Exhaustiveness of known constructions for ideal Chern bands

Determine whether the union of generalized Landau levels and vortexable Chern bands described in the literature exhausts the set of all ideal Chern bands, where an ideal Chern band is defined as any Chern band whose projected effective two-body interaction—obtained by projecting the real-space density–density interaction into the band and expanding in the generalized pseudopotential basis—has a finite projected interaction range r_c (i.e., the expansion coefficients λ_{m,n, k1, k2} vanish for all m greater than some finite integer r_c).

Background

The paper introduces a universal interaction-based framework to analyze Chern bands by projecting real-space density–density interactions into a single band and expanding the resulting effective interaction in a complete basis of generalized pseudopotentials. Within this framework, the authors define the projected interaction range r_c as the largest relative angular momentum index m for which the generalized pseudopotential expansion has nonzero coefficients, and classify bands with finite r_c as ideal Chern bands.

Known classes—generalized Landau levels and vortexable Chern bands—are shown to be examples of ideal Chern bands with finite r_c. The open question concerns whether these known constructions cover all possible ideal Chern bands under the proposed interaction-based perspective, which would amount to a completeness result for ideal Chern bands in this framework.

Resolving this would clarify the scope of the interaction-based classification and whether additional, currently unidentified mechanisms can produce ideal Chern bands beyond generalized Landau levels and vortexable constructions.