The Pennsylvania State University The Graduate School QUANTUM MONTE CARLO STUDIES OF STRONGLY CORRELATED TWO-DIMENSIONAL ELECTRON SYSTEMS

Transcription

1 The Pennsylvania State University The Graduate School QUANTUM MONTE CARLO STUDIES OF STRONGLY CORRELATED TWO-DIMENSIONAL ELECTRON SYSTEMS A Dissertation in Physics by Kenneth L. Graham c 2005 Kenneth L. Graham Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2005

2 The thesis of Kenneth L. Graham was reviewed and approved by the following: Jainendra K. Jain Professor of Physics Thesis Advisor, Chair of Committee James B. Anderson Professor of Chemistry and Physics Co-advisor Vincent H. Crespi Professor of Physics Nitin Samarth Professor of Physics Peter Jurs Professor of Chemistry Jayanth Banavar Chair, Department of Physics Signatures are on file in the Graduate School.

3 Abstract Strongly interacting electron systems, where the interaction between electrons cannot be ignored or treated perturbatively, are among the most complicated systems found in nature. This thesis focuses on three quantum Monte Carlo studies of strongly interacting two-dimensional electron systems. First, using variational quantum Monte Carlo, the nodal structure of Jain s composite fermion wave function is studied by calculating correlation functions between particles and wave function nodes. The density matrix and correlation function between distinguishable particles are also derived from first principles and a node of the wave function is found to lie preferentially between a pair of composite fermions when that pair is closer than an average interparticle separation. Next, the fractional exchange statistics of fractional quantum Hall states, of which a microscopic calculation has been lacking, is presented. In particular, the statistics parameter and Berry phase charge of Jain s composite fermions are calculated and found to agree with theoretical expectation. Finally, the first application of exact cancellation Green s function Monte Carlo to circular quantum dots in zero magnetic field is also presented, as well as a proof that Green s function Monte Carlo converges to the true ground state. iii

4 Table of Contents List of Figures List of Tables Acknowledgments vii x xi Chapter 1 Introduction Composite fermions and the fractional quantum Hall effect The integer quantum Hall effect The fractional quantum Hall effect Composite fermions and wave function nodes Exchange statistics of composite fermions Quantum dots and Green s function Monte Carlo Chapter 2 The Composite Fermion Wave Function Introduction The disk geometry The spherical geometry Chapter 3 Nodal Structure of the Composite Fermion Wave Function Introduction When is a wave function zero a vortex? Finding wave function nodes Correlation Functions Binning the sphere iv

5 3.5 Results and Conclusion Zero-zero Particle-zero Particle-particle-zero (ppz) Discussion Chapter 4 Fractional Exchange Statistics of Composite Fermions Introduction Calculating fractional statistics Fractional statistics and Berry s phase Composite fermion quasiparticle wave functions Results The sign puzzle Discussion Composite fermions: Fermions or anyons? Constraints on possible observation of fractional statistics.. 55 Chapter 5 Exact-cancellation Green s Function Monte Carlo Study of Circular Quantum Dots Introduction Numerical Method Green s function Monte Carlo Exact-Cancellation GFMC Averages of Physical Observables Model Circular quantum dots Trial wave functions Results and Discussion Appendix A Correlation Functions and the Density Matrix 65 A.1 Introduction A.2 Correlation Functions between Distinguishable Particles A.2.1 Case I: m > n A.2.2 Case II: m n v

6 Appendix B The Convergence of Green s Function Monte Carlo 69 B.1 Introduction B.2 Green s Functions for Hermitian Eigenvalue Problems B.2.1 Solving the integral equation iteratively B.2.2 Convergence of the iterative series B.3 Green s functions for Simple Time-Independent Hamiltonian Systems 72 Bibliography 74 vi

7 List of Figures 1.1 Composite view of the integral and fractional quantum Hall effects. The Hall and longitudinal resistances are shown. This figure is from J. P. Eisenstein and H. L. Störmer, Science 248, 1461 (1990) Positions of the zeroes seen by each particle. Panels (a)-(d) correspond to systems with (ν, N) = (2/5, 20), (3/7, 21), (4/9, 16), and (5/11, 25), respectively. The particle positions are shown with large, filled circles, and the zeroes with unfilled symbols (different symbols for the zeroes of different particles). The sphere is plotted in polar coordinates. The north pole is the origin, with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The equator is given by the dashed line. Distance is measured in units of the magnetic length, l 0. The particles and zeroes far from the north pole are not shown because of the significant distortions in the distances while depicting the spherical geometry in a plane. Note that the distortion is not severe enough to destroy the qualitative behavior of off-particle zeroes clustering between nearby particle pairs, as seen in panels (a), (b), and (d) Pair-distribution functions for three FQHE states at ν = n/(2n+1), corresponding to n filled Landau levels of composite fermions. N = 20, 21, and 16 particles have been used for n = 2, 3, and 4, respectively. (a) shows the probability of having two zeroes separated by distance r, (b) shows the probability for a particle and a zero being separated by distance r, and (c) shows the probability for a particle and an off-particle zero. Following Ref. [19], we choose the unit of length for this plot as k 1 F = (4πρ) 1/ vii

8 3.3 g ppz for 20 particles at ν = 2/5, with d = 1.0 r pp and r pp = 4l 0, l 0 being the magnetic length. The sphere is plotted in polar coordinates. The north pole is the origin; with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The white arrows represent the positions of the two fixed particles. The contours are shaded as follows. Black represents zero probability per unit area, the shade of grey toward the edges is a probability per unit area of one, with lighter shades representing probability per unit area greater than one, white being highest. The grainy detail is numerical noise. Distance is measured in units of l g ppz for 20 particles at ν = 2/5, with d = 0.5 r pp g ppz for 20 particles at ν = 2/5, with d = 0.35 r pp Cross-section of g ppz, for 20 particles at ν = 2/5, along the arc passing through the fixed particle positions. In this and subsequent figures, the origin is the north pole and the fixed particle positions are represented by arrows. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.35 r pp, respectively. The numbers appearing beside the central peaks in (b) and (c) give the height of the peak Cross-section of g ppz for 20 particles at ν = 2/5, along the arc passing through the origin in a direction perpendicular to the arc joining the fixed particle positions. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.35 r pp, respectively g ppz for 21 particles at ν = 3/7, with d = 1.0 r pp. The sphere is plotted in polar coordinates. The north pole is the origin; with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The white arrows represent the positions of the two fixed particles. The contours are shaded as follows. Black represents zero probability per unit area, the shade of grey toward the edges is a probability per unit area of one, with lighter shades representing probability per unit area greater than one, white being highest. The grainy detail is numerical noise. Distance is measured in units of l g ppz for 21 particles at ν = 3/7, with d = 0.5 r pp g ppz for 21 particles at ν = 3/7, with d = 0.25 r pp Cross-section of g ppz, for 21 particles at ν = 3/7, along the arc passing through the fixed particle positions. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.25 r pp, respectively Cross-section of g ppz, for 21 particles at ν = 3/7, along the arc passing through the origin in a direction transverse to the arc joining the fixed particle positions. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.25 r pp, respectively viii

9 3.13 g ppz for 20 particles at ν = 4/9, with d = 1.0 r pp. The sphere is plotted in polar coordinates. The north pole is the origin; with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The white arrows represent the positions of the two fixed particles. The contours are shaded as follows. Black represents zero probability per unit area, the shade of grey toward the edges is a probability per unit area of one, with lighter shades representing probability per unit area greater than one, white being highest. The grainy detail is numerical noise. Distance is measured in units of l g ppz for 20 particles at ν = 4/9, with d = 0.5 r pp g ppz for 20 particles at ν = 4/9, with d = 0.25 r pp Cross-section of g ppz, for 20 particles at ν = 4/9, along the arc passing through the fixed particle positions in the upper figure and along the arc perpendicular to the arc joining the fixed particle positions in the lower figure. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.25 r pp, respectively The excess charge density relative to the ground state in the presence of one CFQP on top of the N = 50, ν = 2/5 state. The CFQP is localized about the origin (η/l = 0) and the disk size is R d = l 2N/ν The Berry phase charge Φ /Φ e for a single CFQP at ν = 1/3 (upper panel) and ν = 2/5 (lower panel) as a function of η. Here N is the total number of composite fermions, l 0 is the magnetic length, and Φ e 2πBA/φ 0 is the Aharanov-Bohm phase of an electron moving in the external magnetic field. The error bars from Monte Carlo sampling, which are smaller than the symbol size, are not shown explicitly. The deviation at the largest η/l 0 for each N comes from the CFQP being bear the edge The statistical angle θ for the CFQP s at ν = 1/3 (upper panel) and ν = 2/5 (lower panel) as a function of d η η. Here N is the total number of composite fermions, and l 0 is the magnetic length. The error bar from Monte Carlo sampling is not shown explicitly when it is smaller than the symbol size. The deviation at the largest d/l 0 for each N is due to proximity to the edge ix

