A neural‑network variational Monte Carlo study of the Aharonov‑Casher Hamiltonian— a continuum model for moiré materials—maps a rich phase diagram at fractional fillings. The work identifies Laughlin‑like fractional Chern insulators (FCIs) at filling factor ν = 1/3 and Jain‑like FCIs at ν = 2/3, together with three distinct generalized Wigner crystal (GWC) phases that break discrete translational symmetry. The authors employ two physics‑informed ansatzes: a composite‑fermion (CF) flux‑attachment wave function that captures topological order and a Slater–Jastrow (LSJ) ansatz that captures crystalline order. By comparing variational energies, they pinpoint phase crossings between CF and LSJ states as the non‑uniformity parameter B₁ is varied.

For a system of 36 electrons in 108 unit cells at ν = 1/3 and rs = 9.8, the CF energy is lower for B₁ near zero but the LSJ energy decreases linearly with |B₁|, crossing the CF line at larger |B₁|. This signals a transition from a Laughlin‑like FCI to a crystalline state. A similar crossing occurs at ν = 2/3, for 24 electrons in 36 unit cells with rs = 6.9: a Jain‑like FCI gives way to a honeycomb‑shaped electron GWC when B₁ becomes sufficiently large.

The study is limited to the two filling fractions above and to a specific range of band‑mixing (rs) and modulation (B₁). Behavior at other fractional fillings or in extreme limits of rs is not explored.

The diagram in the article shows the phase boundaries as a function of B₁ and rs for ν = 1/3, confirming the existence of the FCI phase, two distinct crystalline phases, and the transitions between them.