Quantum Group Symmetry and Quantum Hall Wavefunctions on a Torus

We find a quantum group structure in two-dimensional motion of nonrelativistic electrons in a uniform magnetic field on a torus. The representation basis of the quantum algebra is composed of the quantum Hall wavefunctions proposed by Haldane-Rezayi at the Landau-level filling factor ν = 1/m (m odd). It is also shown that the quantum group symmetry is relevant to the degenerate Landau states and the deformation parameter of the quantum algebra is given by the filling factor. ———————————————————————– † Fellow of the Japan Society for the Promotion of Science E-mail address : hsato@jpnyitp. yukawa.kyoto-u.ac.jp There have been many discussions in the study of quantum groups (quantum universal enveloping algebras [1, 2]). Quantum group structures are found in (2+1)-dimensional topological Chern-Simons theories [3] as well as in rational conformal field theories and integrable lattice models [4]. Although the abelian Chern-Simons theory does not possess a quantum group structure in the literature [3], it might be possible to exhibit one in some other senses. There have been also interesting investigations of condensed matter problems such as the fractional quantum Hall effect [5] making use of the abelian ChernSimons theory. We have considered the two-dimensional planer motion of a nonrelativistic particle in a uniform magnetic field in the previous paper [12] and we derived the quantum group algebra Uq(sl(2)) acting within each Landau level. It can be understood from the result of that paper that the wavefunctions form a representation basis of the Uq(sl(2)) algebra and that the deformation parameter q is related to the filling factor ν = 1/(2j + 1) by q = exp(2πiν). These observations motivate us to discuss a relation between quantum groups and the quantum Hall effects. In this report we consider a Ne-electron system in the lowest Landau level on a square torus of side L. We show that this case also exhibits a quantum group symmetry similarly as the planer case and the deformation parameter q is given by the filling factor ν = 1/m (m odd integer). We use the results of the previous paper and thus start with reviewing some formulae. In the system of a charged particle in a constant magnetic field B perpendicular to the x-y plane, the generators of the quantum group algebra are realized by the magnetic translation operators Tα [7] T(α1,α2) = exp( i h̄ α · β) , (1) where βi = pi − e c Ai − eB c ǫijx j , (2) ǫ11 = ǫ22 = 0, ǫ12 = −ǫ21 = 1, (3)