10 List of Tables 5.1 Comparison of ground state energies (in H ) for N = 2, 4 quantum dots computed by VQMC, exact-cancellation GFMC and DMC. The DMC results are from Ref. [44]. The frequency of the confining potential is ω = 0.28 H. The orbital and spin angular momentum of each state is also shown. The number in parentheses is the statistical uncertainty in the last digit x

11 Acknowledgments I am indebted to several people. First, my advisor Jainendra Jain, who welcomed me into his research group while I was a refugee from experimental physics, which I am simply not made for. Since then, I have benefitted from his perspicuity, professional guidance, and general kindness. Thanks to him, I know how research is done. My co-advisor from the Chemistry Department, Jim Anderson, has shown me another flavor of quantum Monte Carlo methods. The lessons in the uses of Monte Carlo methods and other numerical methods, as well as responsible conduct of research, are invaluable. I would also like to thank my friends and fellow martial artists at Titan Fitness (formerly The State College Martial Arts Academy) for helping me learn the Art of Expressing the Human Body. And to my family, thank you for supporting me and having my back in all things. xi

12 Dedication This dissertation is dedicated to my father. written in 2002, on New Year s Eve. What follows is an original poem It was the winter of 1995, scant days before Christmas. I was home for the holidays, and my father was coming to visit. I drove to the airport to pick him up. A little late, but full of excitement, I raced toward the security checkpoint, eschewing the escalator and taking the stairs. Imagine my disbelief when, having crested the stairs, I see my father in a wheelchair; a gaunt shadow of himself. Seeing me, he struggled to rise. His traitorous body abandoned him, its gelatinous muscles trembling with exertion like unset custard. I was taken aback, nonplussed as ever. I must have entered the twilight zone, sans eerie sound effects and suspense-filled storyline. xii

13 Looking back, I wonder if he sensed my shock. Did he see the muscles of my face slacken, my broad smile dissolving into nothing. xiii

14 Chapter 1 Introduction Strongly interacting electrons, when constrained to occupy two spatial dimensions, generate some of the most interesting phenomena in nature. In particular, the fractional quantum Hall effect [1] and quantum dots stand out. A quantum dot, which is commonly thought of as having zero spatial extent, is a small collection of electrons which can sometimes behave as an artificial atom. While quantum dots are sometimes placed in a magnetic field, dethe fractional quantum Hall liquid occurs in the presence of high-magnetic field, low temperature, and low disorder; it is the most strongly correlated state of matter. By strongly correlated, we mean that the kinetic energy is considerably less important than the electron-electron interaction, i.e the interaction determines the physics. We are interested in the ways in which quantum Monte Carlo (QMC) methods can be used to study the properties of these strongly correlated electron systems. We here present a brief overview of circular quantum dots and the composite fermion theory of fractional quantum Hall systems.

15 1.1 Composite fermions and the fractional quantum Hall effect The integer quantum Hall effect When studying a system where quantum mechanics is relevant, one begins with the hamiltonian operator Ĥ and the Schrödinger equation ĤΨ = EΨ. For N electrons confined to the xy-plane in the presence of a magnetic field in the ẑ direction, the Hamiltonian can be written as Ĥ = 1 2m N j=1 ( ˆp j + e ) 2 c A( r e 2 1 j) + ɛl x j x k, (1.1) j<k where m is the electron s effective mass, ɛ is the semiconductor s dielectric constant, l = c/eb is the magnetic length, and A is the magnetic vector potential. If the electron system has a high mobility and is cooled below liquid helium temperatures, the electron s kinetic energy is considerably larger than the strength of the Coulomb interaction. Another way to look at this is in terms of the electron density. In this regime, the electron density is relatively high, making the kinetic energy larger than the strength of the Coulomb interaction. In this case the interaction can be safely ignored and the problem is exactly solvable. The resulting energy levels are called Landau levels. The Landau level energy spectrum is E nm = ω c (n + 1/2), where n = 0,1,... and ω c = eb/m c. Landau level (LL) filling can be described by a parameter ν, called the filling factor, which tells you how many LLs have been filled. ν is the ratio of electron number to the number of magnetic flux quantum piercing the system [2]. When ν is an integer, a strange thing happens. As Klaus von Klitzing discovered in 1980, instead of varying linearly with applied magnetic field B, the transverse resistivity has strange plateaus centered precisely on integer filling factors ν = 0, 1, 2,.... The longitudinal resistivity also vanishes precisely at each plateau. This phenomenon is called the integer quantum Hall effect (IQHE) [3]. The presence of plateaus is explained neatly by appealing to the disorder in the two-dimensional electron gas (2DEG). If the 2DEG were disorder-free then the density of states versus energy would consist of a sequence of delta functions, with 2

16 3 each delta function occurring at the LL energies. In the presence of disorder each delta function broadens into an energy band, allowing localized, non-conducting states to exist in the gap and extended states toward the center of the band. If the electron (or hole) density is fixed, then the filling factor depends solely on the magnetic field. If the magnetic field is increased from zero, the electrons begin entering the lowest Landau level (LLL), filling up extended states which contribute to the 2DEG s conductivity. Thus the transverse resistivity has ohmic behavior. Once the LLL is filled, electrons begin filling the localized states in the gap. Since the localized states do not contribute to the conductivity, there is a plateau in the transverse resistivity. Minima in the longitudinal resistivity arise from dissipationless electron transport at the sample edge [4] The fractional quantum Hall effect This strangeness does not end with the IQHE. If the two-dimensional electron system is made to have even higher mobility, the magnetic field is made stronger, and the temperature is lowered still, the degeneracy of the LLL approaches infinity and all electrons can fill states inside the LLL. In 1983, Tsui, Störmer, and Gossard discovered this phenomena. They observed a plateau in the transverse resistivity and a minimum in the longitudinal resistivity. Unlike the IQHE, the plateaus and minima were now centered about fractional filling factors ν = 1/3! Despite being utterly suprising, there were two important clues: the 1/3 state was incompressible, i.e. it had a gap and its density was uniform. This is the fractional quantum Hall effect (FQHE) [1]. The theoretical explanation of the FQHE proved much more difficult. When all electrons live in the LLL, the kinetic energy becomes an irrelevant constant and can be discarded. The Coulomb interaction is then the only energy in the problem. Perturbation theory is then ruled out. Fortunately, certain things can be carried over from the IQHE. In the LLL, using the disk geometry (see appendix), single electron states have the form z n e z 2 /4, where z = x iy denotes the position of an electron as a complex number and distance is measured in magnetic length l. These functions form a Hilbert space, the Segal-Bargmann space of analytic functions. All wave functions describing physics of the LLL must then be analytic

17 4 in z [6]. With this constraint, the generic wave function for N electrons at filling factor ν is [ ] Ψ = F A [{z j }] exp 1 z l 2, (1.2) 4 with F A [{z j }] an antisymmetric analytic polynomial of the z s. Using the fundamental theorem of algebra, the number of zeroes of the wave function for a given filling factor ν is known. Given ν, there are N/ν single particle states occupied, implying that the largest power of z j in the polynomial part of the wave function is N/ν, which, according to the fundamental theorem of algebra, is also the number of zeroes of the wave function. Here, we have neglected O(1) corrections. For a finite system, the number of zeroes can be determined exactly [7, 8]. For LLL wave functions, if we assume that each CF-LL is filled with N/n electrons (where n is the number of CF-LL s), then the largest power of z j in the wave function is N/ν (2p + 1). With the Hilbert space defined and the wave function functionally constrained, an important first step in understanding the FQHE was provided by Laughlin, who later shared a Nobel prize with Tsui and Störmer. Laughlin wrote a variational wave function meant to describe FQH states with fillings ν = 1/m, with m an odd integer: Ψ 1/m = [ ] j z k ) j<k(z m exp 1 z l 2. (1.3) 4 Laughlin also provided wave functions for excitations, the Laughlin quasihole and quasiparticle [9, 20]. The excitations, which require a finite energy, provide the energy gap needed for Ψ 1/m to be incompressible. Laughlin s wave function for a quasihole at position η in the ν = 1/m state is [9] l l Ψ qh 1/m = j (z j η)ψ 1/m. (1.4) The nodal structure of Laughlin s wave function is simple. All zeroes are located at electron positions [10]. If all particles are fixed, save one, the mobile particle sees a zero at every other particle (actually, an mth order zero). A wave function node of this type is termed a vortex, since the phase of the wave function changes by 2π when a closed path encloses it. Thus, at ν = 1/m, there are m vortices on

18 5 Figure 1.1. Composite view of the integral and fractional quantum Hall effects. The Hall and longitudinal resistances are shown. This figure is from J. P. Eisenstein and H. L. Störmer, Science 248, 1461 (1990). each electron. This property of no-free-zeroes is not possible at other fillings. For instance, the ν = 2/5 wave function must have 2.5 zeroes per electron, and one zero must be a vortex bound to the electron, per the Pauli exculsion principle. Where are the other 1.5 zeroes? To date, a host of other fractions have been observed, including those with fractional fillings greater than one. Figure (1.1), found in Ref. [5], shows the similar phenomenology of the integral and fractional quantum Hall effects. Laughlin s theory does not apply to the majority of the FQH states shown in Figure (1.1) Composite fermions and wave function nodes As new fractions were observed (ν = 2/5, 3/7, 4/9,... ) it became apparent that more work was needed. These and other fractions can be grouped into a sequence n 2pn + 1 = 1 3, 2 5, 3 7, 4 9, 5 11,..., (1.5) first noticed by Jain [11], who recognized that any explanation of the FQHE should include the observed fractions, predict fractions not yet observed, explain why

