The physics of the state at filling factor ν = 1/2 remained a puzzle for many years. Attention was redrawn to it inlate 1989/early 1990 by the work of Jiang et al., who reported a deep resistance minimum at ν = 1/2 incertainhigh-quality samples, and by certain anomalies at ν = 1/2 insurface acoustic wave absorption observed by Willett et al.. This time, with composite fermions available, rapid progress was made.

The lowest Landau level problem has no kinetic energy. When electrons transmute into composite fermions, the interelectron interaction energy transforms, in the simplest approximation, into a “kinetic energy” of composite fermions. (In general, not all of the Coulomb interaction transforms into kinetic energy, which leaves behind a residual interaction between composite fermions.) The CF kinetic energy manifests dramatically through the quantized Λ levels and the FQHE at ν = n/(2pn ± 1). These sequences terminate into ν = 1/2p in the limit of n → ∞. Should composite fermions exist in this limit, the magnetic field experienced by them vanishes. Motivated by the experiments mentioned in the preceding paragraph, Halperin, Lee, and Read (also see Kalmeyer and Zhang) made the striking proposal that composite fermions form a Fermi sea here, called the CF Fermi sea:

an infinite number of filled Landau levels = Fermi sea, (10.1)

an infinite number of filled Λ levels = CF Fermi sea. (10.2)

When electrons are confined to two dimensions, cooled to near absolute zero temperature, and subjected to a strong magnetic field, they form a quantum fluid that exhibits unexpected behavior, for example, the marvelous phenomenon known as the fractional quantum Hall effect. These properties result from the formation of a new class of particles, called “composite fermions,” which are bound states of electrons and quantized microscopic vortices. The composite fermion quantum fluid joins superconductivity and Bose–Einstein condensation in providing a new paradigm for collective behavior.

This book attempts to present the theory and the experimental manifestations of composite fermions in a simple, economical, and logically coherent manner. One of the gratifying aspects of the theory of composite fermions is that its conceptual foundations, while profoundly nontrivial, can be appreciated by anyone trained in elementary quantum mechanics. At the most fundamental level, the composite fermion theory deals directly with the solution of the Schrödinger equation, its physical interpretation, and its connection to the observed phenomenology. The basics of the composite fermion (CF) theory are introduced in Chapter 5. The subsequent chapters, with the exception of Chapter 12, are an application of the CF theory in explaining and predicting phenomena. Detailed derivations are given for many essential facts. Formulations of composite fermions using more sophisticated methods are also introduced, for example, the topological Chern–Simons field theory.

This chapter introduces the basic principles of the composite fermion theory. It should really be called the “composite fermion model” or the “composite fermion hypothesis” in this chapter. The extensive scrutiny and testing that elevate it to the status of a “theory” are topics of subsequent chapters.

The great FQHE mystery

Not often does nature present us with a mystery as well defined as the phenomenon of the fractional quantum Hall effect. The questions that theory is challenged to answer could not be more sharply posed.

• What is the physics of this quantum fluid? The appearance of precise quantum numbers and dissipationless transport in a dirty solid state system containing many electrons is a signature of cooperative behavior. What are the correlations in the FQHE state and why do they manifest themselves in such a rich, yet stunningly simple fashion? Why do gaps open at certain fractional fillings of the lowest Landau level? Experiments point to something unique and special about ground states at certain special filling factors. What order brings about this uniqueness?

• A tremendous amount of factual information is contained in the fractions that are observed and the order in which they appear. That imposes rigorous constraints on theory. With the proliferation of fractions, certain striking patterns have emerged. Many fractions are conspicuous by their absence. For example, the simplest fraction f = 1/2 has not been observed. In fact, no sub-unity fractions with even denominators have been observed. Another remarkable aspect is that fractions are not isolated but belong to certain sequences. For example, in Fig. 2.7, the fractions 1/3, 2/5, 3/7, 4/9, 5/11, … follow the sequence f = n/(2n+1). These observations lead to the following questions: Why are some fractions seen but not others? Why do they appear in sequences?

• […]

The appearance of the Planck constant h in the formula for RH is an indication of the inherently quantum mechanical nature of the effect. In this chapter, we study a single electron confined to two dimensions and exposed to a magnetic field. This problem was solved exactly soon after the invention of quantum mechanics (Darwin; Fock; Landau), because it is merely a one-dimensional simple harmonic oscillator problem in disguise. The most remarkable aspect of the solution is that the electron kinetic energy is quantized. The discrete kinetic energy levels are called “Landau levels.”

