F I E L D T H E O R E T I C A L QUANTITIES IN T H E F R A C T I O N A L Q U A N T U M H A L L E F F E C T B y S tephanie H y t h e C u r n o e B . Sc. (Phys ics ) U n i v e r s i t y of T o r o n t o , 1991 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y W e accept th is thesis as c o n f o r m i n g to the requi red s t anda rd T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A J u l y 1997 © Stephanie H y t h e C u r n o e , 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia; I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada V6T 1Z1 Date: Abstract This thesis studies two models of the fractional quantum Hall effect (FQHE), the boson-ic (Chern-Simons-Landau-Ginzburg) description and the fermionic (composite fermion gauge theory) description. The bosonic theory attempts to describe the F Q H E states at filling fractions v = while the fermionic theory attempts to describe the states at v = ... p,, and the metallic states in between. 2np±l Within the bosonic theory, the fractionally charged quasiparticles of the F Q H system are vortices which appear during the breakdown of the uniform quantum Hall state. The energetics of a single vortex state are studied whereby it is shown how the system may become unstable to the formation of vortices. Numerical vortex profiles are computed by minimising the Hamiltonian. Using the fermionic theory of composite fermions interacting with gauge fluctuations, we consider two important field theoretic quantities, the self-energy and the thermody-namic potential in a finite magnetic field. We find that the conventional Luttinger-Ward treatment of the oscillatory behaviour of the thermodynamic potential is not applicable in two dimensions, for any kind of interaction. Instead we propose a new formulation which omits all crossed graphs and which necessarily includes the oscillatory self-consistent self-energy. To second order in perturbation theory, the oscillatory self-energy is calculated by retaining Landau level quantisation on the internal fermion line. The low energy form of the self-consistent self-energy is obtained by means of a new iterative procedure which is introduced here. This procedure makes use of the structure introduced by Landau level quantisation. We also investigate the structure induced in the analogous two dimensional electron-phonon problem, in order to assist our understanding of the composite fermion n self-energy. In the low energy l imit , it is found that the renormalised form of the compos-ite fermion Green's function is of the same form as the unrenormalised Green's function. Therefore we argue that the principal effects of interactions may be accounted for using a field-dependent renormalised mass. The iterative procedure for finding the self-consistent self-energy is used to evaluate the renormalised gap between the Fermi energy and the first excited states, which rapidly converges in a few iterations. We find a significant departure from the asymptotic result obtained by ignoring Landau level quantisation in the regime of experimentally relevant values of the parameters. We compare our findings with measurements of the gap in fractional Ha l l states near v = 1 / 2 . 111 Table of Contents Abstract ii Table of Contents iv List of Tables viii List of Figures ix Acknowledgements xiv 1 Introduction 1 1.1 Physics in Two Dimensions 3 1.2 The Quantum Hall Effect 5 1.2.1 The Integer Effect 6 1.2.2 The Fractional Effect and the Laughlin Theory 9 1.3 Chern-Simons Field Theories and the Transmutation of Statistics . . . . 11 1.4 Overview 13 1.4.1 Vortices 14 1.4.2 A Novel State of Matter? 14 1.4.3 Magnetic Oscillations - the Old Meets the New 15 2 Vortices and the Bosonic Description of the FQHE 17 2.1 Superfiuid Analogy 18 2.1.1 Off-Diagonal Long Range Order 19 iv 2.1.2 The Excitation Spectrum 19 2.1.3 Experiments 20 2.2 Vortices 22 2.3 Numerical Vortex Configurations 28 2.4 Discussion of Results 35 2.5 Summary 38 3 Composite Fermions, Gauge Fluctuations and the Gap 39 3.1 Composite Fermions . . . 42 3.1.1 Experimental Evidence for Composite Fermions 42 3.1.2 The Composite Fermion Gauge Theory 44 3.2 The Self-Energy at AB = 0 48 3.3 The Self-Energy for AB ^ 0 50 3.4 The Renormalised Gap . 55 3.5 Phonon Interactions 59 3.5.1 Debye Phonons 60 3.5.2 Einstein Phonons 65 3.6 Iterative Self-Consistent Results for E(e) and Aw* 68 3.7 The Effective Mass: Comparison to Experiment 72 3.8 Summary 76 4 Oscillatory Quantities 78 4.1 Non-Interacting Approximation 79 4.1.1 The Chemical Potential 79 4.1.2 The Compressibility Measurement 84 4.1.3 The Average Energy 85 4.1.4 Measuring the Average Energy 88 v 4.1.5 Thermodynamic Potential and its Derivatives 91 4.2 Gauge Interaction Effects : . . . 92 4.2.1 Chemical Potential 93 4.2.2 The Second Order Corrections to fi 96 4.3 The Oscillatory Behaviour of £1 101 4.3.1 The Oscillatory Behaviour of f>('2) 102 4.3.2 Non-Perturbative evaluation of fi 105 4.4 Summary 110 5 Conclusion 111 Symbols 114 Bibliography 116 A Phonon Interactions in Finite Magnetic Field 122 A.l Debye Phonons 122 A.2 Einstein Phonons 124 B Self-Energy Corrections in a Finite Magnetic Field 126 C Proofs Involving the Thermodynamic Potential 131 C.l Proof of (4.25) 131 C.2 Proof of (4.81) 131 C.3 Proof of (4.84) 134 D Oscillatory Corrections to D(u, q) 135 E Oscillatory Corrections to ft<2) 137 vi Oscillatory Expansion of Q in 2-D 139 F . l P r o o f of (4.86) :. , 139 F . 2 P r o o f of (4.87) 140 vii List of Tables 2.1 Energy and size (TO) of vortices and antivortices for n — 3, 5 and 7. f.t.cr is a naive estimate of the value of the chemical potential at which a condensate of these configurations is expected to form (/i0 = !0~2eV). /.icr is more carefully described in the text 36 2.2 The density pv± and the corresponding filling fraction v\ at which the condensate of vortices is expected to become dense. (See text for a precise definition.) 36 2.3 Summary of numerical results for quasi-particle and quasi-hole creation energies and gap energy. Units are ^ 37 3.1 Four experiments which measure the SdH oscillations in the fractional quantum Hall regime and their corresponding self-energy coefficients. K-2 is determined from the given experimental values for e, ne and ra& using Eq. (3.27) 72 v i i i List of Figures 2.1 T h e funct ion h(r) corresponding to a vortex for v = 1 / 3 , 1 / 5 a n d 1/7. . 32 2.2 Dens i ty profile of the vortex configuration for v = 1 / 3 , 1 / 5 and 1/7. . . . 33 2.3 T h e funct ion h(r) corresponding to an antivortex for v — 1 / 3 , 1 / 5 a n d 1/7. 34 2.4 Dens i ty profile of the antivortex configuration for v — 1 / 3 , 1 / 5 and 1/7. 35 3.1 Schemat ic drawing of the geometry used in the magnet ic focusing exper-iment of G o l d m a n , Su and J a i n . E a c h arc is a cyc lo tron orbit (after • G o l d m a n et al . [63]) 43 3.2 F e y n m a n d iagrams representing the first order corrections, K^^q, u), to the gauge propagator 46 3.3 F e y n m a n d i a g r a m representing the self-energy, n a n d n are L a n d a u level indices. T h e straight line represents the fermion (either electron or c o m -posite fermion) and the wavy line represents the boson (either a p h o n o n or a gauge f luctuation) 51 3.4 R e a l part of the self-energy of composi te fermions. s — 2 corresponds to C o u l o m b i c electron-electron interactions: 5 = 3 corresponds to short-ranged screened interactions (see text). T h e energies are in units of Auc. 53 3.5 Imaginary part of the self-energy of composi te fermions. s = 2 corresponds to C o u l o m b i c electron-electron interactions: s = 3 corresponds to short-ranged interactions and the energies are in units of Auc, as before. . . . 54 3.6 G r a p h i c a l so lut ion to (3.34) for different values of n. T h e crosses show the poles between which lies the renormal i sed gap 56 ix 3.7 First order perturbation results for the effective gap, Aw*, as a function of coupling, K-2, for p — 50. The solid curve is |1 — ^ log(2p + 1) | _ 1 and points are numerical solutions to Eq. (3.34). The inset shows the same data plotted as a function of n e , which is related to K2 via eq. (3.27). . 57 3.8 Imaginary part of the Green function of composite fermions in a finite magnetic field with n = 0 and p = 50. The upper figure has a coefficient K-2 = 0.31 and the lower one has A'2 = 6.3. The energy is in units of Acuc. 58 3.9 The real and imaginary parts of the self-energy for electrons interacting with phonons with a Debye spectrum (uD = 20uc) in a small magnetic field. All energies are in units of LOC\ the coupling is Ko = 1 61 3.10 The derivative with respect to energy of the real part of the self-energy of electrons interacting with Debye phonons is a finite magnetic field (too = 20cuc). The energy is in units of uic, and Ko = 1 63 3.11 The imaginary part of the Green function of electrons interacting with Debye phonons for KD = .001, .1 and .25 with n = 0 (see the text for definition of the coupling KD) 64 3.12 Real part of the self-energy of electrons interacing with phonons with an Einstein spectrum (tog = 20o;c). Both axes are in units of uc; the coupling is KB = 1 66 3.13 Rainbow diagram contributions to the self-energy 69 3.14 The effective mass, m* as a function of B and v for composite fermions. The lower x— axis has been obtained by assuming an electron density ne = 2.23 x 10 ncm~ 2 . The crosses are numerical calculations which include corrections coming from E ( 2 ) (the self-consistent self-energy). The curves are a guide for the eye 71 x 3.15 Four experiments which measure the effective mass from SdH measure-ments, a) Du et al. [30], b) Coleridge et al [32], c) Leadley et al. [57] d) Manoharan et al. [31]. Note that in all cases the vertical axis is in units of me, whereas in Fig. 3.14 it is drawn in units of m .^ 73 3.16 The effective mass m* as a function of electron density. These are the same results as in Fig. 10 except that the .T-axis has been obtained using ne = (2A""'e~ ) with e = 13 and m = .07me. The crosses are numerical calculations which include corrections coming from S^2) (the self-consistent self-energy) 74 3.17 Experimental results of Leadley et al. [57] showing the effective mass as a function of electron density 75 4.1 The chemical potential of composite fermions as a function of filling frac-tion calculated for m* = 20mt and B = 13T using Eqs. (4.7) and (4.8). This plot shows only the oscillatory component of /(, which oscillates about its values at v = \ •. . 82 4.2 Experimental results of Eisenstein et al. [55] showing a) the compressibility of a 2-D electron gas and b) the chemical potential, which is the integral of the compressibility. Note that the horizontal axis is the same on both plots , 83 4.3 The compressibility as a function of filling fraction calculated for m* = 20m& and The y-axis has been scaled by the interaction strength, ^ (see text) 86 xi 4.4 A schematic illustration of the splitting of the lowest Landau level. When there are no interactions the lowest Landau is not split. The mean field theory of composite fermions splits the lowest Landau level into sublevels separated by Auc. Gauge fluctuations renormalise the mass, so that in the end the lowest Landau is subdivided into levels separated by Aw*, but still centered around 87 4.5 The average energy as a function of magnetic field at T = 0. The scaling of the it/-axis has been chosen to match the experimental parameters of Kukushkin et al. (see text) 88 4.6 The experimental results of Kukushkin et al. [54] showing b) The average energy as a function of magnetic field at T = 0 and c) its derivative. . . 89 4.7 The derivative of the average energy with respect to the magnetic field as a function of magnetic field at T = 0. The discontinuities are related to the gap as explained in the text 90 4.8 The self-energy when the highest composite fermion Landau level is half-filled at T = 0 (solid) and for a finite temperature with a small, positive shift of the chemical potential (dashed), e is measured with respect to the chemical potential. The vertical scale is arbitrary (it depends on the coupling A 2 ) 94 4.9 Feynman diagram representing the second order correction to the thermo-dynamic potential. The straight lines represent composite fermions and the wavy line represents a gauge fluctuation 97 4.10 The zeroth, first and second order oscillatory contributions to 9S2K The wiggly cuts indicate that the oscillatory component of the fermion line is to be evaluated 103 xii Some F e y n m a n d i ag rams represent ing four th order c o n t r i b u t i o n s to Q — f2 0 . T h e first shows a d i sconnec ted g raph , the second is the 0 ( A 4 ) ske le ton g r a p h and the t h i r d is an example of a skele ton g r a p h w i t h a self-energy in se r t i on 132 x i i i Acknowledgements First of all, I would like to-thank Professor Philip Stamp for supervising this Thesis and for many hours of thought provoking discussion. I would like to thank Professor Nathan Weiss for introducing me to this topic and for encouragement throughout the years. I acknowledge the good fortune I have had by embarking on a post secondary ed-ucation at a time when higher education was perceived as a valuable social endevour in Canada. I have enjoyed opportunities that have included four years of postgraduate funding from the Natural Sciences Engineering Research Council (NSERC) of Canada, affordable tuition, and good summer jobs at Ontario Hydro and with the Summer Un-dergraduate NSERC program. I am very grateful to my colleague, friend and companion, Martin Dube, for his affection during this time. Filially, I would like to acknowledge the support of my family thoughout south-western Ontario, especially my mother, Lynda Grace Curnoe. xiv Chapter 1 Introduction A sys t em of e lectrons confined to two d imens ions i n the presence of a s t rong m a g n e t i c field has m a n y of the features tha t character ise p rob lems i n m o d e r n condensed m a t t e r phys ics . It is a s t rong ly corre la ted sys tem, thanks to the presence of s t rong C o u l o m b i c e lec t ron-e lec t ron in te rac t ions . It is an idea l example of a lower d i m e n s i o n a l sy s t em, where i t s conf inement to two d imens ions gives rise to f rac t iona l q u a n t u m numbers . Its phase d i a g r a m consis ts of states w i t h v a n i s h i n g r e s i s t iv i ty w h i c h are a l t e rna ted w i t h i n s u l a t i n g phases as a func t ion of magne t i c field, g i v i n g rise to the p h e n o m e n o n k n o w n as the f r ac t iona l q u a n t u m H a l l effect. F i n a l l y , the sys tem appears to be desc r ibab le as a gauge theory w i t h a novel quas ipa r t i c l e , bu t is essent ia l ly non-pe r tu rba t ive thanks to the s t rong e lec t ron-e lec t ron in te rac t ions . T h e f r ac t iona l q u a n t u m H a l l effect ( F Q H E ) is a p h e n o m e n o n that occurs at very l o w tempera tu res when electrons confined to two d imens ions at the interface be tween two semiconduc to r s are exposed to a s t rong magne t i c f ie ld. T h e magne t i c f ie ld is a p p l i e d p e r p e n d i c u l a r to the two-d imens iona l e lec t ron gas. T h e current response to an e lec t r ic f ield is measured , f rom w h i c h the r e s i s t iv i ty is de t e rmined . T h e r e s i s t i v i t y is , i n fact , a t w o - d i m e n s i o n a l tensor, by v i r tue of the usua l H a l l effect, s ince the e lect rons exper ience a force due to the magne t i c field w h i c h is pe rpend icu l a r to the i r ve loc i ty . T h e F Q H E is the e x p e r i m e n t a l observa t ion of p l a t e a u x i n the transverse c o m p o n e n t of the r e s i s t i v i t y o c c u r r i n g w h e n the lowest energy level of the sys tem has a f r ac t iona l o c c u p a n c y [1]. T h e l o n g i t u d i n a l componen t of the r e s i s t iv i ty is observed to v a n i s h at these same values . 1 Chapter 1. Introduction 2 This behaviour is exactly analogous to the integer quantum Hall effect (IQHE), which is the observation of the same phenomenon when an integer number of energy levels are completely occupied [2]. While the features of the IQHE may be accounted for with a theory of non-interacting electrons in the presence of a magnetic field, the same does not hold for the features of the F Q H E , which are contingent on the presence of strong electron-electron interactions. The ongoing theoretical challenge of the F Q H E has been the search for a description that is both physically correct and that illuminates the essential physics of the problem. Laughlin solved this system for the cases when the fractional occupancy of the lowest energy level is v = ^ - j - by his discovery of a variational ground state wavefunction for a system of interacting electrons in a magnetic field [3]. Extensive numerical work on small systems has proven that Laughlin's wavefunction has an extremely large overlap with numerical solutions [4], and is an exact solution for point interactions [5, 6]. Laughlin's discovery goes a long way towards illuminating the nature of the ground state at these special occupancies. In particular, it explains the presence of a gap in the excitation spectrum. The main shortcoming of the Laughlin solution is that it is not readily ex-tendible to the other tilling fractions. Furthermore, the strongly interacting theory from whence the Laughlin wavefunction is derived is essentially non-perturbative, and thus its utility for making physical predictions is limited by the lack of viable approaches to attack this non-perturbative problem. There have been two other approaches to understanding the F Q H E , both of which were motivated by the desire to gain a better physical insight into the problem. The bosonic theory (also known as the Chern-Simons-Landau-Ginzburg (CSLG) theory) was invented in order to exploit the similarities between the F Q H E and superfluidity [7]. The fermionic theory (composite fermion (CF) theory) was originally conceived by Jain as an explanation of the remarkable similarity between the F Q H E and the IQHE [8]. This Chapter 1. Introduction 3 approach provides an explanation for much of the phenomenology associated with the F Q H E and also serves as a starting point for a field-theoretic approach. This thesis is concerned with several aspects of these models of the F Q H E . In this Introduction we describe the physics of two dimensions, followed by a discussion of basic ideas of the integer and fractional QHE. In Chapter 2 we elaborate on the system at the filling fractions v = -j—• using the CSLG theory. The special role of vortices is described and numerical solutions for vortex profiles are presented. Chapters 3 and 4 are concerned with the composite fermion theory of the F Q H E . In these Chapters we examine two important field theoretic quantities, the self-energy and the thermodynamic potential. The self-energy yields information about the low energy single particle exci-tations, in particular, the gap. The thermodynamic potential is the theoretical origin of several quantities of both theoretic and experimental interest, including the compress-ibility, chemical potential and magnetisation. In the presence of a magnetic field, these quantities are ''"oscillatory", that is, they oscillate as a function of an applied magnetic field. In Chapter 3 the self-energy is calculated self-consistently in full oscillatory form and is used to determine the effective mass of composite fermions for a range of physically realistic parameters. In Chapter 4 we discuss the oscillatory quantities associated with the F Q H E and compute the thermodynamic potential. 1.1 Physics in Two Dimensions "I call our world Flatland, not because we call it so, but to make its nature clear to you, my happy readers, who are privileged to live in space." [9] This thesis will make extensive use of ideas that are especially pertinent to the physics of lower-dimensional systems. In particular, we need to understand the characteristics of a system of identical particles in two dimensions. It is well known that in three or Chapter 1. Introduction 4 more dimensions the symmetry of the wavefunction is required to be S = +1 or — 1, corresponding to bosons or fermions respectively. As an example of this we consider exchanging two particles and then returning them to their original configuration: single valuedness implies that S2 = 1. This is a consequence of the fact that in D > 3 all the paths in configuration space that connect two identical initial and final states may be continuously deformed into a single-path (i.e., they belong to a single homotopy class). This is not necessarily so in two dimensions: in general there will be many homotopy classes, corresponding to the number of times the particles encircle each other. In this case the homotopy classes are isomorphic to elements of the braid group [10]. In general this leads to a symmetry Sn = 1 and the possibility of fractional statistics. Given the possible existence of fractional statistics, we proceed to investigate the scenarios which may give rise to it. Using a differential geometry approach, Leinaas and Myrheim showed that paths on a generalised 2-dimensional surface enclosing a singularity (such as the tip of a cone) in general acquire an additional phase [11]. Our interests lie with another particular case, where phases are acquired due to the presence of a gauge field. Gauge fields always appear in the context of charged particles as the mediators of the force between them. They are subject to a symmetry known as gauge invariance which is related to conservation of charge current. This symmetry is ij)(x) -> ^(.r)exp[zA(:c)] (1.1) A(x) A{x) + -VA(x) (1.2) e where ip(x) is the electron wavefunction and A(x) is the gauge field. It is obvious from these relations that gauge fields may also influence the statistics of particles. The most famous example of this is the Aharonov-Bohm effect, in which particles in a 1-D closed geometry acquire a phase when magnetic flux is adiabatically introduced through the Chapter 1. Introduction 5 ring [12]. However, we are concerned with the 2-D case. In this situation a term of the form ^lx.y.\AIJ.dvAx ^ 7T may be added to the Lagrangian, which is gauge invariant up to a surface term. This term gives rise to particles which upon exchange acquire a phase of S = exp [13]; such particles are known as "anyons". There is one other feature related to the fact that we are confined to two dimensions that is of considerable' importance to the Hall effect. In the special situation where a magnetic field is applied perpendicular to the plane, we find that the kinetic energy is quenched by the magnetic field. The energy spectrum collapses into highly degenerate Landau levels and the momenta kx,ky together are no longer good quantum numbers. The energy is determined completely from the Landau level index number. It is the large gaps between the Landau levels and their large degeneracy that set the stage for the integer quantum Hall effect. 1.2 The Quantum Hall Effect The integer quantum Hall effect was discovered in 1980 by von Klitzing, Dorda and Pep-per [2], following a suggestion of Ando, which was to examine the Hall conductivity of a two dimensional electron gas in a large magnetic field when the Fermi level was between two Landau levels [14]. As predicted, they observed that electrons in the impurity bands between Landau levels are localised and that the Hall (transverse) current arises from electrons in all of the Landau levels. However, no one had anticipated the most astonish-ing feature of the results: a robustly quantised Hall conductance whenever u, the number of filled Landau levels, is an integer. The fractional quantum Hall effect was discovered only two years later by Tsui, Chapter 1. Introduction 6 Stormer and Gossard [1], completely unanticipated by theorists. They originally observed a plateau in the Hall conductance at v = \ y Improved samples have led to observations of the F Q H E at filling fractions with denominators as large as v = j | [15]. The device used to measure the Q H E is a semiconductor heterojunction, typically GaAs-AlGaAs. A thin layer of electrons is trapped at the interface, when excess electrons from AlGaAs move into the lower energy conduction band of GaAs. An electron gas of thickness ~ 50A and carrier density ne « 1 0 n c m - 2 is formed. With this density, a magnetic field B ~ 4T is required to achieve a filling factor of v ~ 1. With a band mass vib ~ .07me, this corresponds to a cyclotron frequency uc = ^ ~ 6 x 1 0 - i e V ~ 70K, which sets the temperature limit for observation of the IQHE. 1.2.1 The Integer Effect The integer effect is now well understood. The system consists of electrons confined to two dimensions inside a magnetic field which points in the third direction, and is described by the Hamiltonian H = - ^ ( i V - eAf ( 1 . 4 ) 2 m where A satisfies V x A = B. The solution to the single particle Hamiltonian consists of harmonic oscillator eigenstates with an energy spectrum of Landau levels at En = (n + T})^C- The spacing between Landau levels is given by the cyclotron energy uc = In an infinite, disorderless system the energy density of states is a series of delta functions, 6(E — E„). Disorder and finite size cause the delta functions to spread out and introduce localised states between the Landau levels. The localised states between Landau levels do not contribute to the conductivity. In order to see the IQHE a large B-iie\d must be used, so that the spacing between Landau levels is large and the levels do not overlap even when they are broadened by disorder. Chapter 1. Introduction 7 T h e f i l l i n g f r ac t ion v is defined as where ne is the e lec t ron densi ty (per un i t area) and $ 0 = 1 S the m a g n e t i c f lux q u a n t u m . 