Incommensurate crystals and stripe phases at filling fraction ν=1/5

Determine the competition among an incommensurate generalized Wigner crystal, a stripe phase, and the Laughlin liquid at filling fraction ν=1/5 in Aharonov–Casher bands, whose periodicity is not commensurate with that of the magnetic field.

Background

The variational ansatz used in the paper assumes a crystal lattice commensurate with the moiré lattice and therefore does not capture crystalline states with incommensurate periodicity. The authors specifically identify filling fraction ν=1/5 as a case where an incommensurate crystal and a stripe phase may compete with the Laughlin liquid.

Resolving this problem would extend the phase-diagram analysis beyond the commensurate generalized Wigner crystals treated in the paper and would determine whether additional translational-symmetry-breaking phases intervene in the liquid–crystal competition.

References

(There are several possible scenarios for how the system evolves away from these discrete values of $\nu$---an issue we leave for future work.)

— Entropy-driven transitions between extended integer and fractional quantum Hall regimes  (2609.16483 - Kim et al., 15 Sep 2026) in General Considerations, paragraph beginning “In this work, we will treat situations…”

We did not address crystalline ground states whose periodicity is incommensurate with that of the magnetic field, as such states are not captured by our ansatz. This situation arises, for instance, at filling $\nu=1/5$, where an incommensurate crystal as well as a stripe phase will compete with the Laughlin liquid. We leave this interesting scenario for future work.

— Non-uniform quantum geometry stabilizes generalized Wigner crystals  (2609.04149 - Morales-Durán et al., 3 Sep 2026) in Discussion section