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
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ISBN-13 978-0-521-86232-5 hardback ISBN-10 0-521-86232-9 hardback
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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. 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 inChapter 5. The subsequent chapters, with the exception of Chapter 12, are anapplication 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 canbe 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 tim 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 presentationreflectmy views onwhat 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 inwhich I have participated appears more oftenthanit 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 incomplet 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,AlexeiAbrikosov, PhilAllen, PhilAnderson, 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, Shankar, Mansour Shayegan, Chuntai Shi, Boris Shklovskii, Steve Simon, Shivaji Sondhi, Doug Stone, Horst Stormer, Aron Szafer, David Thouless, Csaba Toke,˝ 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 Toke,˝ 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 oncomposite 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 b But the experience has been instructive and also greatly rewarding. I hope that the reader will find the book useful. Symbols and abbreviations
b Chern–Simons magnetic field B external magnetic field B∗ effective magnetic field experienced by composite fermions also denoted Beff , BCF or B inthe literature CF composite fermion 2pCF composite fermioncarrying2 p vortices CFCS composite fermion Chern–Simons CF-LL level CS Chern–Simons e∗ local charge of anexcitation eCS Chern–Simons electric field dielectric function of the background material ( ≈ 13 for GaAs) FQHE fractional quantum Hall effect ωc cyclotronenergy ω∗ c cyclotronenergy 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 ∗ ma activationmass of composite fermion mb electronbandmass ( mb = 067 me inGaAs) me electronmass invacuum ∗ mp polarizationmass 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 xvi Symbols and abbreviations
φ0 flux quantum (φ0 = hc/e) ∗ ν∗ wave function of noninteracting electrons at ν PLLL lowest Landau level projection operator ν wave function of interacting electrons at ν Q monopole strength Q∗ effective monopole strength QHE quantum Hall effect RH, ρxy Hall resistance 2 RK the von Klitzing constant (RK = h/e ) RL, ρxx longitudinal resistance RPA random phase approximation ρ two-dimensional density SMA single mode approximation TL Tomonaga–Luttinger TSG Tsui–Stormer–Gossard 2DES two-dimensional electron system √ e2 VC Coulomb energy scale (VC ≡ ≈ 50 B[T] K for GaAs) Vm interaction pseudopotentials CF Vm interaction pseudopotentials for composite fermions z positionin2D ( z ≡ x − iy) 1 Overview
The past two and a half decades have witnessed the birth and evolution of a new quantum fluid that has produced some of the most profoundly beautiful structures in physics. This fluid is formed when electrons are confined to two dimensions, cooled to near absolute zero temperature, and subjected to a strong magnetic field. In this overview chapter, results and ideas are stated without explanation; they reappear later in the book with greater elaboration.
1.1 Integral quantum Hall effect The field beganin80 with the discovery of the integralquantumHall effect (IQHE), when vonKlitzingobserved (Fig. 1 ) plateaus in the plot of the Hall resistance as a function of the magnetic field. The Hall resistance on these plateaus is precisely quantized at
h R = , (1) H ne2 where n is aninteger, h is the Planck constant, and e is the electron charg Quantizations in physics are as old as quantum mechanics itself. What is remarkable about the quantization of the Hall resistance is that it is a universal property of a complex, macroscopic system, independent of materials details, sample type or geometry; it is also robust to variation in temperature and disorder, provided they are sufficiently small. Furthermore, concurrent with the quantized plateaus is a “superflow,” , a dissipationless current flow in the limit of zero temperatur It has been known since the early days of quantum mechanics that a magnetic field quantizes the kinetic energy of an electron confined in two dimensions. These quantized kinetic energy levels are called Landau levels (LLs), which are separated by a “cyclotron energy” gap. The number of filled Landau levels, which depends on the electron density (ρ) and the magnetic field (B), is called the filling factor (ν), givenby
ρhc ν = . (2) eB
2 The RH = h/ne plateau occurs in the magnetic field range where ν ≈ n.
1 2 1 Overview
Fig. Left: Discovery of the integral quantum Hall effect by K. von Klitzing (February 5, 80). Right: The first high precisionmeasurement(February 12, 80). Courtesy: K. vonKlitzing. (Reprinted with permission.)
