Generalized Pseudopotentials for the Anisotropic Fractional Quantum Hall Effect

Haldane pseudopotentials have played a key role in the study of the fractional quantum Hall (FQH) effect as they allow an arbitrary rotationally-invariant interaction to be expanded over projectors onto the two-particle eigenstates of relative angular momentum. Here we introduce a more general class of pseudopotentials that form a complete basis in the cases where rotational symmetry is explicitly broken, e.g., due to tilted magnetic field or anisotropic band structure. Similar to the standard isotropic pseudopotentials, the generalized pseudopotentials are also parametrized by a unimodular metric, which groups the effective interactions into equivalence classes, and is particularly useful in determining optimal model Hamiltonians of the anisotropic FQH fluids. We show that purely anisotropic pseudopotentials lead to new types of bound ``molecular" states for a few particles in an infinite plane. We discuss realizations of the generalized pseudopotentials in systems with tilted magnetic field, as well as fractional Chern insulators harboring intrinsic anisotropy due to the underlying lattice structure.