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In this work we propose a parton state as a candidate state to describe thefractional quantum Hall effect in the half-filled second Landau level. The wavefunction for this parton state is $\mathcal{P}_{\rm LLL}\Phi_{1}^3[\Phi_{2}^{*}]^{2}\sim\Psi^{2}_{2/3}/\Phi_{1}$ and in the sphericalgeometry it occurs at the same flux as the anti-Pfaffian state. This state hasa good overlap with the anti-Pfaffian state and with the ground state obtainedby exact diagonalization, using the second Landau level Coulomb interactionpseudopotentials for an ordinary semiconductor such as GaAs. By calculating theentanglement spectrum we show that this state lies in the same phase as theanti-Pfaffian state. A major advantage of this parton state is that its wavefunction can be evaluated for large systems, which makes it amenable tovariational calculations. In the appendix of this work we have numericallyassessed the validity of another candidate state at filling factor $\nu=5/2$,namely the particle-hole-symmetric Pfaffian (PH-Pfaffian) state. We find thatthe proposed candidate wave function for the PH-Pfaffian state is particle-holesymmetric to a high degree but it does not appear to arise as the ground stateof any simple Hamiltonian with two-body interactions.

Parton construction of a wave function in the anti-Pfaffian phase