19 6 certain other fractions are not observed, and unite the IQHE and FQHE, since they have very similar signatures. In addition, a microscopic description was desirable, complete with ground and excited state wave functions in the LLL. Taking the nodal structure of Laughlin s wave function as a point of departure, Jain introduced the composite fermion theory of the FQHE [11, 8]. While all zeroes are vortices bound to the electrons for fillings ν = 1/m, at other fillings all zeroes need not be so constrained. Jain used the adiabatic principle to help explain the FQHE. Imagine a 2DEG in the IQH regime, and then begin lowering the the filling factor ν until the 2DEG is in the FQH regime. At this point, the kinetic energy becomes constant and the coulomb interaction becomes important. What happens to the electrons? Jain has shown that the electrons best minimize their interaction energy by morphing into fermionic quasiparticles having the same spin and intrinsic charge as electrons, similar to electrons transforming into Landau quasiparticles in a Landau fermi liquid. These new particles are called composite fermions. A composite fermion can be heuristically viewed as an electron bound to an even number of magnetic flux quanta. The accurate definition is that a composite fermion (CF) is an electron bound to an even number of vortices of the many-body wave function. CFs feel a reduced effective magnetic field B = B 2pρφ 0, where ρ is the electron areal density, 2p are the (even) number of vortices bound to each electron, and φ 0 = hc/e is the magnetic flux quantum. The effective magnetic field results from the phases associated with the vortices partially canceling the Aharanov-Bohm phases from the external magnetic field. Since the vortices also help screen the Coulomb interaction (see Chapter 4), CFs are more weakly-interacting than bare electrons. In fact, CFs can be considered non-interacting, to a good approximation. These non-interacting CFs are then in a situation similar to electrons in the IQHE, and can be viewed as occupying composite fermion quasi-landau levels (CF-LLs). Thus CFs can undergo an IQHE. This heuristic picture allowed Jain to write wave functions describing the FQHE of electrons at filling factor ν = n/(2pn + 1) as the IQHE of CFs at filling factor n, and 2p is the even integer representing a CF s vorticity. These wave functions for general FQH states are written as [11] Ψ n/(2pn+1) = j<k(z j z k ) 2p Φ n, (1.6)

20 7 where Φ n is the wave function for n fully occupied Landau levels. The factor j<k (z j z k ) is called the Jastrow factor and is how the CF theory of the FQHE binds vortices to electrons. Careful inspection of Eq. (1.6) will reveal that Ψ n/(2pn+1) does not always reside in the LLL. Away from n = 1, Φ n ceases to be analytic. To properly describe the FQHE when CFs fill more than one CF-LL, the right-hand side of Eq. (1.6) must be projected onto the LLL [11, 8, 12, 13]. Since the wave function in Eq. (1.6) is nonanalytic, we can make few exact statements about its nodal structure. We know there are two vortices explicitly bound to each electron. We also know that the Slater determinant Φ n has at least one vortex bound to each electron, per the Pauli exclusion principle; with more nodes and antinodes possible, provided fermionic statistics are preserved. Fortunately, there is no ambiguity in the LLL. The LLL-projected CF wave function for general FQHE states is [11, 8, 12] Ψ LLL n/(2pn+1) = P LLL (z j z k ) 2p Φ n, (1.7) where P LLL is the LLL projection operator. The procedure of LLL projection is well-defined and is discussed in Refs. [6, 12, 8]. We can now ask certain questions about the LLL wave function s nodal structure. Where are the nodes? Since LLL projection leaves only one vortex bound to each electron, are other nodes located near electron positions? Can some kind of bound state of electrons and zeroes be identified? j<k Exchange statistics of composite fermions All fundamental particles found in nature are either fermions or bosons. This is an empirical fact. It is also known that the configuration space of a collection of N hard-core particles is multiply connected [33]. The topological nature of exchange trajectories for identical particles determines their exchange statistics. For example, in three spatial dimensions, any trajectory which corresponds to two exchanges of two identical particles can be shrunken to a point, which is equivalent to no exchange. These sorts of exchanges are described by the symmetric group. This is why, if one considers how exchanges modify the wave function, the wave

21 8 function can only change by a sign upon particle exchange [33]. If identical particles are somehow constrained to move in two spatial dimensions, the exchanges are no longer governed by the permutation group. This is because any two trajectories corresponding to different particle exchanges cannot be continuously deformed into each other. They are topologically distinct trajectories. This can be seen clearly in the case of two identical particles. In this case, the configuration space is the circumference of the circle, provided points where the particles have identical positions ( r 1 = r 2 ) are omitted and the relative coordinate is fixed ( r 1 r 2 ). An exchange path is then a loop around the circle with winding number one. For two particle exchanges, the path is a loop of winding number two, i.e. the path wraps around the circle twice. Since these two paths have distinct winding numbers, they are distinct and cannot be continuously deformed into one another. This analysis tells us that the N body wave function no longer acquires a ±1 upon exchange of identical particles. Instead, exchanges are described by the braid group, making the wave function acquire a phase upon exchange. The value of the phase determines the statistics of the particles, which interpolate between fermions and bosons [34]. These exchange statistics are called fractional statistics. Given the empirical fact that all fundamental particles are either bosons or fermions, any discussion of fractional statistics may appear academic. Fortunately, that is not the end of the story. There is no a priori principle precluding certain emergent quasiparticles of a two-dimensional condensed matter system from exhibiting fractional statistics; indeed, the emergence of such phenomena would be an interesting display of novel concepts emerging [32] in a many-body system. Fortunately, nature has provided a candidate system namely, the fractional quantum Hall fluid [1]. Because the CF theory of the FQHE provides an accurate account of the low energy physics, including incompressibility at certain fractional fillings, it must also contain the physics of fractional statistics, which indeed is the case. The fractional statistics can be derived heuristically in the CF theory as follows [25]. Composite fermions are bound states of electrons and an even number (2p) of vortices. When a composite fermion goes around a closed path encircling an area A, the total

22 9 phase associated with this path is given by Φ = 2π(BA/φ 0 2pN enc ), (1.8) where N enc is the number of composite fermions inside the loop and φ 0 = hc/e is the flux quantum. The first term on the right hand side is the usual Aharonov- Bohm phase for a particle of charge e going around in a counterclockwise loop. The second term is the contribution from the vortices bound to composite fermions, indicating that each enclosed composite fermion effectively reduces the flux by 2p flux quanta. (A note on convention: We will take the magnetic field in the +z direction, the electron charge to be e, and consider the counterclockwise direction for the traversal of trajectories.) Through this line of reasoning, it is clear that fractional statistics keeps track of how the effective magnetic field experienced by composite fermions is affected by local fluctuations in the density, as obtained, for example, by creation of a localized excitation. 1.2 Quantum dots and Green s function Monte Carlo Quantum dot systems were first studied with quantum Monte Carlo (QMC) in 1989 [50]. Since then, many QMC studies of quantum dots have been added to the quantum dot literature [51, 52], with much Monte Carlo work being devoted to quantum dot systems in the presence of a magnetic field. For certain magnetic field strengths these quantum dots are considered to be in the fractional quantum Hall regime; then the composite fermion theory can be applied [53]. To date, Green s function Monte Carlo (GFMC) methods have not been applied to the quantum dot problem. GFMC was originally introduced to solve the N-boson Schrödinger equation for its ground state [37, 38], which is symmetric. If we consider a fermion state that is spatially symmetric, then we may also use the GFMC method to find the fermion ground state. GFMC can also tackle spatially antisymmetric fermion ground states or boson excited states. In this case, one needs to worry about the fermion sign

23 10 problem. Fortunately, there are modifications one can make to GFMC [39, 40, 41]. The most computationally efficient of these methods, exact-cancellation GFMC (XGFMC), was introduced by Anderson [41]. We use XGFMC to study circular quantum dots. By calculating ground state energies and comparing them to diffusion Monte Carlo studies, we demonstrate that XGFMC provides quantum dot energies lower and more precise than any other QMC method which has been applied to quantum dots (in zero magnetic field).

