Numerical exploration of trial wave functions for the particle-hole-symmetric Pfaffian

We numerically assess model wave functions for the recently proposed particle-hole-symmetric Pfaffian (`PH-Pfaffian') topological order, a phase consistent with the recently reported thermal Hall conductance [Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $\nu=5/2$ quantum-Hall plateau. We find that the most natural Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $\nu=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak $p$-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.