How low can the fractional quantum Hall effect go

While it is generally believed that a crystal state is favored at filling factors $\nu< 1/6$ in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences $\nu=n/(6n\pm 1)$ and $\nu=n/(8n\pm 1)$. To address this paradox, we investigate the nature of the state at filling factors $\nu=1/7$, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.