Candidate local parent Hamiltonian for the 3/7 fractional quantum Hall effect

Although a parent Hamiltonian for the Laughlin 1/3 wave function has been long known in terms of the Haldane pseudopotentials, no parent Hamiltonians are known for the lowest-Landau-level projected wave functions of the composite fermion theory at n/(2n+1) with n≥2. If one takes the two lowest Landau levels to be degenerate, the Trugman-Kivelson interaction produces the unprojected 2/5 wave function as the unique zero energy solution. If the lowest three Landau levels are assumed to be degenerate, the Trugman-Kivelson interaction produces a large number of zero energy states at ν=3/7. We propose that adding an appropriately constructed three-body interaction yields the unprojected 3/7 wave function as the unique zero energy solution and report extensive exact diagonalization studies that provide strong support to this proposal.