The Pennsylvania State University The Graduate School INTERACTING COMPOSITE FERMIONS. A Thesis in Physics by Chia-Chen Chang. c 2006 Chia-Chen Chang

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1 The Pennsylvania State University The Graduate School INTERACTING COMPOSITE FERMIONS A Thesis in Physics by Chia-Chen Chang c 2006 Chia-Chen Chang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2006

2 The thesis of Chia-Chen Chang was reviewed and approved by the following: Jainendra K. Jain Erwin W. Mueller Professor of Physics Thesis Advisor, Chair of Committee Gerald D. Mahan Distinguished Professor of Physics Stéphane Coutu Associate Professor of Physics Associate Professor of Astronomy and Astrophysics James B. Anderson Evan Pugh Professor of Chemistry Jayanth R. Banavar Professor of Physics Head of Physics Department Signatures are on file in the Graduate School.

3 Abstract Systems containing a collection of strongly interacting particles are at the frontier of physics. Many-particle physics is challenging in that many of the usual tools for quantitative calculation are not applicable, and new concepts and techniques are usually required. Theoretical breakthrough in many-body physics is usually accompanied by the identification an emergent particle derived from the original interacting system. A good illustration is provided by the composite fermion (CF) theory of the fractional quantum Hall effect (FQHE). The composite fermion theory of FQHE transcends any other theory in the field of FQHE. It provides a mapping between the strongly interacting electrons and a system of weakly interacting composite fermions. The CF theory unifies the FQHE with the well understood integral quantum Hall effect, allowing a single particle description to be utilized. The non-interacting CF theory has successfully explained the physics of FHQE at the fillings ν = n/(2pn ± 1) where p and n are integers. However, new physics can occur due to the weak interaction between CFs, which is the primary concern of this thesis. (i) Recently, experiments show signatures of new FQHE states, for example 4/11 and 5/13, which cannot be understood in a model of non-interacting composite fermions. By including the residual interaction of CFs, these new fractions are understood as the fractional quantum Hall effect of composite fermions. (ii) An extremely accurate description of few electrons in a semiconductor quantum dot at high magnetic field is developed based on correlated basis of the CF theory. (iii) At low filling factors, we show, by directly comparing with exact eigenstates, that the crystal state, resulting from the interacting between particles, is a topological quantum crystal of composite fermions. (iv) Motivated by the CF theory for electrons, we study the mapping of interacting bosons in a rapidly rotating trap at filling factors n/(n+1) onto non-interacting fermions at fillings n. This mapping provides a good account of the behavior of bosons for small n. (v) The competition iii

4 between the CF-crystal and CF-liquid orders at filling factor 1/5 is investigated. In the thermodynamic limit, the liquid state prevails. A variational combination of the crystal and liquid states provides an extremely accurate description for small systems. iv

5 Table of Contents List of Figures List of Tables Acknowledgments viii xiii xvi Chapter 1 Introduction 1 Chapter 2 Quantum Hall Effects and Composite Fermion Theory Two-dimensional electron system The Hall Effect: Classical and Quantum Classical Hall Effect Integral Quantum Hall effect Fractional Quantum Hall effect Composite Fermion Theory Overview Composite Fermions Microscopic Wave Functions Remaining Thesis Plan Chapter 3 The next generation FQHE Partially Spin Polarized Fully Spin Polarized v

6 Chapter 4 Composite fermion theory of correlated electrons in semiconductor quantum dots in high magnetic fields 35 Chapter 5 Microscopic verification of topological electron-vortex binding in the lowest Landau-level crystal state 44 Chapter 6 Composite-fermionization of bosons in rapidly rotating atomic traps Introduction The Hamiltonian Composite fermion theory ν = 1/ ν 1/ ν = Calculation Quantitative comparisons Conclusion Chapter 7 Competition between composite-fermion-crystal and liquid orders at ν = 1/5 72 Appendix A Exact Diagonalization 81 A.1 Coulomb matrix elements A.1.1 Disk geometry A.1.2 Spherical Geometry A.2 Haldane Pseudopotentials A.2.1 Disk Geometry A.2.2 Spherical Geometry A.3 Algorithm A.4 Some Exact Quantities A.4.1 Pair Correlation Function and Density A.4.2 Angular momentum Appendix B Variational Monte Carlo 91 vi

7 Appendix C Projection of the Hatree-Fock crystal wave function 94 Bibliography 98 vii

8 List of Figures 2.1 (a) Schematic diagram of a Si-MOSFET. The 2DES resides at the interface between silicon and silicon oxide. (b) Schematic diagram of a modulation-doped GaAs-AlGaAs heterostructure. Electrons are held against the AlGaAs by the electric field set up by the positively charged silicon dopants in AlGaAs. (c) Energy band diagram of a modulation-doped GaAs-AlGaAs heterostructure. The deformed energy band in the GaAs side is the result of the electrostatic potential from silicon dopants. Electrons are trapped in the triangular potential at the interface. The electron motion along the z-axis is quantized. At low temperature, only the lowest energy state is occupied. The figure is taken from Ref. [Stö99] Geometry of the Hall measurement. The Hall resistance R H and longitudinal resistance R are measured as a function of magnetic field B and current I. V H denotes the Hall voltage across the current path. V represents the longitudinal voltage drop. The figure is taken from Ref. [Stö99] IQHE data for a 2DES in GaAs-AlGaAs. The Hall resistance forms well-defined plateaus at filling factors ν = 1, 2, 3, and 4 (See Eq. (2.4) for the definition of ν). The longitudinal resistance also develops deep minima at these fillings. The figure is taken from Ref. [Stö99] (a) Density of states n(e) of Landau levels in a clean system. Each level is a sharp δ-function. The energy gap is E = ω c. (b) Density of States n(e) of a Landau level in a disordered system. A band of extended states is located at the center of each level with localized states in between viii

9 2.5 Original data of the first observation of the ν = 1/3 FQHE. The measurement was done on a modulation-doped GaAs/AlGaAs sample. At T = 0.48K, the Hall resistance ρ xy has a well-defined plateau at the B field corresponding to the 1/3 filling. ρ xx, the magnetoresistance has a minimum at the same B field. This figure is taken from Ref. [TSG82] The FQHE in ultrahigh mobility modulation-doped GaAs-AlGaAs 2DESs. The most prominent fractions form a series ν = n/(2n ±1), n an integer. The figure is taken from Ref. [Stö99] The comparison of the FQHE and the IQHE. The top (bottom) panel shows the IQHE (FQHE) of electrons. (The IQHE and FQHE data are obtained from Ref. [CNU + 86] and [ST97], respectively). The correspondence between prominent features in the longitudinal resistance are clearly shown. This figure is taken from Ref. [Jai00] The transformation from strongly interacting electrons (left panel) to weakly interacting composite fermions (right panel). Electrons are represented by filled circles, magnetic flux is depicted by arrows. After the transform, each electron is bound to 2 flux quanta, resulting in a reduced effective magnetic field The energy spectrum at ν = 4/11 for N = 6, 10, 14, and 18 particles, respectively. It is assumed that the state has partial spin polarization. The single particle energy E includes the interaction with the uniform, positively charged background. The quantity l 0 = c/eb is the magnetic length, and ǫ is the dielectric constant of the background material. The error bars show one standard deviation in the Monte Carlo simulation Neutral excitation gap and the ground state energy at ν = 4/11 as a function of N 1, N being the number of electrons The energy spectrum at ν = 7/19 for (a) N = 11 and (b) N = 18 particles and at ν = 6/17 for (c) N = 8 and (d) N = 14 for a partially spin polarized system ix

10 4.1 Schematic depiction of Slater determinant basis states for N = 6 electrons at L = 95, which maps into L = 5 with 2p = 6. The single electron orbitals at L = 5 are labeled by two quantum numbers, the LL index n = 0, 1,..., and the angular momentum l = n, n+1,... The x-axis labels n+l and the y-axis n. The dots show the occupied orbitals forming the Slater determinants Φ L =5 α relevant up to the first order. The state shown at the top left has the lowest kinetic energy (if the kinetic energy is measured relative to the lowest Landau level, then, in units of the cyclotron energy, the total kinetic energy of this state is two). The other nine states have one higher unit of kinetic energy. The basis states Ψ L=95 α are obtained according to Eq. (4.2), through multiplication by j<k (z j z k ) 6, which converts electrons into composite fermions carrying six vortices. That is shown schematically by six arrows on each dot. The single state at the top is relevant at the zeroth order, and all ten basis states are employed at the first order. (In fact, there are a total of 12 linearly independent states Φ L α at the first order for L = 5, but they produce only ten linearly independent states Ψ L α at L = 95.) Comparison of the exact excitation gaps ( ) for N = 6 with the gaps obtained in the first-order CF theory ( ) Pair correlation function for N = 6 electrons at L = 99. The position of one particle is fixed on the outer ring, coincident with the position of the missing peak. The ground state wave function used in the calculation is obtained from (a) exact diagonalization; (b) the zeroth-order CF theory; (c) the first-order CF theory; (d) the second-order CF theory. The noise in (a) and (d) results from the relatively large statistical uncertainty in Monte Carlo because of the more complicated wave function x

11 5.1 The correlation energy of the optimal CF crystal, i.e., the percent of deviation of its Coulomb energy from the Coulomb energy of the uncorrelated electron crystal, for N = 6 particles. The superscript 2p on 2p CFC indicates the vortex quantum number of composite fermions. The energy of the electron crystal for L > 400 is taken from Yannouleas and Landman[YL03, YL04]. The deviation of the exact energy from the electron crystal energy is also shown for L 145; for larger angular momenta, where the exact energy is not available, we show an accurate approximation, V (2) CF (explained in the text), as an independent reference. For 2p > 6, the number of vortices carried by composite fermions is shown in brackets near the diamond. The energy difference per particle between the electron and the CF crystals is given in the inset, quoted in units of e 2 /ǫl 0, where l 0 is the magnetic length and ǫ the dielectric constant of the host semiconductor The pair correlation functions for the CF crystal (solid circles), the electron crystal (empty squares), and Laughlin s wave function (empty triangles) on a circle of radius R = l 0 for six particles at ν = 1/7. The solid line shows the exact pair correlation function The low-energy spectrum of (a) N = 9 and (b) N = 10 interacting bosons at ν = 1/2, interacting with a delta function interaction with strength g. Dashes represent exact results, while the dots show predictions of the composite fermion theory. Spherical geometry is used in the calculation, and L is the total orbital angular momentum. In this and the subsequent two figures, the statistical uncertainty from Monte Carlo sampling (not shown) is smaller than the symbol size The low-energy spectrum of (a) N = 10 and (b) N = 12 interacting bosons at ν = 2/3. Various symbols have the same meanings as in Fig The low-energy spectrum of (a) N = 9 and (b) N = 12 interacting bosons at ν = 3/4. See the caption of Fig. 1 for the definition of various symbols Angle view and contour plot of the pair correlation function for (a) exact, (b) mixed, (c) CFC, and (d) liquid state for N = 6 particles at ν = 1/5. See text for the definition of the pair correlation function. All figures are plotted on the same scale. The missing peak depicts the position R of the fixed particle in Eq. (7.8). The exact and the liquid plots were shown earlier in Ref.[YL04] xi

