# Pairing of Composite-Electrons and Composite-Holes in $\nu_{T}=1$ Quantum
Hall Bilayers

Luca Rüegg<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Gaurav
Chaudhary<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Robert-Jan
Slager  TCM Group, Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

Motivated by recent experimental indications of preformed electron-hole pairs
in $\nu_{T}=1$ quantum Hall bilayers at relatively large separation, we
formulate a Chern-Simons (CS) theory of the coupled composite electron liquid
(CEL) and composite hole liquid (CHL). We show that the effective action of
the CS gauge field fluctuations around the saddle-point leads to stable
pairing between CEL and CHL. We find that the CEL-CHL pairing theory leads to
a dominant $s$-wave channel in contrast to the dominant $p$-wave channel found
in the CEL-CEL pairing theory. Moreover, the CEL-CHL pairing is generally
stronger than the CEL-CEL pairing across the whole frequency spectrum.
Finally, we discuss possible differences between the two pairing mechanisms
that may be probed in experiments.

Introduction.— Quantum Hall (QH) bilayers exhibit many phenomena that are
absent in the QH monolayers Eisenstein  _et al._ (1992); Eisenstein (2014). At
the total filling fraction $\nu_{T}=\nu_{+}+\nu_{-}=1$ ($+/-$ refer to the
upper/lower layer), when the two layers have almost equal carrier densities
and small layer separation $d$ (characterised by the ratio: $d/\ell$, where
$\ell=\sqrt{\hbar/(eB)}$ is the magnetic length), the system exhibits a
remarkable exciton condensate (XC) phase Spielman  _et al._ (2000); Eisenstein
and MacDonald (2004). This superfluid phase of excitons was first observed in
a Josephson-like zero-bias peak of the tunneling current between the two
layers Spielman  _et al._ (2000), and was later confirmed by a vanishing
counterflow Hall resistance Kellogg  _et al._ (2004) and a perfect Coulomb
drag Nandi  _et al._ (2012).

In the $d\rightarrow 0$ limit, the system can be viewed as a two-component
monolayer QH system, which is an incompressible state described by the
Halperin $(1,1,1)$ wavefunction Halperin (1983). The enriched physics
associated with the layer degree of freedom is manifested in a broken symmetry
Fertig (1989) and spontaneous interlayer phase coherence Wen and Zee (1992,
1993); Eisenstein and MacDonald (2004). In the opposite limit of large
interlayer separation, the two layers can be described by a compressible state
of decoupled composite Fermi liquids (CFLs) Halperin  _et al._ (1993). The
intermediate layer separation, where the possible transition between the two
phases occurs is not entirely understood. There exist many different
possibilities: a potential phase transition Shibata and Yoshioka (2006); Zhu
_et al._ (2017), a composite boson (CB) exciton condensate Lian and Zhang
(2018), phase coexistence between CBs and CFs Simon  _et al._ (2003), and
several interlayer pairing instabilities Morinari (1999); Kim  _et al._
(2001); Möller  _et al._ (2008); Sodemann  _et al._ (2017); Bonesteel (1993);
Bonesteel  _et al._ (1996); Cipri and Bonesteel (2014); Isobe and Fu (2017);
Wagner  _et al._ (2021).

More insight into the intermediate region is provided by two recent
experiments Eisenstein  _et al._ (2019); Liu  _et al._ (2022). In an
interlayer tunneling experiment, Eisenstein et. al. Eisenstein  _et al._
(2019) report that the tunneling pseudogap in widely separated layers is
suppressed at interlayer distance larger than what is expected for the
transition to the XC phase. In a temperature dependent Coulomb drag and
counter-flow experiment on QH bilayers of graphene, Liu et. al. Liu  _et al._
(2022) report that a significant fraction of excitons are present at
temperatures higher than the transition temperature $T_{c}$ associated with
the formation of the XC phase. The main features of these experiments indicate
a region of preformed pairs of electrons and holes which go over a smooth BCS-
BEC-like crossover to the XC phase. The BCS-BEC crossover picture is also
supported by a large overlap between numerical exact-diagonalization and a
trial BCS wave function for interlayer CF electron-hole pairs Wagner  _et al._
(2021).

Figure 1: (a) Schematic of the CEL-CHL bilayer system. The solid spheres with
arrows are the CEs and the void spheres are CHs. The dashed arrows stand for
the gauge fluctuations $\tilde{a}^{s}$. (b) Feynman diagram for the effective
interaction between CEs and CHs mediated by the effective gauge propagator.

In this Letter, we formulate a CS theory of a bilayer of CEL in one layer and
CHL in the other as schematically shown in Fig. 1 (a). Microscopically, the
two layers are coupled by Coulomb interactions of bare particles. Starting
from a large separation limit, within the random-phase-approximation (RPA)
Bohm and Pines (1953); Fetter  _et al._ (1991), we arrive at an effective
action for the fluctuations of the CS gauge fields around their mean-field
value. The CS gauge fields mediate the effective interactions between the CFs
in the two layers as shown in Fig. 1 (b). This approach was previously used
for the bilayers where both layers are a CEL Bonesteel (1993); Bonesteel  _et
al._ (1996); Cipri and Bonesteel (2014); Isobe and Fu (2017), and emergence of
pairing instabilities between the two CELs was predicated. We show that in the
CEL-CHL theory the effective interactions also lead to an interlayer pairing
instability (albeit in the CE-CH channel) and facilitate the formation of
interlayer “composite excitons (CX)". These CXs are thus expected to start
forming at large layer separation and can be thought of as the particles that
undergo the BCS-BEC crossover as the layers are brought closer. We also show
that the dominant pairing occurs in the $s$-wave channel of the CXs, thus
theoretically justifies the picture in Ref. Wagner  _et al._ (2021). This is
in contrast to the CEL-CEL theory where $p$-wave pairing instability is
favoured Möller  _et al._ (2008, 2009); Isobe and Fu (2017). This difference
is rooted in the density-current coupling, which in the CEL-CEL theory breaks
time reversal symmetry (TRS) to favour a particular $p$-wave channel. In our
CEL-CHL theory, the density-current coupling has opposite sign in the two
layers and drops out in the Cooper channel, leading to an effective TRS.

Finally, we show that when the full frequency dependence of the effective
interactions is taken into account, the $s$-wave CEL-CHL pair is more tightly
bound than the $p$-wave CEL-CEL pair, thus favouring a BCS-BEC crossover
picture in this theory. Furthermore, when the two layers have slightly
mismatched densities, i.e.
$\nu_{T}=(\frac{1}{2}-\delta)+(\frac{1}{2}+\delta)$, the two theories have
different pair-breaking mechanisms, which also favours larger pair binding
energies in the CEL-CHL theory. These can be possibly used in experiments to
further confirm BCS-BEC crossover of CXs.

CS Theory.— For the CEL-CHL CS theory, we first need to consider the theory of
holes in the lowest Landau level (LLL). For our purpose, we consider the
Lagrangian theory of holes in the LLL formulated in Ref. Barkeshli  _et al._
(2015) (after setting $\hbar=c=e=1$):

$\displaystyle\begin{split}\mathcal{L}_{\text{hole}}=\,&h^{\dagger}\left(i\partial_{t}-A_{t}+\mu_{h}\right)h-\frac{1}{2m_{h}}h^{\dagger}\left(i\boldsymbol{\nabla}-\boldsymbol{A}\right)^{2}h\\\
&-V(h)+\frac{1}{2\pi}A_{t}\boldsymbol{\nabla}\wedge\boldsymbol{A},\end{split}$
(1)

with the Coulomb interaction potential

$V(h)=\frac{1}{2}\int
d^{2}\boldsymbol{y}\,h^{\dagger}h\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}h^{\dagger}h,$
(2)

where $h$ is the hole field, $e$ is the elementary charge, $A$ is the external
vector potential, $\mu_{h}$ is the hole chemical potential, and $m_{h}$ is the
hole effective mass. Importantly, the hole field has opposite charge in the
external gauge field $A$ compared to the electron field. The CS term
associated with the external field in the Lagrangian incorporates the effects
of the filled LLL. Functional variation of the action with respect to the
external field $A$ yields

$\displaystyle n_{e}+n_{h}$ $\displaystyle=\frac{B}{2\pi},$ (3)
$\displaystyle(\boldsymbol{j}_{e})_{i}+(\boldsymbol{j}_{h})_{i}$
$\displaystyle=\frac{\epsilon_{ij}}{2\pi}E_{j},$ (4)

where $\epsilon_{ij}$ is the Levi-Civita symbol. The first equation means that
electrons plus holes fill the LLL. The second equation is the Hall
conductivity. While, at the level of the Hamiltonian, the hole theory in the
LLL can be obtained by a simple particle-hole transformation on the electron
theory, in the Lagrangian incorporation of the extra CS term above puts
important physical constraints. This extra CS term does not change the
Hamiltonian Lopez and Fradkin (1991).

We transform the holes $h$ into composite holes (CHs) $\chi$ by attaching two
magnetic flux quanta to each [see Fig. 1 (a)]. In the Lagrangian, this is
achieved by including the dynamical gauge field $a$ Barkeshli  _et al._
(2015):

$\displaystyle\begin{split}\mathcal{L}_{\text{CH}}=\,&\chi^{\dagger}\left(i\partial_{t}+a_{t}-A_{t}+\mu_{\chi}\right)\chi-V(\chi)\\\
-\frac{1}{2m_{\chi}}&\chi^{\dagger}\left(i\boldsymbol{\nabla}+\boldsymbol{a}-\boldsymbol{A}\right)^{2}\chi+\frac{1}{2\pi}A_{t}\boldsymbol{\nabla}\wedge\boldsymbol{A}-\frac{1}{4\pi}a_{t}\boldsymbol{\nabla}\wedge\boldsymbol{a},\end{split}$
(5)

where the interaction potential is the same as above since the density of
holes and CHs is the same. The equation of motion of the dynamical gauge field
stems from the second CS term and reads:
$\boldsymbol{\nabla}\wedge\boldsymbol{a}=4\pi\chi^{\dagger}\chi$; which again
has opposite sign compared to the case of the CF field $\psi$ and its
associated dynamical gauge field:
$\boldsymbol{\nabla}\wedge\boldsymbol{a}=-4\pi\psi^{\dagger}\psi$. The
Halperin-Lee-Read theory and hence the CF Lagrangian break the particle-hole
symmetry Halperin  _et al._ (1993). The CHL theory described by the above
Lagrangian cannot be obtained by a simple particle-hole transformation of the
CEL Lagrangian. Instead, as argued in Ref. Barkeshli  _et al._ (2015), this
CHL at $\nu=1/2$ is a topologically distinct state to the CEL at $\nu=1/2$.
Pairing between these two topologically distinct CFLs is our main motivation.

