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

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<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### 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.

Introduction.— Dualities constitute a powerful approach in theoretical
physics. Ranging from the Kramers-Wannier duality Kramers and Wannier (1941);
Kogut (1979) to the celebrated AdS/CFT correspondence Maldacena (1999); Witten
(1998), they offer a versatile tool to analyze the full content of a variety
of theories and systems Beekman  _et al._ (2017), including disordered spin
systems and strongly interacting conformal field theories. As dualities in
essence are non-perturbative, they find particular applications in strongly
interacting systems, often mapping strongly coupled field theories to their
dual weakly coupled counterparts Seiberg  _et al._ (2016). As such, they
provide a deeper insight into the novel interplay of topology and strong
correlations in fractional quantum Hall physics Seiberg  _et al._ (2016); Wang
and Senthil (2015); Sodemann  _et al._ (2017).

Quantum Hall bilayers (QHBs) at filling
$\nu_{T}=\nu_{\uparrow}+\nu_{\downarrow}=\frac{1}{2}+\frac{1}{2}$
($\uparrow/\downarrow$ is the upper/lower layer index) are an intriguing
example of strongly correlated physics. At zero layer separation $d$,
interlayer Coulomb interactions ensure that every electron from one layer is
aligned with a hole from the other layer. These macroscopically large numbers
of excitons occupy the unique Halperin (111) state Halperin (1983) to form an
exciton condensate (XC) phase Spielman  _et al._ (2000); Eisenstein and
MacDonald (2004). At $d\gg\ell$, where $\ell$ is the magnetic length
$\sqrt{\hbar/(eB)}$, the two layers decouple and are believed to form two
isolated $\nu=\frac{1}{2}$ composite Fermi liquids (CFLs) Halperin  _et al._
(1993). The phase of the QHB between these two limits is a long-debated topic
with several possibilities discussed in the literature Bonesteel (1993);
Bonesteel  _et al._ (1996); Morinari (1999); Kim  _et al._ (2001); Simon  _et
al._ (2003); Shibata and Yoshioka (2006); Doretto  _et al._ (2006); Zhu  _et
al._ (2017); Lian and Zhang (2018). It was realized quite early that an
effective interlayer attraction exists for composite fermions (CFs) at all
layer separations Bonesteel (1993); Bonesteel  _et al._ (1996). This was later
extended to analytically Cipri and Bonesteel (2014); Isobe and Fu (2017) and
numerically Möller  _et al._ (2008, 2009) show that the dominant BCS pairing
occurs in the $l=1$ angular momentum channel, leading to a $p+ip$ pairing
instability 111In the Supplemental Material, we show that the dominant channel
rather corresponds to $l=-1$. This is, however, only a semantical distinction
and we agree with the results of the aforementioned references..

Alternatively, it was recently hypothesized that in the weak-coupling limit,
the bilayers form $s$-wave pairs of a composite electron liquid (CEL) in one
layer and a composite hole liquid (CHL) in the other layer L.  _et al._
(2022). This point of view is particularly appealing to understand the
experimental signatures of preformed excitons before the onset of the XC phase
L.  _et al._ (2022); Eisenstein  _et al._ (2019), since it creates intuitively
correct quasiparticles in the weak-coupling limit that can be linked to the
strong-coupling XC phase via a BCS-BEC crossover. An excellent numerical
agreement with a trial wave function for an $s$-wave CEL-CHL (CEL-CHL-s)
paired state was also found Wagner  _et al._ (2021). Further, using Eliashberg
theory, it was analytically shown that the $s$-wave is the dominant pairing
channel in the CEL-CHL description Rüegg  _et al._ (2023). Indeed, the BCS-BEC
crossover to the XC phase was argued in Ref. Sodemann  _et al._ (2017), albeit
using the $p+ip$ paired CEL-CEL (CEL-CEL-p) in the weak-coupling limit.

Interestingly, the two weak-coupling scenarios, the CEL-CHL-s state and the
CEL-CEL-p state, possess nearly identical overlap with the exact
diagonalization (ED) ground state Wagner  _et al._ (2021). Furthermore, at
zero temperature, these two paired states have the same pairing strength Rüegg
_et al._ (2023). A recent modification of the Eliashberg theory further showed
that the CEL-CHL-s and CEL-CEL-p pairing strength can be made close to
degenerate, or under the inclusion of a further term exactly degenerate Lotrič
and Simon (2023). These observations suggest a close relation between the CEL-
CEL-p and CEL-CHL-s paired states. As noted before, a topological equivalence
between the weak-coupling CEL-CEL-p state and the strong-coupling Halperin
(111) state has already been argued before Sodemann  _et al._ (2017).
Similarly, is the strong-coupling XC phase dual to the weak-coupling CEL-CHL-s
state? Moreover, how are the two weak-coupling descriptions CEL-CHL-s and CEL-
CEL-p dual to each other, given their different pairing structures? In this
work, we address these questions.

Figure 1: Diagram of the main dualities. Solid lines stand for topological
equivalence, and dashed lines are for particle-hole dualities.

Here, we establish a weak-coupling correspondence between the two pictures
based on field-theoretical arguments. We start with the Dirac CF theory Son
(2015) which contrary to the Halperin-Lee-Read (HLR) theory is manifestly
particle-hole (PH) symmetric Halperin  _et al._ (1993). Nonetheless, in some
cases the low-energy predictions of the HLR theory have been shown to have an
emergent PH symmetry Wang  _et al._ (2017); Cheung  _et al._ (2017),
indicating a deeper connection between these two theories. We consider a
bilayer Dirac CF theory with $s$-wave layer-triplet spinor-singlet pairing. By
introducing a mass in the Dirac particles and taking their non-relativistic
limit, the CF Fermi surface projected paired state is mapped to a CEL-CEL-p
state with layer-triplet pairing. We show that in the bilayer Dirac CF theory
when one of the layers is PH transformed, the pairing term becomes singlet in
the layer pseudospin and triplet in the spinor, while keeping its $s$-wave
structure. Taking the non-relativistic massive limit and projecting over the
CF Fermi surface, we arrive at the CEL-CHL-s state with interlayer-singlet
pairing. Finally, for the CEL-CHL-s state, we write down an effective
topological field theory which is equivalent to the XC theory described by the
Halperin (111) $K$-matrix Wen and Zee (1992, 1993); Sodemann  _et al._ (2017).
A summary of these dualities is presented in Fig. 1. We further present wave
function arguments that circumstantiate these results.

Weak-coupling duality.— Eliashberg calculations under a random-phase
approximation in the CEL-CEL theory show that the leading order dependency of
the pairing strength on the angular momentum channel $l$ goes like $l^{2}-2l$
Isobe and Fu (2017). Hence, the $l+1$ and $-l+1$ channels are degenerate, and
the $l=1$ channel has the strongest pairing. The same calculations in the CEL-
CHL theory show that there the leading order of the pairing strength goes like
$l^{2}$ Rüegg  _et al._ (2023), thus exhibiting an effective time-reversal
symmetry (TRS). As a result, $\pm l$ pairing channels are degenerate, and
$l=0$ leads to the most dominant pairing instability. At zero temperature
(Matsubara frequency $\omega_{m}=0$), the CEL-CEL $l+1/-l+1$ and the CEL-CHL
$\pm l$ channels form a degenerate quadruplet. This indicates a PH-duality for
interlayer-paired ground states, where the pairing symmetry increases by one
when going from the CEL-CHL to the CEL-CEL description.