24 Chapter 2 The Composite Fermion Wave Function 2.1 Introduction The beauty of the composite fermion theory lies in the simple elegance with which it unites the integral and fractional QHEs. This unity is apparent in Jain s composite fermion (CF) wave function [11]. The accuracy of the CF wave function has been consistently confirmed since its introduction. It has stood up to intense scrutiny from experiments [13] and to comparison with numerous exact diagonalization studies [8, 12, 13, 53, 54]. In many cases, CF theory produces overlaps with the exact ground state which are better than 99%. 2.2 The disk geometry Jain s wave function describes N electrons at Landau level (LL) filling factor ν = n/(2pn + 1), with n = 1,2,... and 2p an positive even integer. The electrons live in the xy plane and are under the influence of a strong magnetic field in the z direction. The magnetic field is considered to infinitely strong and the temperature considered to be absolute zero. In this extreme quantum limit, all electrons are confined in the lowest electronic Landau level (LLL). This is a crucial point. Because the wave function must lie in the LLL, it must also be an analytic

25 12 function of the electron positions [6]. This places strong constraints on possible forms of the wave function and also constrains its nodal structure. Since the FQHE is observed in planar systems, it is natural to consider a disk geometry. In this geometry, an electron s position on the xy plane is given by a complex coordinate z = x iy, with distance measured in magnetic length l 0 = c/eb. Jain s CF wave function, in the disk geometry is written as Ψ n/(2pn+1) = P LLL J 2p Φ n, (2.1) where Φ n is the wave function for n fully occupied LLs, J = j<k (z j z k ) is the Jastrow factor, and P LLL is the LLL projection operator [11]. Since CF theory describes the FQHE of N strongly interacting electrons at filling factor ν = n/(2pn + 1) as the IQHE of N weakly-interacting CFs at integer CF Landau level (CF-LL) filling factor n, Φ n describes the IQHE of CFs. Φ n is of the form and the η s are given by η 1 (z 1 ) η 1 (z 2 ) η 1 (z N ) η Φ n = 2 (z 1 ) η 2 (z 2 ) η 2 (z N ) η N (z 1 ) η N (z 2 ) η n (z N ) (2.2) n ( ) η nm (z j ) = N nm e 1 4l Pk 2 z k 2 n + m z n ( 1) k k z k+m, (2.3) n k 2 k k! k=k 0 with n = 1, 2,... labeling the CF-LLs, m = n, n + 1,..., n labeling the orbital angular momentum, k 0 = max(0, m), and N nm = n!/(2π2 m (n + m)!). The radius of the disk is R = l 2N/ν. 2.3 The spherical geometry The problem of N electrons in two dimensions in a perpendicular magnetic field has also been formulated in the spherical geometry [15, 14]. The magnetic field is radial and is produced by a magnetic monopole of charge Q at the center of the sphere. The magnetic flux through the sphere s surface is 2Qφ 0 and the magnetic

26 13 field strength is B = Qφ 0 /(2πR 2 ), where φ 0 = hc/e is the magnetic flux quantum. The monopole charge Q may be an integer or half-integer, in accordance with Dirac s quantization condition. This allows us to map the problem of N strongly interacting electrons at flux 2Q into N weakly-interacting CFs at flux 2q = 2Q 2p(N 1), where flux is now measured in units of φ 0. interacting electrons has the standard form [8, 11] The wave function for Ψ Q = P LLL Φ 2p 1 Φ q, (2.4) where Φ 1 describes the fully-filled lowest Landau level and Φ q represents Slater determinant wave functions of non-interacting electrons at q. The single-particle basis states that constitute Φ q are the monopole harmonics [15]: Y q,n,m (Ω j ) = N q,n,m ( 1) q+n+m e iqφ j u q m q+m j v j n ( )( ) n ( 1) s 2q + n (vj v j ) n s (u s q + n + m s ju j ) s (2.5) s=0 where n = 0, 1,... is the Landau level index, m = l, l + 1,..., l 1, l labels the degenerate states in the nth LL, and l = q + n. The normalization coefficient is given by ( ) 1/2 (2q + 2n + 1)(q + n + m)!(q + n m)! N q,n,m =. (2.6) 4πn!(2q + n)! Ω j represents the position of the jth electron on the surface of the unit sphere, and u j cos(θ j /2) exp( iφ j /2) (2.7) v j sin(θ j /2) exp(iφ j /2). (2.8) The wave function Φ q is the constructed as a Slater determinant of the monopole harmonics: Det[Y i (Ω j )]. (2.9) It has previously been shown [12, 8] that the process of multiplying Φ q by Φ 2 1 and then projecting onto the LLL to construct the CF wave function is not a

27 14 unique LLL projection and one can instead write the CF wave function as [8] J Det[Y i (Ω j )], (2.10) in which Y q,n,m and J are defined as: Y q,n,m (Ω j ) = [ ] (2q + p(n 1) + 1)! (2q + p(n 1) + n + 1)! N q,n,m ( 1) q+n+m u q+m q m j v j n ( )( ) n ( 1) s 2q + n s q + n m s s=0 [ ] u s jv n s j U s j V n s j 1, (2.11) where U j p k v k u j v k v j u k + u j, (2.12) and u k V j p +, (2.13) u j v k v j u k v j j,k j k J ( ) ip (u j v k v j u k ) p exp 2 (φ j + φ k ). (2.14) We use the above method of LLL projection when studying the nodal structure of CF theory since it allows us to work with relatively large systems with ease. We use a direct projection method and the above projection method to test the robustness of the fractional statistics concept to LLL projection.

28 Chapter 3 Nodal Structure of the Composite Fermion Wave Function 3.1 Introduction Many condensed matter theorists have wondered why the composite fermion (CF) wave function so accurately captures the essence of lowest Landau level (LLL) physics. As we know from various Monte Carlo studies over the years, the success of a variational wave function lies in how accurately it describes the nodes of the state it is trying to model. The CF wave function s success stems from its nodal structure. This chapter, which discusses work published in collaboration with Jainendra K. Jain and Sudhansu S. Mandal [7], investigates the nodal structure by finding the nodes of the CF wave function and calculating various correlation functions between zeroes and electrons. We begin by discussing an important distinction between the on-particle and off-particle zeroes of the wave function; also, electron and particle will be used interchangebly in this chapter. Section 3.3 discusses the method used to find the wave function zeroes and Section 3.4 defines the correlation functions. Section 3.5 contains results for the various correlation functions for filling factors ν = 1/3, 2/5, 3/7, and 4/9, for up to 30 electrons.

29 When is a wave function zero a vortex? Is each zero of the wave function a vortex? Since a zero is a location where an electron cannot be, is there a positive charge associated with a zero? Is there a general principle which determines the location of a wave function zero? These questions, while simple, strike to the heart of the CF wave function s nodal structure. Since the CF wave function is fermionic, it obeys the Pauli exclusion principle. Since the CF wave function is also analytic, its nodal structure can be discovered by first fixing all electrons except one, the N th, and decomposing the wave function according to the fundamental theorem of algebra Ψ CF = j (z j z (0) j ), (3.1) where the z (0) j are coordinates of zeroes and the product is over the number of zeroes. Since the wave function is fermionic, at least N 1 of the zeroes must lie on the N 1 fixed electrons. If there is more than one zero on any electron, the multiplicity must be odd to preserve antisymmetry. This leads to a corollary to the Pauli principle for many-body wave functions: An odd number of zeroes must be found at each electron position. Furthermore, the zeroes will be positioned such that the Coulomb interaction energy is minimized. Given that the zeroes ensure maximal electron-electron repulsion, tempered by the wave functions angular momentum, it is natural to conjecture that the zeroes might act as positively charged particles. Since electrons avoid zeroes, a correlation hole might be created, allowing an effective attraction between electrons and zeroes. We first discuss the charge of a zero. Following Ref. [16], we find it important to make a distinction between zeroes which are on or off the particles. A vortex in the wave function at η is created by multiplication by the factor j (z j η), which ensures that all electrons avoid η. It clearly has a charge deficiency associated with it. The off-particle zeroes of a given particle are not zeroes for the other particles (this can be explicitly seen in Fig. 3.1, which shows the positions of the zeroes of all particles for a given starting configuration), and therefore are not avoided by all particles. No charge deficiency is associated with the off-particle zeroes of a given particle because they

30 17 are not vortices. The zero bound to a particle behaves differently. It is a zero for all particles (except, of course, the particle to which it is bound). The on-particle zero creates a correlation hole around the particle, and has a positive charge associated with it. Another way to test whether a zero at ηis vortex is to calculate the winding phase of the wave function around η. If the phase change is an integer multiple of 2π, then η is a vortex. To emphasize this distinction, we will reserve the use of the term vortex for the on-particle zeroes, such as the ones that arise from the Jastrow factors in Eqs. (1.3) and (1.6). The off-particle zeroes, which do not have a direct physical significance, will be called either nodes or simply zeroes. How about the attraction between zeroes and particles? Consider the zeroes of a test particle. The test particle cannot be attracted toward its own zeroes. First consider the on-particles zeroes. Since the on-particle zeroes are the other electrons in the system and the test particle is repelled by the other electrons, the test particle cannot be attracted to the on-particle zeroes. The off-particle zeroes are locations where, if the test particle were placed there, the wave function vanishes. Since any electron configuration for which the wave function vanishes is impossible, a configuration where the test particle is coincident with an off particle zero is impossible. Moreover, as the test particle moves closer to a zero, the wave function decrease linearly to zero, per the fundamental theorem of algebra. Thus, the particle is in fact repelled by its zeroes. This repulsion, while not literally correct, is useful intuitively and is conceptually similar to Pauli repulsion. Furthermore, the pair-distribution function of the test particle and its zeroes should show evidence of short-range repulsion and the lack of long-range order characteristic of a liquid. One may also ask if there may be some attraction between the zeroes of a test particle and other particles. Such an attraction is not analogous to the attraction between two oppositely charged objects, but can possibly be induced by the fact that the zeroes of the test particle repel it, so having them close to the other particles will ensure that the test particle avoids them as well. The reality, in general, is much more complex than this simple-minded single-particle argument might suggest, because the zeroes in the wave function are distributed so as to ensure that all particles optimally avoid one another, not just the test particle. In the case of ν = 1/m, the zeroes of the test particle sit on other particles because such a binding is allowed by Pauli principle, and also guarantees repulsion between

