Ground states at the filling factors $\nu=7/3$ and $8/3$ in the second Landau level

The Laughlin state successfully describe the fractional quantum Hall state at $\nu=1/3$ in the lowest Landau level. However, it is known that the Laughlin wavefunction has little overlap with the ground state wavefunction at $\nu=7/3$ in the second Landau level. The ground states at $\nu=7/3$ and $8/3$ are still unknown.To determine the ground states at these fillings, we use the exact diagonalization method and density-matrix renormalization group (DMRG) method. We calculate overlaps between the ground state and the trial wavefunctions, the ground state energies, and the ground-state pair-correlation functions. We find that the ground state wavefunction at $\nu=8/3$ have very high overlap between the parafermion state, and the ground state energy of the parafermion state is lower than that of the Laughlin state. Further, the short-range structures of pair-correlation functions are significantly different from that of the Lauglin state.From these results, we consider that the parafermion state is a strong candidate of the ground state at $\nu=7/3$ and $\nu=8/3$.