Chern-Simons Modified RPA-Eliashberg Theory of the $\nu =\frac{1}{2}+\frac{1}{2}$ Quantum Hall Bilayer

The $\nu =\frac{1}{2}+\frac{1}{2}$ quantum Hall bilayer has been previsously modeled using Chern-Simons-RPA-Eliashberg (CSRPAE) theory to describe pairing between the two layers. However, these approaches are troubled by a number of divergences and ambiguities. By using a “modified” RPA approximation to account for mass renormalization, we can work in a limit where the cyclotron frequency is taken to infinity, effectively projecting to a single Landau level. This, surprisingly, controls the important divergences and removes ambiguities found in prior attempts at CSRPAE. Examining BCS pairing of composite fermions we find that the angular momentum channel $l=+1$ dominates for all distances $d$ between layers and at all frequency scales. Examining BCS pairing of composite fermion electrons in one layer with composite fermion holes in the opposite layer, we find the $l=0$ pairing channel dominates for all $d$ and all frequencies. The strength of the pairing in these two different descriptions of the same phase of matter is found to be almost identical. This agrees well with our understanding that these are two different but dual descriptions of the same phase of matter.

Figure 1

The Eliashberg anomalous coupling constant ${\lambda }_{\varphi }^l({\omega }_m)$ calculated using CSMRPAE for interlayer spacing $d/l_B=1$. The coupling constant is shown for different pairing channels $l$ and for both composite-fermion-electron-composite-fermion-electron pairing (CE-CE) and for composite-fermion-electron-composite-fermion-hole pairing (CE-CH). We see that CE-CH pairing with $l=0$ and CE-CE pairing with $l=1$ are most attractive (most negative) and are very nearly equal to each other (the two lowest curves almost precisely overlap). The inset shows an enlargement of the difference between the two lowest curves ($l=0$, CE-CH pairing and $l=0$ CE-CE pairing). The difference is a factor of order 100 smaller than the difference to other pairing channels. Other values of $d/{\ell }_B$, and larger range of ${\omega }_m$ are shown in the Supplemental Material [53] Sec. VII.