitum? Exercises

201 201 203 206 206 213 213 213 214 214

8

Incompressible ground states and their excitations 8.1 One-particle reduced density matrix 8.2 Pair correlation function 8.3 Static structure factor 8.4 Ground state energy 8.5 CF-quasiparticle and CF-quasihole 8.6 Excitations 8.7 CF masses 8.8 CFCS theory of excitations 8.9 Tunneling into the CF liquid: the electron spectral function Exercises

217 217 218 220 222 224 226 237 245 245 250

9

Topology and quantizations 9.1 Charge charge, statistics statistics 9.2 Intrinsic charge and exchange statistics of composite fermions 9.3 Local charge 9.4 Quantized screening 9.5 Fractionally quantized Hall resistance 9.6 Evidence for fractional local charge 9.7 Observations of the fermionic statistics of composite fermions 9.8 Leinaas–Myrheim–Wilczek braiding statistics 9.9 Non-Abelian braiding statistics 9.10 Logical order Exercises

253 253 254 255 263 264 266 268 269 279 281 281

x

Contents

10 Composite fermion Fermi sea 10.1 Geometric resonances 10.2 Thermopower 10.3 Spin polarization of the CF Fermi sea 10.4 Magnetoresistance at ν = 1/2 10.5 Compressibility

286 287 297 301 301 305

11 Composite fermions with spin 11.1 Controlling the spin experimentally 11.2 Violation of Hund’s first rule 11.3 Mean-field model of composite fermions with a spin 11.4 Microscopic theory 11.5 Comparisons with exact results: resurrecting Hund’s first rule 11.6 Phase diagram of the FQHE with spin 11.7 Polarization mass 11.8 Spin-reversed excitations of incompressible states 11.9 Summary 11.10 Skyrmions Exercises

307 308 309 311 314 324 326 331 337 347 348 360

12 Non-composite fermion approaches 12.1 Hierarchy scenario 12.2 Composite boson approach 12.3 Response to Laughlin’s critique 12.4 Two-dimensional one-component plasma (2DOCP) 12.5 Charged excitations at ν = 1/m 12.6 Neutral excitations: Girvin–MacDonald–Platzman theory 12.7 Conti–Vignale–Tokatly continuum-elasticity theory 12.8 Search for a model interaction Exercises

363 363 366 368 370 372 377 384 386 389

13 Bilayer FQHE 13.1 Bilayer composite fermion states 13.2 1/2 FQHE 13.3 ν = 1: interlayer phase coherence 13.4 Composite fermion drag 13.5 Spinful composite fermions in bilayers Exercises

394 395 400 403 408 409 411

14 Edge physics 14.1 QHE edge = 1D system 14.2 Green’s function at the IQHE edge 14.3 Bosonization in one dimension 14.4 Wen’s conjecture

413 413 414 417 429

Contents

14.5 Experiment 14.6 Exact diagonalization studies 14.7 Composite fermion theories of the edge Exercises

xi

432 437 438 441

15 Composite fermion crystals 15.1 Wigner crystal 15.2 Composite fermions at low ν 15.3 Composite fermion crystal 15.4 Experimental status 15.5 CF charge density waves

442 442 446 449 452 455

Appendixes A Gaussian integral B Useful operator identities C Point flux tube D Adiabatic insertion of a point flux E Berry phase F Second quantization G Green’s functions, spectral function, tunneling H Off-diagonal long-range order I Total energies and energy gaps J Lowest Landau level projection K Metropolis Monte Carlo L Composite fermion diagonalization References Index

458 460 462 463 465 467 477 482 486 490 499 502 504 540

Preface Odd how the creative power at once brings the whole universe to order. Virginia Woolf

When electrons are confined to two dimensions, cooled to near absolute zero temperature, and subjected to a strong magnetic field, they form a quantum fluid that exhibits unexpected behavior, for example, the marvelous phenomenon known as the fractional quantum Hall effect. These properties result from the formation of a new class of particles, called “composite fermions,” which are bound states of electrons and quantized microscopic vortices. The composite fermion quantum fluid joins superconductivity and Bose–Einstein condensation in providing a new paradigm for collective behavior. This book attempts to present the theory and the experimental manifestations of composite fermions in a simple, economical, and logically coherent manner. One of the gratifying aspects of the theory of composite fermions is that its conceptual foundations, while profoundly nontrivial, can be appreciated by anyone trained in elementary quantum mechanics. At the most fundamental level, the composite fermion theory deals directly with the solution of the Schrödinger equation, its physical interpretation, and its connection to the observed phenomenology. The basics of the composite fermion (CF) theory are introduced in Chapter 5. The subsequent chapters, with the exception of Chapter 12, are an application of the CF theory in explaining and predicting phenomena. Detailed derivations are given for many essential facts. Formulations of composite fermions using more sophisticated methods are also introduced, for example, the topological Chern–Simons field theory. To keep the book within a manageable length, many developments are mentioned only briefly, but my hope is that this book will at least serve as a useful first resource for any reader interested in the field. It can be used as a textbook for a graduate level special topics course, or selected portions from it can be used in the standard graduate course on condensed matter physics or many-body theory. Many simple exercises have been included to provide useful breaks. Disclosure: Personally, the most difficult aspect that I have faced while writing this book has been my own long and intimate involvement with composite fermions, which, one might hope, would enhance the probability that the exposition is sometimes well thought out, but makes it difficult to ensure the kind of objectivity that can come only with distance, both in space and in time. Fortunately, much is known that is indisputable, which allows one to distinguish opinions and speculations from facts. The selection of topics, the emphasis, and the logic of presentation reflect my views on what is firmly established, what is important, and how it should be taught. For the theoretical part, my preference has been for concepts and formulations that directly relate to laboratory and/or computer experiments. If work in which I have participated appears more often than it deserves, it is because that is what I know and understand best. My sincere apologies are extended to those who might feel xiii