7v is the n u m b e r of filled L a n d a u levels: i t is also the number of e lectrons per m a g n e t i c flux q u a n t u m of the ex te rna l magne t i c field. T h e I Q H E effect occurs whenever v is an integer . W h e n th is happens the sys tem behaves as an i n su l a to r and the l o n g i t u d i n a l c o n d u c t i v i t y falls to zero e x p o n e n t i a l l y as a func t ion of t empera ture . A t the same t i m e , w h e n an e lec t r ic f ield is a p p l i e d , every free e lect ron gains a net m o m e n t u m i n the transverse d i r e c t i o n , c r ea t ing a H a l l current that is p r o p o r t i o n a l to the dens i ty o f e lect rons , a n d a c o n d u c t i v i t y w h i c h tel ls us tha t w h e n a current is in jected t h rough the sample a H a l l vo l tage (perpen-d i c u l a r to the current ) appears but there is no vol tage i n the d i r e c t i o n o f the current . L o o k i n g at the above fo rm for the c o n d u c t i v i t y , i t is not at a l l apparent w h y the c o n d u c t i v i t y s h o u l d increase i n quan t i sed steps as a func t ion of v. T h e p l a t e a u x i n the c o n d u c t i v i t y are due to the presence of i m p u r i t i e s w h i c h t rap e lect rons i n to l o c a l i s e d states w h i c h occur i n between the L a n d a u levels. These states can a c c o m m o d a t e e x t r a e lectrons as they are added to the sys tem w i t h o u t c o n t r i b u t i n g to the cur rent , a n d i n th is way the sys t em stays i n the q u a n t u m H a l l state even as v is va r i ed away f r o m the integer values. (1.6) The. r e s i s t i v i t y is the inverse of this m a t r i x , (1.7) Chapter 1. Introduction 8 The single particle eigenstates may be written as linear combinations of either ex-tended or localized harmonic oscillator wavefunctions which are the solutions to (1.4). The set of solutions depends on the gauge: the gauge Ax = Q,Ay = Bx gives extended eigenstates, and the gauge Ax — —H^-,Ay = 4p gives localized eigenstates. Clearly one solution may be expressed as a linear combination of solutions of the other type. The (unnormalised) extended states are written as where Hn(x.) is the nth order Hermite polynomial. (One could equally well interchange x and y here). The localized states are written (using complex coordinates z = x + iy) as In both cases n is the Landau level index. The important difference between the two is that the extended wavefunctions are also y-momentum eigenstates and the localised states are angular momentum eigenstates (with quantum number ???.). Therefore, it is preferable to use the localised eigenstates when the electron is in the vicinity of an impurity since the perturbation caused by the impurity is localised. Likewise, near the edges of the sample it is more appropriate to perturb about the extended solutions. When an electric field is applied (in the .T-direction) the Hamiltonian is still solvable if the gauge yielding extended eigenstates is used. Laughlin [16] showed that the electric field causes the centre of the wavefunctions to be shifted by an amount Ax = mczE jeB1. The current is calculated for each eigenfunction using the covariant derivative, Dtp = (id — eA)ip which yields a gauge invariant result. The current in the ^-direction, Jx = i>k,n(x>y) = e ,kyHn(x)e-x2/2 (1.8) (1.9) is the Landau level index. Chapter 1. Introduction 9 1.2.2 The Fractional Effect and the Laughlin Theory The fractional quantum Hall effect owes its existence to interactions between the elec-trons. A new ground state is formed, but only at very low temperatures, lower than the interaction energy between the electrons. When this happens the conductivity of the system behaves in a manner that is remarkably similar to the integer system, except that it occurs when the lowest Landau level is fractionally filled. This cannot be understood in terms of individual electrons: it must be considered as a many-body phenomenon. The Ar-particle Hamiltonian is given by where the last two terms are the background potential and the electron-electron interac-tion respectively. For a single particle (N = 1) the last term vanishes and the background is assumed to be uniform, and so the second term is constant. Then the solutions are the same as those found for the integer effect. There is no analytic many-body solution when N > 1: the solution discovered by Laughlin [3] is a variational solution which has been shown numerically to be a very good approximation to the actual solution. The Laughlin many-body solution minimizes H at filling fractions v = ^, where m is an odd integer. The oddness of m forces i> to be antisymmetric in the z,-. This wavefunction is comprised of states from the lowest Landau level. This may be seen by comparing it with (1.9) - it only depends on z, not and 1, therefore the Landau level index is n = 0. The wavefunction is also an eigenstate of angular momentum, with eigenvalue A ' ( A ' .~ 1 ) "', as required by the symmetry of H with respect to a rotation about z. A notable feature of this wavefunction is that it is incompressible [17]. The incompressibility is a consequence of the fact that, in order to expand or contract the system, particles must be (1.10) Mzir • • , ZN) = EI (z.i ~ zkTl e xP 3<k N (1.11) Chapter 1. Introduction 10 in jec ted in to the sys tem, a n d this costs a finite amoun t of energy. T h e gap is the energy needed to p roduce pa i rs of f r ac t iona l ly charged quas ipar t i c les ( Q P ' s ) a n d quas iho les ( Q H ' s ) . T h e size of the gap also ind ica tes the re la t ive s t a b i l i t y o f the state. T h e m a n y -b o d y f o r m of the Q H wavefunct ion was also conjectured by L a u g h l i n [18] by c o n s i d e r i n g how the g r o u n d state wavefunct ion w o u l d evolve as one magne t i c f lux q u a n t u m was added to the sys t em ad i aba t i c a l l y , a n d then removed by a gauge t r ans fo rma t ion . T h e f o r m of the Q H wavefunc t ion loca ted at ZQ is (1.12) w h i c h L a u g h l i n argued h a d a charge of ^ . L a u g h l i n ' s fo rm of the Q H wavefunct ion is genera l ly accepted: however several dif-ferent forms for a Q P have been p roposed [4, 18, 19, 20, 21]. Here we s i m p l y quote two of the t r i a l wavefunct ions , and we note tha t a l l of t h e m have a different f o r m f r o m that w h i c h w i l l be cons idered i n C h a p t e r 2. T h e Q P wavefunc t ion p roposed by L a u g h l i n is « - n ( ^ - ^ ) n t e - ^ r « p ( - i E W ) . <"3) J a i n ' s Q P wavefunc t ion l oca t ed at the o r i g i n is [20] i l ^ - ^ f c r - ' e x p f - j E N 2 ) - (1-14) j<k \ ^ i I L i k e the g r o u n d state and the Q H state, the L a u g h l i n Q P state does not i n c l u d e any-m i x i n g in to h igher L a n d a u levels. In contras t , th is m i x i n g appears e x p l i c i t l y i n J a i n ' s p roposed f o r m , as i n d i c a t e d by the presence of zi,z~2 • • •• J a m a rgued tha t the na tu re 1 1 1 '1 ~2 .,2 ./2 2 'cl ~2 <3 Chapter 1. Introduction 11 of the F Q H E was fundamentally related to the IQHE and that this relation could be exploited by explicitly including structure from the higher Landau levels into his proposed many-body states [23]. More will be said about Jain's theory in Chapter 3. 1.3 Chern-Simons Field Theories and the Transmutation of Statistics Although Laughlin solved the F Q H E for v — it soon became evident that a dif-ferent approach was needed to account for other aspects of the theory. Girvin [7] first expounded the need for a theory based on an order parameter in order to formalise what was already evident, i.e., that the F Q H system at fractional fillings shares with super-fluids the important property of vanishing resistivity [22]. This led to the development of the "Chern-Simons-Landau-Ginzburg" (CSLG) approach to the F Q H E in which the order parameter is derived from bosonic wavefunctions [19]. Another unresolved aspect of the theory was the so-called "hierarchy" problem, the experimental observation that certain series of filling fractions (v — = \ v 4, | . . . ) had greater stability than oth-ers [y = 4^i-= \i |) • • • )• This problem was addressed by several authors [6, 24] who proposed various schemes for the generation of the "daughter" states by conden-sation of the quasiparticles of the Laughlin states, and also by Jain [8] in his theory of "composite fermions" (CF's). Jain's idea may also be extended to a theory of fermions interacting with gauge fluctuations [25, 26]. Despite the fact that they both describe the same phenomenon, the CSLG and the C F theories have completely different perspec-tives. However, both theories are derived using the same trick, which makes use of the properties of two dimensional systems, allowing for the addition of a Chern-Simons term in the Lagrangian. We first consider the situation described bv the CSLG theory, i.e., v = T T V T - In this Chapter 1. Introduction 12 case we have a magnetic field satisfying eB (2n+l )n e (1.15) $ 0 which means that there are exactly 2n + 1 magnetic flux quanta of the external magnetic field for each electron. This is described by a Lagrangian C = -iip\chdo)4> + 2^|(&V - ^ A ) V f + j dy^{y)<iP(y)V(x - y)^{x)^(x) (1.16) where C6Q = dt. The trick is to attach to each electron 271+1 magnetic flux quanta oriented oppositely to the external applied field by means of a (singular) gauge transformation [27]: 1>{x) -+ # r ) e x p (^27?, + l)YJ%J = ]>{x) (1.17) A(x) - A(x) + ( 2 n + V % = A(x) + a(x) (1.18) e f#j where % is the angle between electrons at positions i and j. a(x) is known as the "statistical'"' gauge field. Let us evaluate the average value of V x a(x) using Stokes's theorem, / V x Z ) WijdA = y VJ V^.-j • riS. (1.19) The angle may also be expressed as Imlog(z,- — Zj). The only non-zero contributions to the integral come from angles associated with particles inside the closed path. Thus the result is simply 2~KN , and so the average value of V x a is just b = V x a = (2n + 1 )$one. The new many-body wavefunction ip(x) is bosonic, as may be seen by exchanging any two indices, which corresponds to exchanging particles: ip(x) is fermionic, therefore exchange of two particles yields a minus sign; as for the phase, the angle transforms as 9j.j —»• Oji + 7T, which yields exp/(2??. + 1 )7r = — 1. It is clear that if 2??. + 1 was replaced by 2rc, ip would be fermionic. Chapter 1. Introduction 13 The constraint that there are 2??. + 1 flux quanta attached to each particle is imple-mented by the introduction of a Chern-Simons term in the Lagrangian, £ = -^^t(c^^5o-^e(a() + .4o))i' + ^ | ( ^ ^ V - ^ ( A + a))^|2 (1.20) — t > A a " c > V + J' dyj>HyW(y)V(x - y)^(x)iP(x) 1.21] 2*0(2n + Varying with respect to a 0 yields the equation of motion = = S ^ f i ) ( 1 ' 2 2 ) which is the required constraint. The important thing to notice is that when the system is at v — ^ - j - there is an exact cancellation between b and B, so that what remains is a system of free bosons in zero magnetic field. The fermionic case is similar except that the cancellation occurs at even fillings, p = The consequences of this will described in much greater detail in the following chapters. 1.4 Overview The F Q H E has attracted a great deal of theoretical interest in recent years as a practical application of the intriguing mathematical ideas just presented. The bosonic theory gave physicists an alternative view of the nature of the quantum Hall states. The fermionic theory described the quantum Hall states observed at all filling fractions and the metallic states in between. Very recently it has been shown' [76] that the metallic states (for example, at v = \) are the first physical example of a non-Fermi liquid. The fermionic theory also serves naturally as a mean field starting point for the describing the oscillatory phenomenology of the F Q H E . This thesis explores some of these issues. Chapter 1. Introduction 14 1.4.1 Vortices W e have jus t shown that the H a l l sys tem at v — c an be m a p p e d in to a s y s t e m of i n t e r ac t i ng bosons i n zero magne t i c f ie ld. It was shown by L a u g h l i n [17] tha t the p r i m a r y exc i t a t ions are pa i rs of f r ac t iona l ly charged Q P ' s a n d Q H ' s w h i c h are assoc ia ted w i t h the l o c a l a d d i t i o n of a s ingle magne t i c f lux q u a n t u m . B o t h the Q H a n d the Q P occu r n a t u r a l l y i n the boson ic desc r ip t ion , and the fact that each e lec t ron is t i e d to 2n + l f lux q u a n t a ensures tha t there is a charge ± < ^ j i n the v i c i n i t y of a s ingle f lux q u a n t u m . In C h a p t e r 2 we show that w i t h i n the bosonic theory the charge conf igura t ions o f b o t h the Q P a n d Q H are vor t ices . T h e f r ac t iona l charge associa ted w i t h the vor t ices has the i m p o r t a n t p r o p e r t y tha t i t is local If the vor t ices are p i n n e d , then the f r ac t iona l charge does not con t r i bu t e to the b u l k current . T h i s is the m e c h a n i s m by w h i c h the f r ac t iona l q u a n t u m H a l l effect is m a i n t a i n e d even as the f i l l i n g f rac t ion is va r i ed away f rom v — ^p[, ana logous to the role p l a y e d by i m p u r i t i e s i n the integer effect. G i v e n the size of a vor tex , we m a y de t e rmine a c r i t e r i o n for the b r e a k d o w n of the q u a n t u m H a l l state. F r o m this we set l i m i t s o n the size of the p l a t e a u x i n the H a l l conduc t i v i t y . 1.4.2 A Novel State of Matter? L a n d a u [28] conjec tured that a sys t em of i n t e r ac t i ng fermions (a F e r m i l i q u i d ) c o u l d be m a p p e d in to a sys t em of w e a k l y i n t e r ac t i ng fermions w i t h r eno rma l i s ed parameters . T h e dressed electrons are k n o w n as "quas ipar t ic les ," a n d the p r i m a r y exc i t a t i ons are jus t quas ipa r t i c l e s w i t h a m o m e n t u m greater t han the F e r m i m o m e n t u m . T h e pa ramete r s i n ques t ion inc lude quant i t ies such as the effective mass and the i n t e r a c t i o n pa ramete r s (the famous f-funct ion) , w h i c h are defined i n p e r t u r b a t i o n theory, a n d m a y be measu red i n exper imen t s . Chapter 1. Introduction 15 Using the composite fermion theory, the Hall system at v = ^ is described as a system of fermions in zero magnetic field. They are coupled through a long-range Coulomb force which may be expressed as an interaction with a gauge field. The transverse component of the gauge field gives rise to perturbative corrections which appear to invalidate the Landau quasiparticle picture [26]. This is apparent in several closely related features, the most striking of which is an apparent divergence of the effective mass. Slightly away from v = r, the cancellation between the external magnetic field and the fictitious "statistical" field is no longer exact. In this situation the composite fermions are in a small magnetic field and so naively we can predict that the system will have a gap associated with the cyclotron frequency of the effective magnetic field. The gap can thus be parametrised in terms of an effective mass, Aw* = ^r. Perturbative calculations show that this mass also diverges in the limit v —»• \ [29]. Even more astonishing, this result has been qualitatively confirmed by experiment [30, 31, 32]. Until now, a quantitative comparison has been lacking because of the extreme limits analysed in the initial calculations. Furthermore, quantitative results based on a perturbative approach are questionable because the coupling in this system is not small. In Chapter 3 we address both of these issues in our search for a self-consistent determination of the effective mass away from v — \. 1.4.3 Magnetic Oscillations - the Old Meets the New Magnetic oscillations occur in two or three dimensions as the filling factor of a system is varied. This may be achieved by either varying ne or jj\ in most experiments it is the latter. For example, consider what happens to the system as the magnetic field is increased. The spacing between Landau levels grows and the number of states in each Landau level increases until there is a sudden drop in the chemical potential. This produces oscillations in the density of states near the Fermi surface and every other Chapter 1. Introduction 16 measurab le q u a n t i t y (such as the res is t iv i ty , m a g n e t i s a t i o n a n d c o m p r e s s i b i l i t y ) w i l l va ry i n the same manner . T h e existence o f such osc i l l a t ions was first no ted i n theory by L a n d a u i n 1930 [33] a n d observed la ter tha t year by Shubn ikov a n d de H a a s [34] i n r e s i s t i v i t y measurements and by de H a a s a n d van A l p h e n [35] i n m a g n e t i s a t i o n measurements 1 . T h e c o n n e c t i o n between theory and exper iment was es tab l i shed i n 1933 by Pe ie r l s [37]. T h e d H v A effect, w h i c h an o s c i l l a t i o n of the magne t i s a t i on , p roved to be a very useful t o o l for m a p p i n g out the F e r m i surface. T h e o r e t i c a l a c t i v i t y i n th is area reached i ts p r i m e i n the 1960's w h e n different techniques were used to o b t a i n formulae d e s c r i b i n g the a m p l i t u d e s of the var ious types of osc i l l a t ions [38, 39, 40, 41]. W h a t is s u r p r i s i n g about the f rac t iona l H a l l sys t em is that the p e r i o d o f o sc i l l a t i ons depends o n not jj. T h i s suggests that the f r ac t iona l H a l l sys t em m a y be ana lysed u s i n g k n o w n techniques w i t h a s imp le s u b s t i t u t i o n of AB for B, and th is appears to be the case w h e n a non- in t e rac t ing sys tem of compos i t e fermions w i t h a r e n o r m a l i s e d , field-dependent mass is used. However , when we a t t empt to account for in te rac t ions i n p e r t u r b a t i o n theory, we f ind tha t a p p r o x i m a t i o n s deve loped i n the 1960's are not r e a d i l y app l i cab l e to the H a l l sys t em because of the confinement to two d i men s i o n s in s t ead of three. T h i s is true for any in te rac t ions , not jus t gauge f luc tua t ions . In C h a p t e r 4 we discuss the o sc i l l a t o ry quant i t i es of non- in t e rac t ing compos i t e fe rmions w i t h an effective mass , a n d we re-examine the formulae der ived for three d imens ions , a n d show w h y they are not v a l i d i n two d imens ions . A n interesting historical account of magnetic oscillations may be found in [36]. Chapter 2 Vortices and the Bosonic Description of the FQHE In the boson ic theory of the F Q H E the electrons i n a q u a n t u m H a l l s ta te at filling-f rac t ion v — are t r ans formed in to bosons by a t t ach ing an o d d n u m b e r of m a g n e t i c f lux q u a n t a to each one. H i s t o r i c a l l y , there have been two different m o t i v a t i o n s for th is . F i r s t , there was the desire to express the theory w i t h an order pa ramete r , to make a f o r m a l ana logy between the H a l l sys t em and other sys tems w i t h v a n i s h i n g res i s t iv i ty , namely , s u p e r c o n d u c t i v i t y and superf luids [7]. Secondly , the i n t r o d u c t i o n of the phase assoc ia ted w i t h the b o u n d f lux q u a n t a was precisely w h a t was needed to give the theory off -d iagonal l o n g range order [43]. T h i s i m p o r t a n t p h y s i c a l p rope r ty is not at a l l ev ident i n the L a u g h l i n theory. In b o t h cases a very specific n u m b e r of b o u n d f lux q u a n t a is necessary - i t mus t be chosen so that on average i t cancels the b a c k g r o u n d m a g n e t i c f ie ld . In o ther words , the electrons are b o u n d to ^ = 2n + l f lux quan ta . F o r the purposes o f th i s C h a p t e r , our interests l ie w i t h the first m o t i v a t i o n since a theory w i t h an order pa rame te r opens the p o s s i b i l i t y of p e r f o r m i n g v iab le ca l cu l a t i ons for large number s of pa r t i c les a n d hence possesses p red ic t ive capab i l i t i e s . In the first sec t ion of th is C h a p t e r we descr ibe the c o n s t r u c t i o n of the order pa rame te r a n d the c a l c u l a t i o n of the e x c i t a t i o n s p e c t r u m . T h e lowest energy exc i t a t i ons have been the subject o f intense expe r imen ta l sc ru t iny ; some o f these expe r imen t s are d i scussed here. Vor t i c e s are conf igura t ions i n w h i c h the order pa ramete r has a s i n g u l a r i t y at some po in t (i .e., i t vanishes) a n d are thus t opo log i ca l l y d i s t inc t f rom the u n i f o r m g r o u n d s tate 17 Chapter 2. Vortices and the Bosonic Description of the FQHE 18 associated with the fillings v — j ^ - - As we shall see, the vortices of this theory possess the same properties as the Laughlin quasiparticles and quasiholes which are described in the Introduction. They have a localised fractional charge ± - ^ - j - positioned where an additional magnetic flux quantum has been introduced. These properties are explained in a natural and intuitive way within the bosonic theory of the F Q H E . The vortex states are of interest for two reasons. First, they may account for the existence of the plateau in the resistivity during the transitions between Hall states in the fractional regime. Second, a vortex/anti-vortex pair is the lowest energy neutral excitation occurring on at least part of the excitation curve. The final sections of this Chapter are a study of single vortex states using a simple Landau-Ginzburg model. By establishing appropriate boundary conditions and min-imising the Hamiltonian, we numerically construct vortex and anti-vortex profiles. Using these profiles, the creation energy of a vortex/anti-vortex pair is calculated and compared with results obtained by other methods. Within our model, which neglects interactions between vortices, we also discuss the energetics of the formation of a vortex lattice and its role in the transition between F Q H E states. 2.1 Superfluid Analogy The bosonic theory is an attempt to capture the essential physics of the F Q H E at and near the fillings v = 9^3; , where the vanishing resistivity suggests that the ground state is a novel kind of condensate. The first step in formulating the superfluid analogy is the construction of an order parameter which follows from the bosonisation of the theory. Chapter 2. Vortices and the Bosonic Description of the FQHE 19 2.1.1 Off-Diagonal Long Range Order In this Section we review the arguments given by Girvin and MacDonald [43, 42] which demonstrated that the bosonic wavefunctions, formed by attaching 2n + 1 flux quanta to each electron at v = possess off-diagonal long range order (ODLRO). In superfiuids the presence of this property indicates that there is a long range correlation of the one-body density matrix, i.e. p(z, z) = N I d2z2,... <PzNil>\z, z2,... zN)ip{z',z2,... zN) (2.1) —» \z — z'\~'8 for large \z — z] (2.2) for some power /?>(). The Laughlin wavefunction (1.11) does not possess O D L R O . Using the singular gauge transformation (1.17) and (1.18) which attaches 2n + 1 magnetic flux quanta to each electron, Girvin and MacDonald showed that that Laughlin wavefunction becomes ip(zl: z 2 , . . . , zN) = \zi - ^ | 2 n + 1 exp ( - W \ z A (2.3) i<j V 4 k . J and that for large \z — z'\ the one-body density matrix goes as p(z, z ) \z — z | ^ + T . (2.4) This demonstrates that the system of electrons with 2?i + l flux quanta attached possesses algebraic ODLRO. 2.1.2 The Excitation Spectrum Using Feynman's theory of superfiuid He 4 [44] as a starting point, Girvin, MacDonald and Platzman [22] derived the excitation spectrum of the quantum Hall states at v = By direct application of Feynman's method, they found that in the long wavelength limit the mode at the cyclotron energy, A(0) = TILUC, completely saturates the oscillator Chapter 2. Vortices and the Bosonic Description of the FQHE 20 weight, in agreement with Kohn's theorem1. In the F Q H E it is expected that the lowest energy excitations should occur within the lowest Landau level: i.e., that there should be intra-Landau level excitations. Girvin, MacDonald and Platzman sought the spectrum of this type of excitation by projecting their result onto the lowest Landau level. Because of Kohn's theorem, the weight of this lower energy mode must vanish faster than k'2 as k -> 0. Their results showed that the system has a gap for any value of k. In the limit k —> oo the excitations are free Laughlin quasiparticle quasihole pairs. There is a minimum which occurs at the inverse magnetic length (/ = —}= « 10 _ 6cm) known as the magnetoroton minimum which is associated with a bound state of a QP-QH pair. As k —> 0 there is a crossover to a magnetoplasmon mode2. Similar curves were derived within the composite fermion theory (described in the next Chapter) for the filling fractions v = 0n^±l [46]. These curves are characterised by having p minima. 2.1.3 Experiments A great deal of effort has been devoted to the experimental verification of the excitation spectrum. The earliest experiments that measured the gap were performed using thermal activation [47]. In these experiments the longitudinal resistivity is measured as a function of temperature and the gap A is determined using the relation [48] pxx oc exp ( T T ^ ) • ( 2 - 5 ) A is always measured to be less than predicted gap energies, even when finite sample width and L .L . mixing have been included in the calculations, because of impurities which 1 Kohn's theorem states that the cyclotron mode has all of the weight of the oscillator strength in the limit k —» 0 and that all other modes vanish faster than fc2 [45]. 2 Zhang has suggested another possibility: that in the limit k —> 0 the lowest energy excitation may consist of a bound state of two QP's and two QP's. [19] Chapter 2. Vortices and the Bosonic Description of the FQHE 21 broaden the energy levels. Later experiments [31, 49] showed that there exist a series of filling fractions (such as v = | , | , y,...) with decreasing gap sizes, implying decreasing stability. This was the basis for complicated '''hierarchy" schemes [6, 24] and for Jain's theory of composite fermions [8]. There are two types of experiment that probe the k —• 0 limit of the excitation spec-trum. The first is light-scattering measurements [50]. Typically, light with wavevector k ~ 104 c m - 1 «C j is used in a back-scattering geometry. A temperature activated peak is observed at the energy of the excitation. Another version of this experiment [51] was performed using samples in which there was a density modulation imposed on the elec-tron gas with wavevector qo ~ j (corresponding to the magnetoroton minimum). This provides a means for photons to couple to the 2-D electron gas at wavevectors mq<j, for m an integer. The resulting spectra are a series of peaks at different energies corresponding to wavevectors mq0. The other type of experiment that probes the k —» 0 limit is thermodynamic mea-surements in which the gap is determined indirectly from some kind of discontinuity as one of the parameters (either the magnetic field or the electron density) is varied to bring the system through a quantum Hall state. These measurements include photolumines-cence [52, 53, 54], which measures the free energy of the electron gas, compressibility [55] from which the chemical potential may be determined, and resistivity (SdH oscillations) [30, 31, 32, 56, 57]. Much more will be said about these measurements in Chapters 3 and 4. Finally, ballistic phonons have been used to probe the excitation spectrum at k = j [58]. The point where the phonons have the strongest interaction with the electron gas is where the structure factor is a maximum, which occurs at the magnetoroton minimum. Chapter 2. Vortices and the Bosonic Description of the FQHE 22 2.2 Vortices Vortices play an essential role in the transition between quantised states. By breaking the translational symmetry of the system, their, role is analogous to the role played by impurities in the integer case. Physically, they are singularities which consist of a localised excess/deficit of charge. In this way the system can accommodate slight local deviations of charge density away from the special filling fractions v — while maintaining the bulk characteristics. This leads to a finite width in the steps of the quantised Hall conductivity. It has been shown that single vortex solutions exist and can be found by the addition (or removal) of particles to the system [19]. On the other hand, we can consider varying the filling fraction by making a small, uniform change in the external magnetic field B. In this case a finite number of vortices per unit area would be needed to accommodate the excess magnetic field and maintain the quantum Hall state. Thus, in this scenario we might reasonably expect that the ground state would be a lattice of vortices. However, each vortex costs a finite amount of energy to create. Because of this fact, it is obvious that a lattice of vortices would not be the ground state of a system with an infinitesimal change in B, since an infinitesimal change in B would give rise to an infinitesimal change in the energy of the uniform state. A ground state consisting of a vortex lattice is only possible when the change in the external magnetic field is finite, as will be shown below'3. As described in Chapter 1, the theory of interacting electrons in an external magnetic field B = V x A is transformed into a theory of bosons via a unitary (singular) transfor-mation which attaches an odd number of flux quanta to each electron. The boson field is represented by a complex function ip. The theory is described by the Lagrangian C = -i^[chdo-ie(a0 + AQ)U + ^~\[hV - - ( A + a)]^|2 2m c 3 T h e details of tins section appear in [59]. Chapter 2. Vortices and the Bosonic Description of the FQHE 23 + 2$0(2n + l ) £ / w A a ^ a A + 2 ~ ( V W ~ ^ ( 2 ' 6 ) where A is the externally applied gauge field, a is the statistical magnetic field (the flux attached to each electron), A is a repulsive potential which is used as an approximation to the Coulomb interaction and p is the chemical potential. Following the approach of [60], we begin by writing down the equations of motion obtained from (2.6) by varying ip, a 0 and a,- respectively: D2 i(do + ieoo)tj} + -^i> - X(^ip)ip + pip = 0 (2.7) w,U £l>diCii = _ (2 8) V ^ „ ( 2 n + 1) ~ * „ ( 2 r c + l ) ^ D r f - (D,vH] = ^ + + djao) = M2l+1fi* (2-9) where D{ — <9,; — ie(Aj + a,;) is the covariant derivative and e?