To establish a duality between the CEL-CEL-p and CEL-CHL-s weak-coupling
scenarios, we take the relativistic Dirac CF theory as our starting point. The
non-relativistic HLR theory Halperin  _et al._ (1993) (denoted as CEL here)
and anti-HLR theory (denoted as CHL) Barkeshli  _et al._ (2015) are not
manifestly PH symmetric Girvin (1984); Kivelson  _et al._ (1997); Rezayi and
Haldane (2000); Barkeshli  _et al._ (2015); Balram and Jain (2016); Geraedts
_et al._ (2016). At the mean-field level at $\nu=\frac{1}{2}$, non-
relativistic CFs are charge-neutral and have a quadratic dispersion; obscuring
the notion of PH symmetry. The relativistic Dirac CF theory is manifestly PH
symmetric Son (2015). The microscopic starting point for the Dirac CF theory
are relativistic Landau levels at half-filling (charge neutrality). The Dirac
CF theory is then derived by using the 2+1D fermionic particle-vortex duality
Seiberg  _et al._ (2016). The CF nature of the emergent quasiparticles can be
seen from the fact that under the particle-vortex transformation particle
density and magnetic field are interchanged. Similar to the HLR theory, the
Dirac CF theory describes a system with non-zero density but a vanishing
magnetic field. Indeed, the HLR Lagrangian can be obtained from the Dirac CF
Lagrangian by introducing a PH-symmetry-breaking mass term and integrating out
the valence band of the massive Dirac spectrum in the limit of large chemical
potential in the conduction band Son (2015); sup .

As discussed in Ref. Sodemann  _et al._ (2017), consider a bilayer of Dirac
CFs with an interlayer BCS pairing of the type
$i\Delta\psi\sigma_{y}\tau_{x}\psi$, where
$\psi=(\psi_{\text{t}},\psi_{\text{b}})$ denotes the Dirac CF fields in the
top/bottom layer, and $\tau_{i}/\sigma_{i}$ are the Pauli matrices acting on
the layer/spinor space. The gap function $\Delta$ is homogeneous with $s$-wave
symmetry. The choice of singlet pairing structure in the internal pseudospin
of the Dirac field requires a triplet structure in the layer pseudospin to
ensure full fermionic anti-symmetry of the CF Cooper pairs. The explicit
Bogoliubov-de Gennes (BdG) Hamiltonian is block-diagonal with two identical
blocks that are

$\mathcal{H}_{\text{BdG}}^{\text{tb}}=\begin{pmatrix}\boldsymbol{k}\cdot\boldsymbol{\sigma}-\mu&-i\Delta\sigma_{y}\\\
i\Delta^{\ast}\sigma_{y}&\boldsymbol{k}\cdot\boldsymbol{\sigma}^{\ast}+\mu\end{pmatrix}$
(1)

in the basis
$(\psi_{\text{t/b}}(\boldsymbol{k}),\psi^{\dagger}_{\text{b/t}}(-\boldsymbol{k}))^{\intercal}$,
and $\mu$ is the chemical potential. Here, we assume that each layer is
exactly at the LL filling factor $\nu=\frac{1}{2}$ and for now ignore the
minimal coupling of the momentum $\boldsymbol{k}$ to the statistical gauge
fields $\boldsymbol{a}_{\text{t/b}}$. Diagonalizing the single-particle part
and then projecting onto the Fermi surface, assuming $\mu\gg|\Delta|$, the
pairing term becomes $\Delta
e^{i\theta_{\boldsymbol{k}}}\,\psi^{\dagger}_{\text{t}+}(\boldsymbol{k})\psi^{\dagger}_{\text{b}+}(-\boldsymbol{k})$,
where $\theta_{\boldsymbol{k}}$ is the polar angle, and $+$ denotes the upper
Dirac eigenstate sup . Therefore the pairing projected over the Fermi surface
acquires a $p_{x}+ip_{y}$ form with a triplet structure in the layer
pseudospin. This is identical to the celebrated Fu-Kane model for chiral
topological superconductivity on the surface of a topological insulator Fu and
Kane (2008). In the present case, the only differences are: (i) the
superconductor describes a pair of CFs, and (ii) it has a double degeneracy
due to the presence of an identical BdG copy under reversing the layer degree.
Now, we can take the non-relativistic limit by introducing a Dirac mass in the
CFs and integrating out the lower single-particle band Son (2015); sup . The
underlying CF theory becomes the conventional HLR theory of CEL, while the
pairing structure remains invariant. As a result, the CEL-CEL-p state is
recovered starting from the $s$-wave paired Dirac CFs.

Now, from the same starting point of paired Dirac CFs, we obtain the CEL-CHL-s
state. For this purpose, we PH transform the bottom layer which has the
following effect on the pairing term sup

$i\Delta\psi\sigma_{y}\tau_{x}\psi\quad\rightarrow\quad
i\Delta\psi\sigma_{0}\tau_{y}\psi.$ (2)

The interlayer BdG Hamiltonian is

$\mathcal{H}_{\text{BdG}}^{\text{tb}}=\begin{pmatrix}\boldsymbol{k}\cdot\boldsymbol{\sigma}-\mu&-\Delta\sigma_{0}\\\
-\Delta^{\ast}\sigma_{0}&\boldsymbol{k}\cdot\boldsymbol{\sigma}^{\ast}+\mu\end{pmatrix}.$
(3)

At the Fermi surface the pairing term becomes
$-\Delta^{\ast}\,\psi^{\dagger}_{\text{t}+}(\boldsymbol{k})\psi^{\dagger}_{\text{b}-}(-\boldsymbol{k})$
sup . We now take the non-relativistic limit by introducing a Dirac mass,
while keeping in mind that the PH-transformation in the bottom layer switches
the sign of the mass term $\bar{\psi}_{\text{b}}\psi_{\text{b}}$. For the non-
relativistic limit, we now have $\psi_{\text{b}-}$ for the Dirac fermions
above the Fermi surface because the PH transformation inverts the Dirac cone
in the bottom layer. Thus, we obtain the desired $s$-wave pairing along with
the underlying CF theory described by the CEL in the top layer and the CHL in
the bottom layer sup . This mechanism can also be understood from the Berry
phase argument. For the Berry curvature
$\mathcal{F}_{\text{b}}(\boldsymbol{k})$ in the bottom layer we have:
$\mathcal{CT}\,\mathcal{F}_{\text{b}}(\boldsymbol{k})\,(\mathcal{CT})^{-1}=-\mathcal{F}_{\text{b}}(-\boldsymbol{k})$.
Upon adding this to the Berry curvature of the top layer particle, the
relative motion degree of freedom acquires no total Berry phase upon
encircling the Dirac cone; contrary to the CEL-CEL case where the $\pi$ Berry
phase can be seen to induce the $p$-wave pairing symmetry.