For our bilayer system, we take CFs as the degrees of freedom in one layer and
CHs in the other; and accordingly denote $\psi_{+/-}=\psi/\chi$. The total
Euclidean Lagrangian is sup

$\displaystyle\begin{split}\mathcal{L}=\,&\sum_{s=+/-}\bigg{[}\psi^{\dagger}_{s}\left(\partial_{\tau}+ia^{s}_{\tau}+siA_{\tau}-\mu_{s}\right)\psi_{s}\\\
&+\frac{1}{2m_{s}}\psi^{\dagger}_{s}\left(i\boldsymbol{\nabla}+\boldsymbol{a}^{s}+s\boldsymbol{A}\right)^{2}\psi_{s}+\frac{si}{4\pi}a^{s}_{\tau}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s}\\\
&+V(a^{s})\bigg{]}+\frac{i}{2\pi}A_{\tau}\boldsymbol{\nabla}\wedge\boldsymbol{A}+V_{+-}(a^{+},a^{-}),\end{split}$
(6)

with the interactions

$\displaystyle V(a^{s})$
$\displaystyle=\,\frac{1}{2}\int\frac{d^{2}\boldsymbol{y}}{(4\pi)^{2}}\,\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s}\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s},$
(7) $\displaystyle V_{+-}(a^{+},a^{-})$
$\displaystyle=\int\frac{d^{2}\boldsymbol{y}}{(4\pi)^{2}}\,\boldsymbol{\nabla}\wedge\boldsymbol{a}^{+}\frac{1}{\sqrt{|\boldsymbol{x}-\boldsymbol{y}|^{2}+d^{2}}}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{-}.$
(8)

In the expressions for the interactions above we already enforce the equations
of motion for the dynamical gauge fields.

Effective Interaction.— The Lagrangian in Eq. (6) possesses the mean-field
solution $a^{s}=-sA$. We denote fluctuations in the dynamical gauge fields
around this saddle-point as $\tilde{a}^{s}$. Furthermore, in the Coulomb
gauge, the spatial part can be written as
$\boldsymbol{\tilde{a}}(\boldsymbol{q},i\omega_{m})=\tilde{a}_{1}(\boldsymbol{q},i\omega_{m})\hat{z}\wedge\boldsymbol{\hat{q}}$,
where $\omega_{m}=2\pi mT$ is a bosonic Matsubara frequency. Within the RPA,
the effective action for the CS gauge field fluctuations up to second order is
Bonesteel (1993); Bonesteel  _et al._ (1996); Cipri and Bonesteel (2014);
Isobe and Fu (2017); sup

$\displaystyle\begin{split}S_{\text{eff}}=\,&\frac{T}{2}\sum_{\omega_{m}}\int\frac{d^{2}\boldsymbol{q}}{(2\pi)^{2}}\sum_{\mu\nu,ss^{\prime}}\\\
&\tilde{a}_{\mu}^{s}(\boldsymbol{q},i\omega_{m})D_{\mu\nu,ss^{\prime}}^{-1}(q,i\omega_{m})\tilde{a}_{\nu}^{s^{\prime}}(-\boldsymbol{q},-i\omega_{m}),\end{split}$
(9)

where $D(q,i\omega_{m})$ is the gauge propagator and $\mu,\nu$ are temporal
and spatial coordinates denoted by $0$ and $1$ respectively. We are interested
in the low-energy, long-wavelength modes, i.e. the regime
$\frac{\omega_{m}}{\epsilon_{F}}\ll\frac{q^{2}}{k_{F}^{2}}\ll 1$ and $qd\ll
1$. The most singular terms in the gauge propagator are then sup

$\displaystyle\begin{split}&D_{11,ss}(q,i\omega_{m})\approx-\frac{1}{\chi
q^{3}}\left(\frac{m_{+}m_{-}}{2\pi(4\pi)^{2}}q+\frac{m_{+}m_{-}k_{F}}{(2\pi)^{3}}\frac{|\omega_{m}|}{q}\right),\\\
&D_{11,s(-s)}(q,i\omega_{m})\approx\frac{1}{\chi
q^{3}}\frac{m_{+}m_{-}}{2\pi(4\pi)^{2}}q,\end{split}$ (10)

where
$\chi=\frac{m_{+}m_{-}k_{F}}{2(2\pi)^{4}}\frac{|\omega_{m}|}{q^{3}}+\left(\frac{m_{+}+m_{-}}{192\pi^{4}}+\frac{2dm_{+}m_{-}}{(4\pi)^{4}}\right)$.
These are interpreted as current-current correlations. Importantly, the intra-
and interlayer terms are equally singular.

After summing over the spatio-temporal indices, the CS gauge field mediates an
effective interaction within the CEL/CHL, and between the CEL and the CHL,
given by their respective matrix elements Isobe and Fu (2017); sup

$\displaystyle\begin{split}V_{ss^{\prime}}^{\text{eff}}(\boldsymbol{k},\boldsymbol{k}^{\prime},\boldsymbol{q},i\omega_{m})=-ss^{\prime}\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})(\boldsymbol{\hat{q}}\wedge\boldsymbol{k}^{\prime})}{2m_{s}m_{s^{\prime}}}\\\
\times\left[q^{2}\left(\frac{m_{+}+m_{-}}{6\pi
m_{+}m_{-}}+\frac{d}{4\pi}\right)+\frac{|\omega_{m}|}{q}\frac{k_{F}}{\pi}\right]^{-1}.\end{split}$
(11)

The dominant contribution in the interlayer Cooper channel
$V_{c}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})=V_{+-}^{\text{eff}}(\boldsymbol{k},-\boldsymbol{k}-\boldsymbol{q},\boldsymbol{q},i\omega_{m})+V_{-+}^{\text{eff}}(\boldsymbol{k},-\boldsymbol{k}-\boldsymbol{q},\boldsymbol{q},i\omega_{m})$
is attractive. Thus,to this point, the pairing instability in the CEL-CHL
theory is similar to the one obtained for the CEL-CEL pairing theory Bonesteel
(1993); Bonesteel  _et al._ (1996); Cipri and Bonesteel (2014); Isobe and Fu
(2017). Our starting theory in Eq. (6) differs from the CEL-CEL theory, not
only via the additional CS term of the external gauge field, but also
regarding the charges of the CFs. However in the mean-field solution, the
attached flux cancels the external field for the electrons and holes alike.
The remaining fluctuations near the mean-field take both positive and negative
values. Thus,the electrons and holes experience like charges in their
respective CS gauge field fluctuations and resultant like charge currents.
Since the dominant pairing contributions stem from the current-current
coupling, to this level, there is yet no difference between CEL-CEL and CEL-
CHL pairing. Further analysis shows that this attraction can be associated
with the out-of-phase interlayer current fluctuations
$b_{1}^{-}=\frac{\tilde{a}^{+}_{1}-\tilde{a}^{-}_{1}}{\sqrt{2}}$ sup . As we
will see below, the difference between the two theories appears in the
density-current coupling of the CS gauge fields, which leads to different
pairing symmetries.

Pairing Symmetry.— Within Eliashberg theory Marsiglio (2020), the inverse
Green’s function in the Nambu space spanned by
$(\psi_{+}(\boldsymbol{k}),\psi_{-}(\boldsymbol{k}),\psi^{\dagger}_{+}(-\boldsymbol{k}),\psi^{\dagger}_{-}(-\boldsymbol{k}))^{\intercal}$
is

$\displaystyle
G^{-1}(\boldsymbol{k},i\epsilon_{n})=\begin{pmatrix}(i\epsilon_{n}Z_{n}^{s}-\xi_{\boldsymbol{k}}^{s})\delta_{ss^{\prime}}&\hat{\phi}_{n}(\boldsymbol{k})\\\
\hat{\phi}_{n}(\boldsymbol{k})^{\dagger}&(i\epsilon_{n}Z^{s}_{n}+\xi_{\boldsymbol{k}}^{s})\delta_{ss^{\prime}}\end{pmatrix},$
(12)

with $Z_{n}^{s}$ the quasiparticle residue, $\hat{\phi}_{n}(\boldsymbol{k})$
the anomalous self-energy,
$\xi_{\boldsymbol{k}}^{s}=\frac{\boldsymbol{k}^{2}}{2m_{s}}-\mu_{s}$, and
$\epsilon_{n}=2\pi(n+1)T$ a fermionic Matsubara frequency. $Z_{n}^{s}$ and
$\hat{\phi}_{n}(\boldsymbol{k})$ get corrections from the exchange and Cooper
channel interaction respectively. Going beyond the dominant current-current
terms, these are written as Isobe and Fu (2017); sup

$\displaystyle\begin{split}&V_{ex}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})=-\frac{1}{2}\sum_{\mu\nu,ss^{\prime}}\begin{pmatrix}1&-i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s^{\prime}}}\\\
i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s}}&\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})^{2}}{m_{s}m_{s^{\prime}}}\end{pmatrix}_{\mu\nu}\\\
&D_{\mu\nu,ss^{\prime}}(q,i\omega_{m})\left[\delta_{s,+}\delta_{s^{\prime},+}+\delta_{s,-}\delta_{s^{\prime},-}\right],\end{split}$
(13)

and

$\displaystyle\begin{split}&V_{c}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})=\frac{1}{2}\sum_{\mu\nu,ss^{\prime}}\begin{pmatrix}1&i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s^{\prime}}}\\\
i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s}}&-\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})^{2}}{m_{s}m_{s^{\prime}}}\end{pmatrix}_{\mu\nu}\\\
&D_{\mu\nu,ss^{\prime}}(q,i\omega_{m})\left[\delta_{s,+}\delta_{s^{\prime},-}+\delta_{s,-}\delta_{s^{\prime},+}\right].\end{split}$
(14)

