Pairing of composite fermions in half-filled Landau level state is reexamined
by solving the BCS gap equation with full frequency dependent
current–current interactions. Our results show that there
can be a continuous transition from the Halperin–Lee–Read state to
a chiral odd angular momentum Cooper pair state for short-range
contact interaction. This is at odds with the previously established
conclusion of first order pairing transition, in which the low frequency
effective interaction was assumed for the entire frequency
range. We find that even if the low frequency effective interaction
is repulsive, it is compensated by the high frequency regime, which
is attractive.Weconstruct the phase diagrams and show that ℓ = 1
angular momentum channel is quite different from higher angular
momenta ℓ ≥ 3. Remarkably, the full frequency dependent analysis
applied to the bilayer Hall system with a total filling fraction
ν = 1
2
+ 1
2 is quantitatively changed from the previously established
results but not qualitatively.