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

While a parent Hamiltonian for Laughlin $1\mathrm{/}3$ 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 $n\mathrm{/}(2n+1)$ with $n\ge 2$. 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 $\nu =3\mathrm{/}7$. We propose that adding an appropriately constructed three-body interaction yields the unprojected $3\mathrm{/}7$ wave function as the unique zero energy solution, and report extensive exact diagonalization studies that provide strong support to this proposal.