Competing states for the fractional quantum Hall effect in the 1/3-filled second Landau level

We study the nature of the fractional quantum Hall state in the 1/3-filled second Landau level at filling factor 7/3 via exact diagonalization. We show a series of transitions in the energy spectrum from a Laughlin-type spectrum, to an intermediate compressible spectrum, to a reentrant incompressible spectrum, and to a compressible spectrum with decrease of the Haldane pseudopotential. To search for a trial state describing the 7/3 state, we compute the overlap of the exact 7/3 ground state with various competing states including the Laughlin state, the particle-hole conjugate of the ${\mathrm Z}_4$ parafermion state, the fermionic Haffnian state, the antisymmetrized product state of two composite fermion seas (CFSs) at 1/6 filling, and the antisymmetrized correlated state of two CFSs at 1/4 filling, which are obtained as an antisymmetrized projection of the bilayer quantum Hall states. Specifically, we prove that the fermionic Haffnian state is equivalent to the antisymmetrized projection of the Halperin (551) state.