Statistics of Composite Systems and Anyons in the Fractional Quantum Hall Effect

177 Progress of Theoretical Physics, Vol. 107, No. 1, January 2002 Statistics of Composite Systems and Anyons in the Fractional Quantum Hall Effect Hitoshi Ito∗) Department of Physics, Faculty of Science and Engineering, Kinki University, Higashi-Osaka 577-8502, Japan (Received June 28, 2001) The commutation relations of composite fields are studied in the 3, 2 and 1 spatial dimensions. It is shown that the field of an atom consisting of a nucleus field and an electron field satisfies, in the space-like asymptotic limit, the canonical commutation relations within the sub-Fock space of the atom. The field-particle duality in the bound state is discussed from a statistical point of view. Then, the commutation relations of the scalar object in the Schwinger (Thirring) model are briefly discussed and they are shown to be consistent with its interpretation as the Nambu-Goldstone boson. The composite anyon fields are shown to satisfy the proper anyonic commutation relations with additive phase exponents. Then, quasiparticle picture of the anyons is clarified under the restriction of this additivity. The difference between field and particle characteristics becomes more prominent in the 2-dimensional space. It is argued that the hierarchy of the fractional quantum Hall effect can be rather simply understood by utilizing the quasiparticle characters of the anyons in the case that the background-boson gauge is assumed. To contrast with this case, the composite fermion theories are critically reviewed. §1. Introduction More than forty years ago, composite systems in quantum field theory were investigated. In such a study, one starts from a local scalar field, say A(x), and assumes the existence of discrete eigenvalues m2 and M 2 of the 4-momentum squared, P 2 , where m is the mass of the original field A. If
A(x)A(y) | P 2 = M 2  =  0, there may be a composite (bound) state of mass M . Then, one defines a bilocal field B(x, ε) = T A(x + ε)A(x − ε) (1) representing this state, where T denotes the time-ordered product. Zimmermann showed, with some mathematical assumption, that the asymptotic (t → ±∞) field of B satisfies the proper commutation relation in the limit ε → 0, if it is suitably normalized. 1) One can infer from this construction that composite systems can be described by the local field operators, and there are no differences between elementary and composite particles in constructing the S matrix elements. The idea of boot strapping (nuclear democracy) emerged from this observation. However, the successes of gauge theories have changed drastically the framework of elementary particle theory. As a result, the quantum fields have been revived, and the hierarchy structure of the gauge interactions has gradually been recognized. Field theory in its present form is an effective theory for each level of classes in the hierarchy. ∗) E-mail: 178 H. Ito There is another hierarchy of compositeness in nature, 2) which has been revealed through the success of the composite models of elementary particles. The hierarchy here consists of the classes of quarks, hadrons, nuclei, atoms, and so on. The level of a class is specified by the energy scale, indicating the limit of applicability of the theory governing it. We, further, believe that the theories describing these classes are quantum field theories, and there exist elementary fields for each class, which are constructed from the elementary fields of a deeper class. The most instructive example of this interpolating mechanism may be provided by considering the class of an atom. An atom is a composite system that consists of a nucleus field and an electron field interacting through a photon field. Then, we construct the atom field as a composite field of elementary fields of a deeper class. In this respect, we are especially interested in the statistical properties of the composite system, since we feel it sometimes difficult to understand the statistical properties in the framework of particle quantum mechanics. For example, the nucleus of the hydrogen atom is a fermion and it changes into a boson by acquiring another fermion. This is mysterious from the viewpoint of the particle quantum mechanics.∗) The difference between quantum field theory and many-particle quantum mechanics is more prominent in the world of the 2 spatial dimensions. Its topological structure allows the exotic statistics of the field operators, which cannot always be realized by a (quasi)particle having definite values of the flux and the charge. Only if we can define the attributes of such a composite consistently, we can call it a quasi-particle anyon. The first purpose of the present paper is to give unified consideration to the essential points of the field-particle duality in composite systems. We also emphasize the topological difference between 3 and 2 spatial dimensions. The main issue is to determine how to understand the Chern-Simons (CS) gauge field that governs the statistics of the 2-dimensional space. We consider it to be the field of a boundary condition to be eliminated in the final physical results. The complicated nature of the statistics of anyons sometimes confuses theoretical understanding of the related phenomena. An example is the fractional quantum Hall effect (FQHE). Our second purpose is to clarify the hierarchical structure of the fractional quantum Hall state (FQHS) from the point of view of the CS gauge theory of anyons, which has not always been recognized correctly. We study the field aspects of the hydrogen atom in §2 and derive the canonical commutation relations for the atom field. Section 3 is devoted to studying the anyon fields in 2 spatial dimensions. We show the additivity of the phase exponents in the commutation relations, which restricts possible charges and fluxes of quasiparticles. Some aspects of anyons in the FQHE are sketched on this basis. In 1 spatial dimension, a composite field has commutation relations with rather clear-cut structure, which is shown briefly in §4. In §5 we study the hierarchy structure of the FQHS, placing emphasis on the special nature of the CS gauge field. The composite fermion theory is critically reviewed from this standpoint. Finally, we make some conceptual remarks on the subject in the last section. ∗) A gas of the H atoms undergoes Bose-Einstein condensation. This means that the protons become distributed like bosons, since their positions almost coincide with those of the atoms. Statistics of Composite Systems and Anyons in the FQHE §2. 179 The atom field The symmetries of the many-body wave function were investigated by Ehrenfest and Oppenheimer 3) using interesting considerations without referring to field quantization. We now study the same subject in the framework of quantum field theory, intending to make clear the field aspects of these symmetries. We consider here the simplest case of the hydrogen atom. We first introduce the composite field Ψ through the equation Ψ (x1 , x2 ) = T ψ(x1 )φ(x2 ), (2) where ψ(x1 (...truncated)