12 7.2 The pair correlation function g(θ) of the mixed wave function (solid square), CF crystal (empty square), liquid wave function (cross) and the exact wave function (solid line) at ν = 1/5 for N = 6 particles. We fix a particle at the position (5.430l 0, 0) and plot the pair correlation on a semi-circle of radius R = 5.430l 0 as a function of the radial angle θ xii

13 List of Tables 3.1 The systems studied in this work. N is the number of particles in the reversed spin sector The gaps at ν and ν, determined from CF and exact diagonalization, respectively, at several filling factors. Only the gaps of incompressible states are shown. N is the total number of particles and N is the number of particles in the second LL for the state at ν. The gaps are quoted in units of e 2 /ǫl 0. The statistical uncertainty from Monte Carlo is shown in parentheses The trial wave functions Ψ tr ν and Ψ tr ν are compared with Ψ 0 ν for the finite system incompressible states at three filling factors. (See text for definitions.) The overlaps are defined as O = Ψ 0 ν Ψtr ν and O = Ψ 0 ν Ψ tr ν. The energies E 0, E tr and E tr correspond to the wave functions Ψ 0 ν, Ψtr ν and tr Ψ ν, respectively. E0 were calculated earlier in Ref. [MJ02] Exact ground state energy (V ex ) and the ground state energy obtained from the zeroth (V (0) (1) CF ) and the first-order (V CF ) CF theory for N = 6. The dimensions of the bases diagonalized are D ex, D (0) CF and D (1) CF, respectively. The statistical uncertainty arising from Monte Carlo sampling is given in parentheses Overlaps between exact ground states and CF ground states obtained at the zeroth (O (0) ) and the first order(o (1) ). The statistical uncertainty from Monte Carlo sampling does not affect the first three significant figures Comparison of the second order CF theory with exact results for the L = 99 ground state. D (2) is the dimension of the correlated CF basis, V (2) CF is the CF prediction for the ground state energy, and O (2) is the overlap between the CF and the exact wave functions.. 41 xiii

14 5.1 The last three columns give the overlaps of CF crystal (CFC), electron crystal (EC), and Laughlin s wave function with the exact ground state wave function at several filling factors ν. The overlap is defined as Ψ trial Ψ exact 2 / Ψ trial Ψ trial Ψ exact Ψ exact. The second column gives D, the dimension of the basis space for N = 6 electrons, and L is the total angular momentum of the state Total interaction energies for the exact ground state, the CF crystal (CFC), the electron crystal (EC), and Laughlin s wave function for six particles at several filling factors. The uncertainty in the last digit from Monte Carlo sampling is given in parentheses The squared overlaps O 2 gr of the exact wave functions for the ground state at ν = 1/2, 2/3, 3/4, and 1 with the trial wave functions of Laughlin (ν = 1/2), Moore and Read (ν = 1), and Jain (ν = 2/3 and ν = 3/4) for several particle numbers N. The definition of the overlap is given in Eq. (13). In this and the following tables, the statistical uncertainty in the last two digits is shown in parentheses when it is larger than For ν = 1, the overlap has been evaluated exactly The squared overlaps O 2 ex of the exact wave functions for the first excited state at ν = 1/2, 2/3, and 3/4, with Jain s wave functions for several particle numbers N. L is the orbital angular momentum of the first excited state The overlap of the Coulomb ground state wave function at ν = 1/2, 2/3, 3/4, and 1 with the trial wave functions of Laughlin (ν = 1/2 ground state), Moore and Read (ν = 1 ground state), and Jain (ν = 2/3 and 3/4 ground states), for several particle numbers N. (Table 6.1 dealt with the exact wave functions for a short range interaction.) The statistical uncertainty in the last two digits is shown when it is larger than The overlaps for ν = 1 (last column) are exact xiv

15 7.1 Interaction energy per particle for the exact (E ex ), mixed (E mixed ), CFC (E CFC ), and liquid (E liquid ) wave functions. The energy calculated by the CF-diagonalization is in the column labeled by E CFD. E EC denotes the energy of the electron crystal wave function χ(l ) (The definition of χ(l ) is given is the text). The unit of energy is e 2 /ǫl where ǫ is the dielectric constant of the host semiconductor and l is the magnetic length. Numbers shown in the parentheses are statistical uncertainties. Exact energy for N = 3 was shown earlier in Ref.[GJ83, LYSY84, YL03] and that for N = 6 in Ref.[YL03, YL04, CJJ05]. Ref.[YL04] also calculated the exact energy for N = 7. CFC energy for N = 6 was shown in Ref.[CJJ05]. Liquid energy for N = 6 was reported in Ref.[YL03, CJJ05]. The electron crystal energy for N = 6 and 7 was given in Ref.[YL03, YL04] The overlap of the mixed (O mixed ), CFC (O CFC ), and liquid (O liquid ) state with the exact state. Numbers shown in the parentheses are statistical uncertainties. The overlap of the CFC state with the exact state for N = 6 was calculated in Ref.[CJJ05]. The overlap of the exact state with liquid state was shown for N = 4 in Ref.[Lau83] and that for N = 6 was shown in Ref.[YL03, CJJ05]. Ref.[YL04] calculated the overlap of the exact state with liquid state for N = xv

16 Acknowledgments It was said that a journey is easier when one travels with companions and mentors. This thesis is a collection of my Ph.D. graduate study during the past five years. In this process, I have been luckily accompanied and guided by many people. Here, I wish to express my sincerest appreciation to all of them. Foremostly, I would like to thank my advisor, Professor Jainendra K. Jain, who is not only a great scientist with deep vision but also a very kind person for students. His strong physical intuition and critical approach in solving scientific problems have greatly inspired me. It is my great honor and privilege to have had him as my thesis advisor. I am also most grateful to Professor Gerald D. Mahan, who gave me the opportunity to expand my expertise on the subject of many-body model calculations. His open view and insight in physics taught me a lot. In addition, his enthusiasm, kindness, humor, and strong spirit have made a great impact on me. During the last year of my graduate education, my visit to the Institute of Theoretical Science (ITS) at the University of Notre Dame was a wonderful experience. I had the opportunity to collaborate with Professor Hans Hansson (Stockholm University), Professor Susanne Viefers, and Professor Jon Leinaas (both at University of Oslo). It was enjoyable to work with these outstanding scientists. I also want to thank Professor Boldizsár Jankó, the director of ITS, for providing support during my stay at the ITS. My other Ph.D. thesis committee members: Professor Stéphane Coutu and Professor James Anderson, are greatly acknowledged for their useful comments on the thesis. My thanks also go to Professor Sudhansu Mandal and Professor Gun Sang Jeon for their help in the research and numerical techniques. I thank my colleagues at Penn State: Michael Peterson, Kenneth Graham, Csaba Töke, Chuntai Shi, and Shivakumar Jolad for their friendship that gave me home-like feeling at work. I also thank Dr. Paul Larmart and Dr. Rusko Ruskov for commenting on my thesis and for help with job application. Assistance from the High Performance Computing Group led by V. Argawala, xvi

17 J. Holmes, and J. Nucciarone at The Pennsylvania State University Academic Services and Emerging Technologies is gratefully acknowledged. This made the intensive numerical calculations in this thesis possible. Lastly and most importantly, I am deeply grateful to my wife, Ching-Yi, for her love, support, and encouragement during these years. One of the best experiences of this period was the birth of our daughter, Maggie, who has thence dramatically changed and enriched our life. xvii

18 Chapter 1 Introduction The traditional view of physics is largely concerned with the idea of reductionism. It is believed that once we know the laws of nature that govern the smallest scales, the mysteries of the physical world can be understood completely. A well known example of this philosophy is particle physics. Physicists use massive and humongous accelerators to seek, down to the smallest scale, the most fundamental ingredients of matter. There is yet another notion that fascinates scientists, namely, the idea that to understand nature, one has to understand and study the principles that govern collective behavior of assemblies of matter. For a wide range of purposes, the microscopic laws that are responsible for matter in small scales are well studied. Despite knowing all the ingredients and laws, the understanding of the system emerging from the known constituent particles is far from complete. As an illustration, the structure and properties of a single helium atom can be understood completely with the help of quantum mechanics. Yet to understand how a bucket of helium behaves under extremely low temperatures, we need new principles principles that describe the collective behavior of matter when a large number of helium atoms are brought together. This philosophy is marked in a short paper More Is Different by P.W. Anderson.[And72] In this paper he defined the concept of emergence with the following quote... at each new level of complexity, entirely new properties appear, and the understanding of these behaviors requires research which I think is as fundamental in its nature as any other.

19 2 Anderson s quote form the basis of the modern attitude in condensed matter physics the notion that the study of the collective principles that govern matter is a frontier by itself. The development of condensed matter physics during the past several decades justifies this view. The discovery of high temperature superconductivity, superfluidity and Bose-Einstein condensation illustrate the fact that nature always surprises us with behaviors that depend on the collective properties of systems containing a macroscopic number of particles. The new concepts and new mathematics involved in the understanding of these beautiful phenomena have also pushed forward the development of other fields in physics. One of the most spectacular examples of emergent physics is the composite fermion. It was discovered by Jain[Jai89a, Jai89b] to describe the physics of the fractional quantum Hall effect[tsg82]. The composite fermion results in a system of strongly interacting electrons moving on a two-dimensional plane penetrated by a strong magnetic field. Its existence requires the presence of all the constituent electrons interacting in a complicated way. The composite fermion provides an simple and elegant way for physicists to explore the many aspects of a strongly correlated electron system. In this thesis, the basic aspects of the composite fermion theory are discussed in Chapter 2. We will focus, in this chapter, the composite fermion model that ignores interactions between composite fermions. This model successfully explains much of the physics of fractional quantum Hall effect. In Chapter 3 through Chapter 7, the physics of interacting composite fermions will be addressed.

