Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The $λ_N$-Jellium model

Recent experiments on multilayer graphene systems have rekindled interest in electronic crystal phases in two dimensions -- but now for phases enriched by non-trivial quantum geometry. In this work, we introduce a simple continuum model with tunable Berry curvature distribution and total flux, enabling systematic study of crystallization in geometrically nontrivial bands. In the noninteracting limit, the addition of a C6-symmetric periodic potential yields a rich phase diagram, for which we provide several analytical insights. Notably, we derive a general formula for the Chern number in the weak-potential regime that is broadly applicable to single-band projected models. Removing the periodic potential and treating Coulomb interactions self-consistently at the Hartree-Fock level, the resulting phase diagrams host a variety of crystalline states, including anomalous Hall crystals, halo Wigner crystals in which localized electrons spontaneously acquire orbital angular momentum leading to depleted electron occupation at the zone center, and a novel halo anomalous Hall crystal that combines these properties with a finite Chern number. We identify why these phases are energetically favorable through analytical and energetic considerations. Our results provide insight into the interplay between crystallization and band geometry, while also offering a simple toy model amenable to numerical methods beyond mean-field.