- = —OQCIJ + dja0 is the statistical electric field. The first equation (2.7) is just the Schroedinger equation. Eq. (2.8) is the mean field constraint that the electron density p = ip^t/j is proportional to the number of attached flux, b = V x a. The third equation (2.9) relates the current to the statistical electric field. We seek.solutions to these equations for a(x) and p(x). In the special case where £ = -(2n + l)$oT (2.10) A is constant the equations of motion have a constant solution b -B = i-i . 9 $ 0(2n + l) * 0(2n + l) A' { ' ' Another approach to solving for the eigenstates of the system is to minimize the Hamil-tonian, H= jct2xU = j d'2x ±\dm2 - t^tt2 + 2m 2 (2.12) Chapter 2. Vortices and the Bosonic Description of the FQHE 24 subject to to the constraint Eq. (2.8) that V x o = $ 0(2n + l)p. (2.13) It is clear that (2.11) minimises H. The first term of H is zero and the solution p = f lies in the minimum of the potential given by the last two terms. This solution corresponds to the special filling fraction v = J ^ J . Now we consider the case where B does not satisfy (2.10). It is straight forward to show that there are no covariantly constant solutions, such that D^yj - 0. The proof is as follows: first write yj = £ e j f ! . Now D^yj = 0 implies that <9/(£ = 0 and that QQ, — e(.4|( + aM), so that the combination A + a is a pure gauge. It follows that B = —b. Now Eqs. (2.7) and (2.8) with D^yj = 0 imply that either ip = 0 or that f = ^ o ( - 2 ^ + 1 ) , so that ^ = . ,ZB,. This completes the proof that nonzero covariantlv constant solutions only exist when f = ^ + i y In addition to the lack of solutions with D^ip = 0 there are also no nonzero covariantly static solutions (i.e. solutions with D0ip = 0) unless f = ®0^+1y Consider yj = nel°. Then D0yj = 0 implies that d0?] = 0 and d00 + ea0 = 0. The third equation of motion (2.9) then implies that (eaj + eAj - dj6)if _ -e^d^eaj + eAj + djQ)r) , m ~ $ 0(2n + 1) ' There are three possibilities to consider. First, 1] = 0, in which case yj = 0. Secondly, we could have that e(a,: + A,) = 8,9. Then (b + B) = V x (a + A) = h^djOjO - 0, which, using (2.8), implies that B = —(2n + l)$of • Finally we consider the possibility that e(a,; + Aj) + dj6 varies with time. OQA, = 0 by assumption and d^ttj ^ 0 would imply that OQT] 7^ 0, which contradicts the main assumption that D0yj = 0, so all the time dependence would have to be in 8j9. Then dj8 oc e^dodjO which has no solution. Thus there are no nonzero covariant static solutions unless B — —('2n + l)$o^. Chapter 2. Vortices and the Bosonic Description of the FQHE 25 So instead we seek other static solutions with doip = 0 and a 0 0. This is done by minimizing the Hamiltonian (2.12) subject to the constraint (2.8). If we consider the case —B = (2n + l ) $ 0 / f ¥• (2n + l)$of> then clearly the constant configuration p — f, which minimises the potential in (2.12), is no longer a solution since then the gauge fields A and a will not cancel and the first term in (2.12) will lie infinite. On the other hand, a solution of the form p = ^ does satisfy the equations of motion (2.7-2.9) with 2 eoo = y / ( l - / ) . (2-15) This corresponds to a to shift of the chemical potential, effectively shifting the position of the minimum of the potential in (2.12). A cancellation of the gauge fields A, and re-occurs, and the solution lies at the new potential minimum. Since this solution does not minimize the old potential, it leaves open the possibility that a lower energy solution exists. However, we will show that this can only occur for a finite change of B. In seeking solutions to the equations of motion our final alternative is to examine configurations where p deviates locally towards the direction of the minimum of the potential. If the deviation is infinitesimal, then it will cause an infinitesimal flux / dS • (B + b) to pierce the system. Since it is not an integer amount of flux, it cannot be "gauged away" and thus it gives a diverging contribution to the first term of (2.12). At last we consider the case when there is an integer amount of flux threading the system. This scenario necessarily corresponds to vortex configurations. At large distances there must be a phase associated with I(J that will cancel e(A + a), otherwise the first term in (2.12) is logarithmically divergent. If the wavefunction is to be single-valued, it follows that ift must vanish at the origin. In the case where B — —$0(2?i+l)f, vortex solutions will always have a greater energy than the constant configuration, since, as we have shown, the constant configuration Chapter 2. Vortices and the Bosonic Description of the FQHE 26 minimises H. We wish to know the vortex energy in the situation when B ^ — $o(2n + 1)^. Since the chemical potential has been moved away from the special value, we might conclude that the vortices would lower the energy of the system and thus describe the F Q H E away from the special filling as a collection of these vortices. However, in what follows, we demonstrate that this is not generally the case. The point is that the configurations which are extrema (but not necessarily minima) of the Hamiltonian (2.12) subject to the constraint (2.13) are completely independent of the value of the chemical potential //J We have, in fact, proven this already. Configurations are extrema of (2.12) subject to (2.13) if and only if they satisfy the equations of motion (2.7-2.9). Thus if a particular ip is such an extremum (it may be a constant or, more generally, a multivortex configuration) with a given value pi of p., then it will satisfy the equations of motion for some function a^K The same ip will satisfy the equations of motion for any other value p2 of p but this time with a new ea[>2) = ea\^ + p2 — Pi- It is thus also an extremum of the Hamiltonian with this new value p2 of p. Even though the extremal configurations for differing values of p are the same, the energetics may differ for different values of p. Notice, for example, that for p such that clearly has the lowest possible energy. A single vortex configuration ipv(x) which solves the equations of motion (see [61]) will have a larger energy than the ground state. Let e.v be the excess energy of the vortex with respect to the ground state of the Hamiltonian with p = p,^. Now consider an alternate Hamiltonian with p = pi ^ PQ. The same configuration ipv(x) will still be an extremum of this Hamiltonian but, its energy (which is the difference in energy between the vortex configuration and the configuration with p = f) will differ: t± — x $o(2».+l) (let us call this value of the chemical potential po), the configuration p _ Ho. ~ A ev - (Pi - Po)Nx V I (2.16) Chapter 2. Vortices and the Bosonic Description of the FQHE 27 where N,v = ± M s the "particle number" of the vortex. Note that when p is decreased it is preferable to form a vortex (Nv < 0) whereas if p is increased an antivortex is preferred. Equation (2.16) has the following consequences. For small values of bp = po — pi the vortex configuration increases the energy of the system at p = p\. We thus expect that the constant configuration with p = f - will be the configuration of lowest energy despite the fact that the potential energy is not at its minimum. Thus the system remains at the special filling fraction even after p has been shifted from /.t0 to p\. (We emphasize again that this occurs for small shifts / J 0 — Pi-) It follows that in the mean field approximation = 0 (2.17) for a range of p near p. '— p$. This equation is familiar from the integer quantum Hall effect and is due to the presence of a gap in the spectrum. It implies that a finite change in the chemical potential is required before the density can be modified. Returning to Eq. (2.16) we see that when 6p = \p{) — pi\ > ev/N.v, a single vortex has loiuer energy than the constant configuration \ip\'2 = po/X. A gas of such vortices will have an even lower energy. We thus expect that near this value (ev/Nv) of 8p the lowest energy configuration of the Hamiltonian (2.12) subject to the constraint (2.13) will be a collection of vortices. The vortices are charged, therefore the interactions between the vortices will be repulsive, which implies a higher energy. Thus we expect a lattice of vortices to be formed only when the the energy lost by forming vortices is greater than the interaction energy. The situation is similar when the magnetic field is modified instead of p. If we begin at the special filling fraction with B = — &o(2n + 1 )^ - and change B at fixed /.( by a small amount, the lowest energy configuration of the Hamiltonian will occur at a new density for which p is still equal to ^ o ( /^ 2 «) w n i c n w n ^ n o w n°t D e a t the minimum of the potential. As the magnetic field is increased (or decreased) further (again at fixed p — /t0) the cost in energy of a single vortex (or antivortex) becomes progressively smaller until at Chapter 2. Vortices and the Bosonic Description of the FQHE 28 some critical value of the field it becomes negative. At that point the lowest energy mean field configuration is no longer a constant but rather a finite density of vortices in which will ''condense", whereas if it is decreased the antivortices will "condense".) Thus if -^f-is plotted either as a function of p or as a function of B there is a plateau surrounding the value /<o for which this ratio is constant and equal to ^ ^ j - If, on the other hand, B is varied at fixed density then we move off the plateau and the lowest energy mean field configuration consists of a finite density of vortices . 2.3 Numerical Vortex Configurations In the remainder of this Chapter we look more closely at the vortices of this model which, as we have discussed, will be solutions both for p = p() and for values of //, differing from /.to- We present some numerical solutions for these vortices which will allow us to estimate the value of //, at which a lattice of vortices starts to form. The shape of the vortices will also lead to an estimate of the density at which the collection of vortices becomes non-dilute. The method we have chosen for finding vortices is by considering the Hamiltonian and constraint given by Eqs. (2.12-2.13) at a value of //, = p() = $ o ( f » + 1 ) - We then look for radially symmetric configurations (which necessarily carry an integer number of flux quanta of B + b) which minimize the Hamiltonian. Anticipating the fact that our solution will be a vortex with an integer number of flux quanta we organize a radial ansatz as follows: First write case p is no longer equal to B . (If the magnetic field is increased then the vortices 4>o(2n+l) -ikO (2.18) with k an integer. The Hamiltonian (2.12) can now be written as: - /">-6(f)dr e(r) r Jo J / Chapter 2. Vortices and the Bosonic Description of the FQHE 29 -l->e(r) + ^ 4 ( r ) w i t h the cons t ra in t fir) = b(r) [2.19) (2.20) $ o ( 2 n + l ) ' (No te tha t i f we were cons ide r ing a value of p not equal to p0 we w o u l d s t i l l use the above equa t ion but w i t h p. = po/f so that the m e a n field s o l u t i o n w o u l d be £2 = fp/X.) W e now define the func t ion h(r) v i a the f o r m u l a h'(r) b(r) + B = ( 2 . 2 1 ; where h'(r) = dh/dr. /?,(()) c an be chosen equal to 0 w i t h o u t loss of general i ty . T h e second t e rm i n E q . (2.19) is then p r o p o r t i o n a l to: 'k Br 1 r , , , , x , , \ 2 . 0 / , (k h(r)' (2.22) In order for the in tegra l to be finite at large r we require h(oo) — k (or, more precisely, we require h(oo) to be an integer and we choose k i n E q . (2.18) to be that integer) . ( W e o n l y cons ider the cases k = ± 1 here, since those conf igura t ions have the lowest energy.) F u r t h e r m o r e , as is s t anda rd for a l l vor t ices , £ 2 ( r ) must van i sh at the o r i g i n i n order for the energy to be finite. T h e dens i ty cons t ra in t E q . (2.20) then i m p l i e s b(r) -B + h'(r)/r e(r) - > 0 and 0 as ?• 0. (2.23) $ o ( 2 n + l ) $ 0 ( 2 n + l ) T h i s is the mos t diff icult c o n d i t i o n to i m p l e m e n t i n a n u m e r i c a l scheme i n w h i c h the func t i on h(r) is va r i ed to m i n i m i z e the energy. W i t h the above def in i t ions the H a m i l t o n i a n is g iven by H = 2 „ r r d r \ ± i m 2 + Jo 2m \ dr ) 2m \ r h(r X ?(r) - tf*(r) +-fir (2.24) Chapter 2. Vortices and the Bosonic Description of the FQHE 30 with h(r) chosen so that h(Q) - 0, h(oo) = k and £ 2 ( r ) , defined by Eq. (2.23), is > 0. Notice that these conditions guarantee that the total flux of the vortex is I d2x(b + B) = <$>0k. (2.25) The final step is to subtract, from the energy of the vortex solution of Eq. (2.24), the energy of the mean field solution £ 2 = This results in a vortex energy given by: The procedure at this stage is to search, numerically, through the space of such functions h(r) until the Hamiltonian is minimized. One point which is clear is that the form of the vortex solution (for which /' dA(B+b) > 0) is quite different from that of the antivortex solution. The reason is that £(?•) and thus b(r) must vanish at the origin. As a consequence the density yj2(r) oc b(r) for the vortex solution can be a monotonic function of r which increases from zero at the origin and reaches B at infinity. The antivortex solution must however be zero at the origin then increase to a value greater than B (so that f(B + b) < 0) and then decrease again to attain its asymptotic value B as r —>• oo. We see this behaviour clearly in the numerical solutions shown in Figs. 2.1-2.4 below. For the numerical work we chose some representative values for the parameters of the model4: po = .010 eV A*0/A = 10 u cm~ 2 m = .08 me. (2.27) 4These values were chosen to achieve a density of carriers, chemical potential and effective mass similar to those found in experimental conditions [57, 62, 63]. Chapter 2. Vortices and the Bosonic Description of the FQHE 31 N o t e tha t po/X is the dens i ty of carr iers . In F i g s . 2.1 and 2.2 we present the n u m e r i c a l s o l u t i o n for the v o r t e x ( Q P ) con-figurations ( in w h i c h b is lowered re la t ive to B and h(r) —» — 1 as ?' —• oo) . W e p lo t the func t ions h(r) and the dens i ty p(r) for v = 1 / 3 , 1 / 5 a n d 1/7 respect ive ly . W h e n c h a n g i n g u, the dens i ty p r emains f ixed as the magne t i c field B is va r i ed . F i g s . 2.3 a n d 2.4 c o n t a i n p lo t s of h(r) and p(r) for the same values o f v bu t now for the a n t i v o r t e x ( Q H ) conf igura t ions . In T a b l e 2.1 we present the energies a n d a measure ?'0 of the size of each vo r t ex and an t ivor tex . W e have a r b i t r a r i l y chosen the size of the v o r t e x as the va lue of r at w h i c h the energy dens i ty has reached 99% of i ts t o t a l va lue . W e are now ready to descr ibe q u a n t i t a t i v e l y ( w i t h i n th is m o d e l ) w h a t happens w h e n the c h e m i c a l p o t e n t i a l is va r i ed f rom pq. A S d iscussed i n great d e t a i l i n th is C h a p t e r there is no change i n the dens i ty unless p — po is a p p r o x i m a t e l y equa l to the energy of a vo r t ex t imes 2n + 1 (i.e. the energy per pa r t i c l e of the vo r t ex ) . W e can now see, quan t i t a t i ve ly , how this works f rom E q . (2.26). If / ^ 1 so that p = po/f ^ po t hen the energy o f the vor t ex is s i m p l y where the m i n u s s ign is for a vo r t ex and the p lus s ign for an an t ivor t ex . T h u s for p < po the vo r t ex conf igura t ion has lower energy t h a n the an t ivo r t ex conf igura t ion . W e n a i v e l y expect tha t w h e n p0 — p = nev(po) (or near th is po in t ) the m e a n f ie ld g r o u n d s tate s h o u l d be a condensate (poss ib ly a l a t t i ce ) of vor t ices . Converse ly , w h e n p > p0, the a n t i v o r t e x has lower energy, and when p — po = nev(po)1 we n a i v e l y expect a condensate o f an t ivor t i ces . Unfo r tuna t e ly , for our choice of parameters , ev is qu i te large . T h u s the value pcr or p at w h i c h th is condensate occurs i n the above naive c a l c u l a t i o n a n d w h i c h is s h o w n i n T a b l e 2.1 differs f rom po by an unreasonab ly large amoun t . T h i s leads, i n p a r t i c u l a r , to a negat ive value of pcr for the 2n + 1 = 5 a n d 2n + 1 = 7 vo r t ex (2.28) Chapter 2. Vortices and the Bosonic Description of the FQHE 32 r ((im) Figure 2.1: The function h(r) corresponding to a vortex for v = 1/3,1/5 and 1/7. configurations. In fact, as p is varied from po towards pcr, another critical value p. of p is reached at which §op/\B = l/(2n + 3) well before pcr is reached. At p the system is better described by a Chern-Simons theory with the new value i) = 1/(2??. + 3) of the filling fraction. In light of the above remarks we should try to understand whether in fact one does form a vortex condensate in our model at our chosen values of the parameters. Certainly the vortices must overlap well before p or pcr is reached. This can be better understood by first supposing that such a condensate is formed as p. is lowered. The approximate Chapter 2. Vortices and the Bosonic Description of the FQHE 33 0.01 0.02 0.03 0.04 r (jim) F i g u r e 2.2: D e n s i t y profile of the vor t ex conf igura t ion for v — 1 / 3 , 1 / 5 a n d 1/v. dens i ty pi at w h i c h a desc r ip t ion of this condensate i n te rms o f the s ingle vo r t ex so lu t ions j>resented above fails depends most p r o m i n e n t l y on the size of a vor tex . In T a b l e 2.2 we show the a p p r o x i m a t e dens i ty of vor t ices p„i a n d c h e m i c a l p o t e n t i a l pi at w h i c h the vor t ices b e g i n to " touch" . Fo r a hexagona l l a t t i ce of vor t ices th is w i l l o ccu r w h e n / V i — l / ( 2 \ / 3 r f 2 ) , i .e., where d (the d is tance between vor t ices) equals the size of the vor t ices . T a b l e 2.2 also shows the co r re spond ing values of the filling f r ac t ion . N o t i c e tha t i n mos t cases the vor t ex la t t i ce becomes dense w e l l the before the "next" value of n (i.e. w e l l before ^ = l / ( 2 ( n ± 1) + 1)). W e conc lude f rom th is tha t an a p p r o x i m a t i o n Chapter 2. Vortices and the Bosonic Description of the FQHE 34 Figure 2.3: The function h(r) corresponding to an antivortex for v = 1/3,1/5 and 1/Y. in terms of a dilute gas of vortices breaks clown well before p = pcr. It is thus likely that even for our chosen values of the parameters a lattice of vortices will form in the mean field description. The formation of pinned vortices is what gives rise to the Hall plateaux by allowing the system to continue to behave as if it were in a state v = ^ - j - even after the magnetic field or the chemical potential has been changed to move it away from that value. This is because the (anti)vortices accommodate the localised excess (deficit) of charge. Chapter 2. Vortices and the Bosonic Description of the FQHE 35 r W Figure 2.4: Density profile of the antivortex configuration for v = 1/3,1/5 and 1/7. 2.4 Discussion of Results The vortex antivortex configurations described in the preceding section are identically the Laughlin quasiparticles and quasiholes, which are described in Section 1.6. The energies of these configurations have also been determined by other numerical methods. The usual method involves calculating the energy of a Laughlin type many-body solution for a finite number of particles. These energies are often given in terms of the interaction 2 energy, ^ . A summary of the results of various calculations are shown in Table 2.3. We Chapter 2. Vortices and the Bosonic Description of the FQHE 36 vortex/antivortex n size (pm) energy ( x l O 2eV) Per/Pi) 3 0.027 0.25 0.25 vortex 5 0.018 0.21 -0.05 7 0.015 0.20 -0.4 3 0.035 0.83 3.5 antivortex 5 0.025 0.80 5.0 7 0.020 0.79 6.5 Table 2.1: Energy and size ( r 0 ) of vortices and antivortices for n = 3,5 and 7. pcr is a naive estimate of the value of the chemical potential at which a condensate of these configurations is expected to form (po = 10~ 2 eV) . pcr is more carefully described in the text. vortex / antivortex n Pvi [pm'2) KT 1 3 400 2.6 vortex 5 890 4.1 7 1280 5.7 3 240 3.2 antivortex 5 460 5.5 7 720 7.7 Table 2.2: The density pv\ and the corresponding filling fraction v\ at which the conden-sate of vortices is expected to become dense. (See text for a precise definition.) have assumed e = 13 in the conversion of our results, so that = 0.015eV . There is a close agreement of the Q H creation energies determined by Laughl in and by Mor f and Halperin, because they used the same many-body wavefunction to describe the Q H . However, in the time between the Laughlin theory and the emergence of the Chern-Simons theories, there was a great deal of confusion about the form of the qivAsiparticle state, although it was realised by Laughlin that both the QP ' s and Q H ' s would be formed as the result of a local adiabatic addition of one flux quantum. W i t h i n the Laughl in theory, it was straight-forward to hypothesize that the Q H would have a zero at the Chapter 2. Vortices and the Bosonic Description of the FQHE 37 author quasi-hole quasi-particle gap Laughlin [18] 0.025 0.022 0.047 Haldane and Rezayi [4] 1.05 ± . 0 0 5 Movf and Halperin [64] 0.026 0.073 ± . 0 0 8 .099 Curnoe and Weiss [59] .16 .55 .71 Table 2.3: Summary of numerical results for quasi-particle and quasi-hole creation ener-gies and gap energy. Units are ^j. position of the hole. In the discussions of vortices presented above, we have shown that the addition of one flux quantum to create a QP or a QH necessarily implies that the order parameter vanishes at the position of the excitation. The node in the electron density of the QP is an artifact of the Landau Ginzburg theory and it is one source of the discrepancy with the energies found using many-body wavefunctions. In fact, we have not included effects arising from the binding of QP's to the vortices, (which is beyond the Landau Ginzburg theory); such effects would modify the vortex profile (the local charge density) as well as the energy of the configuration. Another source of discrepancy with our results originates from the simplification of the Coulombic interaction to a point interaction. The chemical potential is assumed to be roughly equal in each system and the interaction strength A we have chosen produces the right number of particles (i.e., the correct filling). This means that for a uniform configuration, our interaction is chosen so that the system will have the same amount of energy per particle as a realistic system, whatever its interaction may be. The other results considered in Table 2.3 used a Coulombic type interaction, which is non-local. If a small amount of localised charge is removed (as for a QH), the energy loss of the local interaction will be higher than the loss of a non-local interaction, which yields the result that our energies are expected to be higher than the energies obtained using a Coulomb interaction. Chapter 2. Vortices and the Bosonic Description of the FQHE 38 2.5 Summary In this Chapter we have studied the mean field behavior of the C S L G description of the F Q H E when the filling fraction deviates from the special filling fractions v = T ^ J . We have shown how the field theoretic description of this model at a fixed chemical potential p and magnetic field B can be studied for a range of p surrounding the value p^ corresponding to the special filling fraction. For small values of \p — pa\ (and at zero temperature) the density is independent of p. As p is decreased (increased) beyond some critical value we have shown how the homogeneous mean field configuration is unstable to the formation of fractionally charged vortices (antivortices). We have presented a numerical example of these vortex and antivortex configurations and we estimated the densities and filling fractions at which the description in terms of a noninteracting system of vortices breaks down. Chapter 3 Composite Fermions, Gauge Fluctuations and the Gap Despite the success of the bosonic theory and the appealing physical insight it provided, it was unable to describe filling fractions other than v = -^-j- in any matter at all. Clearly a similar mechanism is at work for all the filling fractions, as evidenced by the oscillatory behaviour observed in many physical properties, including resistivity [30, 31, 32], magnetisation (extrinsic photoluminescence) [53], intrinsic photoluminescence [65] and compressibility [55]. Composite fermions provide such a mechanism. The composite fermion-gauge theory of the fractional quantum Hall effect is an ef-fective field theory which, at a mean field level, explains the hierarchy of observed F Q H states in a natural and intuitive way. In addition to this, it has the structure of an ordinary fermionic gauge theory and hence lends itself to the possibility of performing calculations of real physical quantities using standard methods of quantum field theory. The catch is that such methods are usually perturbative, while this theory is plagued with a rather large coupling between the fermions and the gauge field. Hence perturba-tive methods may be applied and do in fact reveal qualitative features of the system, but the real challenge has been to seek and apply other methods to understand this theory. This Chapter studies the self-energy of composite fermions and also the gap, which is a physical quantity derived from the self-energy. The self-energy is the correction coming from interactions to the free 1-particle Green's function, and yields corrections to the particles' dispersion relation, in particular the effective mass. The perturbative calcu-lation of this quantity for a gapless system with gauge interactions was first performed 39 Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 40 some t ime ago i n a different context [66]; i t has also appeared i n s tudies o f the gauge theory of h i g h t empera tu re superconduc tors [67] and a s i m i l a r c a l c u l a t i o n is a p p l i c a b l e to the F Q H E at the gapless state, v = \ [26]. W e ex tend this c a l c u l a t i o n to find the gaps of filling f ract ions away f rom v = \ and improve the resul ts u s i n g a self-consistent approach . A s we s h a l l see, the compos i t e f e rmion theory of filling f ract ions away f rom v = \ requires an effective magne t i c f ie ld , w h i c h is the difference between the rea l ex t e rna l f ield a n d the f ield clue to the magne t i c f lux a t tached to each e lec t ron . T h e s tudy o f i n t e r a c t i n g sys tems i n a magne t i c field has a l o n g h is tory , bu t h a d u n t i l the Q H E been m o s t l y l i m i t e d to three d imens ions . In two d imens ions we find that the h i g h l y degenerate dens i ty of states in t roduces s ingu la r i t i e s i n the spec t ra l funct ions , even for o r d i n a r y F e r m i l i q u i d s . F o r fe rmions i n t e r ac t i ng w i t h gauge f luc tua t ions the effects are even more p r o n o u n c e d . A s tudy of an i n t e r ac t i ng sys t em w h i c h accounts for s ingu la r i t i e s has not been done, even for F e r m i l i q u i d s . W e make use of the s ingu la r s t ruc ture o f the self-energy to e m p l o y a new, i t e ra t ive , self-consistent m e t h o d to compu te the gap of the F Q H s y s t e m at fillings A s d iscussed i n C h a p t e r 2, there has been a cons iderable n u m b e r of expe r imen t s w h i c h have measured the gap, u s ing i n most cases f a i r ly d i rec t me thods . However , because the gap decreases r a p i d l y for h igher d e n o m i n a t o r filling f ract ions, these m e t h o d s have not been able to s t u d y the gap b e y o n d v = | . I n recent years an innova t ive set of expe r imen t s has emerged w h i c h c l a i m to be able to de te rmine the effective mass t h r o u g h a large range of f rac t ions , up to v = ^ , by p a r a m e t r i s i n g the S d H osc i l l a t ions of the r e s i s t i v i t y i n te rms of an effective mass . These exper iments show an apparent d ivergence o f the effective mass for f i l l i n g f ract ions v - i n the l i m i t p —> oo [30, 31, 32]. T h i s cor responds to a v a n i s h i n g of the gap i n th is l i m i t , i n agreement w i t h theore t i ca l p r ed i c t i ons [29]. O n e o f our objec t ives has been to compu te the effective mass for a range o f p h y s i c a l l y relevant Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 41 parameters, in order to achieve a direct, quantitative comparison between the composite fermion gauge theory and experiment. We approach this problem with a second order perturbative calculation of the self-energy which retains the Landau level structure of the internal fermion line. This calcu-lation is the first step of a new iterative procedure used to find the self-consistent form of the self-energy, which makes use of the structure introduced by Landau level quan-tisation. We also investigate the analogous two dimensional electron-phonon problem, which has never been looked at before, in order to better understand the peculiar fea-tures of the composite fermion self-energy. This procedure is used to evaluate the gap, which rapidly converges after a few iterations. Our results are in sharp disagreement with experiments: however analysis of the various experiments shows that their results do not agree with each other either. Therefore it remains unclear as to whether or not the composite fermion gauge theory is a quantitatively correct description of the F Q H E . This Chapter is organised as follows. We begin in Section 3.1 with a review of composite fermions, the key experiments that justify the composite fermion construction and the Lagrangian formulation of the composite fermion gauge theory. This is followed in Section 3.2 by a discussion of the self-energy calculation at v = \ performed by Halperin, Lee and Read [26]. Then in Section 3.3 we show how to do the analogous calculation away from v — \, taking into account the Landau level structure arising from the finite effective field AB. We use the self-energy to calculate the lowest order in perturbation corrections to the gap in Section 3.4. In Section 3.5 we compare the composite fermion gauge theory results for the self-energy and the gap to a calculation of the self-energy for a normal Fermi liquid in the presence of a magnetic field, interacting with phonons with Debye and optical spectra. Finally in Section 3.6 we present an iterative self-consistent method for determining the gap and in Section 3.7 we compare our results with experiments. Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 42 3.1 Composite Fermions Composite fermions are electrons to each of which is attached an even number (2n) of magnetic flux quanta. The composite fermions experience a magnetic field AB that is the difference between the external magnetic field B and the field associated with the attached flux, B^_. When the filling fraction is v = 0n^±1 the effective magnetic field is AB = B^_ (3.1) = $ 0 n e ( ^ ± l - 2 n p ) (3.2) = ± * ° ^ (3.3) P (3.4) 1 ± 2pn Eq. (3.3) tells us that composite fermions are in a magnetic field AB with a filling p, i.e., there are p filled CF Landau levels. This is the basis of Jain's observation that the F Q H E of electrons could be interpreted as the IQHE of composite fermions. Eq. (3.4) tells us that the effective magnetic field AB is a fraction of the external field, as is the associated cyclotron frequency Aw c = Thus we can picture that the lowest Landau level has been subdivided into levels separated by an energy Auc, of which p are filled. In the special cases when p is an integer, the system has a gap, which equals Acuc. This provides a qualitative explanation for the significance of the special filling fractions v — 2„p±i a n ( l a l s o f ° r the decreasing stability of the incompressible quantum Hall states for higher values of m and p [8]. 3.1.1 Experimental Evidence for Composite Fermions The F Q H E is the observation of SdH oscillations of the longitudinal resistivity and plateaux occurring in the Hall resistivity at the special filling fractions v = .)n^±1 • In Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 43 addition to the resistivity measurements and other experiments which observe oscillatory behaviour as a function of AB, there is other evidence that composite fermions exist. Various methods have been used to determine the cyclotron radius of the charged quasi-particles away from v = \. These experiments have confirmed that the quasiparticles do in fact behave as if they were in a magnetic field AB, with a diverging cyclotron radius as v —»• \. The experiment of Goldman, Su and Jain [63] is an elegant measurement of the cyclotron radius using magnetic focusing. The 2-D electron system is exposed to a magnetic field while a current is injected at one point of the sample. The current is picked up at a point some distance away (see Fig. 3.1). As the field is varied on one side of v — 7; the intensity of the current oscillates with maxima occurring when the distance is an even integral multiple of the cyclotron radius, R c = CAB' {3-°] On the other side of v = \ there are no oscillations observed since the sense of the cyclotron orbit is reversed. The behaviour of the composite fermions is completely anal-ogous to the behaviour of electrons in an external magnetic field. Figure 3.1: Schematic drawing of the geometry used in the magnetic focusing experiment of Goldman, Su and Jain. Each arc is a cyclotron orbit (after Goldman et al. [63]). Willet et al. [68] observed the cyclotron radius by finding geometric resonances using surface acoustic waves (SAW's). SAW's traversing the sample are attenuated by piezo-electric interactions with the electrons in the 2-D gas, causing a shift in the velocity of the Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 44 SAW. The amount of shift decreases with increasing longitudinal conductivity, which is wave-vector dependent. As a function of magnetic field, a minimum in the velocity shift corresponds to a resonance of the conductance which occurs at a wave-vector q = Two minima are observed symmetrically on either side of v = \, corresponding to res-onances at ±AB. This interpretation of the data is consistent with having AB = 0 at v — | . Furthermore, using (3.5) to extract hp, close agreement was found with the gauge theory prediction, kp = (47m e)2. This is proof that a Fermi surface does exist at v — \. Finally we consider the experiment of Kang et al. [69], who observed cyclotron res-onance of electron transport through an antidot superlattice. An antidot lattice is an array of holes superimposed on 2-D electron gas surface. The dots are 100 — 200nm in size and separated by a period d of about five dots. Minima in the longitudinal resistiv-ity are associated with a localisation of the electrons which occurs when the electronic cyclotron orbits precisely encircle a square array of dots (1, 4, 9 etc). The results show-well defined minima at fields B - ^ and AB = , corresponding to the cyclotron resonances of electrons and of composite fermions. 3.1.2 The Composite Fermion Gauge Theory The first theory to describe the F Q H E , formulated by Laughlin, is one of strongly in-teracting electrons in the presence of a magnetic field. In the Introduction we have shown how to transform this theory into one in which the electrons have been replaced by composite fermions in the presence of a field AB. Experiments have shown that this theory gives a good description of many of the features of the F Q H E . Thus it seems as if the strong electron-electron interactions have manifested themselves primarily through stabilising the binding of the magnetic flux to each electron. Therefore we describe the Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 45 m e a n field c o m p o s i t e f e rmion sys tem by the L a g r a n g i a n c = ^(x)(-idt + aoM-r) + ^(x){ldi ~.^A')2ii< a 0 V x a r 2 V(x - y) : p(x)p(y) : J ^ r X - y ) (3.6) 2 n $ 0 where A . 4 = A — a and V x A . 4 = AB. T h e cons t ra in t tha t there are 2n f lux q u a n t a a t t ached to each e lec t ron comes f rom the equa t ion of m o t i o n that is found by v a r y i n g CIQ, w h i c h is ip^(x)y>(x) = p = T h e second t e r m is the H a m i l t o n i a n for "free" c o m p o s i t e fermions and has the usua l so lu t ions o f a charged pa r t i c l e i n a magne t i c field. T h e fou r th t e r m is the s t rong e lec t ron-e lec t ron in t e rac t ion , w i t h o u t w h i c h there w o u l d be no F Q H E . T h i s t e r m m a y be r ewr i t t en i n te rms of the gauge field a u s ing the cons t ra in t . I n the fou r th t e r m , V(x — y) = j^fzrj is a C o u l o m b i c in t e r ac t ion , w i t h Fou r i e r t r ans fo rm v(q) = • (3-0 eq W e also cons ider a s l i g h t l y different case where the C o u l o m b i c e lec t ron-e lec t ron in ter -ac t ions are screened (wh ich m a y be real ised by p l a c i n g the 2 - D e lec t ron gas near to a c o n d u c t i n g p la te ) : the case w h e n v(q) ~ v arises w h e n the i n t e r a c t i o n is effectively zero-range (of s t rength v, and range less t h a n the magne t i c l eng th I). N o t e tha t there is no F\WF^W t e rm i n C s ince i n the low e n e r g y - m o m e n t u m l i m i t w h i c h we w i l l a lways cons ider , the C h e r n - S i m o n s t e rm a n d the correc t ions A " ( 0 ) (below) w i l l d o m i n a t e i n a l l c o m p o n e n t s o f the gauge propaga tor . T h e next step i n the theory is to a l low for gauge f luc tua t ions a r o u n d the m e a n f ie ld s o l u t i o n , A . 4 = 0, w h i c h we m a y wr i t e as a, = At + da,. T h e f e rmion field yj in te rac t s w i t h the gauge f luc tua t ions i n the usua l way, v i a the covar iant de r iva t ive . T h e q u a d r a t i c par t s i n 6a y i e l d a bare p ropaga to r for the gauge f luc tua t ions : the Fou r i e r t r ans fo rm o f Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 46 this is D ( U , ~ 1 = [ ° * I , • (3.8) where the m a t r i x elements are the l o n g i t u d i n a l and transverse componen t s respect ive ly . T h e effective a c t i o n for the gauge f luc tua t ions m a y be de t e rmined by i n t e g r a t i n g out the c o m p o s i t e f e rmion field ip [25, 26]. In the l ow energy l i m i t , th is is d e t e r m i n e d prec ise ly u s i n g the cor rec t ions K{[))(q, u>) f rom the two 1-loop graphs shown i n F i g . 3.2 [70]. These F i g u r e 3.2: F e y n m a n d i a g r a m s represent ing the first order cor rec t ions , K^°\q,co), to the gauge p ropaga to r . cor rec t ions are also c a l c u l a t e d at AB = 0, and i n the l i m i t of s m a l l q, cu are m* ( iu> 27T V qv[. = + (3-9) < = - - / _ + ^ . (3.10) 127rra kpq T h e n the r eno rma l i s ed gauge f luc tua t ion p ropaga to r takes the f o r m D = (D(i))-1 + K{i)))-1. (3.11) T h e mos t s ingu la r componen t is the transverse componen t , c a l c u l a t e d i n the C o u l o m b gauge at AB = 0 for s m a l l u and q: Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 47 T h e exponent 5 c a n have values between 2 and 3. s = 2 w h e n there are unscreened C o u l o m b i c e lec t ron-e lec t ron in te rac t ions and s > 2 cor responds to screening those i n -te rac t ions (as d iscussed above) . T h e constants are 7 = a n d x — ^ for s = 2 a n d X = - f for s — 3 i n th is r a n d o m phase a p p r o x i m a t i o n [26]. e is the d i e l ec t r i c cons tant . T h i s f o r m of the gauge p ropaga to r has a l o n g h i s t o ry w h i c h predates the Q H E . It was first de r ived i n 1973 by H o l s t e i n , N o r t o n and P i n c u s [66] for the p h y s i c a l l y qu i te different p r o b l e m of m e t a l l i c electrons coup led to o r d i n a r y transverse photons . T h e y showed tha t i t causes 11011-Fermi l i q u i d exponents i n the t empera tu re dependence o f the res i s t iv i ty , heat c a p a c i t y a n d N M R r e l a x a t i o n rate i n meta l s . However , there is an i m p o r t a n t difference between the scenar io s tud ied by H o l s t e i n et a l . and the one presented i n th is thesis . In the ear l ier case the gauge f luc tua t ions are f luc tua t ions of the regular e l ec t romagne t i c field - ie. pho tons , and therefore the self-energy correc t ions w h i c h give rise to the n o n - F e r m i l i q u i d exponents are of order . In the case we are cons ide r ing the gauge field is i n t r o d u c e d to enforce a cons t ra in t on the dens i ty of par t ic les : i n the gauge theory o f h i g h t empera tu re superconduc tors [67] i t is i n t r o d u c e d to prevent doub le site occupancy . In b o t h cases i t causes a s t rong c o u p l i n g between the gauge f luc tua t ions a n d dens i ty f luc tua t ions . T h e result is that cor rec t ions c o m i n g f rom this gauge p ro p ag a t o r are not s m a l l , i n fact they are u sua l ly of order un i ty . T h i s seemed to be a very p laus ib le o r i g i n for n o n - F e r m i l i q u i d b e h a v i o u r genera l ly a n d for a w h i l e i t seemed l ike gauge f luc tua t ions c o u l d be a un ive r sa l m e c h a n i s m respons ib le for the u n u s u a l behav iou r of m a n y s t rong ly cor re la ted sys tems [71]. However , th i s theory s t i l l awai ts e x p e r i m e n t a l c o n f i r m a t i o n and a consistent non-pe r tu rba t ive t rea tment . F o r th is reason, one of the goals of the Sec t ion 3.3 is to der ive quan t i t a t i ve p r ed i c t i ons o f the mass r e n o r m a l i s a t i o n a r i s ing f rom gauge f luc tua t ions . B u t first we cons ider the s i t u a t i o n w h e n AB = 0. Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 48 3 .2 T h e S e l f - E n e r g y a t A i ? = 0 T h e express ion (3.12) was first used to ca lcu la te self-energy cor rec t ions a r i s i n g f r o m in te rac t ions w i t h gauge f luc tua t ions by H o l s t e i n et a l . [66] a n d by others [67, 72, 73] i n the context of h i g h T c superconductors . S i m i l a r results were o b t a i n e d by H a l p e r i n , Lee a n d R e a d at v = \ , i .e., at AB = 0 [26]. T h e self-energy is g iven i n p e r t u r b a t i o n theory as [74] k x q :(k,e) d2q I'00 du (27r) 2 JO TT m lmD(q, cu) where & = i h ^ L and x ' l + nB(u) - nf(Zk-q) nD(u) + e - u> - £k-q + i6 e + u - + id lmD{q, u) = -:3.13) (3.14) 7 2 w 2 + q2sX2 A t zero t empera tu re , i n the l i m i t of s m a l l e eva lua ted at e = E q . (3.13) y i e ld s [26] for s > 2 '(k,e) ~ e \k,e) 2/3 2/s (3.15) (3.16) a n d for s = 2 E (e) ~ e l o g e S"(e) ~ e. ;3.17) ;-3.18) E ' (e ) m a y be used to o b t a i n the effective mass [75] l-d2'(h,e)/de — = V l o g (6 ) for 6 = 2. (3.19) (3.20) Chapter 3. Composite Fermions. Gauge Fluctuations and the Gap 49 This result indicates that there is a logarithmically divergent correction to the effective mass as the Fermi surface is approached as a function of e. An analogous behaviour is predicted to occur as v approaches \ , which will be explained more in Section 3.3. The experiment of Jiang et al. [76] revealed the first physical example of a metallic state with unusual, non-Fermi liquid properties by studying the temperature dependence of the resistivity at v = \. In mathematical terms, Fermi and non-Fermi liquids are distinguished by the analytic properties of the Green's functions. Fermi liquids are char-acterised by the presence of a pole (or, more generally, a branch cut) in the propagator which occurs at the dispersion of a physical particle. In the bare fermion Green's function the pole is a <$-function, G ( 0 )(e,/c) = i — (3.21) e - ik - io indicating that the particle has a well defined energy. The position and strength of the pole are modified by interactions, which is described by the self-energy appearing in the renormalised Green's function, G(e,k) = * (3.22) but the essential feature of Fermi liquid theory is that the analytic properties remain the same. This means that the dispersion relation will have the same functional form, but with the bare mass replaced by an effective mass which is given by (3.19). The analytic properties of the Green's functions of this composite fermion gauge theory are modified by the self-energy given in Eqs. (3.15) - (3.18). The appearance of a logarithm indicates the onset of a branch cut singularity instead of a pole; the fractional power which appears in (3.15) is a continuation of this trend. The important point is that in the low energy limit, the self-energy correction is the dominant term in the renormalised Green's function. In addition, the large imaginary part of the self-energy Chapter 3. Composite Fermions. Gauge Fluctuations and the Gap 50 gives rise to a broad peak in the spectral functions which indicates that the quasiparticle is not well-defined in the usual sense of a Fermi liquid theory. The considerations outlined above show that any theory which attempts to describe the v = \ scenario must be able to address the non-Fermi liquid nature in a non-perturbative way. There have been several different approaches to this problem for the gauge theories, including renormalisation group analysis [70, 77, 78, 79], 1/JV expansion [80, 81, 82], eikonal expansion [83], and bosonisation [79, 84], but these various approaches do not yield equivalent results. It is not even resolved as to whether or not Fermi liquid theory actually breaks down in two dimensional systems with short-ranged interactions1. Using renormalisation group analysis various authors [70, 77, 79] have been able to show-that systems with long-ranged interactions of the form .s = 2 (discussed above) are driven by a novel, non-Fermi liquid fixed point of the action. These results lend credence to the initial approach of Halperin, Lee and Read [26], which, although perturbative, appears to capture the essential physics of the problem. 3.3 The Self-Energy for AB ^ 0 2For a finite magnetic field the self-energy expression (3.13) becomes _ . . 2EF f d2q r°° dco ' M 2 f°° de' T w) / S m J (Z7T I JO VT , . / - o o 7T V 11 =0 x /1 + njj(u) - nf(e) nB(u) + n/(e') y € — € — CO + id e — 6 + u + iS We assume that there are [p filled composite fermion Landau levels ([p = the greatest integer less than p) and that the chemical potential lies in the centre of the gap, at p. = Acoc[p. We will consider a system of CF's in a field AB which is small, using the 1 Opinions have shifted back and forth through the years. Refs. [85] argue that there is a breakdown of Fermi liquid theory, while Refs. [86] argue that there is not. 2 T h e results of this Section appear in [87]. (3.23) Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 51 gauge f l uc tua t i on p ropaga to r at A i ? — 0, g iven by (3.12) ( this is jus t i f i ed i n A p p e n d i x D ) . W i t h i n th is same semi-c lass ica l l i m i t , we adopt a we l l k n o w n a p p r o x i m a t i o n for the over lap m a t r i x element between p lane waves a n d L a n d a u level states [88, 89] (see A p p e n d i x A for more de ta i l s ) : A n , n , 9 2 « — — — 3.24 qn \ 2p J where n, n ~ p are L a n d a u level indices , as shown i n F i g . 3.3. F i g u r e 3.3: F e y n m a n d i a g r a m represent ing the self-energy, n and n are L a n d a u level ind ices . T h e s t ra ight l ine represents the f e rmion (either e lec t ron or c o m p o s i t e fe rmion) and the w a v y l ine represents the boson (either a p h o n o n or a gauge f luc tua t ion ) . It is useful to rewr i te these equat ions sh i f t ing the s u m to s tar t at —p a n d then enforc ing par t i c le -ho le s y m m e t r y by t r u n c a t i n g the s u m at p — 1, the n u m b e r o f filled L a n d a u levels . (In r ea l i ty the p r o b l e m is not par t ic le -hole s y m m e t r i c , bu t we are o n l y in teres ted i n energies w i t h i n uc of the F e r m i surface, where par t i c le -ho le s y m m e t r y is a lmos t exact . In the case of c o m p o s i t e fermions , th i s corresponds to h a v i n g an uppe r cu tof f at the t rue c y c l o t r o n frequency) . U s i n g (3.12) and I m G ^ 0 ) ( e ) = n6(e + p - (n + \_p + | ) A w c ) we find the rea l a n d i m a g i n a r y par t s of the self-energy at T = 0 [89] (see A p p e n d i x B for more de ta i l s ) . F o r s = 2 one has: S (e) = 2^ l og p-i S"(e) = K2Auc^2{0[(-m-l/2)Auc-€]+e[(-m-l/2)Auc + e]) (3.26) m = 0 ( m + 1/2)Auc - e 'm + 1/2) Acoc + e (3.25) Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 52 A 2 = (3.27) e2 v/47r7i e 2me' whilst for 3 > s > 2: sgn(e)Awc cot +Aw c esc I — j A s (-[(m + l/2)Aw c - e] _ a#[(?7} + 1/2) Auc - e] m=0 (3.28) p - i A , A w c YJ ( [ ( - ? 7 i - 1/2)Au c + e]-a9[(-m - 1/2)Awc + e] •m=0 A ; = + [ ( - m - 1/2)Aw c - e]"a0[(-77} - 1/2)Aw c - e]) CSC I — I (3.29) (3.30) where a = (s — 2)/s is positive. Note that the coefficient A'2 is the same as the one calculated by Stern and Halperin [29]. Plots of S'(e) and S"(e) for• .s = 2,3 are shown in Figs. 3.4 and 3.5. The results shown in the figures display divergences at every multiple of Acuc. Later in Section 3.5 we find singular behaviour in the derivative of the self-energy coming from interactions with phonons, but the self-energy itself stays finite. Therefore the divergences appearing in the self-energy shown in Fig. 3.4 are clue to both the singular density of states (Landau levels) and the transverse gauge interaction. Stern and Halperin performed similar calculations but with the sum over Landau levels replaced by an integral, effectively smearing out all of the structure shown here. While this is a valid approximation for energies e <^ Acuc, it is clear that these structures are crucial in any discussion of energies e ~ Auc. This energy range will be important when we consider the renormalised gap. A rather extraordinary feature of the CF propagator appears when we closely exam-ine £"(e) and S'(e) around the divergences - we notice that S'(e) shows only positive Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 53 ~i i r~ a 3 <] -2 -10 10 Figure 3.4: Real part of the self-energy of composite fermions. s = 2 corresponds to Coulombic electron-electron interactions: s = 3 corresponds to short-ranged screened interactions (see text). The energies are in units of Atoc. divergence on both sides of each Landau level. The paradox is that typically one would expect E(e) to have the form lv;.P 3 . 3 1 : e - Er + id in simple perturbation theory if our starting fermion spectrum is composed of discrete (albeit degenerate) levels at energies Er, and Vr is some perturbation. This leads to divergences in E" each time one crosses the energies Er. This sign change occurs for E'(e) in the electron-phonon problem (to follow in Section Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 54 3 < 0.8 0.6 0.4 0.2 0.8 0.6 0.4 0.2 s = 2 i i i i i i i i i i i i i i i i i i i i i -10 10 Figure 3.5: Imaginary part of the self-energy of composite fermions. s = 2 corresponds to Coulombic electron-electron interactions; s = 3 corresponds to short-ranged interactions and the energies are in units of Acoc, as before. 3.5): this is most clearly seen by plotting <9S'(e)/<9e for this problem, where one sees a series of positive divergences around each Landau level, with no sign change, as shown in Fig. 3.10. This is exactly what we would expect from (3.31), since it yields 81. \Vr\2 (3.32) de r ( e ~ Erf The behaviour of HCF(^) is thus exactly the opposite of what one expects. The explanation of this paradox is that although Y!'CF(e) is strongly peaked (indeed divergent) around each Landau level, and does not change sign as one crosses a Landau level, Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 55 nevertheless the peak has such a peculiar shape that its Hilbert transform y,'CF(e) also does not change sign as one crosses a Landau level. Each Landau level r contributes a term ~ (e — r)~a9(e — r) to S^.F(e), with long tails for (e — r) 3> 1. The Hilbert transform of such a function does not change sign as e crosses r, unless a > 1/2; however in the CF gauge theory, 0 < a < 1/3. This explains the paradoxical form of <9E/de for composite fermions - it comes from the very long "tails" which extend out from each Landau level. These have no counterpart in the self-energy of a Fermi liquid, such as the electron-phonon problem to be discussed in Section 3.5. 3.4 The Renormalised Gap : !The self-energy derived above is the first-order correction to the one-particle composite fermion Green's function, yielding the full Green's function, Gn(e) = 1 • (3.33) • e - ( ) i + l /2)Aw c +S(e) We wish to find the poles of G,,.(e), which are the solutions to S'(e) + e = (n + l/2)Aa; c, (3.34) which correspond to peaks in the spectral function. For each value of n there can be many solutions to (3.34); in particular we note that for each value of n there is exactly one pole within the gap |e| < This is shown in Fig. 3.6. Since the imaginary part is zero in this region these poles appear as ^-functions in the spectral function. The gap may be determined by approximating S 'W = ^ ( 3 . 3 5 ) 3 The results of Sections 3.4-3.6 appear in [89]. Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 56 -2 e ( A C J J F i g u r e 3.6: G r a p h i c a l so lu t i on to (3.34) for different values of n. T h e crosses show the poles between w h i c h l ies the r eno rma l i s ed gap. a n d then s o l v i n g (3.34) for n = —1 and n — 0 (for .s = 2): d E ' ( O ) -K2Auc 7T 2A"2Atu, 7T p-1 2 U o g ( 2 p + 1) (3.36) (3.37) where the s u m over ???. has been replaced by an in tegra l . T h e n the s o l u t i o n to (3.34) is 2A"2 . . , \ . A w f 7T l o g ( 2 p + 1 ) ) = ± - ^ -W e find tha t for 5 = 2 the r enorma l i sed gap is 2 A 2 ACJ*C « A w c 1 7T log(2p + I] a n d for 5 > 2 Ao>* « Aur 4 A , _ c 1 —p 3.38) (3.39) (3.40) These results were ob t a ined by K i m et a l . [90] and by S te rn a n d H a l p e r i n [29]. U s i n g i = —, i t is p o i n t e d out by these authors that the divergence i n the effective mass m* Aw, as p —» 00 is c o m p l e t e l y analogous to the divergence near the F e r m i surface at v = 1/2. Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 57 However the use o f E q . (3.35) to get a s o l u t i o n for (3.34) is o n l y v a l i d i f S ' (e) + e varies s l o w l y over e such that | S (e) + e\ < a n d thus (3.35) is a p o o r a p p r o x i m a t i o n w h e n K-2 is i n an in te rmedia te range ( ~ 0.5). Moreove r , the divergence i n (3.39) at K2 = 7 r / ( 2 1 o g ( 2 p + 1)) is u n p h y s i c a l : ins tead we expect the a c t u a l gap Au>* to decrease m o n o t o n i c a l l y w i t h K2. In fact, we es t imate tha t i n the a c t u a l expe r imen t s so far done 2 4 6 K 2 F i g u r e 3.7: F i r s t order p e r t u r b a t i o n results for the effective gap, Aw c * , as a f u n c t i o n of c o u p l i n g , K2, for p = 50. T h e s o l i d curve is |1 - 2 - | 2 l o g ( 2 p + 1 ) | _ 1 a n d po in t s are n u m e r i c a l so lu t ions to E q . (3.34). T h e inset shows the same d a t a p l o t t e d as a f u n c t i o n of ne, w h i c h is re la ted to A " 2 v i a eq. (3.27). [30], K-2 ~ 0.8, w h i c h places i t r ight i n the in te rmedia te range, so i t is necessary to go b e y o n d the es t imates i n (3.39) and (3.40). W e do this first by s o l v i n g (3.34) n u m e r i c a l l y : Chapter 3, Composite Fermions, Gauge Fluctuations and the Gap 58 as an e x a m p l e we have done this for p = 50, as shown i n F i g . 