The weak-coupling duality of the interlayer BCS pairing carries over to higher
angular-momentum symmetries. For bilayer Dirac CFs with interlayer pairing of
the form $\Delta e^{il\theta_{\boldsymbol{k}}}$, where $l$ is the angular
momentum channel, the appropriate pair wave function for even (odd) values of
$l$ is $\Delta e^{il\theta_{\boldsymbol{k}}}\psi i\sigma_{y}\tau_{x(y)}\psi$.
A simple extension of the above BdG calculations shows that projection onto
the Fermi surface for the CEL-CEL case acquires the form $\Delta
e^{i(l+1)\theta_{\boldsymbol{k}}}\psi_{+}\tau_{x(y)}\psi_{+}$ for even (odd)
values of $l$. PH transforming the bottom layer transforms the interlayer
pairing as

$\displaystyle\Delta e^{il\theta_{\boldsymbol{k}}}\psi
i\sigma_{y}\tau_{x(y)}\psi\,$ $\displaystyle\rightarrow\,(-1)^{l}\Delta
e^{il\theta_{\boldsymbol{k}}}\psi i\sigma_{0}\tau_{y(x)}\psi,$ (4)

for even (odd) $l$. This shows the duality between the CEL-CEL $l+1$ and CEL-
CHL $l$ channels. Furthermore, the bilayer Dirac CF theory with one layer PH-
transformed has an effective TRS and thus, the $\pm l$ pairing channels are
degenerate. This proves the degeneracy of the quadruplet of CEL-CEL $l+1/-l+1$
and CEL-CHL $\pm l$.

The above analysis shows a duality between CEL-CEL-p and CEL-CHL-s, which are
two BCS weak-coupling pictures for the quantum Hall bilayers. However, it
relies on obtaining the underlying CEL and CHL theories starting from the
relativistic Dirac CF theory and introducing a PH-symmetry-breaking Dirac
mass. Therefore, more precisely, our analysis shows that if the Dirac CF
theory and the conventional HLR theory are in the same universality class, the
resultant interlayer BCS states CEL-CEL-p and CEL-CHL-s are dual to each
other. Recent developments have indicated that even though the HLR theory does
not have an explicit PH symmetry, its response functions can be PH symmetric,
thus opening a possibility for an emergent PH symmetry Wang  _et al._ (2017);
Cheung  _et al._ (2017). Therefore, the Dirac CF and HLR theory may indeed be
in the same universality class. However, a fully rigours proof of this is
still lacking.

Effective field theory.— We now show a duality between the weak-coupling CEL-
CHL-s and the strong-coupling XC phase. For this purpose, we derive an
effective $K$-matrix in the weak-coupling limit and show its equivalence of
$K$-matrix in the BEC limit Wen and Zee (1992, 1993). Consider the condensate
of Cooper pair bosons made out of the paired CEs and CHs given by the
Lagrangian

$\displaystyle\mathcal{L}=\frac{1}{2M}|iD_{\mu}\phi|^{2}-\frac{\lambda}{4}(\phi^{\dagger}\phi)^{2}-\frac{m^{2}}{2}\phi^{\dagger}\phi-
a_{+}^{\mu}j_{\mu}^{\text{qp}}+\dots.$ (5)

Here, $\phi$ is the CF Cooper pair field representing the CEL-CHL condensate,
$iD_{\mu}=i\partial_{\mu}-2a_{+\mu}$,
$a_{+}=\frac{a_{\text{t}}+a_{\text{b}}}{2}$, and $j^{\text{qp}}$ is the
current of Bogoliubov quasiparticles. Because $a_{+}$ couples similarly to the
minimal coupling to the external magnetic field, it creates vortex excitations
in the condensate. The phase $\varphi$ of the condensate field $\phi$ winds by
$2\pi$ around the vortex. Therefore, in the presence of vortices, we decompose
$\varphi=\varphi_{\text{reg}}+\varphi_{\text{v}}$, where $\varphi_{\text{v}}$
has a singularity at the vortex core. Performing a singular gauge
transformation
$\boldsymbol{a}_{+}\rightarrow\boldsymbol{a}_{+}-\boldsymbol{\nabla}\varphi_{v}/2$
and integrating out $|\phi|$, we obtain

$\displaystyle\mathcal{L}=-\beta_{\mu}(j^{\text{v}\mu}-\frac{1}{\pi}\epsilon^{\mu\nu\gamma}\partial_{\nu}\alpha_{\gamma})+\frac{2m^{2}}{M\lambda}(a_{+\mu}-\alpha_{\mu})^{2}$
$\displaystyle\hskip 56.9055pt-a_{+}^{\mu}j^{\text{qp}}_{\mu}+\dots.$ (6)

Here, the the constraint of the conserved vortex current

$\displaystyle
j^{\text{v}\mu}=\frac{1}{\pi}\epsilon^{\mu\nu\gamma}\partial_{\nu}\alpha_{\gamma},$
(7)

where $\alpha_{\mu}=\partial_{\mu}\varphi_{\text{v}}$, is encoded in the
Lagrange multiplier field $\beta$. Shifting $a_{+}\rightarrow a_{+}+\alpha$
and integrating out the massive $a_{+}$ field, an effective topological field
theory takes the form

$\displaystyle\mathcal{L}_{\text{sc}}=\frac{1}{\pi}\beta
d\alpha-\beta^{\mu}j^{\text{v}}_{\mu}-\alpha^{\mu}j^{\text{qp}}_{\mu}.$ (8)

An $s$-wave BCS condensate has gapped BdG and vortex excitations. The first
term is the BF topological field theory previously discussed for $s$-wave
superconductors Hansson  _et al._ (2004). The BF term encodes the $\pi$-phase
accumulated when a Bogoliubov quasiparticle goes around the vortex or vice
versa.

In a gapped superconductor, the low-energy BdG excitations are an equal
superposition of an electron and a hole, they are almost charge neutral to the
external field Kivelson and Rokhsar (1990). At higher energies, either the
electron or the hole content starts to dominate and the quasiparticles acquire
charge. While the Bogoliubov quasiparticle does not carry a charge current, if
the superconductor is a spin-singlet, it does carry a spin current. Thus, this
leads to a spin-charge decoupling. In the BCS state described by the pairing
of CEL and CHL, the order parameter vortex binds a half-quantized flux of the
$a_{+}$ field. Similar to the superconducting case, the composite BdG
excitations are approximately neutral to $a_{+}$. However, the composite
Bogoliubov quasiparticles have a unit charge with respect to the
$a_{-}=\frac{a_{\text{t}}-a_{\text{b}}}{2}$ field. Based on this observation,
we make the substitutions $\alpha\rightarrow a_{+},\,\beta\rightarrow a_{-}$,
so that the BF term becomes $\frac{1}{\pi}a_{+}da_{-}$. The total topological
field theory is the BF term for the $s$-wave superconductor plus the CS terms
of the two CF fields and the external gauge field

$\displaystyle\mathcal{L}_{\text{top}}=\frac{1}{\pi}a_{+}da_{-}-\frac{1}{4\pi}a_{t}da_{t}+\frac{1}{4\pi}a_{b}da_{b}+\frac{1}{4\pi}AdA.$
(9)

The first three terms combine to exactly cancel to give a gapless $a_{-}$
mode. We recall that the $a_{+}$ gauge field is gapped because of the
Anderson-Higgs mechanism (see Eq. (On dualities of paired quantum Hall bilayer
states at $\nu_{T}=\frac{1}{2}+\frac{1}{2}$)) and there is no remaining CS
term associated with this field. Importantly, the final CS term of the
external field $A$ gives the quantized Hall conductivity associated with
$\nu_{T}=1$. Hence, we obtain the required effective topological field theory
consistent with the $K$-matrix of the strong-coupling XC phase Wen and Zee
(1992).

