Microscopic theory of the quantum Hall hierarchy

TL;DR

This paper provides an exact solution to the quantum Hall problem in a specific limit, revealing a fractal organization of ground states consistent with known hierarchies and presenting wave functions for various observed states.

Contribution

It introduces an exact solvable limit for the quantum Hall problem and constructs wave functions that unify various hierarchy states within a single framework.

Findings

Ground states form a fractal pattern consistent with the Haldane-Halperin hierarchy.

Wave functions for Laughlin, Jain, and recently observed states match the exact solutions.

Proposes an adiabatic continuation from the solvable limit to real experimental conditions.

Abstract

We solve the quantum Hall problem exactly in a limit and show that the ground states can be organized in a fractal pattern consistent with the Haldane-Halperin hierarchy, and with the global phase diagram. We present wave functions for a large family of states, including those of Laughlin and Jain and also for states recently observed by Pan {\it et. al.}, and show that they coincide with the exact ones in the solvable limit. We submit that they establish an adiabatic continuation of our exact results to the experimentally accessible regime, thus providing a unified approach to the hierarchy states.