Quantum Hall hierarchy in a spherical geometry

Abstract

Representative wave functions, which encode the topological properties of the spin polarized fractional quantum Hall states in the lowest Landau level, can be expressed in terms of correlation functions in conformal field theories. Until now, the constructions have been restricted to flat geometries, but in this paper we generalize to the simplest curved geometry, namely that of a sphere. Except for being of interest for numerical studies, that usually are performed on a sphere, the response of the FQH liquids to curvature can be used to detect a topological quantity, the shift, which is the average orbital spin of the constituent electrons. We give explicit expressions for representative wave functions on the sphere, for the full Abelian FQH hierarchy, and calculate the corresponding shifts. These microscopic results, based on wave functions, agree with the predictions from the effective Chern-Simons field theory. The methods we develop can also be applied to the planar case. It gives simpler expressions for states with both quasiparticle and quasihole condensates, and allows us to give closed form expressions for a general state in the hierarchy, rather than finding the wave function on a case by case basis.