The interlayer density-current interactions determine whether the $\pm l$ pair
angular momentum states are degenerate as they are the only terms sensitive to
the direction of the inserted flux quanta Isobe and Fu (2017); sup . Their
contribution to the Cooper channel

$\displaystyle\begin{split}\frac{i}{2}\boldsymbol{\hat{q}}\wedge\boldsymbol{k}\bigg{[}\frac{D_{10,+-}(q,i\omega_{m})+D_{01,-+}(q,i\omega_{m})}{m_{+}}\\\
+\frac{D_{10,-+}(q,i\omega_{m})+D_{01,+-}(q,i\omega_{m})}{m_{-}}\bigg{]}\end{split}$
(15)

vanishes in the low-energy regime because the density-current couplings have
opposite signs in the two layers sup . The density-current term originates
from the CS term because it couples the spatial and the temporal components of
the gauge field. Since the CEL and the CHL have the same charge with respect
to their respective gauge fluctuations, they have the same charge current in
response to the gauge fluctuations. However, since the CEL and the CHL have
mean-field charge densities of opposite sign, the density-current correlators
in the two layers have equal and opposite contributions which cancel to
restore an effective TRS. This effective TRS ultimately leads to the
degenerate $\pm l$ pairing channels. This is in contrast to the CEL-CEL
theory. In the CEL-CEL pairing scenario, since the charge-density as well as
the charge currents in the two identical CFLs have the same sign, they add up
to break the TRS Isobe and Fu (2017).

Having shown that $\pm l$ channels are degenerate in the CEL-CHL pairing, we
next analyse the relative strength of different $|l|$ channels. For this
purpose, we make the effective mass approximation $m_{+}=m_{-}=m^{*}$ which
stands on good theoretical Barkeshli  _et al._ (2015) and experimental Pan
_et al._ (2020) footing. We expect our main results to be unchanged by small
variations in the effective masses. Under this approximation, Eq. (15)
vanishes for any momentum and frequency sup . To assess the stability of the
pairing we consider the effective coupling constants

$\displaystyle\begin{split}\lambda_{Z,m}&=\int\frac{d^{2}\boldsymbol{q}}{(2\pi)^{2}}\delta(\xi_{\boldsymbol{k}+\boldsymbol{q}})V_{ex}(\boldsymbol{k},\boldsymbol{q},i\omega_{m}),\\\
\lambda_{\phi,m}^{(l)}&=\int\frac{d^{2}\boldsymbol{q}}{(2\pi)^{2}}\delta(\xi_{\boldsymbol{k}+\boldsymbol{q}})V_{c}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})\left(1+\frac{q}{k_{F}}e^{il(\theta_{\boldsymbol{q}}-\theta_{\boldsymbol{k}})}\right)^{l},\end{split}$
(16)

where $\theta_{\boldsymbol{q}}$ is the polar angle of $\boldsymbol{q}$, and we
assumed that the pairing happens only at the Fermi surface, i.e.
$|\boldsymbol{k}|=k_{F}$. In the limit of zero frequency and long-wavelengths,
the effective couplings are approximated by their singular terms Isobe and Fu
(2017); sup

$\displaystyle\begin{split}\lambda_{Z,0}&\approx\frac{m^{*}}{k_{F}}\int_{0}^{2k_{F}}\frac{dq}{(2\pi)^{2}}\left(\frac{k_{F}^{2}}{16\pi^{3}q^{2}\chi_{0}}+\frac{\frac{k_{F}^{2}}{12\pi^{3}m^{*}\chi_{0}}-\frac{k_{F}^{2}\chi^{\prime}}{16\pi^{3}\chi_{0}^{2}}}{q}\right),\\\
\lambda_{\phi,0}^{(l)}&\approx\frac{m^{*}}{k_{F}}\int_{0}^{2k_{F}}\frac{dq}{(2\pi)^{2}}\left(-\frac{k_{F}^{2}}{16\pi^{3}q^{2}\chi_{0}}+\frac{\frac{dk_{F}^{2}}{16\pi^{3}\chi_{0}}+\frac{k_{F}^{2}\chi^{\prime}}{16\pi^{3}\chi_{0}^{2}}}{q}\right),\end{split}$
(17)

with $\chi_{0}=\frac{m^{*}}{96\pi^{4}}+\frac{2d(m^{*})^{2}}{(4\pi)^{4}}$ and
$\chi^{\prime}=\frac{1}{144\pi^{4}}-\frac{2(m^{*})^{2}d^{2}}{(4\pi)^{4}}$.
Because the leading term in the anomalous self-energy is negative the CEL-CHL
pairing is stable. This attractive interaction is mediated via the out-of-
phase current fluctuations. The next order term corresponds to the repulsive
in-phase current-current interactions Isobe and Fu (2017); sup .

The terms distinguishing the different pairing channels enter only at $q^{0}$
order:

$\lambda_{\phi,0}^{(l)}-\lambda_{\phi,0}^{(0)}\approx\frac{m^{*}}{k_{F}}\int_{0}^{2k_{F}}\frac{dq}{(2\pi)^{2}}\,\frac{4l^{2}}{128\pi^{3}\chi_{0}}.$
(18)

It is immediately clear from this that the favoured pairing happens in the
$s$-wave channel. This can be explained by the fact that the Cooper channel
interactions are isotropic. Using analytic expressions for the propagators
Cipri and Bonesteel (2014); Isobe and Fu (2017) and numerically performing the
integrals in Eq. (16) sup , we obtain the effective couplings [see Fig. 2
(a)]. It is obvious that $l=0$ is the strongest pairing channel for all
frequencies.

CEL-CHL vs. CEL-CEL Pairing.— Next, we compare the relative stability of the
pairs in the two theories, i.e. $s$-wave CEL-CHL pairs (CXs) to the CEL-CEL
Cooper pairs. In Fig. 2 (b), we therefore consider the difference between the
leading CEL-CHL coupling $\lambda_{\phi,m}^{CEL-CHL(0)}$ and the three
strongest CEL-CEL couplings $\lambda_{\phi,m}^{CEL-CEL(l=0,1,2)}$. The CEL-CHL
$s$-wave channel dominates at all frequencies. Thus,after performing the
Matsubara frequency sum, the CEL-CHL pairing theory leads to CXs that are more
tightly bound than the Cooper pairs in the CEL-CEL theory. This suggests that
CXs are more likely to preform in a large region before the exciton
condensation Eisenstein  _et al._ (2019); Liu  _et al._ (2022). Thus, this
theory has stronger tendency to undergo BCS-BEC crossover than the CEL-CEL
theory. The CEL-CEL theory possibly has stronger tendency towards a first
order phase transition Schliemann  _et al._ (2001); Zou  _et al._ (2010).
However, the contrary opinions also exist for the CEL-CEL theory Sodemann  _et
al._ (2017). We mention in passing that at zero-frequency the coupling
constants for the dominant $l=1$ CEL-CEL pairing channel are degenerate with
the $l=0$ channel of CXs.

We also compare the response of the two theories to a slight density mismatch
in the two layers, while keeping the total filling factor fixed, i.e. when the
filling factors are $\nu_{T}=(1/2-\delta)+(1/2+\delta)$. Since each bare
particle is attached to two fluxes, in each layer the external field is not
exactly cancelled by the flux when it is moved away from half filling of the
LLL. This remnant field is often interpreted to lead to a new set of Landau
levels to explain the fractionalization (“$\Lambda$-levels" of the composite
fermions)Jain (1989). Apart from the $\Lambda$-levels, the density imbalance
also changes the Fermi wave vector of the CFL ($k_{F}=\sqrt{4\pi n}$) Halperin
_et al._ (1993). In the following discussion, we assume that density imbalance
is small enough that the individual layer is still away from a fractional
state. Thus, we can ignore the effect of $\Lambda$-levels and in the isolated
CFL, the change in the $k_{F}$ remains the only effect of density imbalance.
In the CEL-CEL theory, while one of the layers is depleted, the density in the
other layer increases, as a result $k_{F}$ also decreases in one layer and
increases in the other layer. This leads to pair-breaking phenomena similar to
the Pauli-pair breaking in conventional superconductors, where the difference
in the energy of the two opposite spin electrons that form the pairs leads to
eventual suppression of the superconductivity when the magnetisation energy
overcomes the condensation energy Chandrasekhar (1962); Clogston (1962).

Figure 2: (a) Effective couplings $\lambda_{\phi,m}^{(l)}$ as functions of the
frequency. The integrals in Eq. (16) diverge for small $q$ both for zero and
finite frequencies. This is cured by introducing the cutoff
$q_{c}=10^{-5}k_{F}$. We set $k_{F}d=1$. (b) Difference between the effective
couplings for CEL-CEL and CEL-CHL pairing
$\Delta\lambda_{\phi,m}^{(l)}=\lambda_{\phi,m}^{CEL-
CHL(0)}-\lambda_{\phi,m}^{CEL-CEL(l)}$, with $k_{F}d=1$. (c) Effective
coupling $\lambda_{\phi,m}^{(0)}$ when varying the Fermi wavevector $k_{F}$
and keeping distance $d$, effective mass $m^{*}$, and frequency $\omega_{m}$
constant parametrised by $\alpha=2m^{\ast}\omega_{m}d^{2}$.

In contrast, in the CEL-CHL theory, the layer with the increased electron
density is described by the CHL Barkeshli  _et al._ (2015). Thus, the CH and
CE densities are equal sup . As a result, the density imbalance of bare
particle leads to identical depletion of the two layers in the CF description.
Thus, Pauli-like pair breaking phenomena are absent in this theory. However,
since $k_{F}$ decreases with the density depletion, the effective coupling
constants $\lambda_{\phi,m}^{(l)}$ usually decrease for small frequencies [see
Fig. 2 (c)]. This can lead to a different mechanism for the suppression of CX
formation with respect to the density imbalance. Notice in the CEL-CEL theory,
this mechanism is also present along with the the Pauli pair breaking.
Thus,the CXs are also more robust to small density imbalances compared to the
CEL-CEL Cooper pairs. Thus, detailed experimental study with density-
imbalanced states can be used to further verify CX formation.

