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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 thephenomena associated with it. Investigations into this state began in the 1980s with theobservations of integral and fractional quantum Hall effects, which are among the mostimportant discoveries in condensed matter physics. The fractional quantum Hall effect anda stream of other unexpected findings are explained by a new class of particles: compositefermions.

A self-contained and pedagogical introduction to the physics and experimentalmanifestations of composite fermions, this textbook is ideal for graduate students andacademic researchers in this rapidly developing field. The topics covered include theintegral and fractional quantum Hall effects, the composite fermion Fermi sea, geometricobservations of composite fermions, various kinds of excitations, the role of spin, edge statetransport, electron solid, and bilayer physics. The author also discusses fractional braidingstatistics and fractional local charge. This textbook contains numerous exercises to reinforcethe concepts presented in the book.

Jainendra Jain is Erwin W. Mueller Professor of Physics at the Pennsylvania StateUniversity. He is a fellow of the John Simon GuggenheimMemorial Foundation, theAlfredP. Sloan Foundation, and theAmerican Physical Society. Professor Jain was co-recipient ofthe 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 architectand champion. Much gorgeous theory, of course, but also an excellent collection of therelevant experimental data. For the initiated, an illuminating account of the relationshipbetween the composite fermion model and other models on stage. For the novice, a lucidpresentation and dozens of valuable exercises.”Horst Stormer, Columbia University, NY and Lucent Technologies. Winner of the NobelPrize in Physics in 1998 for discovery of a new form of quantum fluid with fractionallycharged excitations.

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Cambridge University Press978-0-521-86232-5 - Composite FermionsJainendra K. JainFrontmatterMore information

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COMPOSITE FERMIONS

Jainendra K. JainThe Pennsylvania State University






CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521862325

© J. K. Jain 2007

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2007

Printed in the United Kingdom at the University Press, Cambridge

ISBN-978-0-521-86232-5 hardback

Cambridge University Press has no responsibility for the persistence oraccuracy of urls for external or third-party internet websites referred to

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To Manju, Sunil, and Saloni






Contents

Preface page xiii

List of symbols and abbreviations xv

1 Overview 11.1 Integral quantum Hall effect 11.2 Fractional quantum Hall effect 21.3 Strongly correlated state 41.4 Composite fermions 51.5 Origin of the FQHE 71.6 The composite fermion quantum fluid 71.7 An “ideal” theory 91.8 Miscellaneous remarks 10

2 Quantum Hall effect 122.1 The Hall effect 122.2 Two-dimensional electron system 142.3 The von Klitzing discovery 172.4 The von Klitzing constant 192.5 The Tsui–Stormer–Gossard discovery 212.6 Role of technology 22

Exercises 23

3 Landau levels 263.1 Gauge invariance 273.2 Landau gauge 283.3 Symmetric gauge 293.4 Degeneracy 323.5 Filling factor 333.6 Wave functions for filled Landau levels 343.7 Lowest Landau level projection of operators 363.8 Gauge independent treatment 37

vii






viii Contents

3.9 Magnetic translation operator 403.10 Spherical geometry 423.11 Coulomb matrix elements 523.12 Disk geometry/parabolic quantum dot 613.13 Torus geometry 653.14 Periodic potential: the Hofstadter butterfly 673.15 Tight binding model 69

Exercises 71

4 Theory of the IQHE 774.1 The puzzle 774.2 The effect of disorder 784.3 Edge states 814.4 Origin of quantized Hall plateaus 814.5 IQHE in a periodic potential 964.6 Two-dimensional Anderson localization in a magnetic field 974.7 Density gradient and Rxx 1014.8 The role of interaction 102

5 Foundations of the composite fermion theory 1055.1 The great FQHE mystery 1055.2 The Hamiltonian 1065.3 Why the problem is hard 1095.4 Condensed matter theory: solid or squalid? 1105.5 Laughlin’s theory 1135.6 The analogy 1155.7 Particles of condensed matter 1165.8 Composite fermion theory 1185.9 Wave functions in the spherical geometry 1385.10 Uniform density for incompressible states 1415.11 Derivation of ν∗ and B∗ 1415.12 Reality of the effective magnetic field 1455.13 Reality of the � levels 1465.14 Lowest Landau level projection 1465.15 Need for other formulations 1535.16 Composite fermion Chern–Simons theory 1545.17 Other CF based approaches 164

Exercises 173

6 Microscopic verifications 1746.1 Computer experiments 1746.2 Relevance to laboratory experiments 1756.3 A caveat regarding variational approach 176






Contents ix

6.4 Qualitative tests 1766.5 Quantitative tests 1816.6 What computer experiments prove 1876.7 Inter-composite fermion interaction 1886.8 Disk geometry 1926.9 A small parameter and perturbation theory 197