31 18 all pairs. Unfortunately, for other filling factors the situation is not so simple. The only sure way to answer this question is by actually calculating various correlation functions involving zeroes and particles. It may also be noted in this context that the zeroes do not have their own independent dynamics. The positions of the zeroes are completely fixed once the positions of the particles are fixed. (The wave functions in Eq. 1.6 have no parameters other than the particle positions.) When calculating various physical quantities, for example the current, only the particles ought to be considered. We discussed above how the off-particle zeroes do not have a well-defined charge associated with them. Consider then, the on-particle zeroes, which are vortices and produce a positively charged correlation hole around each particle. They cannot contribute to the current, despite their associated correlation hole, because the positively charged background is static. 3.3 Finding wave function nodes We find the zeroes by minimizing the modulus of the wave function in a twodimensional subspace. There are several minimization methods to choose from, some fast, others robust. Since the wave function has many zeroes we must use a method that easily finds local minima, and since the wave function is a Slater determinant, calculates the wave function as few times as possible. Slater determinant calculation is costly because each single-cf basis state depends on the coordinates of all N electrons. As a result we cannot use standard O(N) updating techniques for fermions [17] to calculate the determinant and O(N 3 ) operations are needed instead at each Monte Carlo iteration. Projecting onto the lowest Landau level is also increasingly costly as we consider more CF-LLs since, for the FQH state at ν = n/ 2pn ± 1, the cost of projection goes as n 3 [8]. Thus the computation time scales as: T N 3 n 3 N mc SN wf, (3.2) where N wf is the average number of wave function calculations needed to find a single zero and S is the number of zeroes. These considerations rule out methods like conjugate gradient. Instead, we use a two-dimensional Newton-Raphson

32 19 method. The Newton-Raphson method is a three-step process, the first of which is testing whether or not the initial guess is a zero. If the initial point is a zero then you stop and move on to the next guess; if the point is not a zero then take steps in the direction of steepest descent until a zero is found. Steps are computed by first assuming that the guess is close to the zero, which allows one to Taylor expand the modulus of the wave function about the zero. All particles except the Nth are fixed, so we expand in the coordinates (θ, φ) of the trial point (the Nth particle), f(θ + dθ, φ + dφ) = f(θ, φ) + f θ dθ + f (θ,φ) φ dφ (3.3) (θ,φ) 0, where f(θ, φ) = Ψ(θ, φ) and (θ+dθ, φ+dφ) is the position of the new point. Since the step must be in the direction of steepest descent, ( Ψ ) must be parallel to the Newton step. Thus the Newton step is given by f f(θ, φ) θ (θ,φ) dθ = ) 2 (3.4) ( f (θ,φ) and dφ = f(θ, φ) f φ (θ,φ) ) 2. (3.5) sin 2 (θ) ( f (θ,φ) Typically, a sequence of Newton steps will quickly converge to a zero, provided the guess lies reasonably close. If the Monte Carlo step-size is sufficiently small, the zeroes will move only slightly from their previous positions, allowing the zeroes of the previous configuration to be guesses for the current configuration. Since a very small step-size is not optimal for Metropolis algorithm, we slightly move all particles relative to the average interparticle separation (rather than a single particle by a large step) with an acceptance ratio 75%. It is important, however, to realize that the comparative ease (and speed) of finding zeroes depends primarily on the configuration of particles and how much a configuration changes between

33 20 Monte Carlo steps. Thus a somewhat uniformly distributed collection of particles more easily lends itself to such methods. Typically, we have performed calculations on systems containing up to 20 particles. For example, a calculation of g ppz of the ν = 2/5 state for 20 particles with 50,000 Monte Carlo steps takes 50 CPU hours on our workstation (Digital Model 600au, Alpha CPU, 500 MHz). A similar calculation of 21 particles in the ν = 3/7 state takes 67 CPU hours on the same machine. The only problem we have encountered with Newton-Raphson lies in the complexity of the basins of attraction. In Newton s method, for any polynomial with three or more distinct roots, the basins of attraction have disjointed fractal regions [18], and nearby points can converge to far away zeroes. This has generally resulted in a given zero being found more than once, although a guess would sometimes fall into a periodic cycle, never converging to a root within the maximum number of Newton steps (up to five hundred). When this happens we obtain guesses by blanketing the sphere s surface with small plaquettes and examining the way the wave function changes sign as one moves around the edge of the plaquette. If both the real and imaginary parts of the wave function change sign twice as we move around the plaquette, then the plaquette s center is taken as a guess. Since these trial zeroes go into Newton-Raphson, in some rare occasions all zeroes are still not found; we completely ignore such a Monte Carlo iteration and return to the previous configuration, using its zeroes as input for the sums in the correlation functions. This is akin to the way a rejected Monte Carlo step is handled in the Metropolis algorithm. A primary concern of any minimization method is the termination condition. The basic question is how does one know when a zero has been found? How small does a function have to be before it is numerically zero? This maximal function value is called the tolerance. For our purposes it is sufficient to require the modulus of the wave function to be below some maximal tolerance. Since there are trivial zeroes bound to each particle it is advantageous to use the modulus of the wave function near these trivial zeroes to define the tolerance. We calculate the modulus of the wave function at some δ away from each of the first N 1 particles, using the lowest value as the tolerance. As the particles move during Monte Carlo, what is or is not a good tolerance may change. To account for this, we recalculate the

34 21 tolerance during each accepted Monte Carlo step. The shift δ indirectly determines the precison of the zeroes found from Newton-Raphson and needs be no smaller than the smallest bin size used to calculate the correlation functions. We use δ = 10 3 l 0 and are able to consistently resolve the position of the zero to at least four decimal places. Now that we know how to find the zeroes of the CF wave function, we can look at a few sample electron configurations and find their zeroes. In Fig. (3.1), we show the positions of electrons and the zeroes seen by each electron. We consider systems with (ν, N) = (2/5, 20), (3/7, 21), (4/9, 16), and (5/11, 25), where the electron configurations were generated by variational quantum Monte Carlo with Metropolis algorithm. 3.4 Correlation Functions We calculate the pair-distribution functions of zeroes g zz (r), particle-zero pairs g pz (r), particle-off-particle zero pairs g p z (r), and the three-point correlation function g ppz (d, r), by the Monte Carlo method (with Metropolis algorithm) in the spherical geometry [14], all properly normalized to approach unity at large r. To illustrate the method, easily generalized to other correlation functions, we begin by discussing the two-point correlation function g pp (r 1, r 2 ) = N(N 1) ρ 2 0 d 2 r 3 d 2 r N Ψ(r 1,..., r N ) 2, (3.6) which, for a homogeneous system, depends only on r 1 r 2, where the r i are the positions of indistinguishable particles. (Here we stress the indistinguishability because the normalization factors change if one calculates correlations between distinguishable entities. More discussion will follow below.) Since the ground state is homogeneous, one can define the pair correlation function as g pp (r) = 1 A d 2 R g(r, R), (3.7) where r = r 1 r 2 is the distance between the pair of particles (or zeroes) and R = (r 1 + r 2 )/2 is the pair s center of mass. Since the system is isotropic, the pair

35 Figure 3.1. Positions of the zeroes seen by each particle. Panels (a)-(d) correspond to systems with (ν, N) = (2/5, 20), (3/7, 21), (4/9, 16), and (5/11, 25), respectively. The particle positions are shown with large, filled circles, and the zeroes with unfilled symbols (different symbols for the zeroes of different particles). The sphere is plotted in polar coordinates. The north pole is the origin, with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The equator is given by the dashed line. Distance is measured in units of the magnetic length, l 0. The particles and zeroes far from the north pole are not shown because of the significant distortions in the distances while depicting the spherical geometry in a plane. Note that the distortion is not severe enough to destroy the qualitative behavior of off-particle zeroes clustering between nearby particle pairs, as seen in panels (a), (b), and (d). 22

