  The nature of the state at low Landau-level filling factors has been a
longstanding puzzle in the field of the fractional quantum Hall effect. While
theoretical calculations suggest that a crystal is favored at filling factors
$\nu\lesssim 1/6$, experiments show, at somewhat elevated temperatures, minima
in the longitudinal resistance that are associated with fractional quantum Hall
effect at $\nu=$ 1/7, 2/11, 2/13, 3/17, 3/19, 1/9, 2/15 and 2/17, which belong
to the standard sequences $\nu=n/(6n\pm 1)$ and $\nu=n/(8n\pm 1)$. To address
this paradox, we investigate the nature of some of the low-$\nu$ states,
specifically $\nu=1/7$, $2/13$, and $1/9$, by variational Monte Carlo, density
matrix renormalization group, and exact diagonalization methods. We conclude
that in the thermodynamic limit, these are likely to be incompressible
fractional quantum Hall liquids, albeit with strong short-range crystalline
correlations. This suggests a natural explanation for the experimentally
observed behavior and a rich phase diagram that admits, in the low-disorder
limit, a multitude of crystal-FQHE liquid transitions as the filling factor is
reduced.
