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

In this work, we investigate the nature of the fractional quantum Hall state in the 1/3-filled second Landau level (SLL) at filling factor ν=7/3 (and 8/3 in the presence of the particle-hole symmetry) via exact diagonalization in both torus and spherical geometries. Specifically, we compute the overlap between the exact 7/3 ground state and various competing states including (i) the Laughlin state, (ii) the fermionic Haffnian state, (iii) the antisymmetrized product state of two composite fermion seas at 1/6 filling, and (iv) the particle-hole (PH) conjugate of the Z4 parafermion state. All these trial states are constructed according to a guiding principle called the bilayer mapping approach, where a trial state is obtained as the antisymmetrized projection of a bilayer quantum Hall state with interlayer distance d as a variational parameter. Under the proper understanding of the ground-state degeneracy in the torus geometry, the Z4 parafermion state can be obtained as the antisymmetrized projection of the Halperin (330) state. Specifically, while unclear at other momentum sectors, all degenerate copies of the Z4 parafermion state can be obtained by antisymmetrizing those of the Halperin (330) state at the zero-momentum sector, where both states occur as the exact ground states of their respective model Hamiltonians with the same degeneracy. Meanwhile, in the spherical geometry, the Z4 parafermion state is shown to be entirely equivalent to the antisymmetrized Halperin (330) state without any ground-state degeneracy issue. Similarly, it is proved in this work that the fermionic Haffnian state can be obtained as the antisymmetrized projection of the Halperin (551) state. The exact 7/3 ground state is obtained as a function of δV1(1), the variation of the first-moment Haldane pseudopotential V1(1) in the SLL with respect to the pure Coulomb interaction. It is shown that, while extremely accurate at sufficiently large positive δV1(1), the Laughlin state loses its overlap with the exact 7/3 ground state significantly at δV1(1)≃0. At slightly negative δV1(1), it is shown that the PH-conjugated Z4 parafermion state has a substantial overlap with the exact 7/3 ground state, which is the highest among the above four trial states. Around the Coulomb point, the energy spectrum exhibits an intriguing change from the spectrum with the Laughlin-type magnetoroton structure to that with the specific quasidegeneracy of the ground state, which is characteristic to the PH-conjugated Z4 parafermion state.