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.

中文翻译：

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Chern-Simons 修正的 RPA-Eliashberg ν=12+12 量子霍尔双层理论

这$\nu =\frac{1}{2}+\frac{1}{2}$量子霍尔双层先前已使用 Chern-Simons-RPA-Eliashberg (CSRPAE) 理论进行建模，以描述两层之间的配对。然而，这些方法受到许多分歧和模糊性的困扰。通过使用“修改的”RPA 近似来解释质量重正化，我们可以在回旋加速器频率达到无穷大的极限下工作，有效地投影到单个朗道能级。令人惊讶的是，这控制了重要的分歧并消除了在 CSRPAE 先前尝试中发现的歧义。检查复合费米子的 BCS 配对，我们发现角动量通道$我=+1$在所有距离上都占主导地位$d$层之间和所有频率尺度。检查一层复合费米子电子与相对层复合费米子空穴的 BCS 配对，我们发现$我=0$配对通道占主导地位$d$和所有频率。发现同一相物质的这两种不同描述中的配对强度几乎相同。这与我们的理解非常吻合，即这是对同一物质相的两种不同但双重的描述。