Abstract. In this paper, the key ideas of characterizing universality classes of dissipation-free (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification of quantum Hall lattices are stated and their physical implications are discussed. Physically relevant subclasses of quantum Hall lattices are defined and completely classified. The results are carefully compared with experimental data and also with other theoretical schemes (the hierarchy schemes). Several proposals for new experiments are made which could help to settle interesting issues in the theory of the (fractional) quantum Hall effect and thus would lead to a deeper understanding of this remarkable effect. † Present address:
A classification of incompressible quantum Hall fluids in terms of integral lattices and arithmetical invariants thereof is proposed. This classification enables us to characterize the plateau values of the Hall conductivity σ H in the interval (0, 1] (in units where e 2 /h = 1) corresponding to "stable" incompressible quantum Hall fluids. A bijection, called shift map, between classes of stable incompressible quantum Hall fluids corresponding to plateaux of σ H in the intervals [1/(2p + 1), 1/(2p − 1)) and [1/(2q + 1), 1/(2q − 1)), respectively, is constructed, with p, q = 1, 2, 3, (. . .), p = q.Our theoretical results are carefully compared to experimental data, and various predictions and experimental implications of our theory are discussed. † Present address:
We examine general aspects of parity functions arising in rational conformal field theories, as a result of Galois theoretic properties of modular transformations. We focus more specifically on parity functions associated with affine Lie algebras, for which we give two efficient formulas. We investigate the consequences of these for the modular invariance problem.a Chercheur Qualifié FNRS
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.