Paired states in half-filled Landau levels

Abstract: We discuss monolayer and bilayer quantum Hall systems in which each layer is a half-filled Landau level (LL) system. In the mean field approximation of the Son's formalism there is a common pairing structure that underlines the possibilities for paired ground states in both systems. We argue that the particle-hole (PH) Pfaffian state in the (particle-hole symmetric) half-filled LL of a monolayer, and analogous state in the PH symmetric bilayer (in which each layer is half-filled LL) can be considered as {\em critical states} i.e. states that cannot describe a phase under PH symmetry. We point out that the inclusion of a PH symmetry breaking (like LL mixing) may stabilize the PH Pfaffian in a monolayer. In the bilayer case, in numerical experiments on a sphere, by choosing the PH symmetric shift, we can stabilize the interlayer correlated (111) excitonic state or critical state, for any distance between the layers, but in general, with no bias, the evolution of the bilayer includes other phases.