xiv

Preface

their work is not adequately represented or, worse, misrepresented. I have made an effort to supply the original references to the best of my knowledge and ability, but the list is surely incomplete. I have collected, over the years, many nuggets of knowledge by osmosis, and my suspicion is that some of them may have crept into the book without proper attribution. The book is not intended, and ought not to be taken, as a historical account. This book is an account of the collective contributions of too many scientists to name individually. It is a pleasure to acknowledge my profound debt to many colleagues whose wisdom and collaboration have benefited me over the years. These include, but are not limited to, Alexei Abrikosov, Phil Allen, Phil Anderson, Jayanth Banavar, G. Baskaran, Lotfi Belkhir, Nick Bonesteel, Moses Chan, Albert Chang, Chiachen Chang, Sankar Das Sarma, Goutam Dev, Rui Du, Jim Eisenstein, Herb Fertig, Eduardo Fradkin, Steve Girvin, Gabriele Giuliani, Fred Goldhaber, Vladimir Goldman, Ken Graham, Devrim Güçlü, Duncan Haldane, Bert Halperin, Hans Hansson, Jason Ho, Gun Sang Jeon, Shivakumar Jolad, Thierry Jolicoeur, Rajiv Kamilla, Woowon Kang, Anders Karlhede, Tetsuo Kawamura, Steve Kivelson, Klaus von Klitzing, Paul Lammert, Bob Laughlin, Patrick Lee, Seung Yeop Lee, Jon Magne Leinaas, Mike Ma, Allan MacDonald, Jerry Mahan, Sudhansu Mandal, Noureddine Meskini, Alexander Mirlin, Ganpathy Murthy, Wei Pan, Kwon Park, Vittorio Pellegrini, Mike Peterson, Aron Pinczuk, John Quinn, T. V. Ramakrishnan, Sumathi Rao, Nick Read, Nicolas Regnault, Ed Rezayi, Tarek Sbeouelji, Vito Scarola, R. Shankar, Mansour Shayegan, Chuntai Shi, Boris Shklovskii, Steve Simon, Shivaji Sondhi, Doug Stone, Horst Stormer, Aron Szafer, David Thouless, Csaba To˝ ke, Nandini Trivedi, Dan Tsui, Cyrus Umrigar, Susanne Viefers, Giovanni Vignale, Xiao Gang Wu, Xincheng Xie, Frank Yang, Jinwu Ye, Fuchun Zhang, and Lizeng Zhang. I am indebted to Jayanth Banavar, Vin Crespi, Herb Fertig, Fred Goldhaber, Ken Graham, Devrim Güçlü, Gun Sang Jeon, Shivakumar Jolad, Paul Lammert, Ganpathy Murthy, Mike Peterson, Csaba To˝ ke, Giovanni Vignale, and Dave Weiss for a careful and critical reading of parts of the manuscript. Thanks are also due to Wei Pan for making available the trace used for the cover page, to Gabriele Giuliani for bringing to my attention the quotation at the beginning of the Preface, and to the National Science Foundation for financially supporting my research on composite fermions. I am tremendously grateful to Gun Sang Jeon for his help with numerous figures. Finally, I express my deepest gratitude to my family, Manju, Sunil, and Saloni, who patiently put up with my preoccupation during the writing of this book. Little had I realized at the beginning how major an undertaking it would be. But the experience has been instructive and also greatly rewarding. I hope that the reader will find the book useful.

Symbols and abbreviations

b B B∗

Chern–Simons magnetic field external magnetic field effective magnetic field experienced by composite fermions also denoted Beff , BCF or B in the literature CF composite fermion 2p CF composite fermion carrying 2p vortices CFCS composite fermion Chern–Simons CF-LL  level CS Chern–Simons ∗ e local charge of an excitation eCS Chern–Simons electric field  dielectric function of the background material ( ≈ 13 for GaAs) FQHE fractional quantum Hall effect
ωc cyclotron energy
ωc∗ cyclotron energy of composite fermion IQHE integral quantum Hall effect k wave vector √  magnetic length ( =
c/eB) L total orbital angular momentum in spherical geometry, or total z component of the angular momentum in the disk geometry L  level; Landau-like level of composite fermions LL Landau level LLL lowest Landau level m∗a activation mass of composite fermion mb electron band mass (mb = 0.067me in GaAs) me electron mass in vacuum m∗p polarization mass of composite fermion n integral filling factor; LL or -level index N number of electrons/composite fermions ν filling factor of electrons ν∗ filling factor of composite fermions xv