Haldane statistics for fractional Chern insulators with an arbitrary Chern number

In this paper, we provide analytical counting rules for the ground states and the quasiholes of fractional Chern insulators with an arbitrary Chern number. We first construct pseudopotential Hamiltonians for fractional Chern insulators. We achieve this by mapping the lattice problem to the lowest Landau level of a multicomponent continuum quantum Hall system with specially engineered boundary conditions. We then analyze the thin-torus limit of the pseudopotential Hamiltonians, and extract counting rules (generalized Pauli principles, or Haldane statistics) for the degeneracy of its zero modes in each Bloch momentum sector.