Conclusions.— We have studied the pairing of CEL and CHL in the quantum Hall
bilayer at $\nu_{T}=\frac{1}{2}+\frac{1}{2}$ by analysing fluctuations in the
CS gauge field around the mean-field solution within the RPA. These
fluctuations mediate interactions between the CEL and the CHL. In the
interlayer Cooper channel, these interactions are attractive and lead to a
stable CEL-CHL pairing with $l=0$ symmetry due to an effective TRS. This
yields a microscopic understanding for numerical results Wagner  _et al._
(2021), where an $s$-wave CEL-CHL pairing trial wavefunction was found to have
the best overlap with exact diagonalization results. The CXs in our theory are
formed in the large layer separation limit, which appears to be in agreement
with the experimental findings of preformed pairs before the transition to the
XC Eisenstein  _et al._ (2019); Liu  _et al._ (2022). However, the pairs in
our theory are pairs of CEs and CHs, which must be contrasted with the pairs
of bare electrons and holes. It is the latter that condense to form an XC in
closely separated bilayers. We leave this relation between the condensate of
CXs and the bare excitons, along with the theory of the full BCS-BEC crossover
regime for future investigations.

###### Acknowledgements.

R. J. S and G. C. acknowledge funding from a New Investigator Award, EPSRC
grant EP/W00187X/1. R. J. S also acknowledges funding from Trinity College,
University of Cambridge. L. R. and R. J. S acknowledge funding provided by the
Winton programme and the Schiff foundation.

## References

  * Eisenstein  _et al._ (1992) J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, and Song He, “New fractional quantum hall state in double-layer two-dimensional electron systems,” Phys. Rev. Lett. 68, 1383–1386 (1992).
  * Eisenstein (2014) J.P. Eisenstein, “Exciton condensation in bilayer quantum hall systems,” Annual Review of Condensed Matter Physics 5, 159–181 (2014).
  * Spielman  _et al._ (2000) I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Resonantly enhanced tunneling in a double layer quantum hall ferromagnet,” Phys. Rev. Lett. 84, 5808 (2000).
  * Eisenstein and MacDonald (2004) J. P. Eisenstein and A. H. MacDonald, “Bose–einstein condensation of excitons in bilayer electron systems,” Nature 432, 691–694 (2004).
  * Kellogg  _et al._ (2004) M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Vanishing hall resistance at high magnetic field in a double-layer two-dimensional electron system,” Phys. Rev. Lett. 93, 036801 (2004).
  * Nandi  _et al._ (2012) D. Nandi, A. D. K. Finck, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Exciton condensation and perfect coulomb drag,” Nature 488, 481 (2012).
  * Halperin (1983) B. I. Halperin, “Theory of the quantized hall conductance,” Helvetica Physica Acta 56, 75 (1983).
  * Fertig (1989) H. A. Fertig, “Energy spectrum of a layered system in a strong magnetic field,” Phys. Rev. B 40, 1087 (1989).
  * Wen and Zee (1992) X.-G. Wen and A. Zee, “Neutral superfluid modes and “magnetic” monopoles in multilayered quantum hall systems,” Phys. Rev. Lett. 69, 1811 (1992).
  * Wen and Zee (1993) X.-G. Wen and A. Zee, “Tunneling in double-layered quantum hall systems,” Phys. Rev. B 47, 2265 (1993).
  * Halperin  _et al._ (1993) B. I. Halperin, Patrick A. Lee, and Nicholas Read, “Theory of the half-filled landau level,” Phys. Rev. B 47, 7312–7343 (1993).
  * Shibata and Yoshioka (2006) Naokazu Shibata and Daijiro Yoshioka, “Ground state of $\nu=1$ bilayer quantum hall systems,” Journal of the Physical Society of Japan 75, 043712 (2006).
  * Zhu  _et al._ (2017) Zheng Zhu, Liang Fu, and D. N. Sheng, “Numerical study of quantum hall bilayers at total filling ${\nu}_{T}=1$: A new phase at intermediate layer distances,” Phys. Rev. Lett. 119, 177601 (2017).
  * Lian and Zhang (2018) Biao Lian and Shou-Cheng Zhang, “Wave function and emergent su(2) symmetry in the ${\nu}_{T}=1$ quantum hall bilayer,” Phys. Rev. Lett. 120, 077601 (2018).
  * Simon  _et al._ (2003) Steven H. Simon, E. H. Rezayi, and Milica V. Milovanovic, “Coexistence of composite bosons and composite fermions in $\nu=\frac{1}{2}+\frac{1}{2}$ quantum hall bilayers,” Phys. Rev. Lett. 91, 046803 (2003).
  * Morinari (1999) Takao Morinari, “Composite-fermion pairing in bilayer quantum hall systems,” Phys. Rev. B 59, 7320–7322 (1999).
  * Kim  _et al._ (2001) Yong Baek Kim, Chetan Nayak, Eugene Demler, N. Read, and S. Das Sarma, “Bilayer paired quantum hall states and coulomb drag,” Phys. Rev. B 63, 205315 (2001).
  * Möller  _et al._ (2008) G. Möller, S. H. Simon, and E. H. Rezayi, “Paired composite fermion phase of quantum hall bilayers at $\nu=\frac{1}{2}+\frac{1}{2}$,” Phys. Rev. Lett. 101, 176803 (2008).
  * Sodemann  _et al._ (2017) Inti Sodemann, Itamar Kimchi, Chong Wang, and T. Senthil, “Composite fermion duality for half-filled multicomponent landau levels,” Phys. Rev. B 95, 085135 (2017).
  * Bonesteel (1993) N. E. Bonesteel, “Compressible phase of a double-layer electron system with total landau-level filling factor 1/2,” Phys. Rev. B 48, 11484–11487 (1993).
  * Bonesteel  _et al._ (1996) N. E. Bonesteel, I. A. McDonald, and C. Nayak, “Gauge fields and pairing in double-layer composite fermion metals,” Phys. Rev. Lett. 77, 3009–3012 (1996).
  * Cipri and Bonesteel (2014) R. Cipri and N. E. Bonesteel, “Gauge fluctuations and interlayer coherence in bilayer composite fermion metals,” Phys. Rev. B 89, 085109 (2014).
  * Isobe and Fu (2017) Hiroki Isobe and Liang Fu, “Interlayer pairing symmetry of composite fermions in quantum hall bilayers,” Phys. Rev. Lett. 118, 166401 (2017).
  * Wagner  _et al._ (2021) Glenn Wagner, Dung X. Nguyen, Steven H. Simon, and Bertrand I. Halperin, “$s$-wave paired electron and hole composite fermion trial state for quantum hall bilayers with $\nu=1$,” Phys. Rev. Lett. 127, 246803 (2021).
  * Eisenstein  _et al._ (2019) J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Precursors to exciton condensation in quantum hall bilayers,” Phys. Rev. Lett. 123, 066802 (2019).
  * Liu  _et al._ (2022) Xiaomeng Liu, J. I. A. Li, Kenji Watanabe, Takashi Taniguchi, James Hone, Bertrand I. Halperin, Philip Kim, and Cory R. Dean, “Crossover between strongly coupled and weakly coupled exciton superfluids,” Science 375, 205–209 (2022).
  * Bohm and Pines (1953) David Bohm and David Pines, “A collective description of electron interactions: Iii. coulomb interactions in a degenerate electron gas,” Phys. Rev. 92, 609–625 (1953).
  * Fetter  _et al._ (1991) A.L. Fetter, C.B. Hanna, and R.B. Laughlin, “Anyons and superconductivity: Random phase approximation,” International Journal of Modern Physics B 05, 2751–2790 (1991), https://doi.org/10.1142/S0217979291001097 .
  * Möller  _et al._ (2009) G. Möller, S. H. Simon, and E. H. Rezayi, “Trial wave functions for $\nu=\frac{1}{2}+\frac{1}{2}$ quantum hall bilayers,” Phys. Rev. B 79, 125106 (2009).
  * Barkeshli  _et al._ (2015) Maissam Barkeshli, Michael Mulligan, and Matthew P. A. Fisher, “Particle-hole symmetry and the composite fermi liquid,” Phys. Rev. B 92, 165125 (2015).
  * Lopez and Fradkin (1991) A. Lopez and E. Fradkin, “Fractional quantum hall effect and chern-simons gauge theories,” Phys. Rev. B 44, 5246 (1991).
  * (32) “See supplementary material,” .
  * Marsiglio (2020) F. Marsiglio, “Eliashberg theory: A short review,” Annals of Physics 417, 168102 (2020), eliashberg theory at 60: Strong-coupling superconductivity and beyond.
  * Pan  _et al._ (2020) W. Pan, W. Kang, M. P. Lilly, J. L. Reno, K. W. Baldwin, K. W. West, L. N. Pfeiffer, and D. C. Tsui, “Particle-hole symmetry and the fractional quantum hall effect in the lowest landau level,” Phys. Rev. Lett. 124, 156801 (2020).
  * Schliemann  _et al._ (2001) J. Schliemann, S. M. Girvin, and A. H. MacDonald, “Strong correlation to weak correlation phase transition in bilayer quantum hall systems,” Phys. Rev. Lett. 86, 1849 (2001).
  * Zou  _et al._ (2010) Y. Zou, G. Refael, A. Stern, and J. P. Eisenstein, “Clausius-clapeyron relations for first-order phase transitions in bilayer quantum hall systems,” Phys. Rev. B 81, 205313 (2010).
  * Jain (1989) J. K. Jain, “Composite-fermion approach for the fractional quantum hall effect,” Phys. Rev. Lett. 63, 199–202 (1989).
  * Chandrasekhar (1962) B. S. Chandrasekhar, “A note on the maximum critical field of high-field superconductors,” Appl. Phys. Lett. 1, 7 (1962).
  * Clogston (1962) A. M. Clogston, “Upper limit for the critical field in hard superconductors,” Phys. Rev. Lett. 9, 266 (1962).
  * Zhang  _et al._ (1989) S. C. Zhang, T. H. Hansson, and S. Kivelson, “Effective-field-theory model for the fractional quantum hall effect,” Phys. Rev. Lett. 62, 82–85 (1989).