20 Chapter 2 Quantum Hall Effects and Composite Fermion Theory 2.1 Two-dimensional electron system A two-dimensional electron system (2DES) can be fabricated experimentally by a number of methods. For example, electrons can be confined on the surface of liquid helium.[ga79] This is possible because there exists a potential barrier of about 1 ev on the surface of liquid helium which prevents electrons from penetrating into the liquid. On the other hand, the image potential of liquid helium attracts electrons and binds them on the surface. Another example is the Si-MOSFET (metal-oxidesemiconductor field-effect transistor), in which electrons are restricted to the twodimensional (2D) interface between silicon and silicon oxide, as shown in Fig. 2.1 (a). The charge density can be controlled by tuning the voltage of the top gate. Although the above-mentioned methods are able to provide an effective 2D confinement, they are not an ideal choice for the study of the quantum Hall effect because of intrinsic disadvantages in the fabrication methods. In particular, the density of electrons on the surface of liquid Helium is rather low ( 10 9 cm 2 ); the overlap of the wave function between electrons is thus small, making the electrons behave like classical particles. The electron density in a Si-MOSFET, on the other hand, can be tuned to as high as cm 2. However, it is impossible to join the silicon (a crystal) and the silicon oxide (an amorphous) in a seamless way.

21 4 Figure 2.1. (a) Schematic diagram of a Si-MOSFET. The 2DES resides at the interface between silicon and silicon oxide. (b) Schematic diagram of a modulation-doped GaAs- AlGaAs heterostructure. Electrons are held against the AlGaAs by the electric field set up by the positively charged silicon dopants in AlGaAs. (c) Energy band diagram of a modulation-doped GaAs-AlGaAs heterostructure. The deformed energy band in the GaAs side is the result of the electrostatic potential from silicon dopants. Electrons are trapped in the triangular potential at the interface. The electron motion along the z-axis is quantized. At low temperature, only the lowest energy state is occupied. The figure is taken from Ref. [Stö99]. Electrons are scattered by the resulting imperfections when they are traveling along the interface. This scattering due to mismatched materials makes the mobility of electrons in a Si-MOSFET poor even at low temperatures, where most phonons are frozen out. Therefore a better 2DES can be realized if electrons are bound to a clean interface of two crystalline structures having similar lattice constants. These requirements are quite stringent; however, they can be realized with modulationdoped semiconductor heterostructures (semiconductors composed of more than one material). Two popular materials for making a pure 2DES are GaAs and AlGaAs. The

22 5 structure of a modulation-doped GaAs-AlGaAs heterostructure is depicted in Fig. 2.1 (b); high-quality GaAs layer is grown using the molecular beam epitaxy (MBE) technique. It is then covered by a AlGaAs layer. During this process, silicon donors are introduced at a distance of about 0.1 µm from the interface in order to provide free charges. These electrons will wander off to the GaAs side and become trapped because GaAs has a lower band energy than AlGaAs. The positively charged silicon ions set up an electrostatic potential that tends to drive electrons back to the AlGaAs side. This is prevented by the potential barrier due to the difference in band energies. The overall result is that electrons are squeezed against the interface, making the motion of electrons in the direction perpendicular to the interface highly constrained, while the motion along the interface is relatively free. Thus, modulation doping has achieved two benefits: it has separated electrons from their donors to reduce scattering by ionized impurities, and confined electrons to two dimensions. Moreover, the interface is clean because GaAs and AlGaAs have similar lattice constants. The samples made with this technique have extremely high mobilities on the order of 10 7 cm 2 /V sec. The electron densities can also reach as high as cm The Hall Effect: Classical and Quantum Classical Hall Effect The Hall effect was discovered by Edwin Hall in 1879 when he ran a current I through a thin sheet of gold and measured two characteristic voltages (See Fig. 2.2 for the geometry of the experiment). One is the voltage V along the current path I, the other is the voltage V H across the current path. Nothing interesting happened until he applied an external magnetic field B perpendicular to the metal sheet. Hall discovered that the magnetic field induced a nonzero V H which was proportional to B. The origin of this phenomenon is classical electrodynamics. The magnetic field generates a sideways Lorentz force on the electrons, which then accumulate on one side of the metal sample. The accumulation of charge ultimately sets up a voltage across the sample. A simple homogeneous isotropic model gives the Hall

23 6 Figure 2.2. Geometry of the Hall measurement. The Hall resistance R H and longitudinal resistance R are measured as a function of magnetic field B and current I. V H denotes the Hall voltage across the current path. V represents the longitudinal voltage drop. The figure is taken from Ref. [Stö99]. resistance R H, defined as the ratio of V H to the current I, as R H = B ρec, (2.1) where B is the magnetic field, e is the electron charge, c the speed of light, and ρ is the charge density. Note that R H does not depend on any material parameters except the charge density ρ. More remarkably, R H has no dependence on the geometry of the sample. Because of its universality, the Hall resistance measurement has become a standard tool for the determination of the density of electrons in conductors or semiconductors Integral Quantum Hall effect Almost a hundred years after the discovery of Hall, in the year 1980, von Klitzing, Dorda, and Pepper discovered what is now called the integral quantum Hall effect (IQHE) in their Hall measurement on a 2DES in a Si-MOSFET at low temperatures (T 2 K) and high magnetic fields (around 10 T).[vKDP80] Rather than Hall s linear relationship Eq. (2.1), they observed a stepwise dependence of the Hall resistance as a function of magnetic field. Typical experimental measurements of the IQHE are shown in Fig The value of R H at the plateau of steps is

24 7 Figure 2.3. IQHE data for a 2DES in GaAs-AlGaAs. The Hall resistance forms welldefined plateaus at filling factors ν = 1, 2, 3, and 4 (See Eq. (2.4) for the definition of ν). The longitudinal resistance also develops deep minima at these fillings. The figure is taken from Ref. [Stö99]. quantized to a few parts per billion to R H = h ie2, (2.2) where i is an integer and h is Planck s constant. The quantization is so accurate that the value h/e 2 has become the standard of resistance in Concomitant with the quantization in the Hall resistance, the longitudinal resistance drops to zero in the limit of zero temperature. To understand the origin of steps and minima, we must first understand the motion of electrons moving on a two dimensional (2D) plane penetrated by a perpendicular magnetic field. 1 Let us recall that the classical trajectory of an electron moving in a 2D plane penetrated by a perpendicular magnetic field is a circular orbit, and its radius can vary continuously. In quantum mechanics, (assuming the magnetic field with strength B is perpendicular to the 2D plane), 1 The quantum mechanical problem of electrons moving on a 2D plane in a perpendicular magnetic field was first solved by L.D. Landau in 1930 s. Details of the discussion can be found in many quantum mechanics textbooks, for example in the complement E V I of Chapter 6, Ref. [CTDL77]. We will not discuss these details in the thesis.

25 8 if the symmetric gauge is chosen, the radius of the circular orbit is quantized. Its corresponding energy is given as E n = ( n + 1 ) ω c, (2.3) 2 where n = 0, 1, 2,..., ω c = eb/mc is the cyclotron frequency, with m being the electron mass, and = h/2π. The quantized energy ladders E n are called Landau levels (LLs). Each LL is highly degenerate with degeneracy per unit area given by g LL = B/φ 0, where φ 0 = hc/e is a quantum of magnetic flux. The filling factor ν is defined as the ratio of the charge density ρ and the LL degeneracy g LL ν = ρ g LL. (2.4) Using these definitions, the Hall resistance formula Eq. (2.1) can be massaged into the form R H = B ρec = B ρ e hc h e = 1 ( ) h. (2.5) 2 ν e 2 This is exactly the value of the quantized Hall resistance if ν is an integer. Indeed, the above derivation gives the value of the quantized R H at the center of the Hall resistance plateau. The remaining mystery is why R H remains at this value when electrons are added or taken away. The answer to the existence of the plateau lies, surprisingly, in the presence of disorder due to the random potential from impurities or defects at the interface. The disorder brings about two changes. Firstly, it lifts the degeneracy of a LL, turning it into a band of states. Therefore, the excitation gap E, which is welldefined before disorder is introduced, no longer exists. Secondly, single particle states around the center of each LL band E n = (n + 1/2) ω c remain extended. Other than those, all states in between the LLs are localized and do not contribute to the transport. For this reason, it is said that there exists a mobility gap. This process is depicted schematically in Fig Let us now assume that the Fermi energy lies somewhere between two neighboring LL bands, i.e., in a mobility gap. Since localized states play no part in conduction, it follows that, as long as the Fermi level lies in the mobility gap, adding or removing electrons (which can be achieved by adjusting the density ρ

26 9 Figure 2.4. (a) Density of states n(e) of Landau levels in a clean system. Each level is a sharp δ-function. The energy gap is E = ω c. (b) Density of States n(e) of a Landau level in a disordered system. A band of extended states is located at the center of each level with localized states in between. or the field strength B) will not change the value of R H. Only when the number of occupied extended states changes can the Hall resistance increase or decrease. This happens when the Fermi energy crosses the central region of the Landau band. This explains the existence of plateaus in R H Fractional Quantum Hall effect A more surprising discovery was made by Tsui, Störmer, and Gossard in 1982 when they found the totally unexpected result of a Hall resistance measurement for a 2DES,[TSG82] shown in Fig Let us recall that, in magnetic fields, the kinetic energy of an electron is quantized into discrete Landau levels populated by electrons. At extremely high fields, all electrons fall into the lowest Landau level. In this case, the kinetic energy becomes irrelevant and the physics is determined solely by the Coulomb interaction between electrons. It was predicted by Wigner[Wig34] that when the interaction between electrons becomes stronger than their kinetic

27 10 Figure 2.5. Original data of the first observation of the ν = 1/3 FQHE. The measurement was done on a modulation-doped GaAs/AlGaAs sample. At T = 0.48K, the Hall resistance ρ xy has a well-defined plateau at the B field corresponding to the 1/3 filling. ρ xx, the magnetoresistance has a minimum at the same B field. This figure is taken from Ref. [TSG82] energy, electrons form a crystal in order to minimize the energy. This line of reasoning motivated Tsui, Störmer, and Gossard to repeat the IQHE experiment for ν < 1 hoping to observe signs of the electron crystal. In addition to the IQHE, they observed, unexpectedly, a plateau on which the Hall resistance R H is quantized at R H = h e2. (2.6) The filling factor ν = 1/3 is a fractional number; hence this phenomenon is named the fractional quantum Hall effect (FQHE). Phenomenologically, this observation was identical to that of the IQHE except that the Hall resistance was quantized at R H = h/(fe 2 ) with f = 1/3, i.e., the lowest LL is partially occupied. The single particle picture used to explain the IQHE is no longer correct, for the ground state becomes highly degenerate. The

28 11 origin of the FQHE must be due to the electron-electron interaction. However, how a collection of strongly correlated electrons manages to form a unique ground state was unknown at that time. In 1983, Laughlin[Lau83] proposed a trial wave function for the ground state of the FQHE at filling factors ν = 1/m where m is an odd integer. This wave function is written as Ψ = j<k (z j z k ) m e 1 P 4l 2 k z k 2 0, (2.7) where the complex number z k = x k iy k denotes the electron coordinates in a 2D plane, l 0 = c/eb is the magnetic length. It was shown by Laughlin[Lau83] that this wave function has high overlap with the exact ground state obtained by diagonalizing the Coulomb Hamiltonian for a small number of particles. The energy of Ψ is lower than the energy of a charge density wave or of an electron crystal. It describes an incompressible quantum liquid of electrons and its elementary excitation has a fractional charge of e /3. Is this the end of the story? Apparently not. Experimentally, following the progress in semiconductor fabrication techniques, more and more fractions were uncovered. The fractions shown in Fig. 2.6 are only some of them. Theoretically, Laughlin s wave function only explains the FQHE state at ν = 1/m where m is an odd number. Much effort[hal83a] was expended to generalize Laughlin s theory to fractions like 2/5, 3/7, 4/9,.... However, no satisfactory theory was found. Moreover, since the IQHE and FQHE are so similar phenomenologically, is there any connection, at least conceptually, between the two? Is there any theory that can possibly unify the integral and fractional QHE? What is the complete picture? Therefore,[GV05]... it seems fair to say that Robert Laughlin only scratched the surface when he introduced his celebrated wave function for the fractional quantum Hall effect.

