Exact Parent Hamiltonian for Model Fractional Quantum Hall States on Lattice

The recent experimental discovery of 1/2-Laughlin state in cold-atom optical lattice has exhibited the great power of engineering topologically ordered states in lattice systems. However, so far, only on-site Hubbard-like interactions are known to be able to produce exact lattice fractional quantum Hall (FQH) states. In this work, we construct lattice-version Haldane pseudopotentials and produce general exact FQH states in a lattice. A series of well-known FQH states, including the Laughlin, Moore-Read, and Read-Rezayi states, can be generated in this way. Our work provides a framework to systematically explore the interaction dynamics of exact LL states in a lattice, which will be useful in guiding material engineering towards realizing exotic FQH states in lattice systems.