Exercises 199

7 Theory of the FQHE 2017.1 Comparing the IQHE and the FQHE 2017.2 Explanation of the FQHE 2037.3 Absence of FQHE at ν = 1/2 2067.4 Interacting composite fermions: new fractions 2067.5 FQHE and spin 2137.6 FQHE at low fillings 2137.7 FQHE in higher Landau levels 2137.8 Fractions ad infinitum? 214

Exercises 214

8 Incompressible ground states and their excitations 2178.1 One-particle reduced density matrix 2178.2 Pair correlation function 2188.3 Static structure factor 2208.4 Ground state energy 2228.5 CF-quasiparticle and CF-quasihole 2248.6 Excitations 2268.7 CF masses 2378.8 CFCS theory of excitations 2458.9 Tunneling into the CF liquid: the electron spectral function 245

Exercises 250

9 Topology and quantizations 2539.1 Charge charge, statistics statistics 2539.2 Intrinsic charge and exchange statistics of composite fermions 2549.3 Local charge 2559.4 Quantized screening 2639.5 Fractionally quantized Hall resistance 2649.6 Evidence for fractional local charge 2669.7 Observations of the fermionic statistics of composite fermions 2689.8 Leinaas–Myrheim–Wilczek braiding statistics 2699.9 Non-Abelian braiding statistics 2799.10 Logical order 281

Exercises 281






x Contents

10 Composite fermion Fermi sea 28610.1 Geometric resonances 28710.2 Thermopower 29710.3 Spin polarization of the CF Fermi sea 30110.4 Magnetoresistance at ν = 1/2 30110.5 Compressibility 305

11 Composite fermions with spin 30711.1 Controlling the spin experimentally 30811.2 Violation of Hund’s first rule 30911.3 Mean-field model of composite fermions with a spin 31111.4 Microscopic theory 31411.5 Comparisons with exact results: resurrecting Hund’s first rule 32411.6 Phase diagram of the FQHE with spin 32611.7 Polarization mass 33111.8 Spin-reversed excitations of incompressible states 33711.9 Summary 34711.10 Skyrmions 348

Exercises 360

12 Non-composite fermion approaches 36312.1 Hierarchy scenario 36312.2 Composite boson approach 36612.3 Response to Laughlin’s critique 36812.4 Two-dimensional one-component plasma (2DOCP) 37012.5 Charged excitations at ν = 1/m 37212.6 Neutral excitations: Girvin–MacDonald–Platzman theory 37712.7 Conti–Vignale–Tokatly continuum-elasticity theory 38412.8 Search for a model interaction 386

Exercises 389

13 Bilayer FQHE 39413.1 Bilayer composite fermion states 39513.2 1/2 FQHE 40013.3 ν = 1: interlayer phase coherence 40313.4 Composite fermion drag 40813.5 Spinful composite fermions in bilayers 409

Exercises 411

14 Edge physics 41314.1 QHE edge = 1D system 41314.2 Green’s function at the IQHE edge 41414.3 Bosonization in one dimension 41714.4 Wen’s conjecture 429






Contents xi

14.5 Experiment 43214.6 Exact diagonalization studies 43714.7 Composite fermion theories of the edge 438

Exercises 441

15 Composite fermion crystals 44215.1 Wigner crystal 44215.2 Composite fermions at low ν 44615.3 Composite fermion crystal 44915.4 Experimental status 45215.5 CF charge density waves 455

AppendixesA Gaussian integral 458B Useful operator identities 460C Point flux tube 462D Adiabatic insertion of a point flux 463E Berry phase 465F Second quantization 467G Green’s functions, spectral function, tunneling 477H Off-diagonal long-range order 482I Total energies and energy gaps 486J Lowest Landau level projection 490K Metropolis Monte Carlo 499L Composite fermion diagonalization 502

References 504Index 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 unexpectedbehavior, for example, the marvelous phenomenon known as the fractional quantum Halleffect. These properties result from the formation of a new class of particles, called“composite fermions,” which are bound states of electrons and quantized microscopicvortices. The composite fermion quantum fluid joins superconductivity and Bose–Einsteincondensation in providing a new paradigm for collective behavior.This book attempts to present the theory and the experimentalmanifestations of composite

fermions in a simple, economical, and logically coherent manner. One of the gratifyingaspects of the theory of composite fermions is that its conceptual foundations, whileprofoundly nontrivial, can be appreciated by anyone trained in elementary quantummechanics.At the most fundamental level, the composite fermion theory deals directly withthe solution of the Schrödinger equation, its physical interpretation, and its connection to theobserved phenomenology. The basics of the composite fermion (CF) theory are introducedin Chapter 5. The subsequent chapters, with the exception of Chapter 12, are an applicationof the CF theory in explaining and predicting phenomena. Detailed derivations are given formany essential facts. Formulations of composite fermions usingmore sophisticatedmethodsare also introduced, for example, the topological Chern–Simons field theory. To keep thebook within a manageable length, many developments are mentioned only briefly, but myhope is that this book will at least serve as a useful first resource for any reader interested inthe field. It can be used as a textbook for a graduate level special topics course, or selectedportions from it can be used in the standard graduate course on condensed matter physicsor many-body theory. Many simple exercises have been included to provide useful breaks.

