On dualities of paired quantum Hall bilayer states at $\nu_T = \frac{1}{2} + \frac{1}{2}$

Abstract

Density-balanced, widely separated quantum Hall bilayers at $\nu_T = 1$ can be described as two copies of composite Fermi liquids (CFLs). The two CFLs have interlayer weak-coupling BCS instabilities mediated by gauge fluctuations, the resulting pairing symmetry of which depends on the CFL hypothesis used. If both layers are described by the conventional Halperin-Lee-Read (HLR) theory-based composite electron liquid (CEL), the dominant pairing instability is in the $p+ip$ channel; whereas if one layer is described by CEL and the other by a composite hole liquid (CHL, in the sense of anti-HLR), the dominant pairing instability occurs in the $s$-wave channel. Using the Dirac composite fermion (CF) picture, we show that these two pairing channels can be mapped onto each other by particle-hole (PH) transformation. Furthermore, we derive the CHL theory as the non-relativistic limit of the PH-transformed massive Dirac CF theory. Finally, we prove that an effective topological field theory for the paired CEL-CHL in the weak-coupling limit is equivalent to the exciton condensate phase in the strong-coupling limit.