36 23 correlation function depends only on r r : g pp (r) = 1 δ(r r i + r j ). (3.8) ρ 0 N i j We evaluate the pair-distribution function using the Monte Carlo method, generating configurations according to the Metropolis algorithm and determining the distances between all pairs for each configuration. The brackets in Eq. (3.8) denote ground state averaging. On the sphere, after sectioning the surface of the sphere into bins of sufficiently small area (more on this later), Eq. (3.8) may be represented in a form useful for Monte Carlo: g pp (r) = 1 N mc ρ 0 N N mc k=1 N i j=1 Θ(r(k) ij r) a r, (3.9) where Θ(r (k) ij r) is 1 or 0 depending on whether the distance between the pair ij lies inside or outside the bin containing r, a r is the area of the bin, and N mc is the total number of Monte Carlo steps. The preceding equation can be modified straightforwardly for g zz and g pz : g zz (r) = 1 N mc ρ Z N Z N mc k=1 NZ i j=1 Θ(r(k) ij r) a r, (3.10) g pz (r) = 1 2N mc ρ 0 N Z N mc k=1 N i=1 NZ j=1 Θ(r(k) ij r) a r, (3.11) where N Z is the number of zeroes, and ρ Z is their density. Eq. (3.11) may also be used to calculate g p z (r) if N Z is replaced by the number of off-particle zeroes N Z = N Z N + 1. Note the factor of 1/2 in Eq. (3.11). Such unconventional normalization factors will appear when calculating correlations between distinguishable entities, for example a particle and a zero. Usually, the normalization factor of an n-point correlation function of identical particles is the number of n-tuples that can be chosen from N particles. The counting is slightly more complicated for distinguishable particles, where we must calculate some (m, n)-point correlation function,

37 24 choosing m particles of type A from a possible M and n type B particles from a possible N (assume M > N). The normalization factor will then be the number of permutations of m type A particles and n type B particles. The general form of an (m, n)-point correlation function of distinguishable entities will be discussed in Appendix A. The three point correlation function g ppz (d, r) = 1 ρ Z ρ (r) (3.12) is evaluated by determining ρ (r), the density of zeroes at r, with the constraint, indicated by the prime, that two composite fermions are held fixed at (±d/2, 0). This constraint is easily incorporated into the Monte Carlo algorithm. In the spherical geometry, the fixed particles are positioned at (θ 1, φ 1 ) = ( d, 0) and 2R (θ 2, φ 2 ) = ( d, π). Eq. (3.12) can also be recast in a form convenient for Monte 2R Carlo calculations: g ppz (d, r) = 1 N mc NZ i=1 Θ(r(j) i r), (3.13) N mc ρ z a j=1 r where Θ(r (j) i r) is 1 or 0 depending on whether the ith zero lies inside or outside the bin containing r Binning the sphere Significant care must be exercised in binning the sphere. We give here some of the relevant details. Since the pair-distribution functions depend only on r, the surface of the sphere can be divided into infinitesimal spherical frusta, where a spherical frusta is the surface of the sphere between two parallel circular cross-sections. We measure distance between pair ij as chord length r ij = 2R u i v j v i u j ; corresponding to the pair lying in the bin indexed by 1 + (integer part)(d r ij /2R), where D is the number of bins. The mth bin, at distance 2R(m 1/2)/D, has area 4πR 2 (m 1/2) D sin 1 ( m D ) sin 1 ( m 1 ( ) D ) 2 (m 1/2) 1 (3.14) D

38 25 where m = 1,..., D. We have used D = 200. For g ppz, which depends on θ and φ, binning is slightly more complicated. Since we are primarily interested in the behavior of g ppz near the fixed particles, smaller bins are chosen here to capture finer details. The φ bins are generated according to φ m = 2πm/D, with m = 1,..., D. The θ arc from 0 to π, however, is binned into three regions. The first region covers θ ɛ (0, π/a 1 ] and is given by θ n = πn/a 1 D 1, where n = 1,..., D 1 and A 1 = (integer part)(π/2θ 1 ) + 1. This choice of A 1 allows the fixed particles to lie in the middle of region one. The second region covers θ ɛ (π/a 1, π/a 2 ] and is given by θ n = πn/a 3 D 2 + π/a 1, where A 2 = 2, A 3 = (A 1 A 2 )/(A 1 A 2 ), and n = 1,..., D 2. While we have selected A 2 = 2, it is essentially a free parameter, provided A 1 > A 2. Region three covers θ ɛ (π/a 2, π/a4] and is given by θ n = πn/a 5 D 3 + π/a 2, where A 4 = 1, A 5 = (A 2 A 4 )/(A 2 A 4 ), and n = 1,..., D 3. We have selected A 4 = 1, though it s only constrained to be less than A 2. In particular we have chosen D = 200, D 1 = D 2 = 80, and D 3 = 40 (note that D 1 + D 2 + D 3 = D); a large D 1 allows high resolution near the fixed particles. The bin area a r is given by 2πR 2 sin(θ n ) δθ n /D, where δθ n = θ n θ n 1 and θ n lies in the appropriate region. 3.5 Results and Conclusion We have calculated several pair-distribution functions involving zeroes at filling factors ν = 2/5, 3/7, and 4/9. We now show the results and discuss what they signify. The particle-particle pair distribution function at these filling factors has been presented earlier [19] Zero-zero Fig. (3.2a) shows the zero-zero distribution function, g zz (r), which gives the probability of finding two zeroes at a distance r. It approaches unity at long distances, indicating that the zeroes themselves form a liquid. At short distances, the zeroes avoid one another.

39 26 Figure 3.2. Pair-distribution functions for three FQHE states at ν = n/(2n + 1), corresponding to n filled Landau levels of composite fermions. N = 20, 21, and 16 particles have been used for n = 2, 3, and 4, respectively. (a) shows the probability of having two zeroes separated by distance r, (b) shows the probability for a particle and a zero being separated by distance r, and (c) shows the probability for a particle and an off-particle zero. Following Ref. [19], we choose the unit of length for this plot as k 1 F = (4πρ) 1/ Particle-zero Fig. (3.2b) shows g pz (r), the probability of finding a zero at a distance r from a particle. The delta function at the origin, due to the single zero at the particle, is not shown explicitly. While forming their own liquid, the off-particle zeroes avoid particles. In fact, a repulsion between zeroes also guarantees a repulsion between the zeroes and particles, for the particles carry zeroes with them. We also plot g p z (r), the probability of finding an off-particle zero at a distance r from an on-particle zero, in Fig. (3.2c). (This is the same as the correlation function for a particle and an off-particle zero.) Note that the curves are normalized differently. The close similarity between the three plots demonstrates that the on-particle zeroes and the off-particle zeroes behave equivalently in the liquid of

40 27 zeroes Particle-particle-zero (ppz) In Fig. (3.1), especially in the first two panels, the zeroes prefer to be between pairs of particles, especially if the particles are closer than the typical interparticle separation. There is a bunching of the zeroes of various particles right in between pairs. This has motivated us to look at the correlations between a pair of particles and the zeroes. For this purpose, we hold two particles at a fixed distance, d, and calculate the probability of finding a zero in the neighborhood. Sufficiently far away, as expected, the probability of a zero is independent of the distance between the two fixed particles. We measure d in units of 4l 0, roughly equal to the interparticle separation, which we call r pp. Figs. (3.3) - (3.5) shows how the probability of finding a zero near the fixed particles evolves as d is varied from 1.0 r pp to 0.35 r pp, for 20 particles at ν = 2/5. When the two fixed particles are at the typical interparticle separation, the particleparticle-zero correlation function is essentially the sum of two independent particlezero correlation functions. When d = 1.0 r pp, in fig. (3.3), we see that finding a zero half-way between the fixed particles is roughly as likely as finding a zero far away. As seen in Figs. (3.4) and (3.5), when d is decreased to r pp /2 and 0.35 r pp, the probability of finding a zero half-way between the fixed particles increases by more than an order of magnitude. As the fixed particles are pushed together, the peak between them increases and narrows. This trend is most apparent in Figs. (3.6) and (3.7), which show the probability of finding a zero along an arc parallel and an arc transverse to the fixed particles (passing through the midpoint), respectively. We also see the emergence of prominent satellite peaks on either side of the two fixed particles, along the line joining the particles. The difference in the height of the central peak coming from the two orthogonal directions is an artifact of the way a grid of bins is generated on the sphere. To ascertain how this behavior evolves with filling factor, we next consider ν = 3/7 and 4/9. Figs. (3.8) - (3.10) shows the d dependence of the probability of finding a zero near the fixed particles, for 21 particles at ν = 3/7. When

41 28 Figure 3.3. g ppz for 20 particles at ν = 2/5, with d = 1.0 r pp and r pp = 4l 0, l 0 being the magnetic length. The sphere is plotted in polar coordinates. The north pole is the origin; with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The white arrows represent the positions of the two fixed particles. The contours are shaded as follows. Black represents zero probability per unit area, the shade of grey toward the edges is a probability per unit area of one, with lighter shades representing probability per unit area greater than one, white being highest. The grainy detail is numerical noise. Distance is measured in units of l 0. d = 1.0 r pp, we find behavior similar to that in Fig. (3.3). This is to be expected, because in both cases the distance between the two fixed particles is a typical interparticle separation and the density of zeroes is essentially a sum of two uncorrelated densities. However, in Fig. (3.9), we see no significant increase in the probability of finding a zero half-way between the fixed particles as d is reduced from 1.0 r pp to r pp /2, although the satellite peaks again appear. The difference in behavior, when compared to ν = 2/5, is further made apparent as d decreases to r pp /4 in Fig. (3.10) and in figs. (3.11) and (3.12). The peak between the fixed particles vanishes; as do the satellite peaks. Instead of satellite peaks, a ring forms around the pair of fixed

