  Recently, generalizations of fractional quantum Hall (FQH) states known as
fractional quantum anomalous Hall or, equivalently, fractional Chern insulators
states have been realized in lattice models. Ideal wavefunctions such as the
Laughlin wavefunction, as well as the corresponding trial Hamiltonians for
which the former are exact ground states, have been vital to characterizing FQH
phases. The Wannier function representation of fractional Chern insulators
proposed in [X.-L. Qi, Phys. Rev. Lett., 126803] defines an approach to
generalize these concepts to fractional Chern insulators. In this paper, we
apply the Wannier function representation to develop a systematic
pseudopotential formalism for fractional Chern insulators. The family of
pseudopotential Hamiltonians is defined as the set of projectors onto
asymptotic relative angular momentum components which forms an orthogonal basis
of two-body Hamiltonians with magnetic translation symmetry. This approach
serves both as an expansion tool for interactions and as a definition of
positive semidefinite Hamiltonians for which the ideal fractional Chern
insulator wavefunctions are exact nullspace modes. We compare the short-range
two-body pseudopotential expansion of various fractional Chern insulator models
at one-third filling in phase regimes where a Laughlin-type ground state is
expected to be realized. We also discuss the effect of inhomogeneous Berry
curvature which leads to components of the Hamiltonian that cannot be expanded
into pseudopotentials, and elaborate on their role in determining low energy
theories for fractional Chern insulators. Finally, we generalize our Chern
pseudopotential approach to interactions involving more than two bodies with
the goal of facilitating the identification of non-Abelian fractional Chern
insulators.