The Landau level is the workhorse of the quantum Hall problem. The integral quantum Hall effect is seen below as a direct consequence of the Landau level formation. The explanation for the fractional quantum Hall effect, which is caused by interactions, requires further insights, but Landau levels again provide the key analogy: the effect arises because the lowest Landau level splits into Landau-like energy levels (called Λ levels).

This chapter deals with Landau levels in two geometries: planar and spherical. Two gauges are used in the planar geometry, the Landau and the symmetric gauges. The spherical geometry considers an electron moving on the surface of a sphere, subjected to a radial magnetic field. Wrapping the plane on to the surface of a sphere is simply a choice of boundary conditions, which should not affect the bulk properties of the state.

Topology is the study of properties of geometric configurations that remain invariant under continuous deformations. It is of relevance in graph theory, network theory, and knot theory. Many physical phenomena have topological content. Perhaps the best known is the Aharonov–Bohm phase for a closed path of a charged particle around a localized magnetic flux, which is independent of the shape and the length of the path so long as it is outside the region containing the magnetic flux. It is a special case of a more general concept, known as the Berry phase. The latter is the phase acquired when the parameters of a Hamiltonian execute a closed loop in the parameter space; the Berry phase does not depend on the rate at which the loop is executed, provided it is sufficiently slow. (There is no speed limit for the Aharonov–Bohm phase. In this case the adiabatic approximation of Berry becomes exact.)

The most dramatic examples of topology lead to macroscopic quantizations, such as the quantization of flux (in units of h/2e; 2e because of pairing) through a superconducting loop or through a vortex in a type-II superconductor. The quantization of flux results from the topological property that the phase of the order parameter can only change by an integral number times 2π around the flux.

Topology enters into the physics of the fractional quantum Hall effect through the formation of composite fermions, one constituent of which, namely the vortex, is topological.

The FQHE was discovered when Tsui, Stormer, and Gossard were looking for the Wigner crystal (WC), i.e. a crystal of electrons. While electrons in the lowest Landau level are now known to form a CF fluid at relatively large fillings, a solid phase is expected at sufficiently small fillings, when electrons are so far from each other that the system behaves more or less classically. This chapter is concerned with the nature of the solid phase. Theory makes a compelling case that the actual solid is not an ordinary, classical Wigner crystal of electrons, but a more complex, topological quantum crystal of composite fermions, in which quantized vortices are bound to electrons. This is called the “CF crystal,” to be differentiated from the electronic Wigner crystal. Experiments have seen indications of the formation of a crystal at low fillings, but further work will be required to ascertain its quantum mechanical character. (The quantum mechanical behavior of 3He and 4He solids has been studied extensively. Quantum mechanics plays a fundamental role in the superfluid behavior of 4He solid, a subject of much current investigation.)

We also briefly mention the possibility of charge density wave states of composite fermions, for example, unidirectional stripes or bubble crystals, which have been proposed theoretically. These have also not been observed yet, but might occur in higher Λ levels (i.e., between regular FQHE states), and are analogous to similar structures believed to occur for electrons in higher Landau levels.

The central postulate of composite fermion theory is that the liquid of strongly interacting electrons in a high magnetic field B is equivalent to an assemblage of weakly interacting composite fermions in an effective magnetic field B*. The effective magnetic field is such a direct, fundamental, and dramatic consequence of the formation of composite fermions that it is taken as the defining property of composite fermions, and its observation is tantamount to an observation of composite fermions themselves. Computer experiments on small systems, described in Chapter 6, demonstrate that the dynamics of interacting electrons in the lowest Landau level resembles that of weakly interacting fermions in an effective magnetic field. The present and the subsequent chapters analyze numerous experimental facts within the CF theory. We begin with the explanation of the FQHE.

Comparing the IQHE and the FQHE

Compelling evidence for composite fermions comes from the simple act of plotting the FQHE data as a function of the effective magnetic field B*, which amounts to rigidly shifting the B-axis by a constant (B – B* = 2pρø0), and then comparing it to the IQHE of electrons, which is a system of weakly interacting fermions. To facilitate comparison for samples with different densities, we plot the resistance trace in the IQHE as a function of 1/ν, which is proportional to B, and compare it with the resistance trace in high magnetic field plotted as a function of 1/ν*, which is proportional to B*.