42 29 Figure 3.4. g ppz for 20 particles at ν = 2/5, with d = 0.5 r pp. particles. A ring is to be expected in the limit d = 0 at all filling factors, because then we have a single charge of 2e at the origin. However, the central peak is much weaker at ν = 3/7 than at ν = 2/5 for any value of d. The zeroes of the 20 particle state at ν = 4/9 behave in a manner similar to that at 3/7. Figs. (3.13) - (3.15) and its cross-sections in Fig. (3.16) shows the zero between the fixed particles being squeezed out as d decreases. In the bottom panel, when the central peak completely vanishes, a ring around the fixed particles emerges Discussion The zeroes of the FQHE states distribute themselves to minimize the Coulomb interaction. The on-particles zeroes are required by the Pauli exclusion principle and do not otherwise have a clear physical meaning, nor do the off-particle zeroes. There is no well-defined charge or any other quantum number associated with the zeroes, the one exception being the zero associated with a vortex excitation. With

43 30 Figure 3.5. g ppz for 20 particles at ν = 2/5, with d = 0.35 r pp. Figure 3.6. Cross-section of g ppz, for 20 particles at ν = 2/5, along the arc passing through the fixed particle positions. In this and subsequent figures, the origin is the north pole and the fixed particle positions are represented by arrows. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.35 r pp, respectively. The numbers appearing beside the central peaks in (b) and (c) give the height of the peak.

44 31 Figure 3.7. Cross-section of g ppz for 20 particles at ν = 2/5, along the arc passing through the origin in a direction perpendicular to the arc joining the fixed particle positions. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.35 r pp, respectively. that warning, we have investigated the nodal structure for several filling factors. Our findings are: (i) The zeroes form a liquid. (ii) There is short range repulsion between the zeroes. This also induces a repulsion between zeroes and particles, for the latter are the carriers of the Pauli zeroes. (For the special case of Laughlin s wave function, all zeroes sit on particles not because the zeroes are attracted to one another or to the particles, but because this is the most efficient way for particles to stay away from one another.) (iii) For some states, especially for ν = 2/5, there is an anomalously large probability of finding a zero right in the middle of a pair of nearby particles.

45 Figure 3.8. g ppz for 21 particles at ν = 3/7, with d = 1.0 r pp. The sphere is plotted in polar coordinates. The north pole is the origin; with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The white arrows represent the positions of the two fixed particles. The contours are shaded as follows. Black represents zero probability per unit area, the shade of grey toward the edges is a probability per unit area of one, with lighter shades representing probability per unit area greater than one, white being highest. The grainy detail is numerical noise. Distance is measured in units of l 0. 32

46 33 Figure 3.9. g ppz for 21 particles at ν = 3/7, with d = 0.5 r pp. Figure g ppz for 21 particles at ν = 3/7, with d = 0.25 r pp.

47 34 Figure Cross-section of g ppz, for 21 particles at ν = 3/7, along the arc passing through the fixed particle positions. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.25 r pp, respectively. Figure Cross-section of g ppz, for 21 particles at ν = 3/7, along the arc passing through the origin in a direction transverse to the arc joining the fixed particle positions. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.25 r pp, respectively.

48 Figure g ppz for 20 particles at ν = 4/9, with d = 1.0 r pp. The sphere is plotted in polar coordinates. The north pole is the origin; with the arc length Rθ used as the radius and the azimuthal coordinate φ as the polar angle. The white arrows represent the positions of the two fixed particles. The contours are shaded as follows. Black represents zero probability per unit area, the shade of grey toward the edges is a probability per unit area of one, with lighter shades representing probability per unit area greater than one, white being highest. The grainy detail is numerical noise. Distance is measured in units of l 0. 35

49 Figure g ppz for 20 particles at ν = 4/9, with d = 0.5 r pp. 36

50 Figure g ppz for 20 particles at ν = 4/9, with d = 0.25 r pp. 37

51 Figure Cross-section of g ppz, for 20 particles at ν = 4/9, along the arc passing through the fixed particle positions in the upper figure and along the arc perpendicular to the arc joining the fixed particle positions in the lower figure. (a)-(c) correspond to d = 1.0 r pp, 0.5 r pp, and 0.25 r pp, respectively. 38

52 Chapter 4 Fractional Exchange Statistics of Composite Fermions 4.1 Introduction The idea of particles living in two-dimensional condensed matter systems possessing fractional exchange statistics was introduced by Leinaas and Myrheim [20]. When identical particles are confined to two dimensions, trajectories in their multiply-connected configuration space that cannot be continuously deformed into one another are topologically distinct. If one considers a two-particle subspace, then the configuration space is a circle, not a disk. Here, exchange paths are labeled by their winding numbers. Viewed in this manner, topologically distinct paths have different winding numbers. This fact allowed Leinaas and Myrheim [20] to define fractional statistics as follows. First choose two particles. Next, allow one particle s path to enclose the other in a closed loop. This changes the wave function s phase by 2πθ. This phase is a topological property since it depends only on the loop s winding number. If θ is a non-integer, then the particles are said to have fractional statistics θ. Even though the explanation of fractional quantum Hall physics follows from the composite fermion theory with no mention of fractional statistics [13], fractional statistics is believed to be a consequence of incompressibility at a fractional filling factor [21, 10, 22]. Fractional statistics may also be experimentally observ-

53 40 able; in fact, the results of several experiments have been published [30]. For Laughlin s quasiholes [9] at ν = 1/m, m odd, the statistics was derived explicitly by Arovas, Schrieffer, and Wilczek [21] in a Berry phase calculation, but a similar demonstration of fractional statistics has been lacking at other fractions, or even for the quasiparticles at ν = 1/m. The need for a microscopic confirmation was underscored by Kjønsberg and Myrheim [23] who showed that, with Laughlin s wave function, the quasiparticles at ν = 1/m do not possess well-defined statistics. The reason for the discrepancy remains unclear, but it illustrates that the fractional statistics is fragile and cannot be taken for granted. The objective of this chapter is to revisit fractional statistics armed with the microscopic composite-fermion (CF) theory of the FQHE [11]. The research supporting this chapter was conducted with Gun Sang Jeon and Jainendra K. Jain and was published as Refs. [26, 27]. The discussion below closely follows those publications. A step in that direction has been taken by Kjønsberg and Leinaas [24], whose calculation of the statistics of the unprojected CF quasiparticle of ν = 1/m, the wave function for which is different from that of Laughlin s, produced a definite value, the sign of which, however, was inconsistent with general considerations. We confirm below that the statistics is robust to projection into the lowest Landau level (LL), and provide a non-trivial resolution to the sign enigma, which has its origin in very small perturbations in the trajectory due to the insertion of an additional CF quasiparticle. The calculation is extended to ν = 2/5 for further verification of the generality of the concept. An interpretation of the meaning of fractional statistics in terms of the effective magnetic field of CF theory is also provided. 4.2 Calculating fractional statistics How does one calculate fractional statistics? Naively, one could attempt to build an N body wave function from single fractional statistics particle basis functions and then swap two of the particles. The phase change of the wave function could then yield the statistics. This approach is doomed, as there are no fundamental fractional statistics particles found in nature. Since there are no fundamental fractional statistics particles in nature, we must

54 41 find emergent particles. Fortunately, the CF theory provides some for us. Because the CF theory provides an accurate account of the low energy physics of the FQHE, including incompressibility at certain fractional fillings, it must also contain the physics of fractional statistics, which indeed is the case. Fractional statistics can be derived heuristically in the CF theory as follows [25]. Composite fermions are bound states of electrons and an even number (2p) of vortices. When a composite fermion moves in a loop of area A, the total phase associated with this path is given by Φ = 2π(BA/φ 0 2pN enc ), (4.1) where N enc is the number of composite fermions inside the loop and φ 0 = hc/e is the magnetic flux quantum. The first term on the right hand side is the Aharonov- Bohm phase for a particle of charge e going around in a counterclockwise loop in the external magnetic field. The second term comes from the vortices bound to composite fermions, indicating that each enclosed composite fermion effectively reduces the flux by 2p. (A note on convention: The magnetic field in the +z direction, the electron charge is e, and only consider counterclockwise motions.) Eq. (4.1) summarizes the origin of the FQHE. The phase in Eq. (4.1) is interpreted as the Aharonov-Bohm phase from an effective magnetic field: Φ 2πB A/φ 0. If we replace N enc by its average value N enc = ρ 0 A, where ρ 0 is the two-dimensional density of electrons, we get B = B 2pφ 0 ρ 0. (4.2) This shows that the FQHE of electrons at ν = n/(2pn+1) is IQHE [3] of composite fermions at CF filling ν = n. At these filling factors, the effective magnetic field is B = B/(2pn + 1). To understand why the Berry phases coming from the vortices in the Jastrow factor effectively cancel the magnetic field, it is instructive to understand the effective magnetic field by eliminating the phases of the Jastrow factor in favor of a vector potential following the standard approach [55, 56]. Consider the Schrödinger