##  Appendix A CEL-CHL CS Theory

We set $\hbar=c=e=1$. The Schrödinger Lagrangian density for a spin-polarised
electron field $c$ in a background gauge field $A$ is Barkeshli  _et al._
(2015)

$\displaystyle\begin{split}\mathcal{L}_{\text{el}}=c^{\dagger}\left(i\partial_{t}+A_{t}+\mu_{e}\right)c-\frac{1}{2m_{e}}c^{\dagger}\left(i\boldsymbol{\nabla}+\boldsymbol{A}\right)^{2}c-V(c),\end{split}$
(19)

where the Coulomb interaction term is

$V(c)=\frac{1}{2}\int
d^{2}\boldsymbol{y}\,c^{\dagger}c\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}c^{\dagger}c.$
(20)

In the CEL picture, we take the CE field $\psi$ as the underlying degree of
freedom. The transformed Lagrangian is Zhang  _et al._ (1989)

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL}}=\psi^{\dagger}\left(i\partial_{t}+a_{t}+A_{t}+\mu_{\psi}\right)\psi-\frac{1}{2m_{\psi}}\psi^{\dagger}\left(i\boldsymbol{\nabla}+\boldsymbol{a}+\boldsymbol{A}\right)^{2}\psi-V(\psi)+\frac{1}{4\pi}a_{t}\boldsymbol{\nabla}\wedge\boldsymbol{a},\end{split}$
(21)

where $a$ is the dynamical gauge field. The CS term yields the equation of
motion $\boldsymbol{\nabla}\wedge\boldsymbol{a}=-4\pi\psi^{\dagger}\psi$,
which is the attachment of the two flux quanta to the CEs.

The $\nu_{T}=1$ charge-balanced bilayer CEL Lagrangian is

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL-
CEL}}=\sum_{s=+/-}\bigg{[}\psi^{\dagger}_{s}\left(i\partial_{t}+a_{t}^{s}+A_{t}+\mu_{\psi}\right)\psi_{s}-\frac{1}{2m_{\psi}}\psi^{\dagger}_{s}\left(i\boldsymbol{\nabla}+\boldsymbol{a}^{s}+\boldsymbol{A}\right)^{2}\psi_{s}-V(\psi_{s})+\frac{1}{4\pi}a_{t}^{s}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s}\bigg{]}-V_{+-}(\psi_{+},\psi_{-}).\end{split}$
(22)

The Coulomb interaction between layers separated by a distance $d$ is

$V_{+-}(\psi_{+},\psi_{-})=\int
d^{2}\boldsymbol{y}\,\psi^{\dagger}_{+}\psi_{+}\frac{1}{\sqrt{|\boldsymbol{x}-\boldsymbol{y}|^{2}+d^{2}}}\psi^{\dagger}_{-}\psi_{-}.$
(23)

Both intra- and interlayer interactions can also be written in terms of the
spatial parts of the dynamical gauge fields by enforcing their equations of
motion

$\displaystyle V(a^{s})$ $\displaystyle=\,\frac{1}{2}\frac{1}{(4\pi)^{2}}\int
d^{2}\boldsymbol{y}\,\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s}\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s},$
(24) $\displaystyle V_{+-}(a^{+},a^{-})$
$\displaystyle=\,\frac{1}{(4\pi)^{2}}\int
d^{2}\boldsymbol{y}\,\boldsymbol{\nabla}\wedge\boldsymbol{a}^{+}\frac{1}{\sqrt{|\boldsymbol{x}-\boldsymbol{y}|^{2}+d^{2}}}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{-}.$
(25)

We now want to describe the $-$ layer in terms of CHs. The non-interacting
part is the same as in Eq. (5) and was derived in Ref. Barkeshli  _et al._
(2015). The intralayer interaction is the same as before. The interlayer
interaction, however, obtains a minus sign when expressed in terms of the CE
and CH fields, which cancels when expressing it in terms of the dynamical
gauge fields:

$\displaystyle\begin{split}V_{+-}(\psi_{+},\psi_{-})&=\,\int
d^{2}\boldsymbol{y}\,\psi^{\dagger}_{+}\psi_{+}\frac{-1}{\sqrt{|\boldsymbol{x}-\boldsymbol{y}|^{2}+d^{2}}}\psi^{\dagger}_{-}\psi_{-},\\\
V_{+-}(a^{+},a^{-})&=\frac{1}{(4\pi)^{2}}\int
d^{2}\boldsymbol{y}\,\boldsymbol{\nabla}\wedge\boldsymbol{a}^{+}\frac{1}{\sqrt{|\boldsymbol{x}-\boldsymbol{y}|^{2}+d^{2}}}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{-}.\end{split}$
(26)

Hence, the total CEL-CHL Lagrangian is

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL-
CHL}}=&\sum_{s=+/-}\bigg{[}\psi^{\dagger}_{s}\left(i\partial_{t}+a_{t}^{s}+sA_{t}+\mu_{s}\right)\psi_{s}-\frac{1}{2m_{s}}\psi^{\dagger}_{s}\left(i\boldsymbol{\nabla}+\boldsymbol{a}^{s}+s\boldsymbol{A}\right)^{2}\psi_{s}-V(a^{s})+\frac{s}{4\pi}a_{t}^{s}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s}\bigg{]}\\\
&-V_{+-}(a^{+},a^{-})+\frac{1}{2\pi}A_{t}\boldsymbol{\nabla}\wedge\boldsymbol{A}.\end{split}$
(27)

In finite-temperature imaginary-time formalism, the above Lagrangian becomes
($\tau=it,\partial_{t}=i\partial_{\tau},dt=-id\tau,A_{t}=-iA_{\tau}$)

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL-
CHL}}=&\sum_{s=+/-}\bigg{[}\psi^{\dagger}_{s}\left(\partial_{\tau}+ia_{\tau}^{s}+siA_{\tau}-\mu_{s}\right)\psi_{s}+\frac{1}{2m_{s}}\psi^{\dagger}_{s}\left(i\boldsymbol{\nabla}+\boldsymbol{a}^{s}+s\boldsymbol{A}\right)^{2}\psi_{s}+V(a^{s})+\frac{si}{4\pi}a_{\tau}^{s}\boldsymbol{\nabla}\wedge\boldsymbol{a}^{s}\bigg{]}\\\
&+V_{+-}(a^{+},a^{-})+\frac{i}{2\pi}A_{\tau}\boldsymbol{\nabla}\wedge\boldsymbol{A}.\end{split}$
(28)

##  Appendix B Saddle-Point Approximation and Effective Action

The CEL-CHL Lagrangian possesses a mean-field solution $a^{s}=-sA$ exactly
analogous to the one for the monolayer CEL where the external flux is
cancelled out. Going beyond this, we want to take fluctuations
$\tilde{a}^{s}=a^{s}+s\times A$ around the saddle-point into account. The
Lagrangian in terms of the fluctuations becomes

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL-
CHL}}=&\sum_{s=+/-}\bigg{[}\psi^{\dagger}_{s}\left(\partial_{\tau}+i\tilde{a}_{\tau}^{s}-\mu_{s}\right)\psi_{s}+\frac{1}{2m_{s}}\psi^{\dagger}_{s}\left(i\boldsymbol{\nabla}+\boldsymbol{\tilde{a}^{s}}\right)^{2}\psi_{s}+V(\tilde{a}^{s})+\frac{si}{4\pi}\tilde{a}_{\tau}^{s}\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{a}^{s}}\bigg{]}+V_{+-}(\tilde{a}^{+},\tilde{a}^{-}),\end{split}$
(29)

where we could replace the dynamical gauge field in the interaction terms with
its fluctuations because only terms quadratic in the fluctuations will
contribute in the effective action (c.f. Eq. (9)); similarly, only quadratic
terms from the CS terms were kept. We write our fields now as functions of
momenta $\boldsymbol{q}$ and bosonic/fermionic Matsubara frequencies
$\omega_{m}=2m\pi T/\epsilon_{n}=2(n+1)\pi T$. Assuming the Coulomb gauge
$\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{a}^{s}}=0$, we can reduce the
spatial components of the fluctuations of the dynamical gauge field to its
transverse part $\tilde{a}_{1}$:

$\tilde{a}_{1}(\boldsymbol{q},i\omega_{m})=\boldsymbol{\hat{q}}\wedge\boldsymbol{\tilde{a}}(\boldsymbol{q},i\omega_{m}),\quad\boldsymbol{\tilde{a}}(\boldsymbol{q},i\omega_{m})=\tilde{a}_{1}(\boldsymbol{q},i\omega_{m})\hat{z}\wedge\boldsymbol{\hat{q}}.$
(30)

We will denote the imaginary time component with the index $0$ from now on.

The Lagrangian in momentum-frequency space is

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL-
CHL}}=\,&\sum_{s=+/-}\bigg{[}-\psi^{\dagger}_{s}(\boldsymbol{q},i\epsilon_{n})i\epsilon\psi_{s}(\boldsymbol{q},i\epsilon_{n})+\psi^{\dagger}_{s}(\boldsymbol{k}+\boldsymbol{q},i\epsilon_{n}+i\omega_{m})i\tilde{a}_{0}^{s}(\boldsymbol{q},i\omega_{m})\psi_{s}(\boldsymbol{k},i\epsilon_{n})\\\
&+\frac{1}{2m_{s}}\psi^{\dagger}_{s}(\boldsymbol{q},i\epsilon_{n})\boldsymbol{q}^{2}\psi_{s}(\boldsymbol{q},i\epsilon_{n})-\frac{1}{m_{s}}\psi^{\dagger}_{s}(\boldsymbol{k}+\boldsymbol{q},i\epsilon_{n}+i\omega_{m})\tilde{a}_{1}^{s}(\boldsymbol{q},i\omega_{m})\boldsymbol{\hat{q}}\wedge\boldsymbol{k}\psi_{s}(\boldsymbol{k},i\epsilon_{n})\\\
&-\frac{1}{2m_{s}}\psi^{\dagger}_{s}(\boldsymbol{k}+\boldsymbol{q}-\boldsymbol{q^{\prime}},i\epsilon_{n}+i\omega_{m}-i\omega_{m}^{\prime})\tilde{a}_{1}^{s}(\boldsymbol{q},i\omega_{m})\tilde{a}_{1}^{s}(-\boldsymbol{q^{\prime}},-i\omega_{m}^{\prime})\boldsymbol{\hat{q}}\cdot\boldsymbol{\hat{q}^{\prime}}\psi_{s}(\boldsymbol{k},i\epsilon_{n})\\\
&-\frac{1}{2}\frac{1}{(4\pi)^{2}}q^{2}\tilde{a}_{1}^{s}(\boldsymbol{q},i\omega_{m})V_{ss}(q)\tilde{a}_{1}^{s}(-\boldsymbol{q},-i\omega_{m})-\frac{s}{4\pi}q\tilde{a}_{0}^{s}(\boldsymbol{q},i\omega_{m})\tilde{a}_{1}^{s}(-\boldsymbol{q},-i\omega_{m})\bigg{]}\\\
&-\frac{1}{(4\pi)^{2}}q^{2}\tilde{a}_{1}^{+}(\boldsymbol{q},i\omega_{m})V_{+-}(q)\tilde{a}_{1}^{-}(-\boldsymbol{q},-i\omega_{m}),\end{split}$
(31)

with the intralayer interaction $V_{++}(q)=V_{--}(q)=\frac{2\pi}{q}$, and the
interlayer interaction $V_{+-}(q)=V_{-+}(q)=\frac{2\pi e^{-qd}}{q}$.

