Stability of the unconventional $\mathbb{Z}_{n}$ parton states at $\nu = 7/3$: The role of finite width

A class of $\mathbb{Z}_n$ parton states has been proposed for the one-third filled second Landau level. These represent superconductivity of bound states of n composite bosons, and support excitations with fractional charge $e/3n$. We consider the feasibility of these states at one-third filled second Landau level as a function of the semiconductor quantum well. We find a phase transition as a function of quantum well width where a parton state is favored for small well widths and the Laughlin state is stabilized beyond a critical well width. We also propose that a $\mathbb{Z}_n$ parton state is relevant at 1/3 filling in bilayer graphene at low magnetic fields. We discuss the role of spin and Landau level mixing, and also possible experimental signatures to distinguish $\mathbb{Z}_n$ parton states from Laughlin’s.