PH Pfaffian intra and inter correlations in the quantum Hall bilayer

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.