29 12 Figure 2.6. The FQHE in ultrahigh mobility modulation-doped GaAs-AlGaAs 2DESs. The most prominent fractions form a series ν = n/(2n ± 1), n an integer. The figure is taken from Ref. [Stö99]. 2.3 Composite Fermion Theory Overview Technically speaking, many-body problems are usually hard, sometimes even impossible, to solve exactly. Indeed, in an interacting system, the constituent particles looses their identity since their motion is determined by the dynamics of other particles. The collection behaves as a whole. However, breakthrough and progress can be made once the true degrees of freedom of new emergent particles of the correlated system are identified. That is to say, in a correlated system, the interaction between particles gives rise to new entites, usually weakly interacting, that are relevant to the problems being pursued. For example, the problem of lattice vibration is complicated at first glances, for the coupling between neighboring atoms makes it almost impossible to solve the equation of motion directly. However, the equations of motion become uncoupled if one makes a transformation to the

30 13 degrees of freedom of phonons. Another example is the quasiparticle concept in Landau s Fermi liquid theory. Being nearly free, Laudau s quasiparticles explore the low energy degrees of freedom and can be used to describe the low energy physics for a collection of strongly interacting electrons in metals. In the problem of FQHE, the bare particles are electrons living in the lowest Landau level. The Hamiltonian consists of just the Coulomb interaction 2 H = i<j e 2 ǫ r i r j, (2.8) where ǫ is the dielectric constant of the host material. Simple as it is, the Hamiltonian H is a difficult one to solve. Perturbation theory can not be used as there is no small parameter around which to expand. The only energy scale is set by the Coulomb interaction itself. For systems with small numbers of electrons, however, H can be numerically diagonalized. The information extracted from exact diagonalization includes the energy spectrum, pair correlation functions, etc. While this information is important in understanding the nature of the system, they do not provide any clue for why some fractions exist but others do not. Moreover, a prediction of the existence of general filling factors cannot be made without diagonalizing the system at those fillings for all possible system sizes. The procedure becomes very difficult and cumbersome due to the fact that, since the basis size grows extremely fast with the number of particles, exact diagonalization is usually limited to a small cluster of particles. Owing to these difficulties, one may ask: what are the true degrees of freedom in the problem of FQHE? Or, equivalently, what are the true particles of FQHE? The answer is provided by the composite fermion theory, which identifies the weakly interacting quasiparticles responsible for the physics of the FQHE. 2 Here we assume that electrons in the lowest LL are spin polarized. Effectively, they can be considered as spinless because the spin degrees of freedom are fixed. The Hamiltonian in Eq. (2.8) is a strict two-dimensional spinless model. In reality, samples have finite extension in the direction along the external magnetic field. It is known that, for a gapped FQHE state, the effect of finite sample thickness is to reduce the size of the gap.[zs86] (The state will be gapless eventually if the sample thickness is too large.) In this thesis, however, we will not consider the finite thickness effect. Also, disorders will not be included explicitly as a quantitative method.

31 HALL RESISTANCE r xy ( h / e 2 ) 14 LONGITUDINAL RESISTIVITY r (k W ) xx 0.5 T=35 mk n= MAGNETIC FIELD B (tesla) 10 LONGITUDINAL RESISTIVITY r (k W ) xx n= MAGNETIC FIELD B (tesla) Figure 2.7. The comparison of the FQHE and the IQHE. The top (bottom) panel shows the IQHE (FQHE) of electrons. (The IQHE and FQHE data are obtained from Ref. [CNU + 86] and [ST97], respectively). The correspondence between prominent features in the longitudinal resistance are clearly shown. This figure is taken from Ref. [Jai00] Composite Fermions In 1989, Jain[Jai89a, Jai89b] made a key postulate: the FQHE of electrons can be viewed as the IQHE of a new type of emergent particles called composite fermions. A crucial observation that led to this assumption was that the phenomenon of the FQHE is very similar to the phenomenon of the IQHE. Both exhibit plateaus in the Hall resistance concomitant with a vanishing longitudinal resistance. In fact, if one were to erase all of the numbers labeling the quantum Hall states in Fig. 2.7, one would be hard pressed to identify whether a given plateau belongs to the integral

32 15 Figure 2.8. The transformation from strongly interacting electrons (left panel) to weakly interacting composite fermions (right panel). Electrons are represented by filled circles, magnetic flux is depicted by arrows. After the transform, each electron is bound to 2 flux quanta, resulting in a reduced effective magnetic field. or fractional quantum Hall effect. By definition, a composite fermion (CF) is an electron bound to an even number 2p (p is an integer) of quantum mechanical vortices. (It has fermionic characteristic because the number of vortices attached is an even integer.) In this sense, a CF is an emergent many-body object: for it to exist, it requires other particles in the system. The bound vortices produce phases that cancel part of the Aharanov- Bohm phases arising from the external magnetic field. The result is that the CFs experience (at the mean field level) an effective field B equal to B = B 2pρφ 0. (2.9) This process is shown schematically in Fig Like the kinetic energy of electrons is quantized into discrete LLs, in the lowest LL the kinetic energy of the CFs is also quantized into CF quasi-lls. To distinguish between the LLs of electrons and the quasi-lls of CFs, the quasi-lls of CFs are named as Λ-LLs. The spacing between two neighboring Λ-LLs is, according to Eq. (2.9), ωc where the effective cyclotron frequency ωc = eb /m c with m being the effective mass of a CF. We note that vortices are topologically equivalent to magnetic flux quanta. An electron moving in a closed loop around a flux quantum or a vortex picks up the same Aharanov-Bhom phase. It is for this reason that a CF is often imagined as the bound state of an electron with 2p flux quanta. However, it should always be borne in mind that the vortices are not flux quanta and that the flux quanta description is only used for explanatory purpose and as an intuitive guide. How does the CF theory relate the FQHE of electrons to the IQHE of CFs?

33 16 Because CFs feel a reduced magnetic field, they have an increased effective filling factor ν = ρφ 0 / B. Using Eq. (2.9), ν can be related to the electron filling factor ν by ν = ν 2pν ± 1. (2.10) The minus sign in the above equation refers to situations when the effective magnetic field is directed opposite to the external field. If ν is an integer, say n, the CFs fill an integral number of Λ-LLs, which produces a gap and hence the IQHE. The non-interacting CF model predicts FQHE at ν = n 2pn ± 1. (2.11) These values are precisely the values where the most prominent FQHE is experimentally observed. For example ν = 2 of CFs corresponds to ν = 2/5 of electrons Microscopic Wave Functions The CF picture described in the previous section provides a unifying scheme for all integral and fractional fillings. However, to test the validity of the picture, microscopic calculations are needed. The CF theory also gives a systematic way of writing down the wave function at electron filling factors ν = n/(2pn ± 1). According to the CF theory,[jai89a, Jai89b] the strongly interacting electron system at filling factor ν, described by a many-body wave function Ψ ν, is mapped into a system of CFs at ν, described by a wave function Φ ν, where Ψ ν = P LLL Φ 2 1 Φ ν. (2.12) Φ 1 is the wave function of non-interacting electrons at filling factor ν = 1, and P LLL is the lowest LL projection operator. In the disk geometry, Φ 1 takes the form Φ 1 = j<k (z j z k ) e 1 P 4l 2 k z k 2 0. (2.13) Φ 1 is called the Jastrow factor because it incorporates pairwise correlations between the electrons. Multiplying Φ 2p 1 with any electronic wave function has the effect of attaching 2p vortices to each electron, turning it into a composite fermion.

34 17 The lowest LL projection operator P LLL is necessary, in general. For example, at ν = n, the wave function Φ n describes CFs filling n Λ-LLs. To obtain the wave function Ψ ν for electrons, which reside in the lowest LL with filling factor ν, one must project out components from higher LLs. The projection procedure goes as follows. It has been shown that the Hilbert space of the lowest LL consists of analytic functions of complex coordinates z j of electrons.[gj84] If Φ ν contains z j, where z j denotes the complex conjugate of z j, then the wave function Φ ν non-analytic. The lowest LL projection operator then moves all the z s to the left of all the z s in Eq. (2.12) and makes the replacement z 2 / z. These derivative operators then act on the remaining part of the wave function including the factor Φ 2p 1. (The derivatives do not operate on the Gaussian part of the wave function.) The process is quite complicated. However, a systematic technique developed by Jain and Kamilla[JK97a] has made it possible to perform the projection for wave functions containing many Λ-LLs. In principle, Eq. (2.12) can be used to create a wave function for electrons at arbitrary ν. In particular, at filling factors ν = n/(2pn ±1) where ν is an integer, the wave function Φ ν (hence Ψ ν ) consists of a single Slater determinant. Away from ν = n/(2pn ±1), there is more than one possibility in writing down the wave function Φ ν. In this situation, one must diagonalize the Hamiltonian within the basis spanned by these correlated CF wave functions. This process is called the composite fermion diagonalization (CFD).[MJ02, JCJ04a, JCJ04b] This method is described and used in many works described in this thesis. To close this section, we would like to stress that the CF wave function is a variational wave function, albeit with no free parameters. Its energy provides a strict upper bound to the exact energy as long as the wave function contains the symmetries of the problem. Fortunately, the CF theory creates variational states that are essentially exact solutions of the Hamiltonian throughout a range of filling factors. is 2.4 Remaining Thesis Plan The plan for the remainder of this thesis is as follows. Recently, experiments using very clean 2DES samples in extremely high magnetic fields promising signs of new

