Composite Fermion Understanding of Geometry Conditions for Fractional Chern Insulators

The mimicry of Lowest Landau Level (LLL) physics in lattice systems has led to the proposal of various geometric conditions to stabilize fractional Chern insulator (FCI) phases. However, these sufficient conditions, often formulated without underlying dynamical reasons, present challenges for experimental realization due to the weak connection with the tunable Hamiltonian. In this work, we derive an "preferred" composite fermion (CF) Hamiltonian for a Chern band system, based on our projective construction of the CF states in FCI. The kinetic part of CF Hamiltonian, originating from not only the electronic band structure, but also from Coulomb interaction, incorporates contributions from band dispersion, Berry curvature, and quantum metric, yielding as a unified description of geometry conditions. This approach reveal the important role of Coulomb interaction in establishment of geometry conditions, and offer applicable insights in experimental realization of FCI states.