The Green’s functions for the fermionic fields are

$G^{s}(q,i\epsilon_{n})=\frac{1}{i\epsilon_{n}-\epsilon^{s}(q)},$ (32)

with $\epsilon^{s}(q)=\frac{q^{2}}{2m_{s}}$. The bare inverse gauge field
propagator is

$D_{\mu\nu,ss^{\prime}}^{(0)}(q,i\omega_{m})^{-1}=\begin{pmatrix}0&-\frac{sq}{4\pi}\delta_{ss^{\prime}}\\\
-\frac{sq}{4\pi}\delta_{ss^{\prime}}&-\frac{q^{2}V_{ss^{\prime}}(q)}{(4\pi)^{2}}\end{pmatrix}_{\mu\nu}.$
(33)

The remaining terms in the Lagrangian generate vertices between the fermionic
fields and the dynamical gauge field. They yield the the one-loop corrections

$\displaystyle\Pi_{00,ss^{\prime}}(q,i\omega_{m})$
$\displaystyle=\braket{a_{0}^{s}(\boldsymbol{q},i\omega_{m})a_{0}^{s^{\prime}}(-\boldsymbol{q},-i\omega_{m})}=\Pi_{00}^{s}(q,i\omega_{m})\delta_{ss^{\prime}},$
(34) $\displaystyle\Pi_{11,ss^{\prime}}(q,i\omega_{m})$
$\displaystyle=\braket{a_{1}^{s}(\boldsymbol{q},i\omega_{m})a_{1}^{s^{\prime}}(-\boldsymbol{q},-i\omega_{m})}=\Pi_{11}^{s}(q,i\omega_{m})\delta_{ss^{\prime}}.$
(35)

Since the vertices are exactly the same as in the CEL-CEL case, the
polarisation diagrams are the same as in Refs. Cipri and Bonesteel (2014);
Isobe and Fu (2017). We calculate the full inverse gauge propagator within the
RPA Bohm and Pines (1953); Fetter  _et al._ (1991):

$\displaystyle D_{\mu\nu,ss^{\prime}}(q,i\omega_{m})^{-1}$
$\displaystyle=D_{\mu\nu,ss^{\prime}}^{(0)}(q,i\omega_{m})^{-1}-\Pi_{\mu\nu,ss^{\prime}}(q,i\omega_{m})$
(36)
$\displaystyle=\begin{pmatrix}-\Pi_{00}^{+}(q,i\omega_{m})&-\frac{q}{4\pi}&0&0\\\
-\frac{q}{4\pi}&-\Pi_{11}^{+}(q,i\omega_{m})-\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)&0&-\frac{q^{2}}{(4\pi)^{2}}V_{+-}(q)\\\
0&0&-\Pi_{00}^{-}(q,i\omega_{m})&\frac{q}{4\pi}\\\
0&-\frac{q^{2}}{(4\pi)^{2}}V_{-+}(q)&\frac{q}{4\pi}&-\Pi_{11}^{-}(q,i\omega_{m})-\frac{q^{2}}{(4\pi)^{2}}V_{--}(q)\\\
\end{pmatrix}_{\mu\nu,ss^{\prime}}.$ (37)

##  Appendix C Low-Energy Long-Wavelength Approximation Propagator

Ultimately, we will be interested in the long-wavelength and low-energy modes
of the system. These will be in the regime
$\frac{\omega_{m}}{\epsilon_{F}}\ll\frac{q^{2}}{k_{F}^{2}}\ll 1$ and $qd\ll
1$. We can then approximate the polarisation diagrams as Cipri and Bonesteel
(2014); Isobe and Fu (2017)

$\Pi_{00}^{s}(q,i\omega_{m})\approx-\frac{m_{s}}{2\pi},\quad\Pi_{11}^{s}(q,i\omega_{m})\approx\chi_{d}^{s}q^{2}-\frac{m_{s}}{2\pi}\frac{\omega_{m}^{2}}{q^{2}}+\frac{k_{F}}{2\pi}\frac{|\omega_{m}|}{q},$
(38)

where $\chi_{d}^{s}=\frac{1}{24\pi m_{s}}$ is the diamagnetic susceptibility.

We calculate the propagator matrix as
$D(q,i\omega_{m})=\frac{1}{\text{det}\left[D(q,i\omega_{m})^{-1}\right]}\text{adj}\left[D(q,i\omega_{m})^{-1}\right]$.
The determinant term is

$\displaystyle\begin{split}\text{det}\left[D(q,i\omega_{m})^{-1}\right]&=\left(\frac{q^{2}}{(4\pi)^{2}}-\Pi_{00}^{+}(q,i\omega_{m})\left[\Pi_{11}^{+}(q,i\omega_{m})+\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)\right]\right)\\\
&\times\left(\frac{q^{2}}{(4\pi)^{2}}-\Pi_{00}^{-}(q,i\omega_{m})\left[\Pi_{11}^{-}(q,i\omega_{m})+\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)\right]\right)\\\
&-\Pi_{00}^{+}(q,i\omega_{m})\Pi_{00}^{-}(q,i\omega_{m})\frac{q^{4}}{(4\pi)^{4}}V_{+-}(q)^{2}\\\
&\approx\frac{m_{+}m_{-}k_{F}}{2(2\pi)^{4}}|\omega_{m}|+q^{3}\left(\frac{m_{+}+m_{-}}{192\pi^{4}}+\frac{2dm_{+}m_{-}}{(4\pi)^{4}}\right)+q^{4}\left(\frac{1}{144\pi^{4}}-\frac{2m_{+}m_{-}d^{2}}{(4\pi)^{4}}\right)\\\
&=\chi q^{3}+\chi^{\prime}q^{4},\end{split}$ (39)

where we defined

$\chi=\frac{m_{+}m_{-}k_{F}}{2(2\pi)^{4}}\frac{|\omega_{m}|}{q^{3}}+\left(\frac{m_{+}+m_{-}}{192\pi^{4}}+\frac{2dm_{+}m_{-}}{(4\pi)^{4}}\right),\quad\chi^{\prime}=\frac{1}{144\pi^{4}}-\frac{2m_{+}m_{-}d^{2}}{(4\pi)^{4}}.$
(40)

The propagators in the different layer sectors are then

$D_{++}(q,i\omega_{m})\approx\frac{1}{\text{det}\left[D(q,i\omega_{m})^{-1}\right]}\begin{pmatrix}q^{3}\left(\frac{2\pi}{(4\pi)^{4}}+\frac{1+\frac{m_{-}}{m_{+}}}{24\pi(4\pi)^{2}}+\frac{m_{-}d}{(4\pi)^{3}}\right)+|\omega_{m}|\frac{m_{-}k_{F}}{16\pi^{3}}&-\left(q^{2}\frac{m_{-}}{(4\pi)^{3}}+q^{3}\frac{1}{48\pi^{3}}+|\omega_{m}|\frac{m_{-}k_{F}}{16\pi^{3}}\right)\\\
-\left(q^{2}\frac{m_{-}}{(4\pi)^{3}}+q^{3}\frac{1}{48\pi^{3}}+|\omega_{m}|\frac{m_{-}k_{F}}{16\pi^{3}}\right)&-\left(q\frac{m_{+}m_{-}}{2\pi(4\pi)^{2}}+q^{2}\frac{m_{+}}{24\pi^{3}}+\frac{|\omega_{m}|}{q}\frac{m_{+}m_{-}k_{F}}{(2\pi)^{3}}\right)\end{pmatrix},$
(41)
$D_{+-}(q,i\omega_{m})=D_{-+}(q,i\omega_{m})^{\intercal}\approx\frac{1}{\text{det}\left[D(q,i\omega_{m})^{-1}\right]}\begin{pmatrix}-q^{3}\frac{1}{2(4\pi)^{3}}&q^{2}\frac{m_{-}}{(4\pi)^{3}}-q^{3}\frac{m_{-}d}{(4\pi)^{3}}\\\
-q^{2}\frac{m_{+}}{(4\pi)^{3}}+q^{3}\frac{m_{+}d}{(4\pi)^{3}}&\frac{m_{+}m_{-}}{2\pi(4\pi)^{2}}\left(q-q^{2}d+q^{3}\frac{d^{2}}{2}\right)\end{pmatrix},$
(42)
$D_{--}(q,i\omega_{m})\approx\frac{1}{\text{det}\left[D(q,i\omega_{m})^{-1}\right]}\begin{pmatrix}q^{3}\left(\frac{2\pi}{(4\pi)^{4}}+\frac{1+\frac{m_{+}}{m_{-}}}{24\pi(4\pi)^{2}}+\frac{m_{+}d}{(4\pi)^{3}}\right)+|\omega_{m}|\frac{m_{+}k_{F}}{16\pi^{3}}&q^{2}\frac{m_{+}}{(4\pi)^{3}}+q^{3}\frac{1}{48\pi^{3}}+|\omega_{m}|\frac{m_{+}k_{F}}{16\pi^{3}}\\\
q^{2}\frac{m_{+}}{(4\pi)^{3}}+q^{3}\frac{1}{48\pi^{3}}+|\omega_{m}|\frac{m_{+}k_{F}}{16\pi^{3}}&-\left(q\frac{m_{+}m_{-}}{2\pi(4\pi)^{2}}+q^{2}\frac{m_{-}}{24\pi^{3}}+\frac{|\omega_{m}|}{q}\frac{m_{+}m_{-}k_{F}}{(2\pi)^{3}}\right)\end{pmatrix}.$
(43)