35 18 FQHE states at fillings between 1/3 and 2/5, for example 4/11 and 5/13.[PST + 02b, PST + 03] These new fractions do not fit into the sequence ν = n/(2pn±1), where n and p are integers. In Chapter 3, by incorporating the residual interaction between composite fermions, we (in collaboration with S.S. Mandal and J.K. Jain) show that these new FQHE states at filling factors between 1/3 and 2/5 can be explained in terms of a fractional QHE of composite fermions. In Chapter 4, we (in collaboration with G.S. Jeon and J.K. Jain) develop a perturbative scheme based on the correlated basis function of the composite fermion theory, that allows a systematic improvement of the wave functions and the energies for low-lying eigenstates. We study systems for which ED results are known, and find that the microscopic CF theory provides an excellent account of the ground state as well as low energy excitations of few interacting electrons in a semiconductor quantum dot at strong magnetic fields. In Chapter 5, by a direct comparison with the exact eigenstate, we (in collaboration with G.S. Jeon and J.K. Jain) demonstrate that the crystal state at low filling factors is not a simple Hartree-Fock crystal of electrons but an inherently quantum mechanical crystal of composite fermions. A system of neutral bosons in a rapidly rotating trap is equivalent to charged bosons coupled to a magnetic field, which opens up the possibility of a FQHE for bosons interacting with a short-range interaction. In Chapter 6, motivated by the CF theory of the FQHE for electrons, we (in collaboration with N. Regnault, T. Jolicoeur and J.K. Jain) test the idea that the interacting bosons map onto non-interacting spinless fermions carrying one vortex each. We study the analogy between interacting bosons at filling factors ν = n/(n + 1) with non-interacting fermions at ν = n for the ground state as well as low-energy excited states. We find that the analogy provides a good account of the behavior at small n, but interactions between fermions become increasingly important with n. Finally in Chapter 7, we (in collaboration with C. Töke, G.S. Jeon and J.K. Jain) study the competition between the composite-fermion-crystal and liquid orders at filling factor ν = 1/5. We show that the state at ν = 1/5 is better described as a composite-fermion crystal than as a Laughlin liquid for small systems, but the latter prevails for systems containing more than 10 particles. A variational combination of the two wave functions provides an extremely accurate wave function

36 19 for small systems. This indicates that the ν = 1/5 fractional quantum Hall liquid is highly susceptible to the formation of a composite fermion crystal in it.

37 Chapter 3 The next generation FQHE 3.1 Partially Spin Polarized In recent years, the physics arising from the interactions between composite fermions has come into focus.[pst + 03, PST + 02b, DPK + 03, MJ02, PJ00, WQ00, LSJ02] 1 The model of noninteracting composite fermions explains the quantization of the Hall resistance[tsg82] at R H = h/fe 2 with f = n 2pn ± 1 (3.1) as the integral quantum Hall effect[vkdp80] of composite fermions.[jai89a, Jai00] (Particle-hole symmetry in the lowest Landau level implies fractional Hall effect also at 1 f or 2 f, for fully or partially spin-polarized systems, respectively.) The weak residual interaction between composite fermions (CFs) often masked by disorder or temperature, much as in the fractional quantum Hall effect (FQHE), a manifestation of inter-electron interactions, is absent in low mobility samples or at high temperatures. However, with improvements in experimental conditions, the physics originating from the interaction between composite fermions is beginning to emerge.[pst + 03, PST + 02b, DPK + 03] A possible manifestation of the interaction between composite fermions will be 1 Section 3.1 was done in collaboration with S.S. Mandal and J.K. Jain. The result was published in Physical Review B 67, R (2003). Section 3.2 was done with J.K. Jain and it was published in Physical Review Letter 92, (2004).

38 21 the appearance of higher order FQHE states in between the above fractions. The possible fractions can be derived straightforwardly.[mj02, PJ00, Jai89a] At the nonintegral values of the CF filling factor given by ν = n + m 2p m ± 1, (3.2) the composite fermions in the topmost partially filled level may capture, as a result of the interaction between them, 2p additional vortices to transform into higher-order composite fermions and exhibit new incompressible states, which will produce FQHE at ν = ν /(2pν ± 1) between the fractions ν = n/(2pn ± 1) and ν = (n+1)/[2p(n+1) ±1]. The situation is analogous to the appearance of FQHE of electrons in partially filled electronic Landau levels. Pan et al.[pst + 03] have reported FQHE at 4/11 and 5/13, and ν = 6/17 in the filling factor range 2/5 > ν > 1/3. The ν = 4/11 minimum is seen in magnetic fields as high as B = 33T, pointing to a fully polarized QHE state. The simplest fractions in the scenario are ν = 1 + 1/3 and ν = n + 2/3, which produce highorder FQHE at ν = 4/11 and ν = 5/13, and ν = 6/17 originates from ν = 1+1/5. While it is encouraging that the desired fractions are obtained, more detailed theoretical investigations do not find new FQHE states for these fractions for fully spin-polarized composite fermions for an idealized model neglecting disorder, transverse thickness, and Landau-level mixing;[mj02, WQ00, LSJ02] the residual interaction between composite fermions in higher CF levels does not appear to be sufficiently strongly repulsive at short distances to cause additional vortices to bind to composite fermions. It is not known at the present which of the neglected effects is responsible for the discrepancy. While only the fully spin polarized states are possible at sufficiently high magnetic fields, where a spin in the wrong direction costs a prohibitively high energy, FQHE states at the fractions of Eq. (3.1) with partial polarizations have been observed experimentally, and transitions between differently polarized states have been studied as a function of the Zeeman energy.[dys + 95, DYS + 97, NLD + 96, KKE99] These studies are well described by the composite-fermion theory including spin.[wdj93, PJ98] Composite fermions with spin are analogous to electrons with spin at ν, with the same Zeeman energy but with an effective cyclotron

39 22 energy. While the Zeeman energy is very small compared to the cyclotron energy for electrons (in GaAs), the two are comparable for composite fermions, thus producing a rich variety of states.[dys + 95, DYS + 97, NLD + 96, KKE99] This raises the question of whether FQHE states with partial spin polarization are possible at fractions like ν = 4/11, which is the subject of this chapter. It is stressed that the present work does not purport to be an explanation for the observation in Ref. [PST + 03] but a prediction for sufficiently low magnetic fields. Following the standard approach, we will neglect in this study the effect of finite thickness and Landau-level mixing. The most reliable method is exact numerical diagonalization, which is not an option for this problem because of the rather large Hilbert space. In this chapter, we carry out diagonalization in a truncated low-energy CF basis of many-body states.[mj02] We will concentrate here on the filling factors range 2/5 > ν > 1/3, that is 2 > ν > 1. The FQHE is most likely at ν = 1 + m/(2m + 1), which corresponds to electron filling factors ν = (3m + 1)/(8m + 3). The positive and negative integral values for m produce ν = 4/11, 7/19,..., and ν = 5/13, 8/21,.... It will be assumed that the up-spin composite fermions fill one level completely, and the down-spin composite fermions have filling ν = m/(2m+1) in the lowest spin reversed band, giving the total CF filling ν = 1 + ν. For our truncated basis, we consider a wave function of the form: Φ α (3m+1)/(8m+3) = Φ 2 1[Φ 1, Φ α m/(2m+1), ]. (3.3) Here, Φ 1, is the fully occupied up-spin lowest Landau level band, and Φ α m/(2m+1), are various orthogonal wave functions (labeled by α) at filling ν = m/(2m + 1) in the down-spin band, obtained by exact diagonalization at ν. (Coupling to higher Landau levels are neglected.) The fully antisymmetric function Φ 1 is one filled Landau level of spinless electrons; the Jastrow factor, as always, converts, through attachment of two vortices to each electron, the ν = 1+m/(2m+1) wave function of electrons in square brackets to the ν = 1 + m/(2m + 1) wave function of composite fermions, which is identified with a basis function for interacting electrons at ν = (3m + 1)/(8m + 3). (The spin part of the wave function is not explicitly shown. The full wave function is obtained by multiplying by the spin part u 1...u N d N +1...d N, where N = N N is the number of up-spin electrons, followed by antisymmetrization.) We will study cases in the below with m = 1

40 23 and 2 (ν = 4/11 and 7/19). The states of the form given in Eq. (3.3) obviously do not exhaust the entire Hilbert space at ν, as they neglect the mixing between Landau levels of composite fermions, but we believe that they span the low-energy Hilbert space. If the system is incompressible, the ground state is likely to be well described by Ψ gr ν = Φ2 1 [Φ 1, Ψ gr, ], (3.4) where Ψ gr ν, is the Coulomb ground state at ν. For ν = (3m + 1)/(8m + 3), the state at ν = m/(2m + 1) is accurately given by the standard wave function PΦ 2 1, Φ m,, where P denotes projection into the lowest Landau level. It is noteworthy that no assumption is made regarding the nature of the state in the reversed-spin sector, and the calculation can, in principle, give either a compressible or an incompressible state. Indeed, a similar study for fully spinpolarized systems at many fractions like ν = 4/11 failed to yield an incompressible ground state[mj02], contrary to what one might have naively expected. An earlier study[pj00] began with the assumption of the partially spin polarized ground state ν Ψ gr ν=4/11 = Φ2 1 Φ 1, [Φ 1, ] 3, (3.5) which is derived from Laughlin s wave function[lau83] [Φ 1, ] 3 in the spin-reversed sector,[hal83b] and considered a trial wave function for its neutral excitation containing a pair of CF particle-hole pair in the reversed-spin sector. It was found that the energy of the excitation remains positive for all wave vectors, indicating that the assumed ground state wave function is stable against excitations. While this study did not eliminate FQHE at 4/11, it did not test whether the ground state is necessarily incompressible, and if so, whether it is well described by the trial wave function in Eq. (3.3). The present study provides a more rigorous (though still not conclusive) test for partially polarized QHE at 4/11. In the following discussion we will employ the spherical geometry,[hal83a, WY76] where we consider N electrons moving on the surface of a sphere in the presence of a magnetic monopole with strength Q at the center. The magnitude of the radial magnetic field B is given by 2Qφ 0 /4πR 2, where φ 0 = he/c is the flux quantum, R is the radius of the sphere, and Q is either an integer or a half-integer due to the Dirac quantization condition. The composite-fermion theory maps the