Disclosure: Personally, themost difficult aspect that I have facedwhile writing this book hasbeen my own long and intimate involvement with composite fermions, which, one mighthope, would enhance the probability that the exposition is sometimes well thought out, butmakes it difficult to ensure the kind of objectivity that can come only with distance, bothin space and in time. Fortunately, much is known that is indisputable, which allows one todistinguish opinions and speculations from facts. The selection of topics, the emphasis, andthe 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 conceptsand formulations that directly relate to laboratory and/or computer experiments. If workin which I have participated appears more often than it deserves, it is because that is whatI 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 tosupply the original references to the best of my knowledge and ability, but the list is surelyincomplete. I have collected, over the years, many nuggets of knowledge by osmosis, andmy 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 nameindividually. It is a pleasure to acknowledge my profound debt to many colleagues whosewisdom and collaboration have benefited me over the years. These include, but are notlimited to,AlexeiAbrikosov, PhilAllen, PhilAnderson, Jayanth Banavar, G. Baskaran, LotfiBelkhir, Nick Bonesteel, Moses Chan, Albert Chang, Chiachen Chang, Sankar Das Sarma,GoutamDev, Rui Du, Jim Eisenstein, Herb Fertig, Eduardo Fradkin, Steve Girvin, GabrieleGiuliani, Fred Goldhaber, Vladimir Goldman, Ken Graham, Devrim Güçlü, DuncanHaldane, 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, SeungYeop Lee, Jon Magne Leinaas, Mike Ma, Allan MacDonald, Jerry Mahan, SudhansuMandal, Noureddine Meskini, Alexander Mirlin, Ganpathy Murthy, Wei Pan, Kwon Park,Vittorio Pellegrini,Mike Peterson,Aron Pinczuk, JohnQuinn, T.V. Ramakrishnan, SumathiRao, Nick Read, Nicolas Regnault, Ed Rezayi, Tarek Sbeouelji, Vito Scarola, R. Shankar,Mansour Shayegan, Chuntai Shi, Boris Shklovskii, Steve Simon, Shivaji Sondhi, DougStone, Horst Stormer, Aron Szafer, David Thouless, Csaba Toke, Nandini Trivedi, DanTsui, 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, KenGraham,

Devrim Güçlü, Gun Sang Jeon, Shivakumar Jolad, Paul Lammert, Ganpathy Murthy, MikePeterson, Csaba Toke, Giovanni Vignale, and Dave Weiss for a careful and critical readingof parts of the manuscript. Thanks are also due to Wei Pan for making available the traceused for the cover page, to Gabriele Giuliani for bringing tomy attention the quotation at thebeginning of the Preface, and to the National Science Foundation for financially supportingmy 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 upwithmypreoccupation during thewriting of this book. Little had I realized atthe beginning howmajor an undertaking it would be. But the experience has been instructiveand also greatly rewarding. I hope that the reader will find the book useful.






Symbols and abbreviations

b Chern–Simons magnetic fieldB external magnetic fieldB∗ effective magnetic field experienced by composite fermions

also denoted Beff , BCF or �B in the literatureCF composite fermion2pCF composite fermion carrying 2p vorticesCFCS composite fermion Chern–SimonsCF-LL � levelCS Chern–Simonse∗ local charge of an excitationeCS Chern–Simons electric fieldε dielectric function of the background material (ε ≈ 13 for GaAs)FQHE fractional quantum Hall effect�ωc cyclotron energy�ω∗

c cyclotron energy of composite fermionIQHE integral quantum Hall effectk 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 fermionsLL Landau levelLLL lowest Landau levelm∗a activation mass of composite fermion

mb electron band mass (mb = 0.067me in GaAs)me electron mass in vacuumm∗p polarization mass of composite fermion

n integral filling factor; LL or �-level indexN number of electrons/composite fermionsν filling factor of electronsν∗ filling factor of composite fermions

xv






xvi Symbols and abbreviations

φ0 flux quantum (φ0 = hc/e)�ν∗ wave function of noninteracting electrons at ν∗PLLL lowest Landau level projection operatorν wave function of interacting electrons at νQ monopole strengthQ∗ effective monopole strengthQHE quantum Hall effectRH, ρxy Hall resistanceRK the von Klitzing constant (RK = h/e2)RL, ρxx longitudinal resistanceRPA random phase approximationρ two-dimensional densitySMA single mode approximationTL Tomonaga–LuttingerTSG Tsui–Stormer–Gossard2DES two-dimensional electron system

VC Coulomb energy scale (VC ≡ e2ε�

≈ 50√

B[T] K for GaAs)Vm interaction pseudopotentialsVCF

m interaction pseudopotentials for composite fermionsz position in 2D (z ≡ x − iy)






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