The IQHE is a consequence of the LL quantization of the electron’s kinetic energy in a magnetic field, and can be understood in terms of free electrons. When the filling factor is aninteger( ν = n), the ground state is especially simple: N electrons occupy the N single particle orbitals of the lowest n Landau levels, described by a wave function that is a single Slater determinant, denoted n. The state is “incompressible,” , its excitations cost a nonzero energy. This incompressibility at integral fillings, combined with disorder-induced Anderson localization, leads to an explanation of the IQHE [368].
1.2 Fractional quantum Hall effect The 82 discovery by Tsui, Stormer and Gossard of the 1 /3 effect, , of a quantized plateau at h RH = , (3) 1 2 3 e (Fig. 2 ) raised the field to a whole newlevel. This plateau is seenwhenthe lowest Landau level is approximately 1/3 full (, ν ≈ 1/3). Laughlin[ 3 ] stressed the correlated nature of the “1/3 state,” and wrote an elegant wave function for the ground state at ν = 1/m: 1 2 m − |zi| 1/m = (zj − zk ) e 4 i . (4) j Fig. Discovery of the fractional quantum Hall effect (FQHE) by D. Tsui, H. L. Stormer and Gossard. October 7, 8 Source: H. L. Stormer, Rev.Mod.Phys. 71 875–889 (99). (Reprinted with permission.) exponent m must be an odd integer to ensure antisymmetry. Laughlin argued that this state is incompressible, and that the excitations have a fractional charge of magnitude e/m. The 1/3 plateau had offered only the first glimpse into an extraordinary new collective state of matter. Subsequent experimentation revealed new phenomena whose breathtaking beauty and richness would have been impossible to anticipate, by any stretch of the imagination, at the time of the first observationof 1 / Many more quantized plateaus, h R = , (5) H fe2 labeled by different fractions f , were observed. The number of fractions rapidly mushroomed over the years with refinements in experimental conditions, and new fractions are being discovered even now. Figure 3 shows a more recent magnetoresistance trac Anexplanationofthe physics of this newcollective state of matter, which rivals superconductivity and Bose–Einstein superfluidity in both scope and elegance of the phenomena associated with it, has been an exciting challenge for, and accomplishment of, the modern many-body condensed matter theory. 4 1 Overview Fig. The magnificent FQHE skylin Diagonal resistance as a function of the magnetic field for a two-dimensional electron system with a mobility of 10 million cm2/V s. A FQHE or anIQHE state is associated with each minimum. Many arrows only indicate the positions of filling factors (for example, 1/2, 1/4, etc.) and have no FQHE associated with them. Source: W. Pan, H. L. Stormer, D. Tsui, L. Pfeiffer, K. W. Baldwin, and K. W. West, Phys.Rev.Lett. 88, 176802 (2002). (Reprinted with permission.) 1.3 Strongly correlated state At high magnetic fields all electrons occupy the lowest Landau level. Their kinetic energy is then constant, hence irrelevant. Interacting electrons in a high magnetic field are mathematically described by the Hamiltonian 1 H = (lowest Landau level), (6) rjk j (i) The no-small-parameter problem The theory contains no parameters. (The Coulomb interaction merely sets the overall energy scal) (ii) The degeneracy problem The number of degenerate ground states in the absence of interaction (with H = 0) is astronomically larg (iii) The no-normal-state problem In some instances, a nontrivial collective phenomenon can be understood as an instability of a “normal state,” which is the state that would be obtained if the 1.4 Composite fermions 5 interaction could be switched off. For the FQHE problem, switching off the interaction does not produce a meaningful stat The FQHE has no normal stat The physics of the FQHE state is “strongly” non-perturbative, in the sense that no weak coupling limit exists in which the solution is close to a known solution. We do not have the luxury of calculating small deviations from a normal state, but must determine “full” answers. We wish to identify the principle responsible for the dramatic physics revealed by experiment, but do not even know where to begin. Yet this has turned out to be a theorist’s dream problem, in which, thanks to an abundance of experimental clues and an incredible amount of luck, a precise and secure understanding has been achieved. A new language has been developed to describe the concepts governing the behavior of this stat Many facets of the FQHE physics, as well as numerous related phenomena, follow directly from a single unifying principle: the formation of “composite fermions.” 1.4 Composite fermions With the observation of many fractions, an analogy with the IQHE could be identified, which has led to an explanation of the FQHE (and more). The eigenfunctions and eigenenergies for the ground and (low-energy) excited states of strongly interacting electrons at an arbitrary LLL filling ν are expressed in terms of the known solutions of the noninteracting electron problem at the LL filling ν∗ as follows: 2p ν = PLLL (zj − zk ) ν∗ , (7) j