##  Appendix D Block-Diagonal Form of the Propagator

Upon making the effective mass approximation $m_{+}=m_{-}=m^{*}$, which
results in $\Pi_{\mu\nu}^{+}=\Pi_{\mu\nu}^{-}=\Pi_{\mu\nu}$, there exists a
different basis for the CS gauge fields in which the propagator becomes block-
diagonal. We define

$\displaystyle b_{0}^{+}$
$\displaystyle=\frac{1}{\sqrt{2}}(\tilde{a}^{+}_{0}-\tilde{a}^{-}_{0}),\quad
b_{1}^{+}=\frac{1}{\sqrt{2}}(\tilde{a}^{+}_{1}+\tilde{a}^{-}_{1}),$
$\displaystyle b_{0}^{-}$
$\displaystyle=\frac{1}{\sqrt{2}}(\tilde{a}^{+}_{0}+\tilde{a}^{-}_{0}),\quad
b_{1}^{-}=\frac{1}{\sqrt{2}}(\tilde{a}^{+}_{1}-\tilde{a}^{-}_{1}).$

The inverse propagator in this basis becomes

$D(q,i\omega_{m})^{-1}=\begin{pmatrix}D^{+}(q,i\omega_{m})^{-1}&0\\\
0&D^{-}(q,i\omega_{m})^{-1}\end{pmatrix},$ (44)

with

$D^{+}(q,i\omega_{m})^{-1}=\begin{pmatrix}-\Pi_{00}(q,i\omega_{m})&-\frac{q}{4\pi}\\\
-\frac{q}{4\pi}&-\Pi_{11}(q,i\omega_{m})-\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)-\frac{q^{2}}{(4\pi)^{2}}V_{+-}(q)\end{pmatrix},$
(45)
$D^{-}(q,i\omega_{m})^{-1}=\begin{pmatrix}-\Pi_{00}(q,i\omega_{m})&-\frac{q}{4\pi}\\\
-\frac{q}{4\pi}&-\Pi_{11}(q,i\omega_{m})-\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)+\frac{q^{2}}{(4\pi)^{2}}V_{+-}(q)\end{pmatrix}.$
(46)

We can invert these as

$D^{\pm}(q,i\omega_{m})=-\frac{1}{\text{det}[D^{\pm}(q,i\omega_{m})^{-1}]}\begin{pmatrix}\Pi_{11}(q,i\omega_{m})+\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)\pm\frac{q^{2}}{(4\pi)^{2}}V_{+-}(q)&-\frac{q}{4\pi}\\\
-\frac{q}{4\pi}&\Pi_{00}(q,i\omega_{m})\end{pmatrix},$ (47)

with

$\text{det}[D^{\pm}(q,i\omega_{m})^{-1}]=\Pi_{00}(q,i\omega_{m})\left[\Pi_{11}(q,i\omega_{m})+\frac{q^{2}}{(4\pi)^{2}}V_{++}(q)\pm\frac{q^{2}}{(4\pi)^{2}}V_{+-}(q)\right]-\frac{q^{2}}{(4\pi)^{2}}.$
(48)

Up to the $-$ sign in the off-diagonal terms, these are exactly the in-
phase/out-of-phase propagators derived in Ref. Isobe and Fu (2017). However,
we have used a different basis and hence when transforming back into the CE-CH
basis, the full propagator and the physical results will have crucial
differences.

##  Appendix E Effective Interaction

The effective CS propagator mediates an interaction between the CEs and CHs.
We define

$\mathcal{V}=\frac{1}{2}\sum_{ss^{\prime}}\psi^{\dagger}_{s}(\boldsymbol{k}+\boldsymbol{q},i\epsilon_{n}+i\omega_{m})\psi^{\dagger}_{s^{\prime}}(\boldsymbol{k}^{\prime}-\boldsymbol{q},i\epsilon_{n}^{\prime}-i\omega_{m})V_{ss^{\prime}}^{\text{eff}}(\boldsymbol{k},\boldsymbol{k}^{\prime},\boldsymbol{q},i\omega_{m})\psi_{s^{\prime}}(\boldsymbol{k}^{\prime},i\epsilon_{n}^{\prime})\psi_{s}(\boldsymbol{k},i\epsilon_{n}),$
(49)

and hence

$V_{ss^{\prime}}^{\text{eff}}(\boldsymbol{k},\boldsymbol{k}^{\prime},\boldsymbol{q},i\omega_{m})=\sum_{\mu\nu}M_{\mu\nu,ss^{\prime}}(\boldsymbol{k},\boldsymbol{k}^{\prime},\boldsymbol{q})D_{\mu\nu,ss^{\prime}}(\boldsymbol{q},i\omega_{m}),$
(50)

with

$M_{\mu\nu,ss^{\prime}}(\boldsymbol{k},\boldsymbol{k}^{\prime},\boldsymbol{q})=\frac{1}{2}\begin{pmatrix}1&-i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}^{\prime}}{m_{s^{\prime}}}\\\
i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s}}&\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})(\boldsymbol{\hat{q}}\wedge\boldsymbol{k}^{\prime})}{m_{s}m_{s^{\prime}}}\end{pmatrix}.$
(51)

If we only consider the most singular terms in $q$, which are
$D_{11,ss^{\prime}}(q,i\omega_{m})$, the effective interaction takes the form

$V_{ss^{\prime}}^{\text{eff}}(\boldsymbol{k},\boldsymbol{k}^{\prime},\boldsymbol{q},i\omega_{m})=-ss^{\prime}\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})(\boldsymbol{\hat{q}}\wedge\boldsymbol{k}^{\prime})}{2m_{s}m_{s^{\prime}}}\left[q^{2}\left(\frac{m_{+}+m_{-}}{6\pi
m_{+}m_{-}}+\frac{d}{4\pi}\right)+\frac{|\omega_{m}|}{q}\frac{k_{F}}{\pi}\right]^{-1}.$
(52)

##  Appendix F Effective Couplings

Just as in the main text we will adopt the effective mass approximation from
now on: $m_{+}=m_{-}=m^{*}$.

Within Eliashberg theory Isobe and Fu (2017); Marsiglio (2020), the effective
couplings for the quasiparticle residue $Z_{n}$ and the anomalous self-energy
$\hat{\phi}_{n}(\boldsymbol{k})$ are defined as

$\displaystyle\begin{split}\lambda_{Z,m}&=\int\frac{d^{2}\boldsymbol{q}}{(2\pi)^{2}}\delta(\xi_{\boldsymbol{k}+\boldsymbol{q}})V_{ex}(\boldsymbol{k},\boldsymbol{q},i\omega_{m}),\\\
\lambda_{\phi,m}^{(l)}&=\int\frac{d^{2}\boldsymbol{q}}{(2\pi)^{2}}\delta(\xi_{\boldsymbol{k}+\boldsymbol{q}})V_{c}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})\left(1+\frac{q}{k_{F}}e^{il(\theta_{\boldsymbol{q}}-\theta_{\boldsymbol{k}})}\right)^{l},\end{split}$
(53)

with the exchange and Cooper channel interactions

$\displaystyle\begin{split}V_{ex}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})&=V_{++}^{\text{eff}}(\boldsymbol{k},\boldsymbol{k}+\boldsymbol{q},\boldsymbol{q},i\omega_{m})+V_{--}^{\text{eff}}(\boldsymbol{k},\boldsymbol{k}+\boldsymbol{q},\boldsymbol{q},i\omega_{m})\\\
&=-\frac{1}{2}\sum_{\mu\nu,ss^{\prime}}\begin{pmatrix}1&-i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s^{\prime}}}\\\
i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s}}&\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})^{2}}{m_{s}m_{s^{\prime}}}\end{pmatrix}_{\mu\nu}D_{\mu\nu,ss^{\prime}}(q,i\omega_{m})\left[\delta_{s,+}\delta_{s^{\prime},+}+\delta_{s,-}\delta_{s^{\prime},-}\right],\end{split}$
(54)
$\displaystyle\begin{split}V_{c}(\boldsymbol{k},\boldsymbol{q},i\omega_{m})&=V_{+-}^{\text{eff}}(\boldsymbol{k},-\boldsymbol{k}-\boldsymbol{q},\boldsymbol{q},i\omega_{m})+V_{-+}^{\text{eff}}(\boldsymbol{k},-\boldsymbol{k}-\boldsymbol{q},\boldsymbol{q},i\omega_{m})\\\
&=\frac{1}{2}\sum_{\mu\nu,ss^{\prime}}\begin{pmatrix}1&i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s^{\prime}}}\\\
i\frac{\boldsymbol{\hat{q}}\wedge\boldsymbol{k}}{m_{s}}&-\frac{(\boldsymbol{\hat{q}}\wedge\boldsymbol{k})^{2}}{m_{s}m_{s^{\prime}}}\end{pmatrix}_{\mu\nu}D_{\mu\nu,ss^{\prime}}(q,i\omega_{m})\left[\delta_{s,+}\delta_{s^{\prime},-}+\delta_{s,-}\delta_{s^{\prime},+}\right].\end{split}$
(55)

Assuming that the gap is much larger than the Fermi energy, the pairing
happens only on the Fermi surface. We can then perform the angular integration
in Eq. (53) and obtain Isobe and Fu (2017)

