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arXiv:cond-mat/9509084 · Journal of Physics A Mathematical and General

Zamolodchikov - Faddeev algebra in two-component anyons

Yue-lin Shen, Mo‐Lin Ge

Published Sep 21, 1996• 1 citation
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Abstract

We investigate Wilczeck's mutual fractional statistical model at the field-theoretical level. The effective Hamiltonian for the particles is derived by the canonical procedure, whereas the commutators of the anyonic excitations are proved to obey the Zamolodchikov - Faddeev algebra. Cases leading to well known statistics as well as Laughlin's wavefunction are discussed.

DOI
10.1088/0305-4470/29/18/035

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We investigate Wilczeck's mutual fractional statistical model at the field-theoretical level. The effective Hamiltonian for the particles is derived by the canonical procedure, whereas the commutators of the anyonic excitations are proved to obey the Zamolodchikov - Faddeev algebra. Cases leading to well known statistics as well as Laughlin's wavefunction are discussed.
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.
The commutation relations of the composite fields are studied in the 3, 2 and 1 space dimensions. It is shown that the field of an atom consisting of a nucleus and an electron fields satisfies, in the space-like asymptotic limit, the canonical commutation relations within the sub-Fock-space of the atom. The composite anyon fields are shown to satisfy the proper anyonic commutation relations with the additive phase exponents. Then, (quasi)particle pictures of the anyons are clarified. The hierarchy of the fractional quantum Hall effect is rather simply nderstood by utilizing the (quasi)particle charactors of the anyons. The commutation relations of the scalar object in the Schwinger(Thirring) model are mentioned briefly.
The complete set of ground state wave functions for N anyons in an external magnetic field on the torus is found. The cases when the filling factor is less than or equal to one are considered. The single valued description of anyons is employed through out by coupling bosons to a Chern-Simons field. At the end, the Chern-Simons interaction is removed by a singular gauge transformation as a result of which the wave functions become multi-component in agreement with other studies. USITP-93-07 1 Anyons [1], particles obeying fractional statistics, have been the subject of intense study in the past few years. Besides the purely theoretical interest in fractional statistics a new motivation has been provided by the Laughlin theory of the fractional quantum hall effect (FQHE)[2]. It was realized some time ago that certain excitations of the Laughlin state are anyonic [3]. Anyons are, therefore, characteristic of the elusive Laughlin state. An understanding of anyons in a magnetic field may help to provide, in turn, an understanding of the rich structure of the Laughlin state. This has been the motivation behind much of the study of anyons in
A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.