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This chapter analyzes the fractional quantum Hall effect in composite fermion systems. The basic idea of composite fermions conceived by Jain and its relation with the Laughlin wave function are presented. The Hall conductivity in a system of composite fermions within Chern–Simons field theory is discussed. The ground state energy of composite fermion systems is found. The metal of composite fermions is also considered. The BCS paired Hall state in the composite Fermi liquid is also discussed.

This chapter discusses the concept of Haldane pseudopotential of interacting electrons in a partially filled Landau level. A particular class of (so-called superharmonic) repulsive pseudopotentials is shown to induce a particular form of Laughlin correlations, characteristic of the Laughlin wave functions and described by Jain's (non-interacting) composite fermion model. This offers justification for composite fermions in some systems (e.g., electrons in a partially filled lowest Landau level) and explains their inadequacy in other systems (e.g., in partially filled higher Landau levels of either electrons or composite fermions themselves). The relation between the form of pseudopotential and the emergence of composite fermions is used to reduce Haldane's hierarchy of incompressible liquids to those predicted by composite fermion theory and observed in fractional quantum Hall experiments. The analysis is extended to multi-component fermion systems, e.g., electrons and trions formed naturally in optical studies of the fractional quantum Hall systems.

This introductory chapter starts with short review of experimental and theoretical description of integral and fractional quantum Hall effects. The standard quantization of states of charged particles moving on the plane in a magnetic field, the so called Landau quantization, is presented. The Laughlin wave function and Jain's idea of composite fermions are introduced and discussed with regard to fractional statistics and statistics transmutation. The Chern–Simons theory is presented as a theoretically complete and effective method for implementation of nonstandard statistics. The Haldane generalization of Pauli exclusion principle for fermions is also reported.

This chapter shows that composite fermions — particles which appear in fractional quantum Hall effect, and are statistically different from fermions — could be described within braid group formalism (looped particles). In the proposed model, some ways of exchange of bare particles of composite particles are restricted and it defines new group describing exchanges of particles. The new group describing exchanges of particles is defined and a unitary one-dimensional representations of this group is described. The configuration space for the system of looped particles is defined as covering space for configuration space of ordinary particles. An explicit form of configuration space for the system of two and three particles on the Euclidean plane is found using Burau representations.

Chapter 12 introduces three composite particles related to excitons: the trion, which is made of one exciton and a carrier; the biexciton, which is made of two excitons; and the exciton-polariton, which is made of one exciton coupled to a photon. Trions are composite fermions, while biexcitons and exciton-polaritons are composite bosons. The chapter also discusses the important role of spin and orbital degrees of freedom in semiconductor trions and biexcitons. The spin wave functions of electron and hole pairs in a trion or biexciton affect their orbital wave function and consequently their energy. In particular, since the orbital part of the ground state must be an even function with respect to fermion exchange, electrons, as well as holes, form a spin singlet state in the trion or biexciton ground state.

This chapter reviews the experimental and theoretical literature on phonon drag thermopower in reduced dimensionality conductors, particularly in the two-dimensional (2-D) case. It emphasizes the relationship between the mobility of electrons due to electron-phonon scattering and phonon drag, which is valid in the case when the electron mobility is dominated by elastic impurity scattering. This relationship applies at low magnetic fields, and also for composite Fermions at even denominator fractional filling factors where the effective magnetic field can be taken to be weak. The chapter also surveys weak and strong electron localization effects, and results in the integer and fractional quantum Hall regimes.

It is explained how plateaux are seen in the Hall conductance of two dimensional electron gases, at cryogenic temperatures, when the magnetic field is scanned from zero to ~10T. On a Hall plateau σ‎_xy = ne²/h, where n is integral, while the longitudinal conductance vanishes. This is the integral quantum Hall effect. Free electrons in such devices are shown to occupy quantized Landau levels, analogous to classical cyclotron orbits. The stability of the IQHE is shown to be associated with a mobility gap rather than an energy gap. The analysis showing the topological origin of the IQHE is reproduced. Next the fractional QHE is described: Laughlin’s explanation in terms of an IQHE of quasiparticles is presented. In the absence of any magnetic field, the quantum spin Hall effect is observed, and described here. Time reversal invariance and Kramer pairs are seen to be underlying requirements. It’s topological origin is outlined.