3.7. T h e n u m e r i c a l resul ts show that for s m a l l values of the coefficient the gap is reduced by a very s m a l l a m o u n t . F o r larger values of K2 the gap decreases r ap id ly . T u r n i n g our a t t en t ion to the spec t r a l func t ions s h o w n i n F i g . 3.8 we see that there is a s i m p l e p h y s i c a l i n t e rp re t a t i on for th is . S m a l l values of the coefficient give rise to very na r row double peaks at each c y c l o t r o n 1 0.8 0.6 -0.4 I o 3 0.2 <l 0 0.15 0.1 0.05 A -4 -2 0 2 . 4 6 ( A « c ) F i g u r e 3.8: I m a g i n a r y par t of the G r e e n func t ion o f compos i t e fe rmions i n a f ini te m a g -ne t ic field w i t h n = 0 and p = 50. T h e upper figure has a coefficient K2 = 0.31 and the lower one has K2 - 6.3. T h e energy is i n un i t s of Acuc. energy i n the spec t r a l func t ion . T h i s is the resul t of m u l t i p l e so lu t ions to (3.34); i n fact there are two for each l o g a r i t h m i c peak i n S ' (e) . T h e ext reme nar rowness of these Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 59 peaks suggests tha t these exc i t a t ions are mere ly L a n d a u level m i x i n g . T h i s c o n c l u s i o n is s u p p o r t e d by the fact ' that for s m a l l values of A 2 the gap is o n l y very s l i g h t l y reduced . A s A 2 is increased the double peaks are reduced i n size a n d even tua l ly give rise to incoherent s t ruc tures . In the example we have been cons ide r ing , p = 50, th i s crossover occurs w h e n A'2 ~ 0.5 w h i c h is also where the gap s tar ts to decrease. W h e n the coefficient becomes large there is no longer any evidence of s imp le L a n d a u level m i x i n g . 3.5 Phonon Interactions In th is Sec t i on we do an analogous c a l c u l a t i o n for electrons i n a finite m a g n e t i c f ie ld i n -t e r ac t ing w i t h phonons , w h i c h , su rpr i s ing ly , has never been p r e v i o u s l y inves t iga ted . O u r m a i n reason for l o o k i n g at the e lec t ron-phonon p r o b l e m is tha t i t is a w e l l - u n d e r s t o o d ex-a m p l e o f a F e r m i l i q u i d , w h i c h nevertheless acquires a n o n - t r i v i a l s t ruc ture w h e n L a n d a u q u a n t i s a t i o n is i n t roduced . W e f ind s t ruc ture associa ted w i t h the L a n d a u leve l degen-eracies, a l t h o u g h i t is not qui te as spec tacu la r as the C F - gauge i n t e r a c t i o n resul t . T h e f o l l o w i n g resul ts are useful for unde r s t and ing the results of Sec t ion 3.3. In zero f ie ld , the spec t ra l funct ions of the coup led e l ec t ron-phonon sys t em were first s t ud i ed by E n g e l s b e r g and Schreiffer [91]. U s i n g o p t i c a l and Debye p h o n o n spec t r a they c a l c u l a t e d the rea l and i m a g i n a r y par ts of the self-energy u s ing conse rv ing a p p r o x i m a -t ions ( o b e y i n g the W a r d ident i t ies ) . T h e y found that the spec t r a l func t ion , y i e l d e d s t ruc tures that they c o u l d ident i fy as we l l defined Q P ' s for two cases: w h e n p is close to the F e r m i surface, and when i t is very large. In the i n t e rmed ia t e reg ime there exis ts o n l y an incoherent "smeared" s t ruc ture . 1 I m G ( e , p) 2' (3.41) Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 60 T h e self-energy express ion is g iven i n p e r t u r b a t i o n theory by 00 de ImG{V(e) - o o 7T 1 + nB(u) - nf(e) nB(u) + nf(e) e — e — u + id e — e + u + id (3.42) where is the bare e lec t ron G r e e n func t ion , ie. , I m C ? f (e) = 7rd(e + (p - 1/2 - n)uc). (3.43) a n d U(q,Lu) depends o n the e lec t ron-phonon c o u p l i n g (see be low) . T h e e lec t ron ic fre-quency e is measured f rom the F e r m i energy, p — puc, and .the energy of the highes t f i l l ed L a n d a u level is (p — l/2)uc: s ince we are d o i n g a quas i -c lass ica l c a l c u l a t i o n , we assume that the F e r m i energy is ha l fway between L a n d a u levels [90, 87]. In the rest of this sec t ion we first ca lcu la te the self-energy a n d the r e n o r m a l i s e d c y c l o t r o n gap energy for the case o f a de fo rma t ion c o u p l i n g to Debye phonons : we then do the same for c o u p l i n g to o p t i c a l phonons . 3.5.1 Debye Phonons In ou r ana lys i s we first cons ider a two d i m e n s i o n a l sys t em of e lectrons i n a s m a l l m a g n e t i c field B w h i c h in terac ts w i t h phonons h a v i n g a Debye s p e c t r u m . In th i s case we w i l l use the p h o n o n p ropaga to r c a l c u l a t e d at B = 0, w h i c h is assumed to have a cutoff at the Debye frequency UD {CS is the speed of sound) . W e w i l l use the same a p p r o x i m a t i o n for the over lap m a t r i x element between p lane waves a n d L a n d a u level states as i n Sec t ion 3.3, g iven by (3.24). (3.44) Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 61 T h e func t i on U(q,u) i n (3.42) is jus t the effective i n t e r a c t i o n between electrons: i n the Debye m o d e l i t is g iven by U(q,u) = f^D(q,u) (3.45) where D(q,u) is the p h o n o n G r e e n func t ion (3.44). T h e e l ec t ron -phonon i n t e r a c t i o n is p a r a m e t r i s e d by the de fo rma t ion p o t e n t i a l , H p , and p is the i o n mass dens i ty of the m a t e r i a l (see A p p e n d i x A for more de ta i l s ) . E v a l u a t i n g the in tegra ls i n E q . (3.42) y i e ld s , -10 -5 0 5 10 £ F i g u r e 3.9: T h e rea l a n d i m a g i n a r y par t s of the self-energy for electrons i n t e r a c t i n g w i t h phonons w i t h a Debye s p e c t r u m (a>p = 20w c ) i n a s m a l l magne t i c f ie ld . A l l energies are i n un i t s of tuc; the c o u p l i n g is A ' p = 1. Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 62 at t empera tu re T = 0: KD IT P-1 Z ( -"D + [(n -p+ l / 2 ) w c - e] lo. .n =p too + in> —P + l / 2 ) w c - f-+ Z ^ 0 + \{n -P+ l/2)uc - e] l og »i =o - p + 1/2)wc - e (•n' — p + 1 / 2 )w c — e — w 0 (n —p + l / 2 ) w c - e (3.46) = K D \ f l ("D ~ £ + [(n -p+ 1 /2K - e\0[(n -p+ l/2)uc]) w =p p-1 \ + Y, ("D + £ + [(-n +p- l/2)uc + e ]0 [ ( -n ' +p- l/2)uc) . (3.47) »'=<) / T h e r ight h a n d side is independent of n , and so we w i l l o m i t the subscr ip t o n S(e) hencefor th . E(e ) is i n un i t s of coc and KD is a d imens ionless cons tant , 3'2Dmu>c 4irc*ap \ 4 7 r n , 1 x 1 / 2 (3.48) S h i f t i n g the s u m to start at —p and t r u n c a t i n g the s u m at p — 1, we find (m + l/2)u>c - e + cu0 E ' ( c ) = ^ i : f [ ( m + l / 2 ) a ; c 7T •m=0 - [ ( m + l / 2 ) w c + e] log € ] l 0 g ( m + 1 /2K - € | m + l / 2 ) w c + e + W£). [m + 1 / 2 > C + e (3.49) p-i S"(e) = A' / j J2 ( K m + 1 / 2 K - e M ( ' m + V 2 ) ^ - e + uD]6[{-m - l / 2 ) w c + e] m=0 + [(m + 1/2)w c + e]0[(m + 1/2)w c + e + c ^ ] 0 [ ( - m - l / 2 ) w c - e]). (3.50) These funct ions are shown together i n F i g . 3.9. T h e s t ruc ture caused by the L a n d a u levels is very weak; i t c an jus t be seen i n E" (e ) ( i f UCJUJD is larger , i t is m u c h more obv ious ) . S"(e) is s i m p l y a s u m of r a m p func t ions , Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 63 110 Figure 3.10: The derivative with respect to energy of the real part of the self-energy of electrons interacting with Debye phonons is a finite magnetic field (uD = 20uc). The energy is in units of u>c, and Ko = 1. coming from each Landau level; as uc —> 0, S"(e) becomes parabolic for f « u c . The logarithmic singularities in S'(e) are quite invisible in Fig. 3.9, but they may be seen in the derivative, shown in Fig. 3.10. We wish to find the poles of Gn(e). Generally there are 0, 1, 2 or 3 roots to the equation S'(e) + € = (n + l/2)o;c, (3.51) which correspond to peaks in the spectral function, lmGn(e). In this expression both Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 64 2 — i i i i I i i i i I 1.5 L a ) 1 0.5 -_ o I 1.5 o i i i i ! i i i i !' 1 ' 1 l i i i i i i i i i i i i >" A 1 1 1 1 I 1 1 1 1 / \ -/ \ / ~ \ 0.3D i i i i 1 i i i i \ 1 / 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.2 - -0.1 l — — — - A -, , , , -10 -5 0 5 10 Figure 3.11: The imaginary part of the Green function of electrons interacting with Debye phonons for lip = .001, .1 and .25 with n — 0 (see the text for definition of the coupling KD). n and e are measured with respect to the Fermi surface. We are only concerned with the low energy behaviour which corresponds to small n and e. When n is small and lip is small there is one solution occurring near e = (n + 1 /'2)uc, which appears as a well defined quasiparticle peak (a ^-function for n = 0). When KQ is large there are three solutions. Two occur well beyond the Debye frequency (outside of the low energy regime) and the third occurs close to e — 0 yielding a ^-function. Intermediate values of KD do not in general yield the very narrow peaks characteristic of a dressed particle, since if Chapter 3. Composite Fermions. Gauge Fluctuations and the Gap 65 so lu t ions to (3.51) do occur they occur at f inite e where the i m a g i n a r y par t o f S is also f ini te . T h e r e m a y also be features a r i s ing f rom incoherent c o n t r i b u t i o n s . Some o f these are shown i n F i g . 3.11. W e remark that the magne t i c field causes s l ight d i s con t inu i t i e s i n the s lope of the spec t ra l funct ions at the L a n d a u level energies. 3.5.2 Einstein Phonons W e n o w cons ider the i n t e r ac t ion w i t h the more s ingu la r E i n s t e i n p h o n o n s p e c t r u m , w i t h p h o n o n p ropaga to r D(q,u>) — uE/(co2 — u%); coE is the " o p t i c a l " p h o n o n frequency. In zero field, th is was first cons idered by Enge l sbe rg and Schreiffer w h o found the self-energy to be , , KEUC , ,e) = l o g 7T e -I- EF + coE e + Ep -cjE E"(e) = KEtoc for|e| > uE (3.52) = 0 for|e| < uE. (3-53) W i t h E expressed i n un i t s of toc, KE is the d imens ionless c o u p l i n g : A.EPbJc T h e effect of the magne t i c f ield is to m o d i f y these equat ions to A W 2 ^ ( 1 1 W e ) TT J^0\(m.+ l/2)Loc + ujE-e (m + l/2)uc + uE + e) p-i S"(e) = KEto\ (6[(m + l/2)uc + coE - e] + 6[(m + l/2)uc + uE + e]). (3.56) m=0 E' (e ) is s h o w n i n F i g . 3.12. T h e r e is no v i r t u a l l y no effect of a s m a l l m a g n e t i c f ie ld on the self-energies for |e| < uE\ b e y o n d this reg ion the s m o o t h func t ions are r ep laced by func t ions w i t h divergences at each L a n d a u level energy. A l l poles i n the spec t r a l f u n c t i o n Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 66 appear as ^-functions since the imaginary part of the self-energy vanishes everywhere except at discrete points, but the weights of the poles vary. The new electron spectrum is determined by summing over all n. Since the n are discrete a renormalised gap can be defined as the energy difference between the highest occupied and lowest unoccupied states, ie., by the difference between the positions of the 5-function peaks of Go(e) and G-i(e). -20 20 Figure 3.12: Real part of the self-energy of electrons interacing with phonons with an Einstein spectrum (uE = 20uc). Both axes are in units of coc\ the coupling is KE = 1. The amount of renormalization of the gap depends on the size of the coefficient of the self-energy, KE. When KE is small the poles lie near (n + l/2)uc for nuc < u E , as in the Debye case. The renormalized gap is approximately uc, only slightly increased due to the interactions. As KE is increased the gap increases further. We determine the gap using the same approximation as in Section 3.4, (3.35) and then solving (3.51) for n = — 1 and n = 0: as'(o) _ - A ' a g 2 de v m^O ( m + l/'2)Uc + UE (3.57) Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 67 2KE ((p+l/2)uc + uB log (3.58) uc/2 + uE J where the sum over m has been replaced by an integral. Then the solution to (3.51) is 1 _ V5± log ((P + 2 / 2 K + WE uc/2 + uE and the gap is = ±- (3.59) = UJr 1 — 2KE f (p+ l/2)u>c + u>E log 7T (3.60) uc/2 + uE This procedure is also applicable to the Debye case when there are well defined quasi-particles for n = —1,0. In this case the renormalized gap is 0J„ = Ur 1 - — [log(2o;p + coc) 7T -(2p+l)log (p+ 1/2)OJC + UD (3.6i: ( P + 1 / 2 K , We may draw some general conclusions about the size of the gap for different coupling strengths using the forms above. Considering first the Einstein case, for small values of the couplings the gap is renormalized to a larger value. This continues to be the case as the coupling is increased. The divergence in the gap seen in Eq. (3.60) is avoided because the gap is always bounded by 2uE- This is due to the presence of the logarithmic peaks in the real part of E(e) which ensures that there will always be a solution to (3.51) for |e| < UJE- For large couplings the gap is reduced. In the Debye case, for small KD there are well defined quasi-particle peaks outside the gap but they are not strictly ^-functions thanks to small but non-zero ImE(e). The finite width maintains the size of the gap to be uc, as shown in Fig. 3.11. For larger values of KD the divergence of the gap in (3.61) is avoided for the same reason, as shown in Fig. 3.11b. For large couplings the Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 68 gap is reduced because of QP poles that lie within ooc, as shown in Fig. 3.11c. As can be seen from Figs. 3.9 and 3.11, the effect of Landau quantization on the quasi-particle properties, for realistic values of UOD,UE and uc, is very small, at least at low energies (it is however worth noting that once e > top in the case of Einstein phonons, a rather obvious singular structure appears in the self-energy, see Fig. 3.1*2). Nevertheless the calculation of these effects may be useful in systems for which coc can be made as large as too or top. In this Section we have examined the structure induced by Landau level quantisation on the two dimensional electron-phonon problem. We find singular, but weak structure which has never previously been investigated. The analysis of this well understood prob-lem has been useful for improving our understanding of the composite fermion Green's function. 3.6 Iterative Self-Consistent Results for E(ej and Aw* As already explained in Refs. [77, 80, 82, 84], the CF problem is essentially a non-perturbative one. The results these references address the // = 1/2 gapless system; an analysis of the s = 2 case for v ^ 1/2 was also given by Stern and Halperin [29], but taking no account of the structure described above. Here we attempt to improve the results just given, by solving iteratively for the self-energy. We have found a way to do this which handles the coherent pole contributions to S(e) in an exact rainbow summation; the incoherent parts must still be treated ap-proximately. We start from the iterative equation X(i+1Ke) = Yfde'D(e-e)G$(e) (3.62) m Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 69 with 1 (3.63) e - (m + 1/2)ACJc + £(')(e) - i6' When the iterations converge (i.e., when E(e) = YJ,,, f de D(e — e')Gm(e')) we will have succeeded in summing over all of the self-energy graphs which have no crossed lines, as shown in Fig. 3.13. These graphs are known as the "rainbow" graphs. Figure 3.13: Rainbow diagram contributions to the self-energy. This kind of calculation is usually difficult but in this case the analysis is simplified by the following. First, because the real part of E^(e) is bounded on either side of the jap e| < by logarithmic peaks, we are guaranteed that there are solutions to (3.51) for all n within the gap. That the imaginary part of E^^e) is zero in the gap implies that there are actually 2p (5-function peaks in Y^m=-PG\n(e) within the gap (exactly one for each m). The distance between the peaks resulting from m — 0 and in = — 1 gives a renormalized gap, Aw[ 1 }. Next we use in (3.62) to generate S ^ . We are particularly interested in the form of E ( 2 ) within the gap because this is where further corrections to the renormalized gap originate. The contributions to ImG ( 1 ) may be split into two parts: a coherent piece, coming from the (^ -function peaks within the gap, and an incoherent piece in the region outside the gap (we use this terminology for convenience- there may in fact be solutions to (3.51) in the latter region which correspond to poles in G^). Upon evaluating R e E ( 2 ) we notice that the coherent parts of ImG ( 1 ) give rise to exactly the same logarithmic Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 70 divergences as seen in S ( 1 \ except that the peaks are located at the poles of I m G ^ and each peak is weighted by a factor - l 2 P(em) = 1 + (3.64) where e,„ are the positions of the poles. However, in most cases the weight coming from the poles within the gap is small compared to the weight of the incoherent part of ImGj,j', which means that there may be significant contributions to ReS ( 2^ that do not come from the poles of I m G ^ . Only the coherent parts of I m G ^ contribute to ImE^(e) for |e| < which vanishes within the renormalized gap, |e| < The corrections arising from the coherent parts may be calculated exactly. However we are forced to examine the incoherent corrections using approximations, beginning with the use of (3.35). This approximation has the effect of smearing out the logarithmic peaks at the Landau levels. As discussed above, these features are a crucial element within the gap: however outside the gap their role is not as important. They may generate quasiparticles peaks as solutions to (3.51) but the weight of these peaks is very small. The artificial smoothing of the Landau level structures would greatly simplify a numerical integration over these parts. Instead of doing this, we make use of the fact that the weight of the incoherent parts may be determined exactly, since Jje|>Au>c/2 delmG1^ (e) = 1 — zp(em). Therefore we approximate (3.62) as where the peak of ImG,„(e) is at emax. This approximation is valid for emax > Auc/2, which occurs when ^ log(2p + 1) « 1. We have used this procedure to calculate the gap as a function of p, keeping only results which appear to have converged after the second step. We find corrections that are as large as 30% of the first order result. These calculations are shown in Fig. 3.14, (3.65) Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 71 F i g u r e 3.14: T h e effective mass , m * as a func t ion of B and v for c o m p o s i t e fe rmions . T h e lower x— ax is has been o b t a i n e d by a s suming an e lec t ron dens i ty ne = 2.23 x 1 0 1 1 c m - 2 . T h e crosses are n u m e r i c a l ca l cu l a t i ons w h i c h i nc lude cor rec t ions c o m i n g f rom E ( 2 ) ( the self-consistent self-energy). T h e curves are a guide for the eye. p l o t t e d as ^ = for var ious coup l ings . W e es t imate the a c t u a l c o u p l i n g , i n the expe r imen t of D u et a l . [30] to be K2 ~ 0.8 (see T a b l e 3.1). W e emphas ize tha t the resul ts for s m a l l values of p are not mean ingfu l because of the a p p r o x i m a t i o n (3.24), w h i c h assumes that p is large, thus there is no over lap w i t h e x p e r i m e n t a l resul ts . Therefore , we d r a w no conc lus ions about the exact r e l a t ion between m* and p. However , there is a range of p where p m a y be cons idered to be large a n d log(2p + 1) is not . In th is range we expect our results to be v a l i d but to have not yet reached the a s y m p t o t i c l i m i t Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 72 o f m*/m = K~2log('2p + 1 ) . T h i s c a l c u l a t i o n shows the effective mass to be far more sens i t ive to the c o u p l i n g t h a n to p, thus i t is diff icult to de te rmine w h a t r e l a t i o n these resul ts have to exper iments w i t h o u t k n o w i n g the c o u p l i n g exac t ly . 3.7 The Effective Mass: Comparison to Experiment In recent years a s m a l l n u m b e r of groups have a t t e m p t e d to measure the effective mass of c o m p o s i t e fermions i n d i r e c t l y us ing the S d H osc i l l a t ions of the l o n g i t u d i n a l r e s i s t i v i t y as a func t ion o f the magne t i c f ield AD. T h i s is achieved by f i t t i ng the e x t r e m a o f the osc i l l a t ions to the D i n g l e f o rmu la w h i c h is pa rame t r i s ed i n terms of an effective mass : Apxx Po = 4 exp -7T A w * r I s i nh £ cos[7r(2p — 1)] 2n2kDT f i A w ! (3.66) (3.67) T h e o n l y o ther u n k n o w n paramete r is the sca t t e r ing t ime r = where Y is the b r o a d -en ing of the C F L a n d a u levels. V is the difference between the true gap and the gap d e t e r m i n e d f rom t h e r m a l a c t i va t i on measurements : the true gap is inferred f rom the S-d H measurements for s m a l l values of p. T h e r e have been four such measurements , w i t h c o r r e s p o n d i n g self-energy coefficients as l i s t ed i n T a b l e 3.1. au tho r e n e ( x l O n c m - 2 ) mh (me) K2 M a n o h a r a n et a l . [31] 13.1 1.6 0.38 0.13 D u et a l . [30] 12.8 2.25 0.068 0.83 L e a d l e y et a l . [57] 12.9 0.63-4.5 0.068 0.45-1.20 C o l e r i d g e et a l . [32] 13 1.27, 1.39 0.067 0.65, 0.68 T a b l e 3.1: F o u r exper imen t s w h i c h measure the S d H osc i l l a t ions i n the f r a c t i o n a l quan -t u m H a l l reg ime and the i r co r re spond ing self-energy coefficients. K2 is d e t e r m i n e d f r o m the g iven e x p e r i m e n t a l values for e, ne and nib u s ing E q . (3.27). Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 73 fWng Factor, v a) - j — n - n • i \ TTT-T 1 : ] : 1 V i : l i _ „ t l i — J c) **** i . • — - • • - — • — i — • — • < .(b) y n I 1 — — • -G902a o G802b * G148 • G627 T . G650 v G640 •» G641 o G646 A 1.1 b) O sampt* B 1.0 o u E \ 0.9 - 1 E 0.8 f t -0.7 0.6 - I 1 c. i Effective Field (T) «* IT J -6 -4 -2 0 2 4 6 d) 1.8 16 1.4 °S 1.2 10 08 » SdH Maxima v SdH Minima • o- - Glp Scaling , (a) B* (teslo) -1.0 -OS 1.0 p F i g u r e 3.15: F o u r exper iments w h i c h measure the effective mass f rom S d H measurements , a) D u et a l . [30], b) C o l e r i d g e et a l [32], c) L e a d l e y et a l . [57] d) M a n o h a r a n et a l . [31]. N o t e tha t i n a l l cases the v e r t i c a l axis is i n un i t s of m e , whereas i n F i g . 3.14 i t is d r a w n i n un i t s o f nib-T h e semi -c las s i ca l a p p r o x i m a t i o n we have used (p > 1) prevents us f r o m m a k i n g a d i rec t , quan t i t a t i ve c o m p a r i s o n w i t h any of the e x p e r i m e n t a l resul ts , w h i c h are a l l i n the range p < 8. However , a l l of the exper iments exh ib i t definite t rends so tha t we m a y ex t r apo la t e thei r results to larger values o f p. T h e i n f o r m a t i o n shown i n F i g . 3.14 has been ca l cu l a t ed u s i n g the f i l l ings s h o w n o n the uppe r ax i s for the different coup l ings shown on the p lo t s . T h i s i n f o r m a t i o n m a y be in te rpre ted i n two ways. F i r s t , we m a y assume that the densi ty, ne, is fixed a n d tha t the Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 74 1.8 1.6 1.4 1.2 -i I , , , i | i , X i/=5/ll-- i/= 13/27 : X : i/= 10/21 - y=41/83 — X / — -/ i>=7/15 - v=20/4l/ i/=50/101 / / / /i/=3/7 -s»• /^Z^^^ " = 2/5 --1^ =1/3 1 1 , , , 1 2 3 n e ( 1 0 u c m ) Figure 3.16: The effective mass m* as a function of electron density. These are the same results as in Fig. 10 except that the rc-axis has been obtained using ne = (2A-"ie~) with e = 13 and m = .07me. The crosses are numerical calculations which include corrections coming from (the self-consistent self-energy). different couplings arise by variations of the other parameters in (3.27). The values of B shown in the lower .r-axis have been determined using ne = 2.2 x 10 1 1 cm - 2 . This allows us to compare to the experimental data of Du et al. [30] - their data is consistent with a coupling of A'2 ~ 1, which is slightly larger than the coupling A " 2 = 0.83 calculated using (3.27). This corresponds to a theoretical prediction for the effective mass that is nearly ten times smaller than the experimental observations. Coleridge et al. [32] also find an effective mass that is ~ 10 times larger than our prediction. Our calculations do Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 75 2.0 M* 1.5 ( m e ) 1.0 0.5 0.0 , .... . -• ' s" _ 1/3 • 2/5 • 2/3 O 3/7 T 3/5 A . . . . 1 . , . . 1 . . . , 0 1 2 3 4 5 n e ( i o 1 5 m - z ) F i g u r e 3.17: E x p e r i m e n t a l results of L e a d l e y et a l . [57] showing the effective mass as a func t i on of e lec t ron dens i ty not agree w i t h the resul ts of L e a d l e y et a l . e i ther , w h o do not observe a mass d ivergence [57], bu t the i r d a t a do not go b e y o n d v = 3 / 7 , w h i c h is not w i t h i n the range o f our a p p r o x i m a t i o n s . (Obv ious ly , we cannot compare the resul ts o f D u et a l . at these values ei ther .) T h e disagreement w i t h the results of D u et a l . [30] m a y be a t t r i b u t a b l e to cor rec t ions c o m i n g f rom d i a g r a m s not i n c l u d e d i n the s u m over r a i n b o w graphs , or more general ly , f rom the non-pe r tu rba t ive na ture of th is c a l c u l a t i o n , thanks to the large c o u p l i n g K2. In general the c o u p l i n g constant w i l l be of order un i ty , bu t the e x p e r i m e n t a l scenar io inves t iga ted by M a n o h a r a n et a l . [31] is a no tab le excep t ion . In the i r expe r imen t the car r ie rs are holes w i t h a ra ther large b a n d mass = Q.38me. T h u s the c o u p l i n g coef-ficient is reduced to A'2 = 0.13, w h i c h is w i t h i n the pe r tu rba t ive reg ime for in t e rac t ions w i t h gauge fluctuations. T h i s is c l ea r ly evident f rom our c a l cu l a t i ons , a n d i n F i g . 3.14 we see that for K2 < 0.25 there is essent ia l ly no mass r e n o r m a l i s a t i o n up to p — 50. T h e resul ts o f M a n o h a r a n et a l . are i n s tark disagreement w i t h th is . T h e i r resul ts suggest a mass divergence as v —> 1/2 and i n the range o f the i r observat ions the mass is u p Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 76 to five times larger than the theoretically determined one. While our results for small values of p may be objectionable because of the semiclassical approximations we have used, we do expect that our results are correct for small coupling and p of order 50. A divergence as v —» \ is still predicted, but the divergence would not become apparent until ~r log(2p + l ) « l , o r p R ! 10'5 {y = 9 x \ ^ + 1 )• It seems extremely unlikely that the measurements of Manoharan et al. would reverse their divergent tendency to meet with our predictions at p = 50 or the asymptotic result, therefore we are forced to conclude that gauge fluctuations are not entirely responsible for the mass divergences observed in the SdH measurements. In support of this, we note that preliminary investigations indicate that density fluctuations strongly affect the measured values of m" 4 . Alternatively, we may assume that each curve in Fig. 3.14 is associated with a different density, ne. Using e = 13 and m = .Q7me in (3.27) we replot the results as shown in Fig. 3.16. The curves show that the effective mass increases with electron density, which agrees with the observations by Leadley et al., but there is no way to make a quantitative comparison for the range of data we have calculated, due to the fact that our approximations are not valid for the range of v studied in the experiment. 3.8 Summary In this Chapter we have computed the second order self-energy of composite fermions interacting with gauge fluctuations while retaining the full oscillatory form coming from Landau level quantisation. We find that Landau level quantisation introduces divergent structures in the composite fermion self-energy. The better understood two dimensional electron-phonon problem is analysed in the same manner, in order to compare with the composite fermion self-energy, and is found to have a weaker but still singular structure. 