Phase diagram of compressible and paired states in the quarter-filled Landau level

Quantum Hall plateaus at quarter fillings occur in GaAs wide quantum wells, hole-doped GaAs, and bilayer graphene. However, the interactions favoring incompressible states over compressible composite-Fermi liquids at such fillings are not well understood. We devise a method for the computation of the trial energies for Haldane pseudopotentials via Monte Carlo sampling. Applying it to the quarter-filled lowest Landau level, we find that tuning the third and fifth pseudopotential values can stabilize the anti-Pfaffian, Moore-Read, and $f$-wave states. The smallest deviations from pure Coulomb interactions are required by anti-Pfaffian, whose presence is indicated by daughter states in recent experiments of bilayer graphene at $\ensuremath{\nu}=\frac{3}{4}$.