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2013

The quantum Hall hierarchy in spherical geometry

Kvorning, Thomas

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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.

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To prove this conjecture, one must show that the phase factors that determine charge and statistics of the quasiparticles, can be read directly form the monodromies of the pertinent conformal blocks, without any additional Berry phases. This has been demonstrated in several cases [ 28 , 29 ] , and we shall assume it to be true also for the hierarchy states although here a proof is so far lacking

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