41 24 ν ν N Q N q Overlap (%) 4/11 1+1/ /19 1+2/ Table 3.1. The systems studied in this work. N is the number of particles in the reversed spin sector. system of interacting electrons at Q to the CF system at q = Q p(n 1). It is convenient to label the wave function by the monopole strength; for example, the wave function at Q is obtained from the electron wave function Φ q Ψ Q = P [Φ 2p 1 Φ q ]. The CF theory fixes the relation between N and Q as follows. The effective q is determined by requiring that N electrons have filling ν = m/(2m + 1): q = N (2m + 1)/2m (m + 2)/2. With N = 2q + 1 and Q = q (N 1), we get Q = 8m + 3 6m + 2 N m2 + 10m + 3. (3.6) 6m + 2 Therefore, at ν = 4/11, where the effective filling ν = 1 + 1/3 with m = 1, the relation is Similarly, for ν = 9/17, where ν = 1 + 2/5 and m = 2, Q 4/11 = 11 8 N (3.7) Q 9/17 = N (3.8) Note that both relations give the desired filling factors at the thermodynamic limit: N ν = lim N. For a given particle number N, the pair (N, Q) is the only input 2Q in the numerical calculation. Table 3.1 gives the systems we have studied below. The energy spectrum is calculated numerically by the Monte Carlo (MC) method, following Ref. [MJ02] Because the low-energy basis states from exact diagonalization are not necessarily orthonormal for a given angular momentum L, we use the standard Gram-Schmidt procedure to obtain an orthonormal basis. by

42 25 E [e 2 /εl 0 ] N=6 E [e 2 /εl 0 ] N= L L E [e 2 /εl 0 ] N=14 E [e 2 /εl 0 ] N= L L Figure 3.1. The energy spectrum at ν = 4/11 for N = 6, 10, 14, and 18 particles, respectively. It is assumed that the state has partial spin polarization. The single particle energy E includes the interaction with the uniform, positively charged background. The quantity l 0 = c/eb is the magnetic length, and ǫ is the dielectric constant of the background material. The error bars show one standard deviation in the Monte Carlo simulation. Some technical details ought to be mentioned here. The Metropolis algorithm employed in our MC calculations has minimum statistical error when the sampling function behaves approximately as the wave function. We find that it is crucial to use several sampling functions with different angular momentum L in order to reduce the error to the desired level. We divide the MC calculations in 10 configurations, with the number of iterations on the order of 10 7 for each configuration. Such large numbers of steps are required to determine extremely small energy differences accurately. To reduce the computation time for large system, for example, N = 14, 18 at ν = 4/11, we place each of the MC configuration on a single node (dual 1GHz Intel Pentium III processor) of a PC cluster. Fig. 3.1 shows the low-energy spectrum at ν = 4/11 for N = 6, 10, 14, and 18, for which there are N = 2, 3, 4, and 5 composite fermions in the spin reversed CF

43 Ε [ e 2 /εl 0 ] /N E [ e 2 /εl 0 ] /N Figure 3.2. Neutral excitation gap and the ground state energy at ν = 4/11 as a function of N 1, N being the number of electrons. Figure 3.3. The energy spectrum at ν = 7/19 for (a) N = 11 and (b) N = 18 particles and at ν = 6/17 for (c) N = 8 and (d) N = 14 for a partially spin polarized system.

44 27 level, respectively. The dimensions of the basis are the same as that of the lowest Landau level Hilbert space of N particles at q. In all cases, the ground state is a uniform state with L = 0. From the analogy to ν = 1/3, it might be expected that the excitation spectrum contains a well-defined branch of composite-fermion exciton,[dj92] containing a single multiplet at L = 2,...,N. This CF exciton branch is identifiable for N = 14 but not at N = 18. Nonetheless, there is a well defined gap in all cases. Fig. 3.2 shows the N dependence of the minimum energy needed for the creation of a single CF exciton. There is a substantial finite size fluctuation s in the value of the gap, because the number of spin-reversed composite fermions is quite small, but we believe that our results indicate that the gap remains finite in the thermodynamic limit, producing incompressibility at 4/11. We will use the gap of the largest system studied as a rough estimate of the thermodynamic gap. The next fraction we consider is ν = 1( ) + 2/5( ), corresponding to a partially polarized state at ν = 7/19. As Fig.3.3 (a) and (b) show, the state here is also incompressible. A thermodynamic extrapolation for the gap is not possible for ν = 7/19, for we have only two results, but the gap for N = 18 is taken as an estimate of the thermodynamic limit. The gaps for ν = 4/11 and ν = 7/19 are estimated to be e 2 /ǫl 0 and e 2 /ǫl 0, respectively, which are roughly two orders of magnitude smaller than the gaps at ν = 1/3 and ν = 2/5, 0.1 e 2 /ǫl 0, and e 2 /ǫl 0, respectively (for the model neglecting transverse thickness). Here l 0 = c/eb is the magnetic length at ν, B is the external magnetic field, and ǫ is the dielectric constant of the host material. The gap at ν = 4/11 is consistent with that quoted in Ref. [PJ00]. It is noted that the gaps are not affected by the Zeeman energy, so long as the partially polarized state is the ground state, because the low-energy excitations of these states do not involve any spin-reversal. (The Zeeman energy is much higher, for typical experimental parameters, than the energy scales considered in this work, making spin-flip excitations irrelevant to the low-energy physics.) Figure.?? estimates the thermodynamic limit of the ground state energy at ν = 4/11 to be e 2 /ǫl 0, which also is in good accord with the value calculated earlier.[pj00] The smallness of the gaps for the higher-order FQHE states confirms that the interaction between the composite fermions in higher CF levels is exceedingly weak

45 28 compared to the Coulomb interaction between electrons that governs the gaps at ν = 1/3 and 2/5. It is remarkable that the composite fermion theory is capable of capturing such subtle quantitative physics, and that experiments have come to a stage where higher order FQHE states are now beginning to reveal themselves. Table 3.1 gives the overlaps of the ground state with the wave function of Eq. (3.4). The overlaps are fairly large, confirming that the trial wave function of Eq. (3.4) describes the ground state effectively; in other words, the physics of the FQHE at ν = 4/11 and ν = 7/19 is related to the ν = 1/3 and ν = 2/5 FQHE in the spin-reversed sector. We have also investigated the possibility of partially polarized QHE at ν = 6/17, which maps into ν = 1( ) + 1/5( ) of composite fermions. The theoretical spectra for 8 and 14 particles at Q 6/17 = (17N 22)/12, shown in Fig. 3.3 (c) and (d), provide an indication of an incompressible state here as well. Surprisingly, the gap for N = 14 is e 2 /ǫl 0, which is of the same order as the ν = 4/11 gap. However, the system sizes are effectively very small, as can be seen by the fact that there are only one or two basis functions at each angular momentum sector, which prevents us from making a reliable assertion regarding the presence of incompressibility at ν = 6/17. At the moment, we are unable to get sufficient accuracy at the next particle number (N = 20). Our study thus predicts that partially polarized high-order FQHE at fractions of the form ν = (3m + 1)/(8m + 3) should be possible in an appropriate range of Zeeman energy and temperature. The temperature scale set by the gap is on the order of 150 mk (at B = 10T) for GaAs, which is an upper limit because corrections due to finite transverse thickness and disorder are expected to suppress the gap substantially. It it at present not possible to ascertain theoretically the relevant Zeeman energy scale, for the lack of a quantitative understanding of the full polarized states at these filling factors. The modifications due to finite thickness, not included above, are also of relevance. Similar considerations may also be useful for the QHE-like features seen previously at ν = 7/11 in very low-density samples.[gs90] The relevance of our results to the Raman experiment[dpk + 03] in the filling factor range 2/5 ν 1/3, where the level structure of composite fermions is observed, also deserves further investigation.

46 Fully Spin Polarized Magnetic field quantizes the kinetic energy of electrons into so-called Landau levels, which is responsible for the phenomenon of the IQHE, namely a quantization of the Hall resistance at R H = h/νe 2, with ν = n being an integer. At very strong magnetic fields, electrons fall into the lowest Landau level (LL), where the physics is entirely governed by the repulsive Coulomb interaction. To minimize their interaction energy, they capture an even number (2p) of quantized vortices each to turn into composite fermions (CF s), which experience an effective magnetic field, and form their own Landau-like levels, termed CF-quasi-Landau levels. The number of occupied CF-quasi-Landau levels, ν, is related to the filling factor of the lowest electron LL, ν, according to the formula ν = ν /(2pν ± 1). In particular, the IQHE of composite fermions (ν = n) provides an explanation of the FQHE of electrons at R H = h/νe 2, with ν = n/(2pn ± 1). The observation of fractions such as 4/11 and 5/13 points to new physics beyond the IQHE of composite fermions. It is natural to ask if the residual interaction between composite fermions can produce such fractions, in the same way as the interaction between electrons produces the FQHE. In other words, do these fractions represent a fractional QHE of composite fermions? For example, consider composite fermions carrying two vortices (p = 1). If they were completely noninteracting, only ν = n/(2n ± 1) would be obtained. However, if an interaction between them could produce FQHE of composite fermions at ν = 1 + ν = 1 + m 2m ± 1, (3.9) (m =integer) that would result in new electron fractions in the range 2/5 > ν > 1/3, given by ν = 3m ± 1 8m ± 3. (3.10) These include the newly observed fractions. (We will consider here only fully spin polarized states, i.e., effectively spinless fermions, so ν = 1+ ν has the lowest LL completely full and a filling ν in the second LL. In actual experiments, the filling factor ν in the second second LL is obtained at the total filling factor ν = 2 + ν.) While this qualitative picture is intuitively appealing, quantitative investiga-

47 30 tions have cast doubt on it. A FQHE at ν requires that the state here be incompressible. (An incompressible state has a uniform ground state with a gap to excitations.) One can ascertain if the state is incompressible using an exact numerical diagonalization for electrons for small systems, or CF diagonalization [MJ02] (outlined below) for larger systems. Extensive studies [MJ02, Rez, WQ00] of 4/11 as a function of the number of electrons, N, have found that the state is incompressible for N = 12 and 20 but not for N = 8, 16 and 24. While the message was mixed, it was on the whole taken to mean that the results do not support, in the thermodynamic limit, a fully polarized FQHE at ν = 4/11. That conclusion, however, is incompatible with experiment [PST + 03], which show a clear evidence for a fully polarized FQHE at ν = 4/11. We explain below the origin of the intriguing behavior for finite systems, why it does not rule out incompressibility in the thermodynamic limit, and then go on to write explicit wave functions that bring out the underlying physics. The calculations below will consider N electrons on the surface of a sphere, moving under the influence of a radial magnetic field B produced by a Dirac monopole of strength Q at the center, which produces a net magnetic flux of 2Q, in units of φ 0 = hc/e, through the surface. (2Q is an integer according Dirac s quantization condition). CF diagonalization refers to determining the low-energy spectrum by numerically diagonalizing the Hamiltonian in the correlated CF basis: {P LLL Φ 2p 1 Φα Q } (3.11) Here {Φ α Q } is an orthogonal basis of N-electron states at Q = Q p(n 1). We will be interested in the filling factor range 2/5 > ν > 1/3, for which the effective filling at ν lies between one and two (with p = 1). We will include all electron basis states at Q which have the lowest LL completely occupied and the second partially occupied. Φ 1 is the wave function for a fully occupied Landau level, and P LLL denotes projection into the lowest Landau level. The basis states in Eq. (3.11) are in general not linearly independent. We extract an orthogonal basis from it following the Gram-Schmidt procedure and then diagonalize the Coulomb Hamiltonian to find eigenstates and eigenenergies. The Hamiltonian matrix elements are evaluated by the Metropolis Monte Carlo method. (All energies will be quoted in units of

