Energy minimization of paired composite fermion wave functions in the spherical geometry

We perform the energy minimization of the paired composite fermion (CF) wave functions, proposed by M\"oller and Simon (MS) [PRB 77, 075319 (2008)] and extended by Yutushui and Mross (YM) [PRB 102, 195153 (2020)], where the energy is minimized by varying the CF pairing function, in the case of an approximate model of the Coulomb interaction in the second Landau level for pairing channels $\ell = -1, 3, 1$ which are expected to be in the Pfaffian, anti-Pfaffian and particle-hole symmetric (PH) Pfaffian phases respectively. It is found that the energy of the $\ell = -1$ MS wave function can be reduced substantially below that of the Moore-Read wave function at small system sizes, however, in the $\ell = 3$ case the energy cannot be reduced much below that of the YM trial wavefunction. Nonetheless, both our optimized and unoptimized wavefunctions with $\ell=-1,3$ extrapolate to roughly the same energy per particle in the thermodynamic limit. For the $\ell = 1$ case, the optimization makes no qualitative difference and these PH-Pfaffian wave functions are still energetically unfavourable. The effective CF pairing is analyzed in the resulting wave functions, where the effective pairing for the $\ell = -1, 3$ channels is found to be well approximated by a weak-pairing BCS ansatz and the $\ell = 1$ wave functions show no sign of emergent CF pairing.