$\displaystyle\begin{split}\lambda_{Z,m}=\frac{1}{(2\pi)^{2}}\frac{m^{*}}{k_{F}}\int_{0}^{k_{F}}dq\bigg{[}&-\frac{1}{\sqrt{1-\left(\frac{q}{k_{F}}\right)^{2}}}\Big{(}D_{00,++}(q,i\omega_{m})+D_{00,--}(q,i\omega_{m})\Big{)}\\\
&-\frac{k_{F}^{2}}{m^{*2}}\sqrt{1-\left(\frac{q}{k_{F}}\right)^{2}}\Big{(}D_{11,++}(q,i\omega_{m})+D_{11,--}(q,i\omega_{m})\Big{)}\bigg{]},\end{split}$
(56)

and

$\displaystyle\begin{split}\lambda_{\phi,m}^{(l)}=\frac{1}{(2\pi)^{2}}\frac{m^{*}}{k_{F}}\int_{0}^{k_{F}}dq\bigg{[}&\frac{1}{\sqrt{1-\left(\frac{q}{k_{F}}\right)^{2}}}\cos\left(2l\arcsin\frac{q}{k_{F}}\right)\Big{(}D_{00,+-}(q,i\omega_{m})+D_{00,-+}(q,i\omega_{m})\Big{)}\\\
&-\frac{k_{F}^{2}}{m^{*2}}\sqrt{1-\left(\frac{q}{k_{F}}\right)^{2}}\cos\left(2l\arcsin\frac{q}{k_{F}}\right)\Big{(}D_{11,+-}(q,i\omega_{m})+D_{11,-+}(q,i\omega_{m})\Big{)}\bigg{]}.\end{split}$
(57)

Contrary to the CEL-CEL case in Ref. Isobe and Fu (2017), for the CEL-CHL
there are no odd-l terms in Eq. (57) because

$D_{10,+-}(q,i\omega_{m})+D_{01,-+}(q,i\omega_{m})+D_{10,-+}(q,i\omega_{m})+D_{01,+-}(q,i\omega_{m})=0.$
(58)

This immediately tells us that the pairing channels $\pm l$ are degenerate.

##  Appendix G Low-Energy Long-Wavelength Approximation Effective Couplings

The pairing stability can be assessed by looking at the effective couplings at
$\omega_{m}=0$, where the low-energy long-wavelength approximation
$\frac{\omega_{m}}{\epsilon_{F}}\ll\frac{q^{2}}{k_{F}^{2}}\ll 1$ and $qd\ll 1$
is valid. We use the approximate forms of the propagator entries from App. C
to write the effective couplings as

$\displaystyle\begin{split}\lambda_{Z,0}=\frac{1}{(2\pi)^{2}}\frac{m^{*}}{k_{F}}\int_{0}^{2k_{F}}dq\Bigg{[}&\frac{k_{F}^{2}}{16\pi^{3}q^{2}\chi}+\frac{\frac{k_{F}^{2}}{12\pi^{3}m^{*}\chi}-\frac{k_{F}^{2}\chi^{\prime}}{16\pi^{3}\chi^{2}}}{q}\\\
&-\frac{2\chi^{\prime\prime}}{\chi}-\frac{k_{F}^{2}\chi^{\prime}}{12\pi^{3}m^{*}\chi^{2}}+\frac{k_{F}^{2}\chi^{\prime
2}}{16\pi^{3}\chi^{3}}-\frac{1}{128\pi^{3}\chi}+O(q)\Bigg{]},\end{split}$ (59)

and

$\displaystyle\begin{split}\lambda_{\phi,0}^{(l)}=\frac{1}{(2\pi)^{2}}\frac{m^{*}}{k_{F}}\int_{0}^{2k_{F}}dq\Bigg{[}&-\frac{k_{F}^{2}}{16\pi^{3}q^{2}\chi}+\frac{\frac{dk_{F}^{2}}{16\pi^{3}\chi}+\frac{k_{F}^{2}\chi^{\prime}}{16\pi^{3}\chi^{2}}}{q}\\\
&-\frac{\frac{d^{2}k_{F}^{2}}{32\pi^{3}}+\frac{-4l^{2}-1}{128\pi^{3}}}{\chi}-\frac{dk_{F}^{2}\chi^{\prime}}{16\pi^{3}\chi^{2}}-\frac{k_{F}^{2}\chi^{\prime
2}}{16\pi^{3}\chi^{3}}-\frac{1}{64\pi^{3}\chi}+O(q)\Bigg{]},\end{split}$ (60)

with

$\chi^{\prime\prime}=\frac{5}{6(4\pi)^{3}}+\frac{m^{*}d}{(4\pi)^{3}}.$ (61)

For $\omega_{m}=0$, $\chi$ goes to
$\chi_{0}=\frac{m^{*}}{96\pi^{4}}+\frac{2d(m^{*})^{2}}{(4\pi)^{4}}$.

##  Appendix H Numerical Integration Effective Couplings

The integrals in Eqs. (59, 60) suffer from divergences as $q\rightarrow 0$. In
Ref. Isobe and Fu (2017) it was shown that it is reasonable to introduce a
cutoff $q_{c}=10^{-5}k_{F}$. To perform the integrals numerically, we use the
exact expressions for the polarisation diagrams
$\Pi_{00}(q,i\omega_{m}),\Pi_{11}(q,i\omega_{m})$ calculated in Ref. Cipri and
Bonesteel (2014):

$\displaystyle\begin{split}\Pi_{00}(q,i\omega_{m})&=F_{1}(q,i\omega_{m})+F_{1}(q,-i\omega_{m}),\\\
F_{1}(q,i\omega_{m})&=2m^{*}f_{1}\left(\frac{2q}{k_{F}},\frac{i\omega_{m}}{\epsilon_{F}}-\frac{q^{2}}{k_{F}^{2}}\right),\\\
f_{1}(y,z)&=\frac{z}{2\pi
y^{2}}\left(1-\left(1-\frac{y^{2}}{z^{2}}\right)^{\frac{1}{2}}\right),\end{split}$
(62)

and

$\displaystyle\begin{split}\Pi_{11}(q,i\omega_{m})&=F_{2}(q,i\omega_{m})+F_{2}(q,-i\omega_{m})+\frac{\epsilon_{F}}{2\pi},\\\
F_{2}(q,i\omega_{m})&=4\epsilon_{F}f_{2}\left(\frac{2q}{k_{F}},\frac{i\omega_{m}}{\epsilon_{F}}-\frac{q^{2}}{k_{F}^{2}}\right),\\\
f_{2}(y,z)&=\frac{z}{4\pi
y^{2}}\left(1-\frac{2z^{2}}{3y^{2}}\left(1-\left(1-\frac{y^{2}}{z^{2}}\right)^{\frac{3}{2}}\right)\right).\end{split}$
(63)

The two polarisation functions are real, unless analytically continued.

Figure H.1: The two polarisation diagrams $\Pi_{00}(q,i\omega_{m})$ and
$\Pi_{11}(q,i\omega_{m})$.

Now we are able to perform the integrals in Eqs. (56, 57). The frequency
spectra are in Fig. H.2 below.

Figure H.2: The quasiparticle residue $\lambda_{Z,m}$ and the anomalous self-
energy $\lambda_{\phi,m}^{(l)}$ for $k_{F}d=1$ and cutoff $q_{c}=10^{-5}$.

The behaviour of the anomalous self-energy $\lambda_{\phi,0}^{(l)}$ as a
function of the layer separation is in absolute values for the $l=0$ channel
and in relative values for $l=1,2$ in Fig. H.3. We see that the pairing
strength decreases with increasing layer separation and that the ordering of
the different channels remains unchanged.

Figure H.3: The anomalous self-energy $\lambda_{\phi,0}^{(0)}$ and the
relative anomalous self-energy $\lambda_{\phi,0}^{(0)}-\lambda_{\phi,0}^{(l)}$
with cutoff $q_{c}=10^{-5}$.

##  Appendix I Density Imbalance

As can be seen from the Lagrangian in Eq. (6), the effective magnetic fields
in the two layers are (taking the charge into account)

$\displaystyle B_{\text{eff}}^{+}$ $\displaystyle=B-4\pi n_{+},$ (64)
$\displaystyle B_{\text{eff}}^{-}$ $\displaystyle=-B+4\pi n_{-}.$ (65)

Using the relation $n_{+}+n_{-}=\frac{B}{2\pi}$, we notice that the CEs and
CHs experience exactly the same effective magnetic field Barkeshli  _et al._
(2015).

Introducing a charge imbalance $\nu_{T}=(1/2-\delta)+(1/2+\delta)$ between the
two layers, we obtain the coupled Lagrangian near the mean-field solution

$\displaystyle\begin{split}\mathcal{L}_{\text{CEL-
CHL}}=&\sum_{s=+/-}\bigg{[}\psi^{\dagger}_{s}\left(\partial_{\tau}+i\tilde{a}_{\tau}^{s}+isdA_{\tau}-\mu_{s}\right)\psi_{s}+\frac{1}{2m_{s}}\psi^{\dagger}_{s}\left(i\boldsymbol{\nabla}+\boldsymbol{\tilde{a}}^{s}+sd\boldsymbol{A}\right)^{2}\psi_{s}+V(\tilde{a}^{s})+\frac{si}{4\pi}\tilde{a}_{\tau}^{s}\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{a}}^{s}\bigg{]}\\\
&+V_{+-}(\tilde{a}^{+},\tilde{a}^{-}),\end{split}$ (66)

where $dA_{\tau}$ denotes the remnant external field that is not cancelled by
the flux attachment. This remnant field makes $B_{\text{eff}}$ in the two
layers non-zero and pointing in opposite directions. This is true regardless
of whether we describe the layers in terms of CEs or CHs. The additional
effect from the density imbalance in the CEL-CHL Lagrangian appears from the
last term above. However, since it is linear in the gauge fluctuations, it
does not effect the theory and we can omit it from the Lagrangian. Thus, so
far the charge imbalance has the same effect in the CEL-CEL as in the CEL-CHL.
However, in the CEL-CHL theory, we should use CEs for the $\nu=1/2-\delta$
layer and CHs for the $\nu=1/2+\delta$ layer Barkeshli  _et al._ (2015). This
means that the Fermi surface shrinks in each of the two layers by the same
amount. This is not the case in the CEL-CEL where the Fermi surface shrinks in
one layer and expands in the other.