This page intentionally left blank 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 80s 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 charg 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 Buckley Prize of the AmericanPhysical Society in200 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 betweenthe composite fermionmodel and other models onstag For the novice, a lucid presentation and dozens of valuable exercises.” Horst Stormer, Columbia University, NY and Lucent Technologies.W inner 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 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/9780521862325 © J. K. Jain 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 ISBN-13 978-0-511-28464-9 eBook (NetLibrary) ISBN-10 0-511-28618-X eBook (NetLibrary) ISBN-13 978-0-521-86232-5 hardback ISBN-10 0-521-86232-9 hardback 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 page xiii List of symbols and abbreviations xv 1 Overview 1 1 Integral quantum Hall effect 1 2 Fractional quantum Hall effect 2 3 Strongly correlated state 4 4 Composite fermions 5 5 Originof the FQHE 7 6 The composite fermionquantumfluid 7 7 An“ideal” theory 9 8 Miscellaneous remarks 10 2 Quantum Hall effect 12 1 The Hall effect 12 2 Two-dimensional electron system 14 3 The vonKlitzingdiscovery 17 4 The von Klitzing constant 5 The Tsui–Stormer–Gossard discovery 21 6 Role of technology 22 Exercises 23 3 Landau levels 26 1 Gauge invariance 27 2 Landau gauge 28 3 Symmetric gauge 4 Degeneracy 32 5 Filling factor 33 6 Wave functions for filled Landau levels 34 7 Lowest Landau level projection of operators 36 8 Gauge independent treatment 37 vii viii Contents 9 Magnetic translation operator 40 10 Spherical geometry 42 11 Coulomb matrix elements 52 12 Disk geometry/parabolic quantum dot 61 13 Torus geometry 65 14 Periodic potential: the Hofstadter butterfly 67 15 Tight binding model Exercises 71 4 Theory of the IQHE 77 1 The puzzle 77 2 The effect of disorder 78 3 Edge states 81 4 Originof quantizedHall plateaus 81 5 IQHE ina periodic potential 96 6 Two-dimensional Anderson localization in a magnetic field 97 7 Density gradient and Rxx 101 8 The role of interaction 102 5Foundations of the composite fermion theory 105 5.1 The great FQHE mystery 105 5.2 The Hamiltonian 106 5.3 Why the problem is hard 1 5.4 Condensed matter theory: solid or squalid? 110 5.5 Laughlin’s theory 113 5.6 The analogy 115 5.7 Particles of condensed matter 116 5.8 Composite fermiontheory 118 5.9 Wave functions in the spherical geometry 138 5.10 Uniform density for incompressible states 141 5.11 Derivationof ν∗ and B∗ 141 5.12 Reality of the effective magnetic field 145 5.13 Reality of the levels 146 5.14 Lowest Landau level projection 146 5.15 Need for other formulations 153 5.16 Composite fermion Chern–Simons theory 154 5.17 Other CF based approaches 164 Exercises 173 6 Microscopic verifications 174 1 Computer experiments 174 2 Relevance to laboratory experiments 175 3 A caveat regarding variational approach 176 Contents ix 4 Qualitative tests 176 5 Quantitative tests 181 6 What computer experiments prove 187 7 Inter-composite fermion interaction 188 8 Disk geometry 2 9 A small parameter and perturbation theory 7 Exercises 9 7 Theory of the FQHE 201 1 Comparing the IQHE and the FQHE 201 2 Explanation of the FQHE 203 3 Absence of FQHE at ν = 1/2 206 4 Interacting composite fermions: new fractions 206 5 FQHE and spin 213 6 FQHE at low fillings 213 7 FQHE inhigher Landaulevels 213 8 Fractions ad infinitum? 214 Exercises 214 8 Incompressible ground states and their excitations 217 8.1 One-particle reduced density matrix 217 8.2 Pair correlationfunction 218 8.3 Static structure factor 220 8.4 Ground state energy 222 8.5 CF-quasiparticle and CF-quasihole 224 8.6 Excitations 226 8.7 CF masses 237 8.8 CFCS theory of excitations 245 8.9 Tunneling into the CF liquid: the electron spectral function 245 Exercises 250 9 Topology and quantizations 253 9.1 Charge charge, statistics statistics 253 9.2 Intrinsic charge and exchange statistics of composite fermions 254 9.3 Local charge 255 9.4 Quantized screening 263 9.5 Fractionally quantized Hall resistance 264 9.6 Evidence for fractional local charge 266 9.7 Observations of the fermionic statistics of composite fermions 268 9.8 Leinaas–Myrheim–Wilczek braiding statistics 2 9.9 Non-Abelian braiding statistics 2 9.10 Logical order 281 Exercises 281 x Contents 10 Composite fermion Fermi sea 286 11 Geometric resonances 287 12 Thermopower 7 13 Spinpolarizationof the CF Fermi sea 301 14 Magnetoresistance at ν = 1/2 301 15 Compressibility 305 11 Composite fermions with spin 307 11 Controlling the spin experimentally 308 12 Violationof Hund’s first rule 3 13 Mean-field model of composite fermions with a spin 311 14 Microscopic theory 314 15 Comparisons with exact results: resurrecting Hund’s first rule 324 16 Phase diagram of the FQHE with spin 326 17 Polarizationmass 331 18 Spin-reversed excitations of incompressible states 337 19 Summary 347 110 Skyrmions 348 Exercises 360 12 Non-composite fermion approaches 363 11 Hierarchy scenario 363 12 Composite bosonapproach 366 13 Response to Laughlin’s critique 368 14 Two-dimensional one-component plasma (2DOCP) 370 15 Charged excitations at ν = 1/m 372 16 Neutral excitations: Girvin–MacDonald–Platzman theory 377 17 Conti–Vignale–Tokatly continuum-elasticity theory 384 18 Search for a model interaction 386 Exercises 389 13 Bilayer FQHE 394 11 Bilayer composite fermionstates 5 12 1/2 FQHE 400 13 ν = 1: interlayer phase coherence 403 14 Composite fermiondrag 408 15 Spinful composite fermions in bilayers 4 Exercises 411 14 Edge physics 413 11 QHE edge = 1D system 413 12 Green’s function at the IQHE edge 414 13 Bosonization in one dimension 417 14 Wen’s conjecture 4 Contents xi 15 Experiment 432 16 Exact diagonalization studies 437 17 Composite fermiontheories of the edge 438 Exercises 441 15Composite fermion crystals 442 15.1 Wigner crystal 442 15.2 Composite fermions at low ν 446 15.3 Composite fermioncrystal 4 15.4 Experimental status 452 15.5 CF charge density waves 455 Appendixes A Gaussian integral 458 B Useful operator identities 460 C Point flux tube 462 D Adiabatic insertion of a point flux 463 E Berry phase 465 F Second quantization 467 G Green’s functions, spectral function, tunneling 477 H Off-diagonal long-range order 482 I Total energies and energy gaps 486 J Lowest Landau level projection 0 K Metropolis Monte Carlo 9 L Composite fermion diagonalization 502 References 504 Index 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 formationof a newclass 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.