Wave functions.— As argued in Ref. Sodemann  _et al._ (2017), an $l=1$
interlayer pairing wave function of CFs of the form

$\det\left[\left(\frac{1}{(z^{\uparrow}_{i}-z^{\downarrow}_{j})^{\ast}}\right)_{ij}\right]\,\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})^{2},$
(10)

is proportional (up to amplitude factors in the relative coordinates) to the
Halperin (111) wave function Sodemann  _et al._ (2017); sup . Here
$z^{\uparrow/\downarrow}=x^{\uparrow/\downarrow}+iy^{\uparrow/\downarrow}$ are
the coordinates of the particles in the upper/lower layer. This can be taken
as further evidence for the small-$d$ and large-$d$ limit to be connected
since the location of zeros in the wave function does not change, only how
they are approached.

On the electron-hole side, the $d=0$ wave function already is an $s$-wave
pairing function Yang (2001); sup

$\psi_{\text{XC}}=\det\left[\left(e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/2\ell^{2}}\right)_{ij}\right].$
(11)

If we now just added the Jastrow factors of the CF construction, the resulting
wave function describing the weak-coupling limit would not have the same
structure of zeros as the strong-coupling $\psi_{\text{XC}}$. To remedy this,
we consider

$\displaystyle\begin{split}&\psi_{\text{CEL-
CHL,s}}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast
2}\times\\\ &\det\left[\left(\prod_{k\neq
i}(z^{\uparrow}_{i}-z^{\uparrow}_{k})^{\ast}\prod_{k\neq
j}(w^{\downarrow}_{j}-w^{\downarrow}_{k})\,e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/2\ell^{2}}\right)_{ij}\right].\end{split}$
(12)

In this way of rewriting, the interlayer pairing symmetry of the expression
inside the determinant is not affected by the presence of Jastrow factors.
Therefore, because of the Jastrow factors outside the determinant, this wave
function describes $s$-wave paired CE and CH. Since, the Jastrow factors in
the determinant cancel the ones in front, the phase structure is the same as
for $\psi_{\text{XC}}$ sup .

These arguments should be taken with a grain of caution though. The wave
functions are all implicitly assumed to be projected into the LLL, i.e.
$z^{\ast}\rightarrow\partial_{z},\,w\rightarrow\partial_{w^{\ast}}$, and all
partial derivatives are acting from the left. In general, LLL projection
introduces additional zeros. For example, $\psi_{\text{CEL-CHL,s}}$ and
$\text{P}_{\text{LLL}}\psi_{\text{CEL-CHL,s}}$ do not have the same phase
structure anymore sup . Physically though, crossover from the CEL-CHL-s state
to the XC state has a much simpler interpretation. As the strong-coupling
limit is approached, the momentum-space pairing becomes a real-space pairing,
resulting in $z^{\uparrow}_{i}\rightarrow w^{\downarrow}_{i}$, and thus the
phase of the Jastrow factors gets cancelled L.  _et al._ (2022). This simple
picture of the BCS-BEC crossover, however, is not possible when starting from
the CEL-CEL-p state in the weak-coupling limit.

Discussion and conclusion.— We have shown a weak-coupling duality between the
CEL-CEL-p and CEL-CHL-s states; two possible descriptions for the BCS pairing
state of CFs for widely separated QHBs at $\nu_{T}=\frac{1}{2}+\frac{1}{2}$.
For this duality, we derive the CEL-CEL-p theory from a bilayer Dirac CF
theory with an $s$-wave pairing by projecting the BdG equations onto the Fermi
surface and then taking the non-relativistic limit. PH-transforming one layer
in the bilayer Dirac CF theory, and applying the same BdG projection and non-
relativistic limit procedure, we arrive at the CEL-CHL-s. This weak-coupling
duality generalizes to a PH-duality between CEL-CEL $l+1$ and CEL-CHL $l$
pairing, and coincides with observed numerical Wagner  _et al._ (2021) and
analytical Rüegg  _et al._ (2023); Lotrič and Simon (2023) results. Next, we
derive an effective topological field theory for the CEL-CHL-s state and show
its equivalence to the $K$-matrix theory of the strong-coupling XC phase in
QHBs. This shows that the weak-coupling CEL-CHL-s state can adiabatically
crossover to the strong-coupling XC phase.

We finally speculate how these dualities could help to understand different
aspects of QHBs. The signatures of a BCS-BEC crossover to the XC phase were
reported in recent experiments Eisenstein  _et al._ (2019); L.  _et al._
(2022). In a full numerical many-body calculation as a function of separation
$d$, such a crossover was seen in both the CEL-CEL-p and CEL-CHL-s as the
starting weak-coupling state Wagner  _et al._ (2021). However, many-body
calculations are often difficult to directly connect with the experimental
observables such as conductivity and tunneling spectroscopy. Therefore, a
reliable mean-field theory is highly desirable for such detailed calculations.
At the mean-field level, similar to the well-known BCS-BEC crossover, the CEL-
CHL-s state can crossover to the XC as interlayer coupling is increased as $d$
is decreased. On the other hand, at this mean-field level, the CEL-CEL-p state
will not go through a crossover, but instead will encounter a phase
transition. This can be readily seen by considering the BdG excitation
spectrum of a $p$-wave pairing state. If we associate the BCS-BEC crossover
with the change in the sign of the chemical potential of CFs that undergo
pairing, as the coupling gets stronger and chemical potential reaches the band
bottom, the $p+ip$ excitation spectrum closes the gap, where it encounters a
phase transition before reaching the BEC regime. Therefore, to capture the
crossover from the CEL-CEL-p as the starting point, one must include some
additional corrections related to the background statistical gauge fields. The
exact nature of these corrections is currently unclear. In fact, there is a
theoretical precedence of such extra considerations in the conventional HLR
theory, in showing its compatibility with an emergent PH symmetry Wang  _et
al._ (2017); Cheung  _et al._ (2017). Although the precise nature of the
crossover, especially in the $p$-wave pairing picture, is challenging, it
might profit from the duality as discussed in this work. This shows the
overall importance of our results. Namely, upon the transformation, a rather
implementable theory emerges, which apart from the fundamental nature of the
discussed relations, would aid in further understanding bilayer quantum Hall
systems. Here, we especially reiterate a natural role of our results in the
context of numerical pursuits.

