Interplay of superconducting, metallic, and crystalline states of composite fermions at $\nu{=}1/6$ in wide quantum wells

Evidence for developing fractional quantum Hall effect (FQHE) at filling fraction $\nu{=}1/6$ and $1/8$ has recently been reported in wide GaAs quantum wells [Wang \emph{et al.}, PRL {\bf 134}, 046502 (2025)]. In this article, we theoretically investigate the nature of the state at $\nu{=}1/6$ as a function of the quantum well width and the density by considering composite-fermion (CF) crystals, CF Fermi sea, and various kinds of paired CF states. The $f$-wave paired state has the lowest energy among the paired CF states. However, for parameters of interest, the energies of the CF crystal, the CF Fermi liquid, and the $f$-wave paired CF state are too close to call. We, therefore, predict that {\it if} the FQHE at $\nu{=}1/6$ is experimentally confirmed, this state would be an $f$-wave paired state of CFs, which can be verified by measurement of its thermal Hall conductance. Exact diagonalization studies for systems with up to 8 electrons show that the ground states at $\nu{=}n/(6n{\pm} 1)$ are incompressible for all widths and densities we have considered and well described by the corresponding Laughlin and Jain states. We propose a phase diagram for large quantum well widths and densities in which at zero disorder, incompressible FQHE states are stabilized at $\nu{=}n/(6n{\pm} 1)$ and $\nu{=}1/6$, but in between these fillings the CF crystal is stabilized. With disorder, which creates a spatial variation in the filling factor, two regimes are identified: (i) for small disorder, when the incompressible states percolate at the special fillings, FQHE with quantized Hall plateaus and vanishing longitudinal resistance should occur; and (ii) for larger disorder, when the CF crystal percolates, the longitudinal resistance rises with decreasing temperature but the domains of FQHE liquid produce minima at the special filling factors. The experiments are consistent with the latter scenario.