  PH Pfaffian topological phase may exist in a uniform system due to strong
Landau level (LL) mixing according to theoretical predictions based on the Son
- Dirac composite fermion theory. Numerical investigations in the presence of
large LL mixing are limited due to numerical complexities, when taking into
account at least one more LL. Because of this, we apply the same field
theoretical approach to the quantum Hall bilayer at total filling factor equal
to one, for which many numerical studies exist. The most recent work in the
torus geometry predicts an intermediate phase (for intermediate distances
between layers) with an even-odd effect. According to our approach, the
intermediate phase represents a mixed negative-flux $p$-wave pairing i.e.
coexisting intra (PH Pfaffian in each layer) and inter (a la PH Pfaffian)
pairing correlations. This again underlines a necessity for strong entanglement
with additional degrees of freedom, i.e. at least one more (additional) LL in
the search for a stable PH Pfaffian phase in a single layer. Based on the
analogy with the bilayer physics we propose a PH Pfaffian wave function that
resides in two LLs.