###### Acknowledgements.

Acknowledgements.— We cordially thank Nigel Cooper and Glenn Wagner for
insightful discussions. 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.

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##  Appendix A Sign of the pairing channel

TRS is explicitly broken in the CEL-CEL due to the external gauge field. Even
though the CEs do not feel a net magnetic field anymore, the sign of the CS
term in the HLR theory is dictated by the direction of the magnetic field. In
the Eliashberg theory for the interlayer CEL-CEL pairing, this means that the
$\pm l$ pairing channels are split and that which of the two is favored is
determined by the sign of the CS term Isobe and Fu (2017). At first thought,
it might seem like there is no qualitative difference between the $\pm l$
pairings. However, because the direction of the magnetic field $B$ and the
charge of the particles $q$ already fix the structure of the Jastrow factors,
holomorphic coordinates for $qB>0$ and anti-holomorphic ones for $qB<0$, it
does make a difference whether the pairing wave function has angular momentum
symmetry $+l$ or $-l$. Here, we present arguments that should settle the
confusion about the sign of the pairing channel.

The starting point is to use the correct Schrödinger Lagrangian for charged
particles in an external gauge field. The corresponding equation of motion for
the field of the charged particle $c$ should be

$i\partial_{t}c=\frac{1}{2m}\left(-i\boldsymbol{\nabla}-q\boldsymbol{A}\right)^{2}c+qA_{t}c.$
(13)

We have set $\hbar=c=e=1$, where $e$ is the elementary charge. The Lagrangian
density is thus

$\mathcal{L}=c^{\dagger}\left(i\partial_{t}-qA_{t}\right)c-\frac{1}{2m}c^{\dagger}\left(i\boldsymbol{\nabla}+q\boldsymbol{A}\right)^{2}c.$
(14)

Rewriting this in terms of the CF field $\psi^{\dagger}=c^{\dagger}e^{iq\int
d\boldsymbol{x}^{\prime}\arg(\boldsymbol{x}-\boldsymbol{x}^{\prime})\rho(\boldsymbol{x}^{\prime})}$
gives

$\mathcal{L}=\psi^{\dagger}\left(i\partial_{t}-qA_{t}+qa_{t}\right)\psi-\frac{1}{2m}\psi^{\dagger}\left(i\boldsymbol{\nabla}+q\boldsymbol{A}-q\boldsymbol{a}\right)^{2}\psi-\frac{q}{4\pi}a_{t}\boldsymbol{\nabla}\wedge\boldsymbol{a}.$
(15)

Assume now that $q$ is positive. We know that the interlayer pairing symmetry
in the case of a negative CS term is $l=+1$ Isobe and Fu (2017). A sensible
BCS pairing wave function that has this symmetry is

$\det\left[\left(\frac{1}{(z^{\uparrow}_{i}-z^{\downarrow}_{j})^{\ast}}\right)_{ij}\right].$
(16)

Here we have used holomorphic complex coordinates $z=x+iy$. Playing the same
game with negative $q$, i.e. the actual CEL-CEL case, the CS term is positive,
and the pairing channel is $l=-1$. Importantly though, the wave function still
has the same form since now we are using anti-holomorphic coordinates.

##  Appendix B PH duality

###  B.1 PH transformation in the Dirac CF theory

The Dirac CF theory proposed by Son Son (2015) is the fermionic particle-
vortex dual theory Seiberg  _et al._ (2016) of Dirac electrons in a background
field. Its Lagrangian density is given by

$\mathcal{L}=i\bar{\psi}\left(\mathrlap{\\!\not{\phantom{\partial}}}\partial+2i\mathrlap{\\!\not{\phantom{a}}}a\right)\psi+\frac{1}{2\pi}Ada+\dots,$
(17)

where the $\dots$ stand for non-universal terms. The adopted convention for
the $\gamma$-matrices is

$\gamma^{0}=\sigma_{z},\quad\gamma^{1}=i\sigma_{y},\quad\gamma^{2}=-i\sigma_{x}.$
(18)

PH transformation in the Dirac CF theory amounts to a concatenation of charge
conjugation $\mathcal{C}$ and time reversal $\mathcal{T}$. The transformation
rules for the different fields are

$\displaystyle\mathcal{CT}\psi(t,\boldsymbol{x})(\mathcal{CT})^{-1}$
$\displaystyle=-i\sigma_{y}\psi(-t,\boldsymbol{x}),$ (19)
$\displaystyle\mathcal{CT}a_{0}(t,\boldsymbol{x})(\mathcal{CT})^{-1}$
$\displaystyle=a_{0}(-t,\boldsymbol{x}),\quad\mathcal{CT}a_{i}(t,\boldsymbol{x})(\mathcal{CT})^{-1}=-a_{i}(-t,\boldsymbol{x}),$
(20) $\displaystyle\mathcal{CT}A_{0}(t,\boldsymbol{x})(\mathcal{CT})^{-1}$
$\displaystyle=-A_{0}(-t,\boldsymbol{x}),\quad\mathcal{CT}A_{i}(t,\boldsymbol{x})(\mathcal{CT})^{-1}=A_{i}(-t,\boldsymbol{x}).$
(21)

Thus, the Lagrangian in the above form remains invariant under $\mathcal{CT}$.
Moreover, a mass term $\bar{\psi}\psi$ or a pure CS term $ada$ change sign
under the PH transformation and break this symmetry. While, a chemical
potential term $\bar{\psi}\gamma^{0}\psi$ remains invariant.

###  B.2 CEL as non-relativistic limit

We will quickly repeat how in Ref. Son (2015) the non-relativistic limit of
the Dirac CF theory is taken. One starts by introducing a
$\mathcal{CT}$-breaking mass term $-m\bar{\psi}\psi$ as well as a chemical
potential $\mu\bar{\psi}\gamma^{0}\psi$ to the Lagrangian. If the mass is
large and the chemical potential lies within the gap, we can integrate out
$\psi$ to obtain as an effective action for the gauge field

$-\frac{1}{2\pi}\frac{m}{|m|}ada.$ (22)

If the chemical potential now lies just above the gap, the gapless fermions
are non-relativistic but still feel the effect of the filled Dirac sea. The
appropriate theory then is

$\mathcal{L}=\psi^{\dagger}(i\partial_{t}-2a_{0})\psi-\frac{1}{2|m|}\psi^{\dagger}(i\boldsymbol{\nabla}+2\boldsymbol{a})^{2}\psi-\frac{1}{2\pi}\frac{m}{|m|}ada+\frac{1}{2\pi}Ada.$
(23)

Assuming a negative mass and making the substitution $2a\rightarrow a-A$, the
resulting theory is

$\mathcal{L}=\psi^{\dagger}(i\partial_{t}+A_{0}-a_{0})\psi-\frac{1}{2|m|}\psi^{\dagger}(i\boldsymbol{\nabla}-\boldsymbol{A}+\boldsymbol{a})^{2}\psi+\frac{1}{8\pi}ada-\frac{1}{8\pi}AdA.$
(24)

This is the usual HLR formulation plus an additional CS term for the external
field which is the contribution of the negative energy states to the Hall
conductivity. We should thus add a $\frac{1}{8\pi}AdA$ CS term to correct for
this.