48 31 e 2 /ǫl 0, where ǫ is the dielectric constant of the background semiconductor, and l 0 c/eb is the magnetic length.) The statistical uncertainty is determined by performing many ( 10) Monte Carlo runs, with iterations in each run. The basis states are, by construction, in the lowest Landau level, so our results provide strict variational bounds on the ground state energy in the limit B. The eigenstates have definite orbital angular momentum, L, with L = 0 for uniform states; the state is compressible for L 0. The ground state obtained by CF diagonalization will be denoted Ψ 0 ν. Details of lowest LL projection and diagonalization can be found in the literature [MJ02, JK97a]. The state at filling factor ν of Eq. (3.10) is obtained at flux values given by [MJ02] 2Q = 8m ± 1 3m ± 1 N 12m ± (m2 + 3) 3m ± 1 (3.12) which ensures lim N N/2Q = ν = (3m ± 1)/(8m ± 1). Our discussion below will establish a close relation between FQHE in the lowest LL at ν (Eq. (3.10)) to FQHE in the second LL at ν = 1+ ν (Eq. (3.9)). Consider the electron state at ν = 1+ ν, with N particles forming a state with filling factor ν in the second LL. Given that the filled lowest LL is inert, one might expect that the state in the second LL at ν is quite similar to the corresponding state at filling factor ν in the lowest LL, but, in reality, there are striking differences between the two [Hal87, dr88]. Consider the example of ν = 1/3. For the 1/3 state in the lowest LL, the system is incompressible for all N, whereas for the 1/3 state in the second LL, the ground state is compressible (L 0) for N = 3 and 5, and almost compressible for N = 7 [dr88]. (The gap for N = 7 is e 2 /ǫl 0, a factor of 37 smaller than the gap at N = 6.) Furthermore, the ground state wave functions at 1/3 in the lowest and second LL s are rather different; the largest overlap between them is obtained for seven particles, which is only 61% [dr88]. In fact, it was initially thought [Hal87] that exact diagonalization studies rule out FQHE at ν = 1/3! Study of bigger systems revived the possibility that the ν = 4/3 state might be incompressible in the thermodynamic limit. (FQHE at 1/3 in the second LL has been observed experimentally [PXS + 99] with a small gap of 100 mk.) As summarized in Table 3.2, the behavior at ν is quite analogous. For example,

49 32 ν ν 2Q N N gap (ν) gap (ν ) /11 4/ (1) (2) /13 5/ (1) /19 7/ (2) Table 3.2. The gaps at ν and ν, determined from CF and exact diagonalization, respectively, at several filling factors. Only the gaps of incompressible states are shown. N is the total number of particles and N is the number of particles in the second LL for the state at ν. The gaps are quoted in units of e 2 /ǫl 0. The statistical uncertainty from Monte Carlo is shown in parentheses. including the electrons in the lowest LL, the state at ν = 4/3 is compressible for N = 8 and 16 particles ( N = 3 and 5) and almost compressible for N = 24 ( N = 7); these match the particle numbers for which ν = 4/11 has been found to be compressible. An implicit assumption was made in the reasoning of Ref. [MJ02]: if a state is incompressible in the thermodynamic limit, then all its finite size incarnations must also be incompressible. There is no known exception to this rule for lowest Landau level fractions of the type ν = n/(2pn ±1), but it is violated for FQHE states in higher (quasi-)landau levels of electrons or composite fermions. The confusion about the next generation FQHE was thus not dissimilar to that encountered in the context of the higher LL FQHE previously, and it is likely that in both cases the state at many fillings is incompressible in the thermodynamic limit, in spite of being compressible for certain finite systems with small N. The connection between the physics of the FQHE at ν and ν is confirmed unambiguously by directly relating, through the CF transformation, their microscopic ground state wave functions. We construct a trial wave function, Ψ tr ν the ground state at ν = (3m ± 1)/(8m ± 3) as follows: for Ψ tr ν = P LLL Φ 2 1Φ ν (3.13)

50 33 ν N O O E 0 E tr E tr 4/ (2) 0.51(1) (9) (4) (4) (1) 0.278(8) (5) (5) (6) 5/ (1) 0.365(3) (9) (9) (3) 7/ (4) 0.009(2) (4) (4) (5) Table 3.3. The trial wave functions Ψ tr ν and Ψ tr ν are compared with Ψ 0 ν for the finite system incompressible states at three filling factors. (See text for definitions.) The overlaps are defined as O = Ψ 0 ν Ψ tr ν and O = Ψ 0 ν Ψ tr E tr correspond to the wave functions Ψ 0 ν, Ψ tr ν and Ψ tr earlier in Ref. [MJ02]. ν. The energies E 0, E tr and ν, respectively. E 0 were calculated where Φ ν is the L = 0 Coulomb ground state at ν = 1+m/(2m±1), obtained by exact diagonalization. Because multiplication by Φ 2 1 attaches two vortices to each electron to convert it into a composite fermion, the wave function Ψ tr ν is interpreted as the FQHE of composite fermions at ν = 1 + m/(2m ± 1). (Although simple to interpret through composite fermions, the actual wave function is extremely intricate.) As another reference point, we consider the trial wave function Ψ tr ν = P LLL Φ 2 1 Φ ν (3.14) where Φ 1+m/(2m±1) is obtained by placing in the second LL the Coulomb ground state at ν = m/(2m ± 1) of the lowest Landau level. Thus, Ψ tr ν is derived from the m/(2m ± 1) state in the second LL, whereas Ψ tr ν is analogous to the m/(2m ± 1) state in the lowest LL. The overlaps and energies in Table 3.3 establish that Ψ 0 ν is well described by Ψ tr ν (and is far from tr Ψ ν ). It is fortunately possible to determine how accurate the CF ground state Ψ 0 ν is: for the non-trivial case of 12 particles at ν = 4/11, exact results are available [Rez], and a comparison shows that the Coulomb energy per particle for Ψ 0 4/11 deviates from the exact energy ( e2 /ǫl 0 [Rez]) by 0.04%. This demonstrates that the CF ground states Ψ 0 ν, and hence Ψtr ν, are extremely accurate representations of the exact ground states. We have thus succeeded in identifying good trial wave functions for the next generation FQHE by analogy to the FQHE at ν. The level of agreement is similar to that achieved for the accepted trial wave functions for the ordinary FQHE at ν = n/(2pn ± 1).

51 34 The gap for the largest available incompressible system at a given filling can be taken as a crude estimate for the thermodynamic gap. For 4/11, the gap for N = 20, 0.006e 2 /ǫl 0, is a factor of 16 (4) smaller than the gap at ν = 1/3 (ν = 4/3) at the same magnetic field, as expected from the fact that the residual interaction between composite fermions at ν is much weaker than the interaction between electrons at ν. The gaps for other fractions are of similar or smaller magnitude. (All gaps being compared are theoretical gaps, without including the effects of finite thickness or disorder.) This explains why the next generation FQHE is so fragile, readily destroyed by disorder or thermal fluctuations. Recent theoretical papers have studied the FQHE of composite fermions using Chern-Simons based approaches [GLS04, LF04]. These approaches do not possess sufficient quantitative accuracy to tell reliably whether the interaction between composite fermions will stabilize a new FQHE state (although they may illuminate certain properties of such a FQHE state assuming it exists). We do not find any evidence for pairing of composite fermions in the second CF quasi-ll, suggested in Ref. [WYQ03] as the mechanism of the next generation FQHE. It is well accepted that the Coulomb interaction does not stabilize the FQHE of electrons at ν = n + m/(2m ± 1) for n 2. The same is presumably true for composite fermions, which would rule out fully spin polarized FQHE at fractions of the form [(2 n+1)m± n]/[4( n+1)m±(2 n+1)] with n > 1. In higher quasi-ll s of composite fermions, charge density waves may be a likely possibility [LSJ02].

52 Chapter 4 Composite fermion theory of correlated electrons in semiconductor quantum dots in high magnetic fields There is a strong motivation for developing theoretical tools for obtaining a precise quantitative description of interacting electrons in confined geometries, for example in a semiconductor quantum dot, 1 because of their possible relevance to future technology. 2 Exact diagonalization is possible in some limits but restricted to very small numbers of electrons, and does not give insight into the underlying physics. For larger systems, one must necessarily appeal to approximate methods. The standard Hartree-Fock or density functional type methods provide useful insight, but are often not very accurate for these strongly correlated systems. The aim of this chapter is to demonstrate that a practically exact quantitative description is possible for a model quantum dot system, facilitated by the ability to construct 1 For reviews on quantum dots and their possible applications, see: L.P. Kouwenhoven, G. Schön, and L.L. Sohn, in Mesoscopic Transport NATO ASI Series E 345 (Kluwer Academic, 1997); G. Burkard and D. Loss, in Semiconductor Spintronics and Quantum Computation, eds. D.D. Awschalom, D. Loss, and N. Samarth, p (2002); S.M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 (2002). 2 The work described in this chapter was done in collaboration with G.S.Jeon and J.K. Jain. This chapter was published in Physical Review B 69, R (2004).

