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 $ ulesssim 1/6$, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $ u=$ 1/7, 2/11, 2/13, 3/17, 3/19, 1/9, 2/15 and 2/17, which belong to the standard sequences $ u=n/(6npm 1)$ and $ u=n/(8npm 1)$. To address this paradox, we investigate the nature of some of the low-$ u$ states, specifically $ u=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.
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