55 42 equation [ ] 1 ( p i + e A( r 2m b c ) 2 i ) + V (z j z k ) 2p Φ ν i j<k = E j<k(z j z k ) 2p Φ ν, (4.3) where V is the interaction. The kinetic energy term will be the important one in what follows. (We note that the unprojected wave function is not an exact eigenfunction of the Hamiltonian. For the sake of the present argument, one may think of Φ ν as an arbitrary wave function rather than the solution of the noninteracting problem at ν ; then, the exact eigenstate in question can always be written in the above form. While performing the actual calculations of the Berry phase, we will of course use the projected wave functions which have a close to 100% overlap with the exact eigenstates.) Display the phases due to the Jastrow factor explicitly: Here (z j z k ) 2p = e i2p P j<k φ jk z j z k 2p. (4.4) j<k j<k φ jk = i ln z j z k z j z k. (4.5) We have been careful above to keep track of the fact that z = re iφ, as appropriate for external magnetic field in the +z direction. The Schrödinger equation can be rewritten as [ 1 2m b i ( p i + e c A( r i )+ e c a( r i) = E j<k ] ) 2+V z j z k 2p Φ ν j<k z j z k 2p Φ ν, (4.6) where the additional vector potential, which simulates the effect of the phases of the Jastrow factor, is given by a( r i ) = 2p 2π φ i 0 φ ij, (4.7) j

56 43 where the prime denotes the condition j i. The corresponding magnetic field is bi = 2pφ 0 ẑ j δ 2 ( r i r j ). (4.8) Thus, the phase of the Jastrow factor is equivalent to each electron seeing a flux tube of strength 2pφ 0 on all other electrons; the minus sign indicates that the flux tube points in the z direction, opposite to the direction to the external field B = Bẑ. This interpretation raises the following questions. (i) The effective vector potential does not take care of all of the phases in the unprojected wave function Ψ up ν, because there are additional vortices and antivortices in Φ ν. What about their effect? (ii) How does the projection into the lowest LL affect the above considerations? The feature that 2p vortices are strictly bound to electrons prior to the projection is lost upon projection into the lowest LL. For example, for ν > 1/3, where composite fermions manifestly carry two vortices prior to projection, only one vortex can be bound to each electron after projection. The projection thus obscures the physics of composite fermions. Is there any way of seeing an effective magnetic field directly with the projected wave functions? Fractional statistics and Berry s phase Fractional statistics is also an immediate corollary of Eq. (4.1). Consider the state with CF filling n < ν < n + 1 where the composite fermions in the topmost partially filled CF-LL are localized in wave packets; their positions are given by η α = x α iy α. Since the CFs are localized, one can imagine a density lump centered at each η α. These localized CFs in the topmost partially filled CF-LL will be called a CF quasiparticle (CFQP). In principle, an effective description in terms of the η α can be obtained by integrating out z j = x j iy j. Let us imagine the winding properties of the CFQP s from the underlying CF theory. Consider two CFQP s, with a separation large enough that the overlap between them is negligible. According to Eq. (4.1) the phase a CFQP acquires when it moves in a loop depends on whether the loop encloses the other CFQP. When it does not, the phase is Φ = 2πeB A/hc. When the other CFQP is enclosed, the phase

57 44 changes by Φ 2p = 2π2p N enc = 2π 2pn + 1 (4.9) because a CFQP has an excess of 1/(2pn + 1) electrons associated with it, relative to the uniform state [giving it a fractional local charge of q = e/(2pn+1)]. With Φ = 2πθ we get the statistics parameter for CFQP s θ = 2p 2pn + 1. (4.10) This value is consistent, if we subtract a one, with those previously published [10, 22]. The goal is to confirm Eq. (4.10) in a microscopic calculation. The calculation begins with the statistics parameter θ = C dθ Ψ η,η i d dθ Ψη,η 2π Ψ η,η Ψ η,η C dθ Ψ η i d Ψη dθ 2π Ψ η Ψ η where Ψ η is the wave function containing a single CFQP at η, and Ψ η,η, (4.11) has two CFQP s at η and η. Here we take η = Re iθ, where C refers to the path with R fixed and θ varying from 0 to 2π in the counterclockwise direction. For convenience, η = 0. Before proceeding, the wave functions for a single CFQP and for two CFQPs must be described Composite fermion quasiparticle wave functions Since a CFQP is a composite fermion placed above the vacuum of ν = n filled CF-LL s, we can build its wave function by analogy with an integral quantum Hall system that has a single electron on top of the vacuum of n filled LL s. We now use CF theory to map the problem at ν = n to the fractional filling ν = n/(2pn ± 1) in a manner that preserves distances (to zeroth order). where We first construct a quasiparticle wave function at B and multiply it by Φ 2p 1, Φ 1 = N j<k=1 (z j z k ) exp [ 1 4l 2 1 ] z i 2 i (4.12)

58 45 and l 2 1 = c/eb 1 = c/eρφ 0. Finally, we project the result onto the lowest electronic LL. This mapping leaves the area of the disk unchanged since, while the Jastrow factor expands the distance between the particles, the Gaussian contracts it in precisely the right amount to cancel the expansion. It is easy to check that the density remains constant in going from ν = n to ν = n/(2pn + 1) in this manner. (See the article by Jain in Ref. [13] for more details.) At ν, the single-particle orbitals in the lowest LL are given by ζ m (z) [ ] z m 2π2m m! exp z 2, (4.13) 4l 2 where l = (2pn + 1) 1/2 l 0 is the magnetic length at B. To put a CFQP at η, we construct the electronic wave function at ν by placing an electron in the relevant Landau level (otherwise empty) at η in a coherent state. The coherent state at η in the lowest LL is given by φ (0) η ( r) = [ ] ηz η 2 z 2 ζ m (η)ζ m (z) =exp. (4.14) 2l 2 4l 2 4l 2 m=0 The coherent state can be elevated to higher Landau levels by repeated application of the LL raising operator a (2 / z z/2)/ 2, which leads to the coherent-state wave function in the (n + 1) st LL, apart from a constant factor: [ ] ηz φ (n) η ( r) = ( z η) n η 2 z 2 exp. (4.15) 2l 2 4l 2 4l 2 It is convenient to define [ ] ηz φ (n) η ( r) = ( z η) n η 2 exp, (4.16) 2l 2 4l 2 so φ (n) η ] ( r) = φ (n) ( r) exp [ z 2. (4.17) η As an example, consider the system at ν = 1/(2p + 1), which is related to the CF filling ν = 1. The electron wave function at ν = 1 with fully occupied lowest 4l 2

59 46 LL and an additional electron in the second LL at η is Φ η 1 = φ (1) η ( r 1 ) φ (1) η ( r 2 ) 1 1 z 1 z 2.. z N 2 1 z N 2 2 ( exp j z j 2 /4l 2 ). (4.18) This leads to the (unnormalized) wave function for a CFQP at ν = 1/(2p + 1): Ψ η 1/(2p+1) Here, we have used = P LLL N i<k=1 φ (1) η ( r 1 ) φ (1) η ( r 2 ) 1 1 z 1 z 2. z1 N 2 z N 2 (z i z k ) 2p exp 1 2p + l 2 l ( j z j 2 /4l 2 0 ). (4.19) = 1, (4.20) l0 2 which is equivalent to Eq. (4.2). The wave function for two CFQP s at ν = 1/(2p + 1), at positions η and η, is similarly given by Ψ η,η 1/(2p+1) φ (1) η ( r 1 ) φ (1) η ( r 2 ) φ (1) η ( r 1 ) φ (1) η ( r 2 ) = P 1 1 LLL z 1 z 2.. z1 N 3 z2 N 3 ( N (z i z k ) 2p exp ) z j 2 /4l 2. (4.21) j i<k=1 These wave functions are different, but similar, to those considered in Ref. [24].

60 47 Figure 4.1. The excess charge density relative to the ground state in the presence of one CFQP on top of the N = 50, ν = 2/5 state. The CFQP is localized about the origin (η/l = 0) and the disk size is R d = l 2N/ν. We can see what this excitation does to the uniform ground state in Fig. 4.1, which shows the excess density δρ( r) = ρ CF QP ( r) ρ 0 ( r) of one CFQP on top of a 50 particle ν = 2/5 state. In a similar way we can construct the wave function for one or two CFQP s of the state at ν = n/(2pn + 1) for arbitrary n and p. For example, the wave function for a single CFQP at ν = 2/(4p + 1) (corresponding to n = 2) is given explicitly