Composite Fermions

The following sections are included:

• CONTENTS

• Preface

The following sections are included:

• Introduction
  • True quasiparticles in condensed matter

• Background and motivation
  • The FQHE Problem
  • Model Hamiltonian
  • Laughlin's wave function
  • The hierarchy scenario

• Basic concepts of the composite fermion theory
  • Heuristic view of composite fermions
  • Nearly-independent CF model (NICFM)
  • Microscopic theory: Beyond NICFM
  • Verifications of the composite fermion theory
  • Fractional charge in the CF theory
  • Models of composite fermions
  • How real are composite fermions?
  • Why are composite fermions born?
  • What is new?
  • Plan of the rest of the article

• Lowest Landau level projection: disk geometry
  • Unprojected CF states
  • LLL Projection
  • Another method for projection
  • Testing wave functions
  • Relevance of disk geometry to FQHE

• Lowest Landau level projection: spherical geometry
  • Single electron states
  • Unprojected composite fermion states
  • Lowest LL projection

• Quantitative study of FQHE states
  • Accuracy of CF wave functions
  • FQHE Ground states
  • CF exciton
  • Instabilities
  • Pair distribution function: Fermi-sea-like correlations
  • Excited bands at general filling factors
  • Spin phase diagram of CF states
  • CF effective masses

• Summary of other studies of composite fermions
  • Chern-Simons theory of composite fermions
  • Gaps
  • Two component systems
  • Residual CF interaction
  • Geometric resonances
  • Thermopower
  • Shubnikov-de Haas oscillations
  • FQHE and disorder
  • Mesoscopics with composite fermions
  • CF-Excitons/ Collective modes
  • Edge states and tunneling
  • CF Transport at ν = 1/2
  • CF-phonon interaction
  • Pairing of composite fermions?

• A heuristic derivation of composite fermion wave functions
  • Basis functions for a single composite fermion
  • Effective magnetic field
  • Many-composite fermion states

• Conclusion

• Acknowledgments

• References

The composite fermion picture has had a remarkable number of recent successes both in the description of the fractional quantized Hall states and in the description of the even denominator Fermi-liquid like states. In this chapter, we give an introductory account of the Chern-Simons fermion theory, focusing on the description of the even denominator states as unusual Fermi liquids.

We review the fermionic Chern-Simons field theory for the Fractional Quantum Hall Effect (FQHE). We show that in this field theoretic approach to the problem of interacting electrons moving in a plane in the presence of an external magnetic field, the FQHE states appear naturally as the semiclassical states of the theory. In this framework, the FQHE states are the unique ground states of a system of electrons on a fixed geometry. The excitation spectrum is fully gapped and these states can be viewed as infrared stable fixed points of the system. It is shown that the long distance, low energy properties of the system are described exactly by this theory. It is further shown that, in this limit, the actual ground state wave function has the Laughlin form. We also discuss the application of this theory to the problem of the FQHE in bilayers and in unpolarized single layer systems.

We present a Chern-Simons (CS) theory for fractional quantum Hall states, in which flux attachment is followed by an enlargement of Hilbert space to include n magnetoplasmon oscillators and n constraints, n being the number of electrons. The simplest approximation yields correlated wavefunctions for fractions ν = p/(2ps + 1). For ν = 1/(2s + 1) we obtain multi-quasihole wavefunctions with correct gaussian and normalization factors, operators that create quaisholes, and a possible Read operator. Our composite fermion and boson operators attach flux as well as the correlation hole to electrons, i.e., bind them to zeros. We perform a further canonical transformation which decouples the the particles and oscillators in the long wavelength limit. The n constraints now act on just the particles, restricting them to the Lowest Landau level (LLL). Their kinetic energy vanishes In the noninteracting limit, they couple to the external field through a magnetic moment e/2m, carry charge e* = e/(2ps + 1) and a dipole moment which alone survives when ν = 1/2. When a Coulombic interaction is turned on, a nonzero 1/m* ≃ (2ν)^2/3e²l₀/6 arises. Density-density correlation functions and Hall conductivity, additive over the particles and oscillators, are calculated within an approximation for ν = 1/2. The formalism makes precise and unifies many prevalent notions regarding composite fermions and bosons, quasiparticles, cyclotronic physics, m and m*.

We introduce the reader to the Büttiker-Landauer theory of edge state transport, and then review its generalization to a composite fermion theory that applies in both the fractional and integer quantum Hall regimes. The composite fermion edge state theory accounts for the observed integer and fractional quantized Hall conductances and dissipationless transport of macroscopic Hall bars. It also accounts for the observed transport properties of nanoscale adiabatic and non-adiabatic devices in the fractional quantum Hall regime, including experiments on selective population and detection of fractional edge channels and the temperature dependence of Aharonov-Bohm conductance resonances. It describes current noise in terms of fractionally charged quasiparticles associated with current fluctuations. We also discuss whether edge states in the fractional quantum Hall regime are an unusual Fermi liquid (as the composite fermion edge state picture suggests) or a chiral Luttinger liquid.

This review will present the experimental findings which both motivated and subsequently have verified aspects of the composite fermion theory of Fermi surface formation at high magnetic fields. Following a brief introduction to two-dimensional systems, early experimental findings predating and influencing the composite fermion theory are examined. The theoretical construction, and in particular the hypothesis of Fermi surface formation by composite fermions are briefly reviewed with respect to experimental predictions. Experiments subsequent to the main theoretical developments are then presented, with emphasis on surface acoustic wave experiments which were responsible for measuring the composite fermion Fermi wavevector and mean-free-path. Further demonstrations of semi-classical motion of the quasiparticles at the Fermi surface are reviewed. Experimental determination of the composite fermion effective mass is examined, considering results from measurements in d.c. transport and surface acoustic waves. Extension of experiments to multiple filling factors in the magnetic field spectrum is also reviewed. Recent experimental results concerning gauge-field variations are presented. Experimental findings are summarized and potential future experiments are finally discussed.

The composite fermion picture has offered experimentalists a concise framework and easy-to-visualize concept to interpret experimental results. This review article surveys DC-transport experiments that rely on the ballistic and semi-classical motion of composite fermions along cyclotron orbits. The transverse magnetic focusing and transport through antidot lattices of composite fermions supports this mean field picture. Even details in the experimental traces can be accounted for when including a proper description of the imposed potential landscape. The spatially dependent effective magnetic field caused by the inhomogeneous density plays a crucial role. Transport through one- and two-dimensional periodic density modulations near half filling of the lowest Landau level is less well understood. For both geometries, the modulation, which is rather weak for electrons at small magnetic fields, induces an unexpectedly strong response for composite fermions.