###  B.3 CHL as non-relativistic limit

As explained above, the $\mathcal{CT}$ transformation only changes the sign of
the mass term. Again making the approximation of having a chemical potential
just above the mass gap then gives the effective theory

$\mathcal{L}=\psi^{\dagger}(i\partial_{t}-2a_{0})\psi-\frac{1}{2|m|}\psi^{\dagger}(i\boldsymbol{\nabla}+2\boldsymbol{a})^{2}\psi+\frac{1}{2\pi}\frac{m}{|m|}ada+\frac{1}{2\pi}Ada.$
(25)

Assuming a negative mass and making the substitution $2a\rightarrow A-a$ gives

$\mathcal{L}=\psi^{\dagger}(i\partial_{t}-A_{0}+a_{0})\psi-\frac{1}{2|m|}\psi^{\dagger}(i\boldsymbol{\nabla}+\boldsymbol{A}-\boldsymbol{a})^{2}\psi-\frac{1}{8\pi}ada+\frac{1}{8\pi}AdA.$
(26)

Adding the correction then leaves us with a total CS term of
$\frac{1}{4\pi}AdA$ for the external gauge field. This is the CS term
accounting for the filled LLL in the hole theory Barkeshli and McGreevy
(2012). The resulting theory is thus the CHL.

###  B.4 Pairing and PH transformation

The CEL-CHL is the non-relativistic limit of a massive bilayer Dirac CFL where
one layer is PH-transformed. We choose to apply the PH transformation in the
bottom layer and the total transformation acts as

$\mathcal{CT}_{\text{b}}\psi(\mathcal{CT}_{\text{b}})^{-1}=\begin{pmatrix}\sigma_{0}&0\\\
0&-i\sigma_{y}\end{pmatrix}\psi,$ (27)

where
$\psi=\begin{pmatrix}\psi_{\text{t}}&\psi_{\text{b}}\end{pmatrix}^{\intercal}$.
This transforms the pairing term as

$\mathcal{CT}_{\text{b}}\Delta\psi^{\intercal}\begin{pmatrix}0&i\sigma_{y}\\\
i\sigma_{y}&0\end{pmatrix}\psi(\mathcal{CT}_{\text{b}})^{-1}=\Delta\psi^{\intercal}\begin{pmatrix}\sigma_{0}&0\\\
0&i\sigma_{y}\end{pmatrix}\begin{pmatrix}0&i\sigma_{y}\\\
i\sigma_{y}&0\end{pmatrix}\begin{pmatrix}\sigma_{0}&0\\\
0&-i\sigma_{y}\end{pmatrix}\psi=\Delta\psi^{\intercal}\sigma_{0}i\tau_{y}\psi.$
(28)

##  Appendix C Wave functions

###  C.1 PH transformation in the LLL

For a wave function $\psi_{\text{particles}}(z_{1},\dots,z_{N})$ of $N$
particles in the LLL with a total degeneracy of $M>N$, its PH-conjugate wave
function $\Theta[\psi_{\text{particles}}]$ in terms of $M-N$ holes is Girvin
(1984); Yang (2001)

$\psi_{\text{holes}}(w_{1},\dots,w_{M-N})=\int\prod_{i}^{N}dz_{i}dz_{i}^{\ast}\left[\prod_{i<j}^{N}(z_{i}-z_{j})^{\ast}\prod_{i<j}^{M-N}(w_{i}-w_{j})^{\ast}\prod_{i}^{N}\prod_{j}^{M-N}(z_{i}-w_{j})^{\ast}\right]\psi_{\text{particles}}(z_{1},\dots,z_{N}),$
(29)

where the integration measure is defined as
$dzdz^{\ast}=e^{-|z|^{2}/2\ell^{2}}\frac{dxdy}{2\pi\ell^{2}}$. It is important
to realize that the coordinates of the holes only appear in the form
$w_{i}^{\ast}$. This is exactly what one should expect because the holes and
the particles are oppositely charged and thus their Aharonov-Bohm phases in
the external field are opposite as well, which manifests itself in having
holomorphic coordinates for one and anti-holomorphic coordinates for the
other. The inverse transform, i.e. starting from the hole wave function
$\psi_{\text{holes}}(w_{1},\dots,w_{M-N})$, is

$\psi_{\text{particles}}(z_{1},\dots,z_{N})=\int\prod_{i}^{M-N}dw_{i}dw_{i}^{\ast}\left[\prod_{i<j}^{N}(z_{i}-z_{j})\prod_{i<j}^{M-N}(w_{i}-w_{j})\prod_{i}^{N}\prod_{j}^{M-N}(z_{i}-w_{j})\right]\psi_{\text{holes}}(w_{1},\dots,w_{M-N}).$
(30)

###  C.2 Exciton condensate at $d=0$

We want to derive an electron-hole wave function for the QHB at $d=0$. The
electron-electron wave function describing this phase is the Halperin (111)
wave function Halperin (1983)

$\psi_{111}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{ij}(z^{\uparrow}_{i}-z^{\downarrow}_{j}).$
(31)

Instead of directly trying to transform this wave function, we take the
commonly used XC wave function Yang (2001)

$\psi_{\text{XC}}=\det\left[\left(e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/2\ell^{2}}\right)_{ij}\right],$
(32)

and transform the holes in the bottom layer $w^{\downarrow}$ back into
particles $z^{\downarrow}$. We obtain

$\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\left[\prod_{i<j}^{N}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i<j}^{N}(w^{\downarrow}_{i}-w^{\downarrow}_{j})\prod_{i}^{N}\prod_{j}^{N}(z^{\downarrow}_{i}-w^{\downarrow}_{j})\right]\psi_{\text{XC}}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{ij}(z^{\uparrow}_{i}-z^{\downarrow}_{j}),$
(33)

where because we are at half-filling $M=2N$, and we have made use of the
Bargmann identity Bargmann (1962) for holomorphic functions $f(z)$

$\int dwdw^{\ast}e^{zw^{\ast}/2\ell^{2}}f(w)=f(z).$ (34)

This shows that PH transforming one layer in $\psi_{111}$ leads to
$\psi_{\text{XC}}$. To see that $\psi_{\text{XC}}$ indeed describes electron-
hole pairs, we multiply the modulus squared of one of the entries in the
determinant with its corresponding exponential factors

$\left|e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/2\ell^{2}}\right|^{2}e^{-|z_{i}|^{2}/4\ell^{2}}e^{-|w_{j}|^{2}/4\ell^{2}}=e^{-|z_{i}-w_{j}|^{2}/4\ell^{2}}.$
(35)

To see that $\psi_{\text{XC}}$ has $s$-wave pairing symmetry we look at the
two-particle pairing function
$e^{z^{\uparrow}(w^{\downarrow})^{\ast}/2\ell^{2}}$. In Fig. C.1, we plot the
phase structure of this pairing function where we fixed the hole as
$w^{\downarrow}=-1$ and the electron $z^{\uparrow}$ is the free variable.
Clearly, as we take the electron around the position of the hole, no phase is
picked up.

