  We report on fixed phase diffusion Monte Carlo calculations that show that,
even for a large amount of Landau level mixing, the energies of the Pfaffian
and anti-Pfaffian phases remain very nearly the same, as also do the excitation
gaps at $1/3$ and $2/3$. These results, combined with previous theoretical and
experimental investigations, indicate that particle hole (PH) symmetry for
composite fermion states is much more robust than a priori expected, emerging
even in models that explicitly break PH symmetry. We provide insight into this
fact by showing that the low energy physics of a generic repulsive 3-body
interaction is captured, to a large extent and over a range of filling factors,
by a mean field approximation that maps it into a PH symmetric 2-body
interaction. This explains why Landau level mixing, which effectively generates
such a generic 3-body interaction, is inefficient in breaking PH symmetry. As a
byproduct, our results provide a systematic construction of a 2-body
interaction which produces, to a good approximation, the Pfaffian wave function
as its ground state.