4P. T. Coleridge, private communication. Chapter 3. Composite Fermions, Gauge Fluctuations and the Gap 77 T h e i n c l u s i o n of L a n d a u level quan t i s a t i on i n the self-energy of c o m p o s i t e fe rmions y i e ld s a s igni f icant depar ture f rom the results o f ca l cu l a t i ons w h i c h ignore th is s t ruc tu re i n the regime of e x p e r i m e n t a l l y relevant parameters . T h e resul ts are then i m p r o v e d u s i n g a new i te ra t ive procedure . T h i s procedure is used to c o m p u t e the effective mass for a range of filling f ract ions and p h y s i c a l l y r ea l i s t i c ' pa r ame te r s . These resul ts are c o m p a r e d w i t h exper iments , three of w h i c h do show an enhancement o f the effective as v —> 1/2. However , of these exper iments , a l l d i sp l ay a m u c h larger enhancement t h a n our theore t i ca l p red ic t ions and i n one case an enhancement is observed that is not pred ic t ed . Chapter 4 Oscil latory Quantities In system of fermions, all experimental quantities are observed to oscillate as a function of applied magnetic field. The best known quantities include the resistivity (SdH oscilla-tions), magnetisation (dHvA effect) and the compressibility. This Chapter is concerned with the latter two quantities, which are both derived from the thermodynamic potential. In Section 4.1 we examine the chemical potential, average energy and thermodynam-ic potential by considering free composite fermions with a field dependent renormalised mass. The various thermodynamic quantities oscillate as a function of AB. This model assumes that the only effect of interactions is to renormalise the effective mass, which greatly simplifies the analysis of these quantities. Under this assumption, we find quali-tatively similar behaviour to experiments: this enables us to extract effective masses from the experimental results and cross-compare them. The effects of gauge fluctuations are discussed in Section 4.2 in order to provide justification of the approach of Section 4.1. We argue that the discontinuity in the chemical potential of composite fermions, which occurs when the topmost composite fermion Landau level is completely filled, is the same as the renormalised gap. The simple model of Section 4.1 is justified by a finite temperature calculation of the compressibility when the filling of CF Landau levels is integral. We also examine the chemical potential when the top C F Landau level is half-filled; this is further confirmation of the simple model of Section 4.1. 78 Chapter 4. Oscillatory Quantities 79 In Section 4.3 we examine the thermodynamic potential Q for two dimensional in-teracting systems. We seek a non-perturbative form for Q since the composite fermion gauge theory is non-perturbative. We examine a construction given by Luttinger [39] which expands Q in powers of ^ and find that it is not applicable in two dimensions, for any kind of interaction. This means that standard techniques for evaluating oscillation amplitudes in three dimensions are not valid in two dimensions. However, we do find a non-perturbative expression for 0 in two dimensions which is valid when all "crossed" diagrams are neglected. This form is used to evaluate the magnetisation of composite fermions interacting with gauge fluctuations. 4.1 Non-Interacting Approximation In this Section we analyse various thermodynamic quantities of free fermions in a mag-netic field. The corresponding quantities for composite fermions are found by a simple substitution of m* for m and Ai? for B. In addition, some care is required to properly account for energy offsets - this is important in the calculation of the average energy. The results of this Section are compared with experiments that measure the compressibility and average energy. 4.1.1 The Chemical Potential The chemical potential is closely related to the problem of the gap, since in the non-interacting system at T = 0 the gap is the same as the discontinuity in the chemical potential of composite fermions. As the density is varied through the series v = the chemical potential jump of electrons is 2p + 1 times higher than that of CF's. To see this, consider what happens when we add one electron to a system where there are exactly p filled C F Landau levels (p an integer). Before adding the electron there are Chapter 4. Oscillatory Quantities 80 Np CF's and A r magnetic flux quanta of AB. Adding the electron is the same as adding one C F and subtracting two flux quanta, so that there are Np + 1 CF's and N — 2 flux quanta. Dividing yields a remainder of 2p + 1, which is the number of CF's that have been promoted to the next CF Landau level. This costs an energy (2p + i)Au*. Similarly, when the magnetic field is varied instead of the density, the chemical potential jump of electrons is pAtu*. To begin, we first consider the chemical potential at T = 0 for electrons in a magnetic field. The density of states is simply a set of degenerate Landau levels, (fQ\e) = J2n He ~ (n + T?)COC), of which the topmost one may be only partially filled. As a function of the filling fraction (or density, when the magnetic field is held fixed), the chemical potential is simply a set of steps, beginning at ^ and increasing in jumps of coc each time the filling-fraction passes through an integer value (i.e., each time a Landau level is completely filled): p=([p+l/2)uc. • (4.1) The T / 0 corrections are derived under the assumption that the chemical potential yields the correct number of particles in the integral over the density of states: p = / deg^(e)nf(e-p) (4.2) ./ —oo oo = ^^([n + l^H- / :? ) (4.3) n=0 p is the filling fraction, and may be a non-integer number. Writing p = [p + dp and p = ([p + l/2)uc + dp and shifting the sum over n to start at — [p, we find p + dp = ]P n.f(n — dp). (4-4) n=-[p Finally, we make the assumption that T <C uc. Within this approximation, we take the value of the Fermi function to be either exactly one or exactly zero for all values of n, Chapter 4, Oscillatory Quantities 81 except for n = 0. T h i s y ie lds [p + 6p =• [p + nf(-dp) (4.5) w h i c h has the s o l u t i o n 4 " = T l o g ( r ^ ) ' ( « ) T h i s resul t gives rise to undes i rab le s ingu la r i t i e s as dp —* 1,0, w h i c h happens to be the reg ion of greatest interest . T o avo id this , we keep two terms i n the s u m (4.4) i n s t ead o f o n l y one. T h e second t e rm is chosen depend ing on w h i c h l i m i t of dp we are nearest to. F o r dp —> 0 we keep n / ( — 1 — dp) and for dp 1 we keep n / ( l — dp), w h i c h y i e ld s 1 ^ f-[l+f(T)]dp ^ [ 1 + f(T)]2dp2 + 4 / ( T ) ( l + Sp)(l - dp dp = / l o g — — — h 2(1+ 6p) 2(1 + dp) J for dp 0 (4.7) _ T l o g (~lf(T) + l}(dp-l) + y/[f(T) + l]2(dp - 1)2 - 4dpf{T)(dp~2 2f(T)dp 2f(T)dp J for dp - ) • 1 (4.8) / ( T ) = e x p ( ^ ) . (4.9) W e p a t c h these so lu t ions together , s w i t c h i n g between (4.7) a n d (4.8) at dp = 0.5. T h i s gives rise to a d i s c o n t i n u i t y at dp — 0.5 w h i c h becomes large at h i g h t empera tu res . O n e c o u l d solve (4.4) w i t h more te rms i n c l u d e d , but th is does not r emedy the p r o b l e m for h i g h tempera tures a n d the express ion becomes u n w i e l d y even w h e n o n l y three te rms are i n c l u d e d . U s i n g the n o t i o n o f compos i t e fe rmions , the ex tens ion to the F Q H E is s t ra igh t fo rward . A l l the a rguments g iven above also a p p l y here, except tha t uc is r ep laced by A w c everywhere . T h e i m p o r t a n t differences arise f rom the dependence of A i ? o n the dens i ty of electrons. W h e n we fix the ex te rna l magne t i c field a n d va ry the densi ty , A i ? also varies a c c o r d i n g to E q . (3.3). 1 E q . (4.7) was derived in [92]. Chapter 4. Oscillatory Quantities 82 Fig. 4.1 shows the chemical potential of composite fermions as a function of filling g o . 5 / /""\ \ "cd • i—i -.en1 - A lical —T= 0.74 K — 0.62 K X^J p g-0.5 — 0.51 K 0.46 K — 0.4 K i i i i i i i i 0.3 0.35 0.4 Fil l ing fraction v Figure 4.1: The chemical potential of composite fermions as a function of filling fraction calculated for m* = 20mb and B = 13T using Eqs. (4.7) and (4.8). This plot shows only the oscillatory component of p, which oscillates about its values at v = \. for various temperatures. In this plot we have subtracted a contribution pAuc, so that what remains is just the oscillatory component. Below v = 1/3 we have made an abrupt change to composite fermions that are electrons with four flux quanta attached. Both types of CF's lead to exactly the same discontinuity at v = \ at the mean field level (and so do the bosonic particles that have three flux quanta). It is evident from the Figure that at the higher temperatures Eq. (4.9) begins to break down, especially for the Chapter 4. Oscillatory Quantities 83 s m a l l e r gaps. T h e co r re spond ing p lo t of electrons has a c h e m i c a l p o t e n t i a l j u m p tha t is 2p + 1 t imes larger t h a n that o f compos i t e fermions at T = 0. T h e effective mass of m* = 20mb, co r r e spond ing to a gap A w * = 3 . 9 K at v = rv used i n these p lo t s was chosen to achieve a g o o d m a t c h to the e x p e r i m e n t a l resul ts of E i s e n s t e i n , Pfeiffer a n d Wes t [55] ( shown i n F i g . 4.2). W e have used this value of m* 0.25 0.30 0.35 0.40 0.45 Filling factor vt F i g u r e 4.2: E x p e r i m e n t a l results of E i sens t e in et a l . [55] s h o w i n g a) the c o m p r e s s i b i l i t y of a 2 - D electron gas a n d b) the chemica l p o t e n t i a l , w h i c h is the in t eg ra l of the compress-i b i l i t y . N o t e tha t the h o r i z o n t a l axis is the same o n b o t h p lo t s . for the ent i re curve, a l t h o u g h i t s h o u l d a c t u a l l y depend on v. E x p e r i m e n t a l resul ts such as those o b t a i n e d by D u et a l . [30] show o n l y a s m a l l v a r i a t i o n i n in* for the first few incompres s ib l e states after v = \ . It is not poss ible to o b t a i n m* f rom our c a l c u l a t i o n i n C h a p t e r 3 of the gap at v — \ because the a p p r o x i m a t i o n s used are not v a l i d for the l ow d e n o m i n a t o r f ract ions . In p r inc ip l e i t is poss ib le to o b t a i n th is va lue f rom other exper imen t s , such as measurements of the S d H osc i l l a t ions pe r fo rmed by L e a d l e y et a l . [57], where they observe in* at f ixed v as a func t ion o f ne. However , they f ind m* ~ H m & at v — \ , w h i c h is cons ide rab ly smal le r . O n the o ther h a n d , the measurements of D u et a l . [30] found an effective mass m* ~ 20mb at v — | w h i c h is i n r o u g h agreement w i t h the c o m p r e s s i b i l i t y measurement . However , the i r exper imen t was pe r fo rmed w i t h a Chapter 4. Oscillatory Quantities 84 density ne = 2.25 x 10 n cm~ 2 , which is roughly twice as large as the density at v — \ for the compressibility measurement (ne = 1.1 x 10 1 1 cm - 2 ) . Our calculations in Chapter 3 (see Figure 3.15) show that the effective mass does depend on the density (this sensitivity was also observed by Leadley et al.), therefore a direct comparison between the effective masses obtained from the compressibility measurement and the measurement of Du et al. may not be valid. For the discussions of the oscillatory quantities to come, it will be helpful to under-stand the behaviour of the chemical potential for free composite fermions as a function of the external magnetic field. For fixed ne and an unrenormalised effective mass, it is: Ai = ( b + l/2)Ao; c = ( b + 1/2)^*2. (4.10) pm This quantity oscillates about p = which is precisely the value of the chemical potential at v = \ in the non-interacting case. Interactions change the effective mass and as before we can guess that there will be a corresponding shift in the chemical potential, so that it will reach the same limit as v —> \ as the non-interacting case. Thus we find that A. = ( b + l / 2 ) A W c * + ^ - ^ . (4.11) where u>* — ^4 . 4.1.2 The Compressibility Measurement Intrinsic magnetoluminescence experiments measure the strength of the luminescence lines resulting from the recombination of electrons in the 2-D gas with holes [65]. Unlike the situation for extrinsic photoluminescence (see Section 4.3.1) the holes are located in the vicinity of the 2-D gas. The strength of the line varies with the compressibility of the 2-D gas. The line is strongest when the electrons are in an incompressible F Q H state Chapter 4. Oscillatory Quantities 85 since then the electrons are comple t e ly unab le to screen the hole a n d the r e c o m b i n a t i o n ra te is a m a x i m u m . T h e r e l a t i o n between screening and i n c o m p r e s s i b i l i t y is the basis of a q u a n t i t a t i v e measurement by E i sens t e in et a l . [55]. In th is case a vol tage is a p p l i e d to a gate above a doub le layer H a l l sys t em. T h e first layer p a r t i a l l y screens the gate vol tage f r o m the second layer. C u r r e n t detected at the second layer is i n d i c a t i v e of the screening a b i l i t y of the first , a n d hence i ts c o m p r e s s i b i l i t y m a y be de t e rmined . W e have der ived the results shown i n F i g . 4.3 u s ing ^ = J ^ - , w h i c h is the de r iva t ive of F i g . 4 .1 . T h e y -ax i s has been scaled by the i n t e r ac t i on s t rength , ^ - w h i c h is 368 K , for B = 1ST and e = 12.8: these are the values used by E i s e n s t e i n et a l . A s ignif icant difference between our theore t i ca l results u s ing c o m p o s i t e fe rmions a n d the e x p e r i m e n t a l results is a negat ive offset i n the c o m p r e s s i b i l i t y tha t appears i n the exper imen t s . T h i s is a h igher energy effect w h i c h or ig ina tes f rom the s t r ong e lec t ron-e lec t ron in te rac t ions at l ow densi t ies [93]. In contras t , the c o m p o s i t e f e rmion theory is a l o w energy theory a n d as such, gives rise to the fine scale s t ruc ture assoc ia ted w i t h the F Q H E but i n f o r m a t i o n conce rn ing higher energy scales is lost i n the t r ea tment tha t we have used, w h i c h has a cut-off at the true c y c l o t r o n frequency uc. 4.1.3 The Average Energy U s i n g the result tha t the compos i t e fermions seem to behave as free pa r t i c l e s w i t h a r e n o r m a l i s e d mass , we m a y proceed to ca lcu la te the average energy i n the same way as the c h e m i c a l po t en t i a l . T h e T = 0 result for p f i l led C F L a n d a u levels w i t h an w i r e n o r m a l i s e d mass is Eav(p) b - i ( n + 1 /2)Auc + 6p( [p + 1 /2)Auc 71 = 0 (4.12) T h e factor ^ is the degeneracy of each L a n d a u level . Chapter 4. Oscillatory Quantities 86 1 1 1 1 1 1 1 , , , 1 1 1 1 1 \ 1 o CD - --0.6 II - T=0.4 K J 11 , 0.46 K T 1 0.4 >, _, ^ H—' • i—1 • I—1 \ 0.51 K ^ (~) CD U P H a o 1 0.62 K A 0.74 K H i A Coi 1 0.3 0.35 0.4 Filling fraction v Figure 4.3: The compressibility as a function of filling fraction calculated for m* = 20mb and The y-axis has been scaled by the interaction strength, ^ (see text). At T ^ 0 some of the CF's will be thermally activated to the next Landau level. In this case the average energy is 00 n Eav(p,T) = Y nf(n + 1/2 - p,)Auc-^. (4.13) In the formulae given above there are implicit assumptions about the splitting of the lowest Landau level into sub-levels. In particular, for an unrenormalised effective mass it is consistent to assume that the energy of the lowest sublevel is 4^ • This picture breaks down when the effective mass increases. As has been discussed at length in previous Chapter 4. Oscillatory Quantities 87 sections, an increase of the effective mass is associated with a decrease in the gap. The chemical potential compensates for this by smoothly increasing while the topmost CF Landau level is being filled, resulting in the formula (4.11) when ne is held fixed. Likewise, the average energy receives an upward shift which reflects the fact that when there is a renormalised effective mass the CF Landau levels are bunched up closely near ^ and the energy of the lowest CF Landau level is somewhat larger than This is illustrated in Fig. 4.4. There is also an offset coming from the ^-factor of the electrons, which shifts Figure 4.4: A schematic illustration of the splitting of the lowest Landau level. When there are no interactions the lowest Landau is not split. The mean field theory of com-posite fermions splits the lowest Landau level into sublevels separated by Au>c. Gauge fluctuations renormalise the mass, so that in the end the lowest Landau is subdivided into levels separated by Aw*, but still centered around ^ . the lowest electronic Landau level down by 2 rr . Taking all of this into account, we plot the average energy as a function of B in Fig. 4.5. The offsets used guarantee that the crossover between the integer and fractional regimes is smooth. The crossover occurs at v = 17 which is considered to be part of both the integer series and the series v = T ^ J . This figure has been drawn in order to compare with the results of Kukushkin et al. [54] shown in Figure 4.6. An electron density ne = 2.4 x 101 1 cm~ 2 and g = 0.42 [94] have been used to scale the y-axis. In the fractional regime, an effective mass m* = 20m& has been used to achieve a reasonable agreement with the experimental results. This is in close agreement with the compressibility measurement at v = \ v although it should be noted that the compressibility measurement had about half the CO c y. coc/2 Chapter 4. Oscillatory Quantities 88 9 i - , , , 1 , , , , , , , , , 1 , r 5 10 15 Magnetic Field (T) Figure 4.5: The average energy as a function of magnetic field at T = 0. The scaling of the it/-axis has been chosen to match the experimental parameters of Kukushkin et al. (see text). electron density (ne = 1.1 x 10 ncm~ 2). It also agrees well with the SdH measurements of Du et al., which were performed at a similar density (n e = 2.25 x 10 ncm~ 2). 4.1.4 Measuring the Average Energy In a series of papers [52, 53, 54] Kukushkin et al. have reported measurements of magne-toluminescence spectra as a function of magnetic field. In these experiments luminescence Chapter 4. Oscillatory Quantities 89 gnetic field (T) F i g u r e 4.6: T h e e x p e r i m e n t a l results of K u k u s h k i n et a l . [54] s h o w i n g b) T h e average energy as a func t ion of magne t i c f ield at T = 0 a n d c) i t s der iva t ive . l ines are the resul t of the r e c o m b i n a t i o n of electrons w i t h holes b o u n d to remote accep-tors. T h e acceptors are i n a mono laye r w i t h dens i ty ~ 5 x 1 0 9 c m - 2 a p p r o x i m a t e l y 40 n m away f rom the 2 - D e lec t ron gas. T h e large dis tances ensures tha t the hole wave-funct ions do not over lap the e lec t ron gas w h i c h means that the electrons w i l l not be able to screen the holes. T h e p o s i t i o n of the luminescence l ine is the in t eg ra l over energy o f the i n t ens i t y o f the l ine . T h i s is re la ted to twice the average energy of the 2 - D e lec t ron gas [95]. A s expec ted , ou r resul ts d i sp l ay the same qua l i t a t i ve behav iou r as the e x p e r i m e n t a l resul ts . In the f r ac t iona l reg ime we see tha t the average energy increases l i n e a r l y w i t h the energy of the lowest L a n d a u level ( y (1 — gj) w i t h s m a l l pe r tu rba t ions a r i s i n g f r o m the f r ac t i ona l states. O n e i m p o r t a n t feature tha t has been o m i t t e d f rom these p lo t s i s the w e i g h t i n g of the L a n d a u levels . F r o m our d iscuss ions i n Sect ions 3.4 and 3.6 we k n o w that a large f r ac t ion o f the weight of the spec t r a l func t ion l ies benea th the the <5-function peaks tha t are the r eno rma l i s ed L a n d a u levels. T h i s w o u l d account for an ove ra l l decrease of the s lope of the average energy i n the f rac t iona l regime. N o t e tha t the e x p e r i m e n t a l resul ts d i s p l a y a s lope that is a l i t t l e less t h a n two t imes larger. A n accura te measurement of Chapter 4. Oscillatory Quantities 90 the slope would help to determine the distribution of the weight in the density of states. Fig. 4.7 shows the derivative of the average energy, of composite fermions (Fig. 4.5) 0.6 > \ H •0 -0.6 10 Magnet ic Field (T) 15 Figure 4.7: The derivative of the average energy with respect to the magnetic field as a function of magnetic field at T = 0. The discontinuities are related to the gap as explained in the text. with respect to the magnetic field. The discontinuities are related to the gap A in both fractional and integer regimes as 'dECF\ 2 A 6 (4.14) clB J Bv Note that in the fractional regime v = v ^ n - Considering real electrons instead of com-posite fermions, the discontinuity is a factor of p larger, in agreement with the formula Chapter 4. Oscillatory Quantities 91 de r ived by A p a l ' k o v and R a s h b a [95], 'dEel dB 2A where the filling is v = ^ . T a k i n g th is in to account , an effective mass o f m* chosen to m a t c h the re la t ive size of the d i s con t inu i t y at v = | a n d v — 1. (4.15) = 2§mb was 4.1.5 Thermodynamic Potential and its Derivatives T h e t h e r m o d y n a m i c p o t e n t i a l is yet another quan t i t y tha t can be eva lua ted u s ing the i d e a of non - in t e r ac t i ng compos i t e fermions w i t h r enorma l i sed effective mass . T h e express ion for th is q u a n t i t y at finite t empera tu re is (kB — 1) (n + l / 2 ) A w * + / n=0 1 + exp T V (4.16) where ^ = ^ is the L a n d a u level degeneracy. F r o m this we m a y der ive the t empera tu re dependent magne t i s a t i on , WB (4.17) -n„. — + l/2]nf([n + 1/2] Aw c * - p) rp OO Qo n = 0 1 + exp '-(n + l / 2 ) A c u * + p T (4.18) N o t e tha t the first t e r m domina t e s i n the low tempera tu re l i m i t . T h e c o m p r e s s i b i l i t y is d2n r (0 ) dp2 m*Au* 2-KT (4.19) $>/(-[ra + l / 2 ] A w c * + A.)n/ ( [n + 1 /2 ]Aw* - p). (4.20) L a t e r i n th is C h a p t e r we sha l l argue that E q . (4.16) does i n fact y i e l d the l e a d i n g order c o n t r i b u t i o n s to the o sc i l l a t o ry behav iou r of the t h e r m o d y n a m i c p o t e n t i a l , a n d consequent ly the re la ted t h e r m o d y n a m i c quant i t ies . Chapter 4. Oscillatory Quantities 92 In contrast to the approximation using free particles given above, the full thermody-namic potential is derived from the Hamiltonian which includes interactions with gauge fluctuations, n - n < ° > = - r i o g ( ~ ) (4.2i) Z = T r e x p ( ^ ) , T r e x p ( ^ < ^ ) . (4.22) The trace is taken over all configurations and numbers of particles. In the non-interacting case this reduces to [96] z<°>=n The thermodynamic potential is thus n<°> = - T £ i o g i = -TTrlogG'^(e J ,6)exp( a O + ) . (4.25) (4.16) follows from this formula for a system of free particles in a magnetic field AB with an effective mass m*. O ( 0 ) in the form (4.25) will be useful later on. The proof of (4.25) is given in Appendix C. 4.2 Gauge Interaction Effects In this Section we provide some justification for the approach of Section 4.1 by evaluating the effects of gauge fluctuations to second order in perturbation theory. We consider the special cases when the filling of CF Landau levels is integral and half integral. The results show that at these fillings the dominant effect of gauge fluctuations is to renormalise the effective mass. The chemical potential is examined in Section 4.2.1 followed by the compressibility in Section 4.2.2. 1 + exp T (4.23) 1 + exp T (4.24) Chapter 4. Oscillatory Quantities 93 4.2.1 Chemical Potential We now consider the effects of gauge fluctuations on the chemical potential, p satisfies the same criterion, (4.2), but in this case the density of states is no longer simply a set of (^ -functions; instead it has the form and in general must be evaluated at finite temperature. E(e) is the self-energy calculated in Chapter 3. We begin by computing the discontinuities in p at T = 0. The goal is to determine what range of p will satisfy (4.2) for p = [p, This is obviously true for all p that lie within the gap. When p. lies outside the gap both the limit of the integral and the form of the density of states change. These two effects enhance each other. For example, a lower value of p will cause the weight of the spectrum to be shifted upwards and the lower cutoff ensures that even less weight will be integrated. Thus at T — 0 the discontinuities in p will be the same as the gap. In between discontinuities we expect p. to increase smoothly and monotonically2. For T 7^ 0 there are corrections coming from the considerations described in Section 4.1.1 and also from the temperature dependence of the spectral function. In our calcula-tions we neglect the latter, so that the final result merely substitutes Au* for Aw c , with a smooth increase of p between the incompressible states. We have just shown that the T = 0 jump in the chemical potential corresponds to the gap as determined in Chapter 3. We now consider the effect of gauge fluctuations at finite temperature when the topmost CF Landau level is approximately half filled. In 2 This picture was described in Ref. [29]. (4.26) (4.27) Chapter 4. Oscillatory Quantities 94 th is case the c h e m i c a l p o t e n t i a l must satisfy /oo denf(e)g(e + p) (4.28) -oo 00 r°° E" (e ) = y / denAe)- v ^ , , s..—--—r (4.29) n ^ J - o c n '[e + b/jL- nAcoc + S ' ( e ) ] 2 + E " ( e ) 2 V ; where bp and bp have the same m e a n i n g as before. T h e self-energy looks s l i g h t l y different t h a n the f o r m cons idered w h e n s o l v i n g for the gap (compare F i g . 4.8 to F i g . 3.4). W h e n e ( A C J C ) F i g u r e 4.8: T h e self-energy w h e n the highest compos i t e f e rmion L a n d a u level is ha l f - f i l l ed at T = 0 (sol id) and for a finite t empera tu re w i t h a s m a l l , pos i t i ve shift o f the c h e m i c a l p o t e n t i a l (clashed), e is measured w i t h respect to the c h e m i c a l p o t e n t i a l . T h e v e r t i c a l scale is a r b i t r a r y (it depends on the c o u p l i n g A' 2 ) . the topmos t L a n d a u level is h a l f filled we expect there to be pa r t i c l e -ho le s y m m e t r y jus t as i n the case w h e n the topmos t L a n d a u level is c o m p l e t e l y filled a n d the c h e m i c a l p o t e n t i a l lies ha l fway between the filled and unf i l l ed levels . Therefore , at h a l f filling the s u m that appears i n the express ion for the self-energy mus t be t r unca t ed at [p. P a r t i c l e -hole s y m m e t r y w i l l not be conserved once the sys tem moves away f rom h a l f filling: th i s is t rue w h e n the cut-off at \jp is used. F o r a g iven value o f n there w i l l be one po le Chapter 4. Oscillatory Quantities 95 of ImCr n ( e ) near e = 0 w h i c h w i l l c a r ry some f rac t ion of the weight : the rest of the weight is d i v i d e d between the uppe r a n d lower par t s o f the spec t r a l f unc t i on . W h e n the spec t r a l funct ions of n and —n are cons idered together the r e m a i n i n g ' ' incoherent" weight is d i v i d e d a p p r o x i m a t e l y equa l ly between the uppe r a n d lower par t s o f the s u m of the two spec t ra l funct ions , since par t ic le -hole s y m m e t r y is a p p r o x i m a t e l y conserved. T h e spec t r a l f unc t i on for n = 0 w i l l have a pole close to e = 0 a n d we assume that i t s incoherent weight w i l l be d i s t r i b u t e d s y m m e t r i c a l l y between pos i t i ve a n d negat ive energies. F o r the purposes of th is Sec t ion , we consider the l i m i t of large l o g p , where we k n o w tha t an a p p r o x i m a t i o n that smears the L a n d a u level s t ruc ture of the self-energy is v a l i d for f i nd ing the r eno rma l i s ed gap (based on the d i scuss ion i n Sec t ion 3.4). In th is l i m i t , the r e n o r m a l i s e d pole pos i t ions are found by l i nea r i s i ng the self-energy i n the v i c i n i t y o f e = 0: therefore the new poles are evenly spaced ( w i t h spac ing that is A w * by def in i t ion) a n d they a l l have the same weight . A s before we cons ider the l i m i t T <C A w * . In th is l i m i t we m a y assume that the F e r m i func t i on is e i ther one or zero except at the L a n d a u level nearest to the F e r m i energy. N o w a l l tha t r emains is to c o m p u t e the pos i t ions of the pole nearest to the F e r m i energy a n d i ts weight . T o do th is we solve e + 6n-nAuc + T,'{e) = Q (4.30) as before, except tha t the self-energy must be eva lua ted for a more genera l c h e m i c a l p o t e n t i a l . T h i s is eas i ly de t e rmined f rom the resul ts de r ived i n C h a p t e r 3: K Lp S'(e) = — E \-nf{n - &p) log(e + 6p - n) + nf(8/j. - n) l og (n - e - Sp)}. (4.31) n=-[p T h i s is shown i n F i g . 