53 36 low-energy correlated basis functions. Our concern will be with the solution of H = j 1 ( p j + e A 2m b c ) 2 j + j m b 2 ω2 0r 2 j + j<k e 2 ǫr jk (4.1) which contains N interacting electrons in two dimensions, confined by a parabolic potential and subjected to a magnetic field. The parameter m b is the band mass of the electron, ω 0 is a measure of the strength of the confinement, ǫ is the dielectric constant of the host semiconductor, and r jk = r j r k. We will consider the limit of a large magnetic field (ω c = eb/m b c ω 0 ), when it is a good approximation to take electrons to be confined to the lowest Landau level (LL). In that limit, the energy eigenvalues have the form E(L) = E c (L)+V (L) where the contribution from the confinement potential is explicitly known as a function of the total angular momentum L: E c (L) = ( /2)[Ω ω c ]L, with Ω 2 = ω 2 c +4ω 2 0, and V (L) is the interaction energy of electrons without confinement, but with the magnetic length replaced by an effective magnetic length given by l /m b Ω. Thus, the problem is reduced to finding the interaction energy V (which will be quoted below in units of e 2 /ǫl) as a function of the angular momentum L. Exact results, known for a range of N and L values from a numerical diagonalization of the Hamiltonian, provide a rigorous and unbiased benchmark for any theoretical approach. Exact studies have shown [SKN96, Mak96, HSN98, RLBZ95, RC99] a correlated liquid like state at small L, but a crystallite at relatively large L, as may be expected from the fact that the system becomes more and more classical as L increases. (The ground state in the classical limit is a crystal [BP94, BR93, HSN02, AMTP04].) Hartree-Fock studies have been performed for the quantum crystallite [MK96, YL03]. We will apply the composite fermion (CF) theory [Jai89a, Jai89b] to quantum dot states [JK95, DJ92, JK97a, CMadGZ98, HY98]. The central idea is a mapping between strongly interacting electrons at angular momentum L and weakly interacting electrons at L L pn(n 1). In particular, a correlated basis {Ψ L α } for the low energy states of interacting electrons at L can be constructed from the trivial, orthonormal Slater determinant basis for non-interacting electrons at L,

54 n 0 0 n+l Figure 4.1. Schematic depiction of Slater determinant basis states for N = 6 electrons at L = 95, which maps into L = 5 with 2p = 6. The single electron orbitals at L = 5 are labeled by two quantum numbers, the LL index n = 0,1,..., and the angular momentum l = n, n + 1,... The x-axis labels n + l and the y-axis n. The dots show the occupied orbitals forming the Slater determinants Φ L =5 α relevant up to the first order. The state shown at the top left has the lowest kinetic energy (if the kinetic energy is measured relative to the lowest Landau level, then, in units of the cyclotron energy, the total kinetic energy of this state is two). The other nine states have one higher unit of kinetic energy. The basis states Ψ L=95 α are obtained according to Eq. (4.2), through multiplication by j<k (z j z k ) 6, which converts electrons into composite fermions carrying six vortices. That is shown schematically by six arrows on each dot. The single state at the top is relevant at the zeroth order, and all ten basis states are employed at the first order. (In fact, there are a total of 12 linearly independent states Φ L α at the first order for L = 5, but they produce only ten linearly independent states Ψ L α at L = 95.) denoted by {Φ L α }, in the following manner: Ψ L α = P (z j z k ) 2p Φ L α. (4.2) j<k Here, z j = x j iy j denotes the position of the jth electron, 2p is the vorticity of composite fermions, and P indicates projection into the lowest LL. (Electrons at L in general occupy several Landau levels.) The symbol α = 1, 2,, D labels the D Slater determinants included in the study. In general, the basis {Ψ L α } is not

55 38 linearly independent, so its dimension, D CF, may not be equal to D (D CF D ). The advantage of working with the correlated CF basis is that D CF is drastically smaller than D ex, the dimension of the lowest LL Fock space at L (which is also the dimension of the matrix that must be diagonalized for obtaining exact results). Fig. 4.1 illustrates some basis functions at L = 95. At a back-of-the-envelope level, one can compare the exact interaction energy at L to the kinetic energy of free fermions at L, with the cyclotron energy treated as an adjustable parameter [JK95, DJ92]. That reproduces the qualitative behavior for the L dependence of the exact energy for small L [JK95, DJ92], but discrepancies are known to appear at larger L [SKN96, Mak96, HSN98, RLBZ95, RC99, YL03]. For a more substantive test of the theory, it is necessary to obtain the energy spectrum by diagonalizing the Hamiltonian of Eq. (4.1) in the correlated basis functions of Eq. (4.2). The CF-quasi-Landau level mixing is treated as a small parameter, and completely suppressed at the simplest approximation, which we refer to as the zeroth order approximation. Here, the correlated basis states at L are obtained by restricting the basis {Φ α } to all states with the lowest kinetic energy at L (with p always chosen so as to give the smallest dimension). Diagonalization in the correlated CF basis is technically involved, but efficient methods for generating the basis functions as well as all of the matrix elements required for Gram-Schmidt orthogonalization and diagonalization have been developed using Metropolis Monte Carlo sampling. We refer the reader to earlier literature for full details [JK97a, MJ02]. A diagonalization of the Hamiltonian in the zeroth level basis produces energies and wave functions for D (0) CF low-lying states. The interaction energy and the wave function for the ground state will be denoted V (0) CF and Ψ(0) CF, respectively. We have carried out [JCJ04b] extensive calculations for a large range of L for up to ten particles, and found that the CF theory reproduces the qualitative behavior of the energy as a function of L all the way to the largest L for which exact results are known. We show in Table 4.1 results for N = 6 electrons in the angular momentum range 79 L 108, which spans both liquid and crystal-like ground states. Ref. [JCJ04b] shows a comparison of the exact pair correlation function with that calculated from Ψ (0) CF for L = 95 (where there is a unique CF wave function);

56 39 L V ex V (0) CF V (1) CF D ex D (0) CF D (1) CF (2) (3) (1) (1) (1) (2) (4) (5) (7) (2) (1) (2) (1) (1) (5) (2) (2) (2) (1) (2) (3) (3) (1) (1) (3) (2) (1) (2) (1) (3) (2) (2) (2) (2) (2) (4) (1) (5) (3) (4) (1) (4) (2) (3) (2) (1) (2) (3) (2) (2) (2) (1) (2) (3) (1) (4) (2) (4) (2) (4) Table 4.1. Exact ground state energy (V ex ) and the ground state energy obtained from the zeroth (V (0) (1) CF ) and the first-order (V CF ) CF theory for N = 6. The dimensions of the bases diagonalized are D ex, D (0) CF and D(1) CF, respectively. The statistical uncertainty arising from Monte Carlo sampling is given in parentheses.

57 40 L O (0) O (1) L O (0) O (1) Table 4.2. Overlaps between exact ground states and CF ground states obtained at the zeroth (O (0) ) and the first order(o (1) ). The statistical uncertainty from Monte Carlo sampling does not affect the first three significant figures. surprisingly, the CF theory, originally intended for the liquid state, automatically produces also a crystallite at large L, even though no crystal structure has been put into the theory by hand [YF98, NMF01]. Table 4.2 gives the overlaps defined as: O (0) (0) Ψ CF Ψ ex / Ψ (0) CF Ψ(0) CF Ψ ex Ψ ex. While the zeroth level description is quite good, the following deviations from the exact solution may be noted. (i) The overlaps are in the range , which are not as high as the overlaps ( 0.99) for incompressible ground states in the spherical geometry. (ii) The energies are within 0.5% of the exact ones, which is quite good but could be further improved. (iii) In the crystallite, the particles are somewhat less strongly localized in the CF wave function than in the exact ground state. See Ref. [JCJ04b]. (iv) The symmetry of the crystallite is predicted correctly with the exception of L = 99, where the CF theory predicts a (0, 6) crystallite [Fig. 4.3(b)], that is, with all six particles on an outer ring, whereas the exact solution shows a (1, 5) crystallite [Fig. 4.3(a)], which has five particles on the outer ring and one at the center. (v) A successful theory must explain not only the ground state but also excited states, especially the low-energy ones. We have considered the gap between the two lowest eigenstates. The zeroth-order theory does not give, overall, a satisfactory account of it. In some instances (e.g., L = 81, 85, 90, 95, 99, 105, 106 for N = 6), the CF theory gives no information on the gap, because the basis contains only a single state here (D (0) CF = 1); in many other cases, the gap predicted by the CF theory is off by a factor of two to three. These discrepancies have motivated us to incorporate CF-quasi-LL mixing per-

58 41 L D ex D (2) CF V ex V (2) CF O (2) (3) Table 4.3. Comparison of the second order CF theory with exact results for the L = 99 ground state. D (2) is the dimension of the correlated CF basis, V (2) CF is the CF prediction for the ground state energy, and O (2) is the overlap between the CF and the exact wave functions. º ¼º¼ ¼º¼¾ ¼ ¼ ¼ Ä ½¼¼ ½¼ Figure 4.2. Comparison of the exact excitation gaps ( ) for N = 6 with the gaps obtained in the first-order CF theory ( ). turbatively. (We stress that CF-quasi-LL mixing implies LL mixing at L, but the basis states at L are, by construction, always within the lowest LL.) At the first order, we include basis states Φ L at L with one more unit of the kinetic energy, which produces a larger basis at L through Eq. (4.2). The basic idea is illustrated for the case of N = 6 and L = 95 (L = 5, 2p = 6) in Fig In a similar way we have constructed correlated basis functions at each angular momentum in the range 79 L 108. As shown in Table 4.1 D (1) CF, the dimension of the basis in the first-order theory is larger than D (0) CF but still far smaller than D ex. A diagonalization of the Hamiltonian in this basis produces the ground state energy V (1) CF and ground state wave function Ψ(1) CF. Leaving aside L = 99, which we shall discuss separately, the following observations can be made: (i) The energies are essentially exact. As shown in Table 4.1, the deviation from the exact energy is reduced to <0.1%, in fact, to <0.05% in most cases. (ii) The overlaps from the first-order theory, O (1) (1) Ψ CF Ψ ex / Ψ (1) CF Ψ(1) CF Ψ ex Ψ ex, are given in Table 4.2. They are uniformly excellent ( ) in the entire L range studied. (iii) The improvement by the first order perturbation theory is also manifest in the

59 42 (a) (b) (c) Figure 4.3. Pair correlation function for N = 6 electrons at L = 99. The position of one particle is fixed on the outer ring, coincident with the position of the missing peak. The ground state wave function used in the calculation is obtained from (a) exact diagonalization; (b) the zeroth-order CF theory; (c) the first-order CF theory; (d) the second-order CF theory. The noise in (a) and (d) results from the relatively large statistical uncertainty in Monte Carlo because of the more complicated wave function. (d) pair-correlation functions, which are now indistinguishable from the exact ones at arbitrary L. That is not surprising, given the high overlaps. (iv) As seen in Fig. 4.2 the first-order theory reproduces the qualitative behavior of the excitation gap as a function of L, and also gives very good quantitative values. The maximum gaps are correlated with the states where a downward cusp appears in the plot of V (L), consistent with the higher stability of these ground states. Finally, we discuss the case of L = 99. Here, the zeroth order CF theory predicts a wrong symmetry for the crystallite [Fig. 4.3(b)]. As seen in Fig. 4.3(c), the first order correction also fails to recover the correct symmetry. That is also reflected in the fact that the modified ground state of the first-order theory yields a relatively small overlap of 0.86, and the energy is off by a relatively large 0.15%. A closer inspection of the correlations in Fig. 4.3(c) reveals a slight broadening of