4.8. Chapter 4. Oscillatory Quantities 96 Linearising Eq. (4.31) about e + 8p gives I<2 y (nf{m-6p)-nf{8p-m)\ + ^ 7T , \ 772 / •m=— [p \ / Clearly the pole ep occurs at ep = —8p with weight A'2 / n/(m — 8p) — iif(8p — m) m=— [p \ The spacing between the poles is given by K-2 LP 1 + E r(777 — ^/.t) — 1lf(6p — mY TV m=- [p m With all of these considerations Eq. (4.29) becomes [p + 8p = [p+—— + zpnf(ep) 1 — w + •v \ 2 4 T Using ep = —(5/./. we find that 4 T ( * p - 1/2) (4.32) (4.33) (4.34) (4.35) (4.36) (4.37) This is equivalent to the expression (4.6) (evaluated for 8p ~ 1/2), except for an overall scaling by a factor of zp. This is exactly what is expected, since the spacing between the poles, which is equivalent to the renormalised gap, is zpAuc = Aw*. 4.2.2 The Second Order Corrections to In Section 4.1.5 we wrote an expression for Q under the assumption that the only effect of interactions was to renormalise the effective mass. We now wish to determine the effect of interactions on the thermodynamic potential in a more rigorous fashion. It is a well known result from quantum field theory that the diagrammatic contributions to Eq. (4.21) consist of all closed, connected diagrams [97]. In this Section we examine the Chapter 4. Oscillatory Quantities 97 Figure 4.9: Feynman diagram representing the second order correction to the thermo-dynamic potential. The straight lines represent composite fermions and the wavy line represents a gauge fluctuation. second order corrections to the thermodynamic potential arising from interactions with gauge fluctuations, represented by the diagram shown in Fig. 4.9. We will rederive some of the results of K i m et al. [90] in a slightly different way, in order to demonstrate that the behaviour of the temperature dependent compressibility at a filled C F Landau level is consistent with the picture of free composite fermions with an effective mass. The second order correction to the thermodynamic potential is given in perturbation theory as fi<2> ~ T ^ G ' 0 ) ( H ) E 1 2 ) ( - 1 U b ) (4.38) i /•CO / denf(e){CrR(e)EA(e)-GA(e)y:R(e)} (4.39) . / — CO 1 r°° = — / denf(t)[GR(e)Z*R(e) - G* f l(e)E«(e)] (4.40) LTXl ./-co = - r dens(e)[CrR{e)y!R{e) - G ^ e ) ^ ) ] (4.41) 7T . / - c o = + (4-42) where G I I { E ) = T T T ^ T I ^ T M A ^ ^ ( 4 - 4 3 ) and 1R{e) is defined by Eqs. (3.25)-(3.27). Using bp = p — (\p + l/2)Au>c as before, ^ i 2 ) = '^E f d ^ ) G R ( e K { e ) (4.44) 1 /'°° 2-7ri ./-co Chapter 4. Oscillatory Quantities 98 = * A 0 S « ( n ' (4.45) P n 2 E nfin — 8p) log \m — n\[iif(5p — m) — iif(m — 8p)]. (4.46) 2TT2 n.m Next we evaluate the derivative to find the density of particles, which is proportional to p when B is held fixed. ri*> = (4.47) _ mI\2Auc y-v ^ | m _ n | j2n^.(^{ _ m)nf(m — 8p)rif(n — 6p) 2TI"T M N — rif(6p — m)'nf(n — 6p)iif(8p — n) +rtf(m — 8p)rif(n — 6p)rif(8p — n)]. (4.48) The compressibility is ,..(2) 2) dp mK-2Au2 ^ — 2 ^ 1 o g | m - n | (4.49) 2 7 T 2 T 2 x [2n (^6/.t — m)n2(m — 6p)rif(n — 8/,i) —4n,f(6p — m)iif(m — 6p)rif(n — 8p)iif(8p — n) —rif(6p — m)nj(n — 6p)rif(6p — n) +rif(8p. — m)nf(n — <5/i)n2(<5/i — n) +rif(m — 6p)nj(n — 6/.i)rif(6p — n) —iif(m — 6p)iif(n — 6p)nj(8p — n) —2n2f(6p — m)nf(m — 6p)nf(n — 8p)]. (4.50) Taking the usual l imit T <C A w c , we evaluate this expression at 8a = keeping only Chapter 4, Oscillatory Quantities 99 the d o m i n a n t te rms (a l l bu t the second), w h i c h go as exp( ^if'')/T2 2mK2Au2 7 T 2 T 2 2K2Acu2 exp 7r2T2 6 X P V 2T -Auc 2T -Acur E i o g m=0 m + 1 l o g p N e x t we evaluate the second t e rm i n (4.42). f ) f = ~Y j denf(e)GR(e)Z'R(e] -neK2Au>c nf(e) pn " ./ e — n + <5/i (4.51) (4.52) (4.53) x [n.f(6p — m)9(e + 6p — m) + n/(ra — 8p)9(m — e — <5/j)]. (4.54) S h i f t i n g e by 8p we f ind ,2) = -neK2Auc y r d n f ( e - 6p) _ m ) e ^ e _ m ) _ n / ( m _ fr.)fl(m-e)]. (4.55) T h e dens i ty o f par t i c les is Pb — dp mK) Aco2 (4.56) /de ——\nf{e - 8p)nf(8p - m)nf(m - 8p)[9{e - m) - 9(m - e)] - - 1,1,11 E N nf(e - 8p)nf(8p - e){nf(6p - m)9(m - e) + nf(m - 8p)9(m - e)]]. (4.57) T h e c o m p r e s s i b i l i t y is ..(2) _ dp mK->Auj2 2TT2T2 " 7 e - n [—2n/-(e — 8p)iif(8p — e)rif(8p — m)rif(m — 8p)[9(e — m) — 9(m — e)] + [nf(e - 8p)n2f(8p - e) nf{6 6p)n}(6p - e)\ Chapter 4. Oscillatory Quantities 100 x[nf(6p — m)9(e — m) + iif(m — 6p)9(m — e)\ + [n/(e — Si.t)[nj(m — 6p)nf(6p — m) — iif(m — 6p)n2f(6p — m)] x[9(e - m) - 9(m - e)}}}. (4.58) O n c e aga in , t a k i n g the l i m i t T « A w c a n d eva lua t ing at 6p = ,..('2) mH^Ato2 ^ 1 /' de 2r.2T2 ^ , ./ 7^7i [—2 exp m=-[p - A w , ~2T~ n f(e - 1 / 2 ) ^ ( 1 / 2 - e)[9(e) - 9(-e) + 9(e - 1) - 9(1 - e) + exp ~ A U c ' (nf(e - 1 / 2 ) ^ ( 1 / 2 - e) - n)(e - 1 / 2 ) ^ ( 1 / 2 - e))[9(e) + 9(1 - e)] + (nf(e - 1/2)4(1/2 - e) - n2(e - 1 /2 )72 / (1/2 - e))[0(e - 1) + 9(-e) A w , + exp 2 T nf(e - 1/2)[0(1 - e) - 0(e - 1) + 0(e) - 0 ( - e ) ] ] . (4.59) T h e r e are four terms i n th is express ion. T h e first and second terms w i l l g a in a d d i t i o n a l e x p o n e n t i a l l y s m a l l factors u p o n in teg ra t ion . T h e four th t e r m survives the i n t eg ra t i on , y i e l d i n g ~ l og n - l / 2 w h i c h vanishes u p o n s u m m a t i o n i n the l i m i t o f large p. O n l y the the t h i r d t e r m ( w h i c h has no exponen t i a l l y s m a l l factor) r emains . T h e largest c o n t r i b u t i o n s to the in t eg ra l come f rom 6 = 0 and e = 1, so the l e ad ing order c o n t r i b u t i o n to KB2) is K mK'zAto2 2n2T2 mh^Au)2 2 v r 2 T 2 exp exp A ? A w 2 exp - A w c 2 T - A w f 2 T A w c b - i E ^ g n = -[p [p-1 n ir2T2 ~"L~ V 2T T h e t o t a l c o m p r e s s i b i l i t y to second order is r ( 2 ) —mluAco2 E i o g ?i=0 l og p . 1 - 1 1 71+ 1 11 - 1 7 T 2 T 2 exp - A w £ 2 T log p . (4.60) (4.61) (4.62) (4.63) (4.64) Chapter 4. Oscillatory Quantities 1 0 1 T h i s is added to the free pa r t i c l e result g iven by E q . ( 4 . 2 0 ) , except tha t here we use AUJC i n s t ead o f A u * . T h e low tempera tu re l i m i t of th is express ion eva lua ted for a c h e m i c a l p o t e n t i a l h a l f way between L a n d a u levels is A w , = — — £ exp . _ T T T 1 V 2 T E q . ( 4 . 6 4 ) gives the correc t ions to th is , so the c o m p r e s s i b i l i t y to second order is ( 4 . 6 5 ) K = mAuc TTT mAtUr exp exp Acoc 2T • A w , K2Auc 1 l o g p 2 A ' 2 , 1 H l o g p 7T ( 4 . 6 6 ) ( 4 . 6 7 ) TTT 1 V 2 T K i m et a l . [90] argued that th is result is consistent w i t h par t i c les h a v i n g an effective mass m m 1 H l o g p 7T 2 A ' 2 7T l o g p ( 4 . 6 8 ) ( 4 . 6 9 ) i n the l i m i t of large p. 4.3 The Oscillatory Behaviour of SI In th i s Sec t i on we examine the osc i l l a t o ry behav iour of the t h e r m o d y n a m i c p o t e n t i a l . W e seek a non-pe r tu rba t ive fo rm, such as that d iscovered by L u t t i n g e r , w h i c h m a y be a p p l i e d to the two d i m e n s i o n a l p r o b l e m o f compos i t e fermions i n t e r a c t i n g w i t h gauge f luc tua t ions . It is found that L u t t i n g e r ' s fo rm, w h i c h is an e x p a n s i o n tha t neglects the o s c i l l a t o r y par t of the self-energy, is not v a l i d i n two d i m e n s i o n s for any k i n d o f i n t e r a c t i o n . Ins tead of th is , we have found a different f o r m w h i c h inc ludes the o s c i l l a t o r y par t of the self-energy and is v a l i d w h e n a l l "crossed" d i a g r a m s are neglected . Chapter 4. Oscillatory Quantities 102 In Sec t i on 4.3.1 we examine the osc i l l a to ry con t r i bu t i ons to the second order pe r tu r -ba t ive co r rec t ion to ft fo l lowed by i n Sec t ion 4.3.2 an ana lys i s of the n o n - p e r t u r b a t i v e express ion for ft. 4.3.1 The Oscillatory Behaviour of ft^2) In any d i a g r a m c o n t r i b u t i n g to the t h e r m o d y n a m i c p o t e n t i a l the o s c i l l a t o r y c o m p o n e n t arises t h r o u g h the s u m over L a n d a u levels associa ted w i t h each f e rmion l ine . I n genera l the c o n t r i b u t i o n f rom each f e rmion l ine m a y be sp l i t in to o sc i l l a t o ry a n d n o n - o s c i l l a t o r y par t s . T h e non-osc i l l a to ry par t is o b t a i n e d by conve r t ing the s u m over L a n d a u levels i n to an in teg ra l . T h i s ope ra t i on has a l ready been discussed i n Sec t ion 3.4, where i t was e m p l o y e d to give an a p p r o x i m a t i o n of the self-energy by essent ia l ly s m o o t h i n g out the s ingu la r s t ruc ture associa ted w i t h the L a n d a u levels . T h e r e m a i n i n g par t m a y be expressed as a Fou r i e r series by us ing the Po i s son s u m f o r m u l a , / 2irkx ^ / ' O O E f(nuc) = / dxf(x l + 2 £ ( - l cos (4.70) B y e x a m i n i n g the second order c o n t r i b u t i o n to ft, Enge l sbe rg a n d S i m p s o n [40] showed that the l e a d i n g osc i l l a to ry behav iou r of ft(2) c o u l d be de t e rmined by neg lec t ing the os-c i l l a t o r y par t of one of the f e rmion l ines . However the i r resul ts h o l d i n three d im e ns i ons o n l y ; we show be low that thei r p roo f does not a p p l y i n two d imens ions . In three d i m e n -sions there is an ex t r a degree of f reedom, the m o m e n t u m i n the .^-direct ion, w h i c h tends to smear the dens i ty of states. In two d imens ions the magne t i c f ie ld quenches a l l of the degrees of f reedom and the dens i ty of states has a very s ingu la r s t ruc ture , w h i c h i n a genera l t rea tment m a y not be "smeared ou t " . In three d imens ions , the second order co r rec t ion to ft is tiw = E E G ^ i i u ^ k ^ ^ k ' ^ D ^ - iuni,fi (4.71) Chapter 4. Oscillatory Quantities 103 where D(iun) is any general interaction. We change variables by defining nuc = 2m ' e = ^ g l - A i and cos20 = 3- Using (4.70) we get n(2) - Q( 2) I Q(2) + p(2) (4.72) where (2) _ 0.3/P — r-x2 E E / ° ° d e d e ' / ' d(cos0)d(cos'0')(e + p) 1 / 2(e' + A0 1 / 2 x G ( 0 ) ( i w n l , e)G (0)(?;wn2, e')D(io» n 2 - iu>ni, <?) /•oo oo <),2; /; = 2T 2A 2 E E / rferie' / d(cos0)d(cos0')(e + /Li)1 / 2(e' + A 0 1 / 2 E ( W„l,W,l2 <f fc=l 0)j G ( 0 ) ( i a ; n i , e ) G ( 0 ) ( ^ n 2 , € X ^ n 2 - iwni,q) /' OO /' 1 ®® % D = 2T 2A 2 E E / rferfe7 <cos0)d(cos0')(e + / . O 1 / 2 ( e ' + A 0 1 / 2 E ( „ ! _ - , „ , . . n X • ' - M - ' " I , , ' (4.73) k C 2irk. (e + /.t)(l — cos i WC ) •1 (4.74) C 27rfc. w 9 , , , \ /2 /Tfc , . 9 » (e + /Li)(l — cos" 0) cos (e + p)(l — cos" 0 Uc J \ LOc xG(0)(tunl, e)G ( 0 ,(iw n 2 , e')D(iun2 - iuni, tf) T h i s is shown d i a g r a m m a t i c a l l y i n F i g . 4.10. (4.75) Figure 4.10: The zeroth, first and second order oscillatory contributions to f>(2). The wiggly cuts indicate that the oscillatory component of the fermion line is to be evaluated. The first term of this expression gives the non-oscillatory contribution to 17(2). The second term gives the oscillatory contribution arising from the Landau level quantisation 3 This is described in detail in [40]. Chapter 4. Oscillatory Quantities 104 o n one of the f e rmion l ines . T h e t h i r d t e rm takes the o sc i l l a t o ry c o n t r i b u t i o n f r o m b o t h l ines . E n g e l s b e r g and S i m p s o n argued that the cos(9 in tegra ls c o u l d be ex tended to ± o o (the s t a t i o n a r y phase a p p r o x i m a t i o n ) since the largest c o n t r i b u t i o n s to the e- integral come f rom the v i c i n i t y e ~ 0 (near the F e r m i surface). T h e n i n t eg ra t i ng over cos 9 y i e ld s a factor ^ i n the second t e r m and (jf) i n the t h i r d , l e ad ing to the c o n c l u s i o n that / \1/'2 the t h i r d t e r m m a y be neglected because i t is of order (^ -J w i t h respect to the second. In the 2 - D case the cos 9 in tegrals are absent and so is the J a c o b i a n factor (e + pfl\e + A0 1 / 2: f22~D = ^ 0 , 2 O ^i,2£> "I" ^2:2D (4.76) where ^0% = ^ E E rded€G^%unl,€)G^%un2,e')D(iunl-iun2,q) (4.77) xG{0)(icunl, e)G(i)\tun2, e')D(iunl - iun2, q) (4.78) xGi0\iunl,e)Gw(iun2, e')D(iunl - iun2, q) (4.79) Therefore i t is obv ious tha t the arguments of Enge l sbe rg a n d S i m p s o n are not a p p l i c a b l e i n th i s case. T h e second a n d t h i r d te rms of the o sc i l l a t o ry expans ion are eva lua ted i n A p p e n d i x E for the 2 - D compos i t e f e rmion sys tem where i t is shown that the t h i r d t e r m is the same order i n ^ as the second. L u t t i n g e r de r ived a m u c h more general , non-pe r tu rba t ive f o r m for Q by s h o w i n g tha t / \ 3/2 the o s c i l l a t o r y par t of the self-energy is a factor (^J sma l l e r t h a n the n o n - o s c i l l a t o r y par t [39]. However this p r o o f also depends c r u c i a l l y o n the presence o f a t h i r d d i m e n s i o n , as we discuss i n the f o l l o w i n g Sec t ion . Chapter 4. Oscillatory Quantities 105 4.3.2 Non-Perturbative evaluation of ii The essentially non-perturbative nature of the composite fermion gauge theory neces-sitates a different approach to computing ii. It is useful to rewrite ii in terms of the full composite fermion Green's function and the proper self-energy, which satisfy the self-consistency relation G _ i = G ( 0 j - i + s _ ( 4 _ 8 0 ^ Note that the proper self-energy has contributions from any closed connected graph that cannot be cut in two when a single fermion line is cut. Following Luttinger and Ward [38], we write the thermodynamic potential as fi = fi = fi0-TETr[log[l + G^(e , )E ( e J )]-E(6 ! )G ( e ; )]+n' (4.81) i = -TTr[logG-1(€,-)-S(e,-)G(€ t-)] + fi' (4-82) where the second line is derived from the first line using (4.25) and (4.80). ii' consists of all closed connected skeleton diagrams (diagrams with no self-energy insertions) with the bare Green's function G ( 0 ) replaced by the full one G. The proof that fj = ii is complicated and the forthcoming arguments depend on this result, therefore we have provided the details in Appendix C. It is also shown in Appendix C that ii is stationary with respect to S, « - a ( . 83 , Luttinger used this to expand ii in powers of the oscillatory part of the self-energy E 0 5 C , H = -TTrf logCV^) - S0(et-)Go(e,-)] + ft'(S0) + 0(Z20SC) (4.84) where S = S 0 + E 0 5 C (4.85) Chapter 4. Oscillatory Quantities 106 and G0-1(e) = e - ( 7 i + l / 2 K + S 0 (4.86) It was shown by Luttinger [39] that there is a cancellation of the second and third terms in (4.84). This is reviewed in Appendix C. In 3-D Lifshitz and Kosevich [98] showed (ignoring dynamic interactions) that / \ 5/2 TTrflogG" 1] ~ (4.87) and Luttinger [39] showed that / x 3/2 losc ~ ( ^ j S 0 . (4.88) Therefore to O ^(^) ^ ^ the last term in (4.84) may be neglected and the leading oscil-latory behaviour of f2 is 9.OSC = - T 53 Tr[log GJ"1 (e,-)] (4.89) = -TX)log[e f--(n + l/2)o;c + /L. + So(el-)] • (4.90) i.n This is the standard expression used to evaluate the magnetisation [91, 99]. However, in 2-D we find that TTrflogG- 1] ~ M l (4.91; and So, c ~ (jj S 0 . (4.92) This is shown in Appendix F. This is very similar to the results of Section 4.4.3, since / \ l / 2 in both cases the factor (^ -J is lost by the omission of the stationary phase integral coming from the third direction. Therefore, in 2-D Eq. (4.84) is no longer a good expansion for 17 since the fourth term is of the same order as the first, so we must Chapter 4. Oscillatory Quantities 107 reconsider E q . (4.82). In general , the last two terms of E q . (4.82), E G + fi', cance l o n l y up to second order i n the c o u p l i n g constant ( 0 ( A 2 ) ) . However , i f the self-energy sa t i s fy ing S = A 2 j GD (4.93) (i.e., i t is the self-consistent s u m over r a i n b o w graphs) is used, then E G = \2GDG cancels w i t h the t e r m i n fi' o r i g i n a t i n g f rom the 0 ( A 2 ) skele ton g r a p h (see F i g . 4.9). T h i s t e r m is s i m p l y A 2 G ( ( ) ) . D G ( 0 ) , w i t h G^ replaced by G . T h e n the o n l y par t s r e m a i n i n g i n fi' o r ig ina te f rom higher order skele ton graphs, w h i c h i n general have "crossed" i n t e r a c t i o n l ines , such as the one shown i n F i g . C . l 1)). Such con t r i bu t i ons are i m p o r t a n t for ve r tex r e n o r m a l i s a t i o n , but we have a lways ignored t hem. Therefore , i t is consis tent w i t h ou r c a l c u l a t i o n s o f C h a p t e r 3 to t r y as a first approach the f o l l o w i n g express ion for fi: fi = - r ^ l o g h - (n + 1 / 2 K + n + S ( e , : ) ] - (4.94) i,n P where the self-consistent self-energy is used. I n c l u d i n g the crossed graphs leads to terms that are of 0 ( A 4 ) or greater . These c o n t r i b u t i o n s w i l l i n general not be classif ied i n powers of ^ , ins tead the i r o rder is d e t e r m i n e d by the scale of the in te rac t ions . Fo r example , i n the case o f Debye phonons the expans ion pa ramete r is [99], w h i l e for s p i n f luc tua t ions the f l u c t u a t i o n energy de te rmines the relevant scale [91, 100]. E q . (4.94) is used to evaluate the m a g n e t i s a t i o n by f o l l o w i n g mos t of the steps i n the d e r i v a t i o n g iven i n [91] a n d [99]. W e beg in by r e w r i t i n g (4.94) as a con tou r in t eg ra l : fi = — - — Y / —rif(x) t a n - 1 — — 4.95 $o * \x-(n + l/2)Au;c + E'(x)J V N e x t we use the Po i s son s u m formula. (4.70) and change var iab les by de f in ing e = nAuc — fi = — / de —nf(x)t&n~l v , Chapter 4. Oscillatory Quantities 108 (4.96) E x t e n d i n g the Uni t s on e to ± o o we integrate by par t s and neglect the end-po in t con t r i -bu t i ons to get -mAuc r°° fdx l"(x) l-Jeicinf{x)] Q O S C 7 T $ 0 - / - o o ^ J 7T " / V ~ Y [x - e + S ' ( . T ) ] 2 + £ " ( * ) 2 x E ^ s i n f ^ i ± i O V ( 4 . 9 7 ) t i k \ Auc ) y T h i s express ion is the same as w h a t w o u l d be found for the n o n - i n t e r a c t i n g case, except tha t the fu l l spec t r a l func t ion replaces the free one. In Sec t ion 3.4 we discussed the f o r m of the spec t ra l func t ion for the case w h e n the self-energy is d e t e r m i n e d to second order a n d i n Sec t ion 3.6 we discussed the case w h e n the self-energy is d e t e r m i n e d self-cons is tent ly . In b o t h cases we showed that i n the v i c i n i t y of the F e r m i surface w h e n the sys t em is gapped the dens i ty of states consists of a series o f weigh ted ^-funct ions w h i c h are the r eno rma l i s ed L a n d a u levels. T h e o sc i l l a t o ry behav iou r o r ig ina tes f r o m the par t s o f the in t eg ra l close to the F e r m i surface: i n essence th is is the reason w h y the d e s c r i p t i o n u s ing free par t i c les w i t h a f ield-dependent r eno rma l i s ed mass works . P e r f o r m i n g the e in tegra l i n E q . (4.97) y ie lds -mAur fdx , N ~ f - l ) f c . (2ixk k=i T h i s express ion has sl ight differences c o m p a r e d to the one de r ived by E n g e l s b e r g a n d S i m p s o n because i t has been done for two d imens ions ins tead o f three. T h e mos t i m p o r -tant difference is that the fu l l o sc i l l a t o ry self-energy appears i n the s p e c t r u m x + S ( . r ) . T h e m a g n e t i s a t i o n is Mosc = (4.99) n o s c = / -nf(x) £ s m ( _ ( * + E ( , ) + (4-98) 1 rdx ~ $ 0 - / vr } t i (-l)k . (2-Kk. , ' Chapter 4. Oscillatory Quantities 109 A w r • cos ( M ( . , + S ( . ) + , ) ; • l ) f c 2 7 T m 9 E ( . r ) / 27rfc 7 di? cos A w , (a: + S ( . r ) + A0 (4.100) T h e second t e r m domina t e s the first by a factor It is the l e a d i n g o s c i l l a t o r y con -t r i b u t i o n to the m a g n e t i s a t i o n der ived by E n g e l s b e r g a n d S i m p s o n . In the 2 - D case the t h i r d t e r m is of the same order as the second. T o show this , we cons ider the rea l pa r t o f the self-energy for compos i t e fermions i n t e r ac t i ng w i t h gauge f luc tua t ions , bu t the resul t is genera l . \x) = K->Au)r IT £ i o g 71 = 0 (n + 1/2) Auc — p — x A w , [nf(p - (n + 1 / 2 ) A w c ) - nf((n + 1 / 2 ) A w c : ^ cos I fc' = 1 \ A0] (4.101) dz'(x) dB Hit de 1 + 7T ./ x l og e - x A w c : / de-* t =i x l o g e - x A w c Aur •2Trk'(e + p) . {2irk'(e + py sin 1 (4.102) m ( A w c ) 2 A . OJr. - e ) - n / ( e ) ] . (4.103) P u t t i n g th i s resul t i n E q . (4.100) we see that the t e r m w i t h ^ is the same order as the second t e r m i n E q . (4.100). If there h a d been a t h i r d d i m e n s i o n then the t h i r d t e r m w o u l d have been suppressed by a factor ( ^ f ^ ) ^ " - T h i s resul t shows tha t the c o n v e n t i o n a l t r ea tment of i n t e r ac t ion effects o n d H v A osc i l l a t ions must be m o d i f i e d w h e n a p p l i e d i n two d imens ions a n d terms i n v o l v i n g the o sc i l l a t o ry self-energy s h o u l d be r e t a ined i n any pe r tu rba t i ve c a l c u l a t i o n . Chapter 4. Oscillatory Quantities 110 4.4 Summary In this Chapter we have studied the thermodynamic quantities of composite fermions interacting with gauge fluctuations which oscillate as a function of Ai?. The chemical potential and related quantities, namely the compressibility and average energy, have been analysed using non-interacting composite fermions with a field dependent effec-tive mass and compared to experiment. It was found that this approximation provides a reasonable description of the experiments, and that the effective masses obtained by analysing the compressibility, free energy and SdH measurements are in rough agree-ment. Explicit second order corrections coming from gauge fluctuations have also been calculated for the compressibility and chemical potential and were found to agree with the approximation using non-interacting particles with an effective mass. . We have also examined the oscillatory behaviour of the thermodynamic potential for interacting systems in two dimensions. Considering both perturbative and non-perturbative forms for fi, we have shown explicitly that an approximation which neglects the oscillatory part of the self-energy is valid in three dimensions but not in two, which means that the conventional Luttinger-Ward treatment of quantum oscillations is not ap-plicable in two dimensions. Instead, we have proposed an alternative, non-perturbative form for fi which neglects crossed graphs. This form contains the self-consistent, oscilla-tory self-energy discussed in Chapter 3. Chapter 5 Conclusion T h i s thesis s tudies two mode l s of the f rac t iona l q u a n t u m H a l l effect, n a m e l y the boson ic ( C h e r n - S i m o n s - L a n d a u - G i n z b u r g ) and the fe rmion ic ( compos i t e f e rmion gauge theory) desc r ip t ions . T h e boson ic theory a t t empts to descr ibe the states at v — ^ - j - u s i n g an order pa ramete r and a [ ^ l 4 po in t i n t e rac t ion . U s i n g this theory we have found n u m e r i c a l f r a c t i o n a l l y charged vor tex a n d an t i -vor tex profiles for v = a n d \ . T h e size a n d energy of each vor t ex m a y be ex t rac ted f rom these results . W e descr ibe the role tha t these parameters p l ay i n the b r eakdown of of the u n i f o r m H a l l s tate w i t h i n th is theory. O u r resul ts are c o m p a r e d w i t h results ob t a ined f rom other n u m e r i c a l me thods . In contras t to the bosonic theory, the fe rmion ic theory s tr ives to descr ibe the f r ac t i ona l q u a n t u m H a l l states at all filling f ract ions as w e l l as the states i n between. T h e m e a n field theory, w h i c h describes the states at v = ^ as F e r m i l i q u i d s i n zero m a g n e t i c field, is r e a d i l y ex tended to an i n t e r ac t i ng gauge theory. Lowes t order pe r tu rba t i ve c a l c u l a t i o n s o f the self-energy a r i s ing f rom in te rac t ions w i t h the gauge field show that the r e n o r m a l i s e d f e r m i o n Green ' s func t ion has a d i s t i n c t l y different a n a l y t i c s t ruc ture c o m p a r e d to i t s m e a n field ( F e r m i l i q u i d ) fo rm, w h i c h is charac ter i sed by a d i v e r g i n g effective mass . N e a r v = ^ the in t e rac t ing fermions exist i n a field AB = B — By/-2 a n d undergo osc i l l a t i ons as a func t ion o f th is f ie ld. W e compu te the self-energy of c o m p o s i t e fe rmions w h i l e r e t a i n i n g L a n d a u level quan t i s a t i on of a l l i n t e rna l fermions l ines . T h e gap m a y be ex t r ac t ed f rom this quan t i ty : i n lowest order p e r t u r b a t i o n theory we find agreement w i t h p rev ious ca l cu l a t i ons o f the self-energy [29] (wh ich have ignored th is s t ruc tu re ) i n 111 Chapter 5. Conclusion 112 the l i m i t v —> \ where one finds a divergence of the effective c y c l o t r o n mass . Howeve r our resul ts u s ing L a n d a u level quan t i s a t i on give rise to different p red ic t ions for the gap for f i l l i n g factors away f rom v = 1/2. W e also c la r i fy the s t ruc ture of the self-energy a n d Green ' s func t ion by c a l c u l a t i n g the analogous quant i t ies for the e l ec t ron -phonon sys t em i n two d imens ions . W e then go b e y o n d p e r t u r b a t i o n theory by means of a new i t e ra t ive p rocedure w h i c h compu te s the self-energy self-consis tent ly ( s u m m i n g over a l l r a i n b o w graphs) . W e c a l -cu la te the effective mass r e su l t ing f rom this procedure for a range of p h y s i c a l l y relevant parameters i n the l i m i t of large p (v = -r^). In th is l i m i t ou r resul ts canno t be d i r e c t l y c o m p a r e d to ex i s t i ng e x p e r i m e n t a l da t a . However , three exper imen t s do observe an en-hancement of the effective mass as v —> 1/2 bu t the enhancement is m u c h la rger t h a n p red i c t ed a n d i n one case an enhancement is observed w h i c h is not p r ed i c t ed . W i t h i n the f e rmion ic theory, we have also e x a m i n e d the o s c i l l a t o r y b e h a v i o u r of the t h e r m o d y n a m i c p o t e n t i a l a n d re la ted quant i t ies . O u r first obse rva t ion is tha t a theory of free c o m p o s i t e fermions w i t h an effective mass (a r i s ing f rom in te rac t ions ) gives a g o o d d e s c r i p t i o n of the e x p e r i m e n t a l l y observed compres s ib i l i t y and free energy. L e a d i n g order cor rec t ions to the u n p e r t u r b e d theory ind ica te tha t the p r i n c i p a l effect of in t e rac t ions is to r enormal i se the effective mass. T h i s is a ref lect ion of the fact tha t the r e n o r m a l i s e d Green ' s func t ion m a i n t a i n s the same low energy s t ruc ture as the u n r e n o r m a l i s e d one. F i n a l l y we demons t ra te the cons is tency of our entire app roach for t r e a t i ng q u a n t u m osc i l l a t i ons i n two d i m e n s i o n a l i n t e r ac t i ng systems, by e x a m i n i n g the gauge inva r i an t t h e r m o d y n a m i c p o t e n t i a l ft. W e f ind that w e l l - k n o w n arguments for three d i m e n s i o n -a l sys tems, w h i c h ind ica t e tha t the l ead ing o sc i l l a t o ry behav iou r comes f r o m the s u m over L a n d a u levels of a s ingle f e rmion l ine i n a F e y n m a n d i a g r a m c o n t r i b u t i n g to the t h e r m o d y n a m i c p o t e n t i a l do not a p p l y i n two d imens ions . T h i s i m p l i e s tha t i n second order p e r t u r b a t i o n theory i t is necessary to keep e x p l i c i t sums over L a n d a u levels i n both Chapter 5. Conclusion 113 f e r m i o n l ines w h e n c o m p u t i n g fi^2). T h i s is consistent w i t h our c a l c u l a t i o n o f the second order self-energy of compos i t e fermions . W e then go o n to examine the general s t ruc ture to a l l orders i n p e r t u r b a t i o n the-ory of the o s c i l l a t o r y t h e r m o d y n a m i c p o t e n t i a l . O u r resul ts show tha t the s t a n d a r d non -pe r tu rba t ive expans ion of the t h e r m o d y n a m i c p o t e n t i a l used to evaluate d H v A a m -p l i tudes is not v a l i d i n two d imens ions ei ther . However , we have found an a l t e rna t ive f o r m w h i c h is v a l i d w h e n crossed d i ag rams are neglected. T h i s f o r m con ta ins the same o s c i l l a t o r y self-consistent self-energy, discussed above, thereby j u s t i f y i n g our p rev ious w o r k o n the self-consistent compos i t e f e rmion gap. T h i s f o r m is v a l i d for the d iscus-s ion of any two d i m e n s i o n a l sys tem i n w h i c h osc i l l a t ions i n the crossed graphs do no t con t r ibu te i n l e ad ing order to the q u a n t u m osc i l l a t ions . W e a p p l y th is resul t to a c a l c u -l a t i o n of the magne t i s a t i on , where i t is shown e x p l i c i t l y tha t the conven t iona l t r ea tmen t of th i s q u a n t i t y (wh ich neglects the o sc i l l a to ry par t of the self-energy) is not v a l i d i n two d imens ions . Chapter 5. Conclusion 114 Symbols a.(x) statistical gauge field (due to attached flux) A(x) external gauge field b statistical magnetic field (due to attached flux) B external magnetic field A gap AB effective magnetic field (AB - B - b) Acuc cyclotron frequency (Auc — ^ r ) Au* effective cyclotron frequency (Aw* = ) e dielectric constant (e = 13 for GaAs) g spin splitting factor (g K 0.42 for GaAs) K-2,KS,KD,K£, coupling constants (defined by Eqs. 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T h e u s u a l forms of the e lec t ron-phonon in te rac t ions for b o t h Debye a n d E i n s t e i n m o d e l s m a y be used. F o r the Debye m o d e l , the e lec t ron-phonon i n t e r a c t i o n has the f o r m Hint = 3D f dh^(r)?pe(r)V • u(r) ( A . l ) where 3D is the de fo rma t ion p o t e n t i a l , u{r) = Y ^ (afe** + ate-**) ( A . 2 ) ' \ l 'MTpV is the d i sp lacement opera tor , w i t h i o n mass dens i ty p, ce l l v o l u m e V = a 3 a n d d i s p e r s i o n u>f = csq: a n d ipe is the e lec t ron wavefunc t ion expressed i n a L a n d a u leve l basis , Mn = XX* {^jf ( ^ r ) 1 / 4 * » ( « / i - « ) « P ( - M ! ^ + ,:;=:,:) [«(,)]'/>. (A.3) In these expressions cuc - eB/m is the c y c l o t r o n frequency, m is the e lec t ron mass a n d / = (Ti/eB)1/'2 is the magne t i c l eng th . T h i s leads to Hint — 3D Y \ n ,n ,k,k.' ,q \ T l - ^ t N t 2ufpV 7e • q\a^ + aJ-)clkcn<k,A(n, n , k, k', q) (A.4) 122 Appendix A. Phonon Interactions in Finite Magnetic Field 123 where [41] \(n,n\k,k\q) = ( ^ r ^ T j ) ^ ( ^ ) ^ y j d\Hn(y/l - kl)Hn,(y/l - k'l)8(z) ( (yll-klf (y/l-k'l)2 . , ',, , . \ / A _N X exp I '- - ^ — ^ + tqyy + i{~k + k + & ) a ; + iqzz J ( A . b ) S ( - f c + fc'-gg) / Q2l*. n-Li'\A (2n'nl\1/2 „ 2 ? 2 * x ( = * ^ J i l l ' - ' (=§^j ( A . 6 ) for n < n a n d g 2 = q2 + g 2 . T h e var iables A;' are of no concern as far as the p roper t i e s o f the e lec t ron Green ' s funct ions are concerned because the energy o f the n o n - i n t e r a c t i n g e lect rons does not depend k a n d the final fo rm of H{nt does not depend o n k, k or qz. F u r t h e r m o r e , i n w h a t fol lows we assume that the p h o n o n opera to r is i s o t r o p i c w i t h respect to the x-y p lane . In the l i m i t o f large n « n a n d ql ^> (n — n)/n the m a t r i x element | A | 2 reduces to | A ( n , g ) | 2 = -r-f) (A.7) nq \ 2nJ 1/2 irq \ 2p J T h e final f o r m o f Hint is (A.8) = Hyj £ V^v^ + A - ? ) 4 c n ' A K n ' ,q). (A.9) n,n ,q S T h e self-energy is der ived f rom the second order c o n t r i b u t i o n o f Hint \ 2c.pa I , , J (2TT)Z J0 7T n = 0 T + n f l (u; ) - n / ( [ n + | ]w c ) | nB(u) + n f ( [ n + \)u)c)\ e — to — (n + \ yjJc + iS e + u — (n + \ )LOc + id J Appendix A. Phonon Interactions in Finite Magnetic Field 124 where the factor 1/a comes from the ^-integral. We now shift the energies by an amount puc which is the chemical potential when it lies halfway between Landau levels. The index is shifted by p. Performing the integrals at T = 0 using ImD(q,co) = yields the result (3.49) and (3.50). When the self-energy and the energy e are expressed in units of the Fermi energy, Ep = pcoc the dimensionless coefficient is KD = 3tlEf = • 3f^m . (A.ll) 4 v / 2 p 7 r c f p a 2uDy/4iTneaM Using typical values for GaAs, ie., uD = (6ir)1/3cs/a = 345K, 3D = -4.4eV, m = .07me, M « I0~°me, a = 3.6.4, ne = 2.25 x 1 0 n c m - 2 and EF « 1.5 x 10"3eV, one has KD « 0.23. Equation (A. l l ) shows that there is no magnetic field dependence in the coefficient. The relation between KD, which is given by (3.48), and-A'p is KD = KD/P- The difference arises because KD is used when the self-energy and the energy e are explicitly given in units of OJC, which is appropriate for solving (3.51). A.2 Einstein Phonons Following the approach of Engelsberg and Schreiffer we choose a coupling that is inde-pendent of q Hint = fj £ K + aL,)4Cn'A(n: U', (l)• (A-12) n,n ,q This leads to a self-energy S„.(e) = fj2J2\A(n,n',q)\2a2 j j ^-lmD(q,u) n x (^ + nB{uj)-nf([n + l2}ujc) | nB[u) + nf{[n + \)uc)\ ^ \ e - u - (n + \)ioc + id e + u - (n + ±)uc +iS J where we have assumed that the normalization of the (/-integral is ( 2 7 r ) 2 / a 2 . Evaluating at T = 0 and using ImD(q,u) = yields Eqs. (3.55) and (3.56). With S and the Appendix A. Phonon Interactions in Finite Magnetic Field energy e expressed in units of Ep, the dimensionless coefficient is </2(2pW2a g2aUTmefl2 A K = ' - ' 4Ep 4EF In this form there is no dependence of the coupling on the magnetic field. Appendix B Self-Energy Corrections in a Finite Magnetic Field The original theory of strongly interacting electrons in a magnetic field takes the form c = 4(x)(-idt)Mx) + 4 ( ^ ) { l d l T 2 i e n A l ) Z U x ) + f d\)4(x)nje(x)V{x - y)4{y)Uy) (B. i ) where B = V x A is the external magnetic field. The electrons are transformed into composite fermions by attaching two magnetic flux quanta to each. The constraint that this additional magnetic field, 6, is proportional to the electron density is b = V x a = Airip^ (x)yj(x) (B.2) and is implemented with the use of a Chern-Simons term in C. In addition, the CF's experience a gauge field that is the difference between the external field A and the "statistical" field a, A — a = AA: C = ,Mx)(-,0, + aoWx) + 4 ( x ) ( l d l " e A A ' Y i P ( x ) a 0 V x a /' 2 V x a(x)V(x - y)V x a(y) + / d 2 / x a W ^ ; 6 ; / ) V X f l ^ (B.3) 4?r where the third term is the Chern-Simons term and we have used the constraint (B.2) to rewrite the last term. In this expression yj is a fermionic operator representing composite fermions. The theory is completed by allowing fluctuations of the gauge field, 6a, C = 4(x)(-idt + a0)4>(x) + 4 ( x ) { l d z ~ e f ^ ~ 6 a ) 2 v ( x ) + 8a1D^8a1. (B.4) 126 Appendix B. Self-Energy Corrections in a Finite Magnetic Field 127 The last term is the effective action of the gauge fluctuations. The second term yields the free CF Hamiltonian, . 4(x)(tdt - eAA,)H{x) HC'F = ~ ( B - 3 J 2m and the CF-gauge fluctuation interaction, 8a,(x)^{x)(idt - eAAj)i>(x) f n r . Hint = B.6) m. in the Coulomb gauge, q-Sa = 0. The eigenfunctions of HQF are the same as (A.3) except that uc is replaced by Aw c . The operator Vj = (idj — eAAi)/m introduces additional complications in the determination of the final form of Hint. We are only considering interactions with the transverse component of 8a = 8ai, corresponding to the transverse component of Vy — cos 9VX + sin 9Vy. Vx and Vy can both be expressed in terms of creation and annihilation operators which act on the harmonic oscillator part of tp: Vx = J ^ ( a + at) (B.7; V 2m Vi = J ^ ( a e * + a t e - " ) . (B.9) V 2m In this case the matrix element will be ACF(n,n',q) = \j-^(^A{ih n - l,q)ei0 + <Jn' + lA(n,n + l,q)e~i0) (B.10) which, in the limit of large n ~ n = p, yields \ACr,Wn\q)\2^^^\A(n,qf. ( B . l l ) The coefficient is 4pAuc 2EF 2m m ;B.I2) Appendix B. Self-Energy Corrections in a Finite Magnetic Field 128 Thus we are left with / 2J7} -, Hint = y T ^ J A ( n ' n \ ? ) K + a-q)Cnc[, , (B.13) therefore the self-energy is ^ , . 2EF f d2q r°° dto f°° de / VTT* f" T I" T S |A (™,n , « ) | 2 I m D ( 9 , . ) I m G » „ , ( e ' / (ZTIY ./() 7T . / - o o VT 7~ m v_.., .. .. , n = 0 ^ A + nB(u) - nf(e') | wJ?(a;) + n /(e')\ ^ y e — to — e + id e + u — e + iS J Just as in the phonon case, a cut-off in the sum at n = 2p is needed. Using ImG°(e) = ix6(e+p — (n + ^ )Auc) and lmD(q, to) = 2 2QJ?)2S /2 Ave calculate the imaginary self-energies at finite temperature [87]. For s > 2 EF feAB\1/2 f°° clq I du j de 2_ =o v y 27r2m V 2p / ./o ./o . / - o o ^ x 9 9 ^ 2 s ^ + A* - ( « + 1/2) Aw c) 7 2w 2 + (/2*x2 x [[nB(w) + ??,f(-e')]<5(e - cu - e') + [ n ^ M + nf(e')]6(€ + w - e')]. (B.15) The (/-integral may be done by scaling: U=ing ( ^ ) ' / 2 = ^ r ™ find /'OO f/^J /-oo ( ( = KsAuc Y / ^777 / RFE ^ + A<< - fa + 1/2] Aw c) x [[nB(u) + ?i/(-e')]^(e - w - e') + [n/j(w) + nf(e')]6(e + u- e')] (B.17) ^ /nj(e - [n + l/2]Acuc + /t) + nf(fi - [n + l/2]Aa; c) x0(e- [n + l/2]Aw c + //,) wB([n + l/2]Aw c - e - /./,) + n7([n + l/2]Auc - /i) + ([n + l / 2 ] A w c - e - A 0 1 _ 2 / a xfl(fa + 1/2]Awc - e - / ( ) ) . (B.18) S e) Appendix B. Self-Energy Corrections in a Finite Magnetic Field 129 Likewise for s = 2 f°° , 070; 7T . / dq , , . ,, = — B.19 therefore S (e) = K 2 A u r y 2 d u j de6(e + n-[n + l/2]Aur) x [fa#(w) + n/(-e')]tf(e - w - e') + [nB(w) + ra/(e')]6(e + u-e')] (B.20) = A 2 A w c 53([71JB(€ - fa + 1/2] Aw c + p) + nf(p - [n + l/2]Aw c)] xO(e - fa + 1/2]Awc + p) + [nB([n + l/2]Acu c - e - p) + nf([n + 1/2]Auc - p)} x0([n+l /2]Aw c -e-p)). (B.21) The real parts are determined using the Kramers-Kronig relation I r & ' ^ i l (B.22) 7T J - 0 0 e — e which for .s > 2 is ^ 1 de 1 1 ^ n = 0 7 --00 e' + (n + 1/2) Aw c — p — e x[nf(p - [n + l/2]Auc)6(e) + n,([n + 1/2]Awc - p)e(-e')} (B.23) A , A w c ]T l(" + 1/2)Awc - p - el2/*"1 x (iif(p - fa + 1/2] Auc) esc ^ ) ^(fa + 1/2] Aw c - p - e) -nf(p - [n + l/2]Aw c)cot 0(e + p - fa + 1/2]Auc) -nf((n + l /2)Aw c - /it) esc 6>(e + / i - fa + 1/2]Auc) +nf([n + 1/2] Aw c - p)cot ( y ) 9([n + 1/2]Awc - p - e)) (B.24) Appendix B. Self-Energy Corrections in a Finite Magnetic Field 130 = sgn(e)cot(^)S;;(e) '•2TT +A; esc (—\ |e + A< - n - 1 ^ l * ^ - 1 x(n/(p - (??, + l/2)Awc)0((rc + 1/2)Awc - ^ - e) -nf((n + 1/2)Awc - /_t)0(e + /:< - (n + 1/2)Awc)) (B.25) For 5 = 2 we have ^ n/(/a - (7i + l/2)Awc)0(e') - n/((n + 1/2) Aw c - p)9{-e) K2Au!c 7T E l o s n = 0 e' + (n + 1/2) Aw c - p - e (n + 1 /2)ALJC — p. — e Aw, (ra/Ou - (n + l/2)Aw c) - nf((n + 1/2)Awc - /-/•)) (B.27; Equations (3.25), (3.26), (3.28) and (3.29) follow in the limit of T = 0. When the self-energy and the energy e are expressed in units of the Fermi energy, Ep = pAuc, the dimensionless coupling is Epe ( 1 X 1 / 2 1/2 (B.28) e^ v47rne, where n 0 = ^ F y J 1.5 x 1 0 n c m - 2 is the density at which K2 = 1. For the experiment of Du et al. [30], one has Ep = 2irne/m,ne = 2.3 x 10 1 1 cm - 2 , m — .07me and e = 13, so one expects K2 « 0.8. Note that equations (3.25)-(3.27) assume that the self-energy has units of Aw c , which is more appropriate for solving the self-consistent equation, (3.51). Appendix C Proofs Involving the Thermodynamic Potential C . l Proof of (4.25) 1 We will derive (4.24) from (4.25). Q0 = -TTr\ogG^\iu>n^l)exp(iunO+) (C.l) = - T £ l o g ( i w n - & ) e x p ( i a ; n O + ) (C.2) n,l We change the sum over n to a contour integral around the lower half plane and then integrate by parts: = 2 ^ X ) ^ d € l o g ( e - 6 K ( e - M ) e x p ( e O + ) (C.3) = - ^ E / ^ l o g ( l + e x p ( ^ ) ) e x p ( ^ ) (C.4) = -TElog ( l + e x p ( ^ g ^ ^ (C.5) C.2 Proof of (4.81) 2 The thermodynamic potential is the sum over all closed connected diagrams, which implicitly suggests a perturbative approach to computing Q. We wish to rewrite f> in a different way. Referring to the connected diagrams shown in Figs. C . l b) and c), we see 1This proof is due to Luttinger and Ward [38] 2We refer the reader to the original paper of Luttinger and Ward [38] and to Section 16.2 of [74]. 131 Appendix C. Proofs Involving the Thermodynamic Potential 132 that a general nth order connected diagram may be written as J : ^ , ^ ^ ^ ) (C .6) r,l E ^ denotes a general nth order self-energy term - it may be proper or improper. E ^ is the sum over all such nth order contributions, proper and improper, so that the total nth order contribution to ft is a , = - £ E ( B W z ) t f ° > ( w r , & ) (C.7) H r.l The factor ^ compensates for an overcounting of diagrams in this procedure. This occurs because in a general nth order diagram for ft there are n fermion lines, therefore there are n ways to write that particular contribution to ft in the form (C.6). a) b) c) Figure C . l : Some Feynman diagrams representing fourth order contributions to ft — QQ. The first shows a disconnected graph, the second is the 0(A 4) skeleton graph and the third is an example of a skeleton graph with a self-energy insertion. Let us assume that the coupling at each vertex is A. A general nth order contribution is proportional to A", and when we consider the form (C.7), all of the dependence is in E . Therefore we may write A 7 \ ' ft-ft0 = E / d A ' G ^ K , 6 ) S K , 6 : A ' ) ^ r (C.8) r.l 7 0 A Appendix C. Proofs Involving the Thermodynamic Potential 133 Here E(w,.,£/:A') is the total self-energy (proper plus improper), which depends on a variable coupling A'. Eq. (C.8) may be rewritten as A ^ = - r E G ( 0 ) K , 6 ) S ( o ; r , 6 ; A ) (C.9) r.l E satisfies a Dyson-like equation S = E + E G ^ E + E G ^ E G ^ E (C 10) = E G ( 0 ) _ 1 G ' (C. l l ) where E is the proper self-energy. Therefore (C:9) is xdn = _T^(ur^r,x)G(ur^r,x) (c.12) r.l We wish to show that (4.82) also satisfies this differential equation. We begin by considering ft'. Recall that ft' consists of the subset of connected diagrams for ft with no self-energy insertions (such as diagram b, but not c), and with all bare propagators G ^ replaced by the full ones G. Thus by definition ft' is a functional of G, so that 6f2' = -T£E(e t - ,6 )6G(e , - ,&) (C.13) i,i Furthermore, G = ( G ( 0 ) _ 1 + S ) _ 1 which.implies that § = § § = T £ 5 ( e , , S ) G 2 ( £ , , 6 ) (C.14) Now we compute the derivative | § ; using (4.82) | ! = - T £ S ( w , , 6 ) G 2 K , 6 ) + ^ (C.15) ^ r,l = 0 (C.16) Now we turn to Because of the stationarity property (C.16) and the fact that G depends on A only through S Ave find that Appendix C. Proofs Involving the Thermodynamic Potential 134 The only other dependence on A comes from the vertices of the skeleton diagrams of ft-', which enables us to use the same reasoning as (C.7-C.12) to obtain A f } = - r E S ( e - 6 ; A ) G ( e , . , 6 ; A ) = A ^ (C.18) Finallv we note that therefore ft(0) = ft0 = ft(0) (C.19) ft = ft. (C.20) C.3 Proof of (4.84) In this section we review Luttinger's derivation [39] of the leading order contributions to ft05c. Given that S o s c ~ (^J So (this is shown in Appendix F), where S = S 0 + Hosc and that TTrflogG" 1] ~ (f) , Eq. (4.82) can be approximated as ft0,c = - T ^ T r l o g ( e , - + S0(e?) - (n + l/2)uc + p) - S0(e,-)G0(et-)) + ft'(So) (C.21) » where Go(ti) = — , , ^ v ( . (C22) e{ + p - (n + l/2)uc + S 0 ( € t ) ft' is evaluated by keeping oscillatory part of each fermion line once while ignoring the oscillatory parts from the rest of the lines. This gives the leading order in the oscillatory behaviour of ft'. This is the same as Tr£,- G0(e7;)F(e,'), where F(e,) is the sum of diagrams obtained by cutting each fermion line in the skeleton graphs once and ignoring oscillatory parts. But this is just S0(e,), and so the second and third terms in Eq. (C.21) cancel. Appendix D Oscillatory Corrections to D(u,q) In this Appendix we compute the oscillatory behaviour of the screened gauge propagator and show that it does not influence the oscillatory self-energy. We consider the Coulomb interaction ( 5 = 2) only. There is no oscillatory dependence in the original unscreened gauge propagator: it arises only as a result of screening due to the presence of the fermion lines which appear in the corrections KQQ and A'}"' (see Fig. 3.2). Only ImA"}^ appears in the low energy limit of the s = 2 version of D(q, u). It appears in the first term of the denominator of D(q,u>), which is i-ftu. Recall that 7 = ^ . We will recalculate ImA'jJ' keeping a sum over Landau level indices on the fermion lines at T = 0. In the absence of a magnetic field A"{^ is given by A u ( w , , ) = _ + y _ ( _ j (u_i^ + i k + iS) (D.i) In a finite magnetic field at T = 0 we find — 6{LJ/COc -n + m)(6(-n + 1/2) - 0( -m + 1/2)) (D.2) = EpfeBV'2 y (d(-n + l/2)-9(-n + u/uc + l/2))6(n-u/uc + [p) (D.3) E <eB 1 / " F ' ^ v " 1 0(n-u/cvc+ \j>) (DA) ^ k * ' \n=-Lp juc[(l/2 + u/uc) (D.5) 135 Appendix D. Oscillatory Corrections to D(cu, q) 136 This shows how ImA"^ oscillates with u. The symbol [x = the greatest integer less than x indicates that it is a stepwise function, so it is not very singular. It appears in the renormalised gauge propagator as q) = -. W U 0 J , , _ 9 (D.6) In the self-energy expression the g-integral scales and there is no dependence left on [to. Therefore, for s = 2 there is no oscillatory contribution from D(u>,q) on the self-energy. Appendix E Oscillatory Corrections to Q.^ In this Appendix we evaluate the second and third terms in the oscillatory expansion of fi(2> f2<2> = Q 0 2 ) + fti2) + l>22). (E.l) These terms are defined by Eq. (4.76). We consider the two dimensional system of composite fermions with 5 = 2 (long-range interactions). u>n,u„ k=l X, , — ; (E.2) iuin — e + f-i iojn + ivn — e + /J fdnU-lf rclecle cos ( ^ ) D ^ q ) 2TE 7T x ((nf(e- u.) - nf(e - n)\ \ € — € — IV„ 'n idq s rdede'cos ((n/(e ~ ^ ~n/(6' • a0) ^ /• d.rn6(.T) / DA(x, q) DR(x, q) \ ^ ^ J 2ni \e — e — x + id e — e — x — id J X ( ^ U ^ - D ^ q ) _ i j t 6 { , _ e _ x ) m ^ q ) + D r { ^ ^ ( E 5 ) idq | rdede' c°s (n)((n/(e - a° -n/(e' - ao) 137 Appendix E. Oscillatory Corrections to VS2"* 138 x r telmD,**,») f ( E . 7 ) . / - o o \ e — e — x ) = ^ ? 4 ^ . / f\5 r ^ c ° s ( U ) ( ( n ' ( e - a° - n / ( e ' -x f <teImD(s, q) f + ~ "] - + nb(e - e)\ ./o V e — e — x e — e + x ) These steps show the following operations. First the sums over Matsubara frequencies are converted to contour integrals: note that the Bose frequency ivn is evaluated along the line ivn = x. Using DA(X) = DB(x) and the Kramers-Kronig relations, the form lmDjt(x, q) is placed everywhere. Finally, we change the limits of the .r-integral so that the g-integral may now be done by scaling, Q i 2 ) = E/^^- E / d e c k ' c o s l*2^—) {(nf(e - p) - nf(e - n)) oo x / dT (-nb(x) + nb(e' ~ e ) _ ~nb(-x) + nb{e - e)\ ^ lo \ e — e — x e — e + x J This expression may be evaluated at T = 0, 4IU 0 0 — Y(-l)k( de de / dx+ de de / dx 7T V./0 J0 Jo . JO Je . / - o o cos(27rA;e'/Aw c) + cos(27rA;e/Aw c) e — e + x The third term in the oscillatory expansion has a similar form, (E.10) cos(27r(fce + k' e')/Au>c) + cos(2n(ke — k' e')/Au>c) e — e + x There are no additional factors of appearing in the third term. (E. l l ) Appendix F Oscillatory Expansion of ft in 2-D F . l Proof of (4.86) In this Section we evaluate Trlog 1 in 2-D at T — 0 for energy levels at En — (n-\-^)u)c This is just 0 „ = o -Tm 1 + exp 'ix - (n + l/2)uc $ o Jo de l + 2 ^ ( - l ) f c c o s fc=i T '2ixke log 1 + exp f.t — e T Evaluated at T = 0, the first term is just ft (o) _ \£^_ ~ 2$ n ' To evaluate the second term we begin by integrating by parts 2m -osc Q o ^ ' 2%k Jo and then evaluate at T = 0, $ 0 j£l V27Tfc m / u;, (2xk/A cos — 1 2» (F.l) (F.2) (F.3) (F.4) (F.5) (F.6) (F.7) , mod — , 1 2$ 0 V2vry L \w c / The last factor gives the oscillations (it is a number that is always less than one). There-fore, (F.8) ftl02 ~ f ^ 1 ft00) A' 139 Appendix F. Oscillatory Expansion of 11 in 2-D 140 Luttinger [39] considered the more general case of Trlog G 1, which he showed could be evaluated using the renormalised quasiparticle spectrum, This expression may be evaluated using the methods of Lifshitz and Kosevich [98] for a general Fermi surface. F.2 Proof of (4.87) In this Section we do a calculation in 2-D in analogy to Luttinger's calculation in 3-D [39] in which he demonstrated that E 0 5 C ~ . We are interested in the quantity p(21) which is related to the self-energy, . (F.9) P(21) = £ n / ( e < - p)^*{x2\r)tj}{xx\r) (F.10) ip(x:r) is the normalised wave-function, ( F . l l ) Use the identity tx 0 < c < 1 (F.13) to get Appendix F. Oscillatory Expansion of Q in 2-D 141 where Q ( f 2 , f i ; t / T ) = £V»*(s2 ;r )e-^/ T V<£i;r) . (F.15) r We evaluate Q(x2,Xi\t/T) separately for 2-D and 3-D: g . ^ ( f 2 , f i ; t / T ) = Y, kx,n) exp (~* (w + 1/2)("c j ^ 2 0 ( f i ; /cx(E)16) $ 0 2 7 r ^ . / f-t[(n + l/2)uc + k2./2m]\ t _ , , . x exp f — ^ ^ U ^ ( f x : A;*, n) (F.17) = y d/c, exp (i,kz{z2 - Z L ) ~ Q2D(X2, XL] t/T) (F.18) = ( S ) M ^ ^ H * ' ™ (P.U, Q(x2,xi,t/T) is evaluated to give (see [39]) Q ( x 2 , f i ; « / T ) = exp ( ^ ( y 2 + ^ 2 ~ X l ) ) F ( f 2 - ^ c ^ t / T ) (F.20) where / T m \ 1 / 2 (Tm?2\ F3D(x:uc:t/T) = ( _ ) exp ( j F 2 C ( f ; u c : t /T). (F.22) This enables us to write p(21) in the form p(21) = exp I J g(x2 - x<:uc:t/T) (F.23) where I rioc+c e W ^ ioc+c (l/7r)sin(7rt)" (/(.T; L J c ; t /T) has poles on the imaginary axis at g W g(x;uc-t/T) = — • dt . F(x:uc:t/T). (F.24) 27H, J-ioc+c (I IT S i l l 7Tt 27n7T tt = / = ± 1 , ± 2 . . . (F.25) Appendix F. Oscillatory Expansion of ft in 2-D 142 which are the poles of F(x\ujc:t/T)\ it has also has poles along the real axis at t = ± n for n = 1,2,3— Still following Luttinger, we deform the contour of the integral and divide it into two pieces, g{x:uc;t/T) = g0(x: uc: t/T) + gosc(x: coc: t/T) (F.26) g0(x:uc:t/T) = - L f dt ^ F(x:coc:t/T) (F.27) 2m Jc (l/7r)sm(7rf) oo' i tfi/T gosc{x-uc-tlT) = E 7^- / dt7-—-—-F(x:uc:t/T) (F.28) 2vr?, 7c, ( I / T T ) sin(7rt) where C is a contour that encircles the negative real axis counter-clockwise and the C; are counter-clockwise circles about the points £/. The sum does not include the point I = 0. The first term of Eq. (F.26), given by (F.27), is the non-oscillatory contribution to the self-energy. It is evaluated by keeping the lowest order contribution in powers of F2l){x:u,.:l/T) = £ 7 exp ("^^^) (F.29) F:w(x:uc:t/T) = {^)" exp j F2D(x; LOc: t/T) (F.30) Therefore, 1 p. f dtT f-(x2 + y2)T\ .90,20 = — — / —pexp (F.31) 2m 2TT JC tl \ 2tp J The second term of Eq. (F.26) is the oscillatory contribution. It is evaluated by first shifting all the contours to t' = t — £/, so that each contour is a little circle about t' = 0, and then taking the limit of small t' as above, F2D(x:uc:(t +tl)/T) = exp ( " ^ / ^ (F.33) Appendix F. Oscillatory Expansion of 0 in 2-D 143 Notice that the first term in the exponent of Eq. (F.34) dominates, therefore F3D(x;uc:{t +tl)/T) = J F2D(x:uc:{t +tl)/T) (F.35) ~ (ury/2F2D(x:uc:(t +t,)/T). (F.36) This leads to the result that 1 * T ftlhi\ rdt ( , x2 + y2\ _N 9™*D = toi^M^lTJtT^V*--^) (F-3° 1 ^ / T V i / 2 ftm\ rdt' (, x2 + y2\ ^ Substituting Eq. (F.25) for ti, we find that {Josc'lD ~ — .90.20 (F.39) 5osc,3£> ~ f y j 0 0 , 3 0 - (F.40) Luttinger argued that the origin of these relations lies in the Fermi function, and that a similar dependence occurs in higher skeleton graphs, yielding the general result that ^osc.lD ~ — So.2£i (F-41) / x 3/2 S ( ) , „ „ ; ~ I j j S 0 , 3 O . (F.42)
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Field theoretical quantities in the fractional quantum Hall effect Curnoe, Curnoe, Stephanie Hythe Stephanie Hythe 1997
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Title | Field theoretical quantities in the fractional quantum Hall effect |
Creator |
Curnoe, Curnoe, Stephanie Hythe Stephanie Hythe |
Date Issued | 1997 |
Description | This thesis studies two models of the fractional quantum Hall effect (FQHE), the bosonic (Chern-Simons-Landau-Ginzburg) description and the fermionic (composite fermion gauge theory) description. The bosonic theory attempts to describe the FQHE states at filling fractions v = [formula] while the fermionic theory attempts to describe the states at v = [formula] and the metallic states in between. Within the bosonic theory, the fractionally charged quasiparticles of the FQH system are vortices which appear during the breakdown of the uniform quantum Hall state. The energetics of a single vortex state are studied whereby it is shown how the system may become unstable to the formation of vortices. Numerical vortex profiles are computed by minimising the Hamiltonian. Using the fermionic theory of composite fermions interacting with gauge fluctuations, we consider two important field theoretic quantities, the self-energy and the thermodynamic potential in a finite magnetic field. We find that the conventional Luttinger-Ward treatment of the oscillatory behaviour of the thermodynamic potential is not applicable in two dimensions, for any kind of interaction. Instead we propose a new formulation which omits all crossed graphs and which necessarily includes the oscillatory self-consistent self-energy. To second order in perturbation theory, the oscillatory self-energy is calculated by retaining Landau level quantisation on the internal fermion line. The low energy form of the self-consistent self-energy is obtained by means of a new iterative procedure which is introduced here. This procedure makes use of the structure introduced by Landau level quantisation. We also investigate the structure induced in the analogous two dimensional electron-phonon problem, in order to assist our understanding of the composite fermion self-energy. In the low energy limit, it is found that the renormalised form of the composite fermion Green's function is of the same form as the unrenormalised Green's function. Therefore we argue that the principal effects of interactions may be accounted for using a field-dependent renormalised mass. The iterative procedure for finding the self-consistent self-energy is used to evaluate the renormalised gap between the Fermi energy and the first excited states, which rapidly converges in a few iterations. We find a significant departure from the asymptotic result obtained by ignoring Landau level quantisation in the regime of experimentally relevant values of the parameters. We compare our findings with measurements of the gap in fractional Hall states near v = 1 / 2 . |
Extent | 6249241 bytes |
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Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-03 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085660 |
URI | http://hdl.handle.net/2429/6776 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1997-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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