Figure C.1: Phase structure of
$e^{z^{\uparrow}(w^{\downarrow})^{\ast}/2\ell^{2}}$ with fixed
$w^{\downarrow}=-1$ and $z^{\uparrow}$ as free variable

###  C.3 CF phase at large-$d$

Coming from the $d\rightarrow\infty$ limit where the two layers are entirely
decoupled, they are described by CFLs. The bilayer CFL has weak-coupling BCS
instabilities: in the $p$-wave channel for a CEL-CEL description Isobe and Fu
(2017), and in the $s$-wave channel for a CEL-CHL Rüegg  _et al._ (2023). A
wave function that captures the CE-CE pairing with $l=1$ angular-momentum
symmetry can be written as Sodemann  _et al._ (2017):

$\psi_{\text{CEL-
CEL,p}}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})^{2}\,\det\left[\left(\frac{1}{(z^{\uparrow}_{i}-z^{\downarrow}_{j})^{\ast}}\right)_{ij}\right].$
(36)

The Jastrow factors in front capture the CF nature and the determinant
incorporates the interlayer two-particle pairing function. Using the
determinant of the Cauchy matrix

$\det\left[\left(\frac{1}{(z^{\uparrow}_{i}-z^{\downarrow}_{j})^{\ast}}\right)_{ij}\right]=\prod_{i,j}\frac{1}{(z^{\uparrow}_{i}-z^{\downarrow}_{j})^{\ast}}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow})^{\ast}\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})^{\ast},$
(37)

the wave function can be rewritten as

$\psi_{\text{CEL-
CEL,p}}=\prod_{i,j}\frac{1}{|z^{\uparrow}_{i}-z^{\downarrow}_{j}|^{2}}\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}|^{2}\prod_{i<j}|z^{\downarrow}_{i}-z^{\downarrow}_{j}|^{2}\,\psi_{111}.$
(38)

From this, as well as field theory calculations, it can be argued that the
large-$d$ phase is topologically connected with the small-$d$ XC. Indeed,
Sodemann et al. Sodemann  _et al._ (2017) hint at the possibility of a BCS-BEC
crossover.

In the same spirit, we consider a wave function for the CE-CH pairing with
$l=0$ angular-momentum symmetry

$\psi_{\text{CEL-
CHL,s}}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast
2}\det\left[\left(\prod_{k\neq
i}(z^{\uparrow}_{i}-z^{\uparrow}_{k})^{\ast}\prod_{k\neq
j}(w^{\downarrow}_{j}-w^{\downarrow}_{k})\,e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/2\ell^{2}}\right)_{ij}\right].$
(39)

We realize that from the $i$th row in the determinant we can extract the
factor $\prod_{k\neq i}(z^{\uparrow}_{i}-z^{\uparrow}_{k})^{\ast}$, and
analogously for $\prod_{k\neq j}(w^{\downarrow}_{j}-w^{\downarrow}_{k})$ in
the $j$th column. Since each Jastrow factor
$(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{\ast}$ features in rows $i$ and $j$, we
can write

$\psi_{\text{CEL-
CHL,s}}=\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}_{j}|^{4}\prod_{i<j}|w^{\downarrow}_{i}-w^{\downarrow}_{j}|^{4}\,\psi_{\text{XC}}.$
(40)

Hence, we obtain wave functions with the same phase structure for the large-
and the small-$d$ limit.

To derive a practical trial wave function from it, we first need to project it
into the LLL. This is done by setting
$z^{\ast}\rightarrow\partial_{z},\,w\rightarrow\partial_{w^{\ast}}$ and then
taking all the partial derivatives to act on the rest of the wave function. We
get

$\text{P}_{\text{LLL}}\psi_{\text{CEL-
CHL,s}}=\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})^{2}\prod_{i<j}(\partial_{(w^{\downarrow}_{i})^{\ast}}-\partial_{(w^{\downarrow}_{j})^{\ast}})^{2}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast
2}\det\left[\left(e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/2\ell^{2}}\right)_{ij}\right].$
(41)

We evaluate this wave function for two particles in the top layer and two
holes in the bottom layer. The comparison of the resulting phase structure
with the one of $\psi_{111}$ is in Fig. C.2. There are two additional zeros
close to the fermionic intralayer zero on the real axis. The introduction of
new zeros in LLL-projected wave functions is a general feature.

Figure C.2: Phase structures of (left)
$\psi_{\text{111}}(z^{\uparrow}_{1},z^{\uparrow}_{2}=2,w^{\downarrow}_{1}=-1,w^{\downarrow}_{2}=-2)$
and (right) $\text{P}_{\text{LLL}}\psi_{\text{CEL-
CHL,s}}(z^{\uparrow}_{1},z^{\uparrow}_{2}=2,w^{\downarrow}_{1}=-1,w^{\downarrow}_{2}=-2)$

###  C.4 PH transformation of $\psi_{\text{CEL-CHL,s}}$

Next, we show our proposed wave function’s structure in the electron-electron
basis. For this, we apply a PH transformation in the bottom layer.

$\displaystyle\begin{split}&(\Theta_{\text{PH}}^{\downarrow})^{-1}\left[\text{P}_{\text{LLL}}\psi_{\text{CEL-
CHL,s}}\right]\\\
=&\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\,\text{P}_{\text{LLL}}\psi_{\text{CEL-
CHL,s}}(z^{\uparrow}_{1},\dots,z^{\uparrow}_{N},w^{\downarrow}_{1},\dots,w^{\downarrow}_{N})\,\psi_{\text{filled
LLL}}(w^{\downarrow}_{1},\dots,w^{\downarrow}_{N},z^{\downarrow}_{1},\dots,z^{\downarrow}_{N})\\\
=&\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})^{2}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\,\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\,\prod_{i<j}(\partial_{(w^{\downarrow}_{i})^{\ast}}-\partial_{(w^{\downarrow}_{j})^{\ast}})^{2}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast
2}\\\
&\sum_{\sigma}\,\text{sign}(\sigma)\prod_{i}e^{z^{\uparrow}_{\sigma_{i}}(w^{\downarrow}_{i})^{\ast}/2\ell^{2}}\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i,j}(z^{\downarrow}_{i}-w^{\downarrow}_{j})\\\
=&\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})^{2}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})^{2}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{3}\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i,j}(z^{\downarrow}_{i}-z^{\uparrow}_{j}),\end{split}$
(42)

where we have eliminated the sum over all permutations by accordingly
permuting all the $w^{\downarrow}_{i}$ and since the total function is anti-
symmetric this also eliminates the $\,\text{sign}(\sigma)$, and we have again
used the Bargmann identity for holomorphic functions. The commuting properties
of the LLL projection and PH transformation can be checked by operating them
in the opposite order to the above and obtain the same result:

$\displaystyle\begin{split}&\text{P}_{\text{LLL}}(\Theta_{\text{PH}}^{\downarrow})^{-1}\left[\psi_{\text{CEL-
CHL,s}}\right]\\\
=&\text{P}_{\text{LLL}}\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\,\psi_{\text{CEL-
CHL,s}}(z^{\uparrow}_{1},\dots,z^{\uparrow}_{N},w^{\downarrow}_{1},\dots,w^{\downarrow}_{N})\,\psi_{\text{filled
LLL}}(w^{\downarrow}_{1},\dots,w^{\downarrow}_{N},z^{\downarrow}_{1},\dots,z^{\downarrow}_{N})\\\
=&\text{P}_{\text{LLL}}\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}_{j}|^{4}\,\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\,\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{2}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast
2}\\\
&\sum_{\sigma}\,\text{sign}(\sigma)\prod_{i}e^{z^{\uparrow}_{\sigma_{i}}(w^{\downarrow}_{i})^{\ast}/2\ell^{2}}\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i,j}(z^{\downarrow}_{i}-w^{\downarrow}_{j})\\\
=&\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})^{2}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})^{2}\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{3}\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i,j}(z^{\downarrow}_{i}-z^{\uparrow}_{j}).\end{split}$
(43)

The resulting wave function is not layer-symmetric. This can easily be
resolved by taking a layer-symmetric combination

$(\Theta^{\downarrow})^{-1}[\psi_{\text{CEL-
CHL,s}}]+(\Theta^{\uparrow})^{-1}[\psi_{\text{CHL-CEL,s}}].$ (44)

Such a layer-symmetrization procedure has also been used in Ref. Lian and
Zhang (2018).

In Fig. C.3, we plot the phases of the LLL-projected PH transformed CEL-CHL
$s$-wave on the left, and we compare it with
$\text{P}_{\text{LLL}}\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}_{j}|^{4}\,\psi_{111}$
in the middle and
$\text{P}_{\text{LLL}}\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}_{j}|^{2}\prod_{i<j}|z^{\downarrow}_{i}-z^{\downarrow}_{j}|^{2}\prod_{i,j}|z^{\uparrow}_{i}-z^{\downarrow}_{j}|^{2}\psi_{\text{CEL-
CEL,p}}$ on the right; the latter corresponds to the $n=2,m=4$ variational
factor in Ref. Sodemann  _et al._ (2017). All wave functions have the
fermionic intralayer zeros, as well as something close to the interlayer zeros
of the Halperin (111) wave function. They also have the identical number of
characteristic additional zeros of LLL-projected wave functions.

Figure C.3: Phase structures of (left)
$\text{P}_{\text{LLL}}(\Theta_{\text{PH}}^{\downarrow})^{-1}\left[\psi_{\text{CEL-
CHL,s}}\right](z^{\uparrow}_{1},z^{\uparrow}_{2}=2,w^{\downarrow}_{1}=-1,w^{\downarrow}_{2}=-2)$,
(middle)
$\text{P}_{\text{LLL}}\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}_{j}|^{4}\,\psi_{\text{111}}(z^{\uparrow}_{1},z^{\uparrow}_{2}=2,w^{\downarrow}_{1}=-1,w^{\downarrow}_{2}=-2)$
and (right)
$\text{P}_{\text{LLL}}\prod_{i<j}|z^{\uparrow}_{i}-z^{\uparrow}_{j}|^{2}\prod_{i<j}|z^{\downarrow}_{i}-z^{\downarrow}_{j}|^{2}\prod_{i,j}|z^{\uparrow}_{i}-z^{\downarrow}_{j}|^{2}\psi_{\text{CEL-
CEL,p}}$

###  C.5 CB phase at intermediate $d$

Ref. Lian and Zhang (2018) proposed an intermediate CB XC phase. Due to the
reduced intralayer effective charge of the CBs the interlayer CB excitons can
become charge-neutral around $d\approx l$. Thus, these CB electron-hole bound
states can then condense. The wave function should be the product of the CB
Jastrow factors and the bosonic condensate part. In Ref. Lian and Zhang
(2018), it was argued that the natural form of the condensate wave function
would be a permanent, the totally symmetric version of the determinant. Thus,

$\psi_{\text{CB
XC}}^{(1)}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast}\text{perm}\left[\left(e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/4\ell^{2}}\right)_{ij}\right].$
(45)

This wave function also shows peaked overlap with ED calculations at $d\approx
l$.

Motivated by our method of absorbing conjugated Jastrow factors into the
pairing function, we propose another wave function for the intermediate CB XC
phase

$\psi_{\text{CB
XC}}^{(2)}=\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast}\,\det\left[\left(\prod_{k>i}(z^{\uparrow}_{i}-z^{\uparrow}_{k})^{\ast}\prod_{k>j}(w^{\downarrow}_{j}-w^{\downarrow}_{k})\,e^{z^{\uparrow}_{i}(w^{\downarrow}_{j})^{\ast}/4\ell^{2}}\right)_{ij}\right].$
(46)

Every Jastrow factor $(z^{\uparrow}_{i}-z^{\uparrow}_{j})$ /
$(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast}$ now only appears in one
row/column which is why there is no double-counting this time. Thus, the
determinant is a bosonic pairing function. This wave function again has the
same phase structure as $\psi_{\text{XC}}$. In Fig. C.4 we compare the phase
structures of $\psi_{\text{XC}}$ on the left, $\psi_{\text{CB XC}}^{(1)}$ in
the middle, and $\text{P}_{\text{LLL}}\,\psi_{\text{CB XC}}^{(2)}$ on the
right. It is interesting to see that the zeros in all three wave functions are
at similar positions, and that the two CB wave functions have almost identical
phase structures.

Figure C.4: Phases of the wave functions (left)
$\psi_{\text{XC}}(z^{\uparrow}_{1},z^{\uparrow}_{2}=2,z^{\downarrow}_{1}=-1,z^{\downarrow}_{2}=-2)$,
(middle) $\psi_{\text{CB
XC}}^{(1)}(z^{\uparrow}_{1},z^{\uparrow}_{2}=2,z^{\downarrow}_{1}=-1,z^{\downarrow}_{2}=-2)$
and (right) $\text{P}_{\text{LLL}}\,\psi_{\text{CB
XC}}^{(2)}(z^{\uparrow}_{1},z^{\uparrow}_{2}=2,z^{\downarrow}_{1}=-1,z^{\downarrow}_{2}=-2)$

###  C.6 PH transformation of $\psi_{\text{CB XC}}^{(2)}$

Next, we apply PH transformation in the bottom layer to see the form of
$\psi^{(2)}_{\text{CB XC}}$ in the electron-electron basis.

$\displaystyle\begin{split}&(\Theta_{\text{PH}}^{\downarrow})^{-1}\left[\text{P}_{\text{LLL}}\psi_{\text{CB
XC}}^{(2)}\right]\\\
=&\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\,\text{P}_{\text{LLL}}\psi_{\text{CB
XC}}^{(2)}(z^{\uparrow}_{1},\dots,z^{\uparrow}_{N},w^{\downarrow}_{1},\dots,w^{\downarrow}_{N})\,\psi_{\text{filled
LLL}}(w^{\downarrow}_{1},\dots,w^{\downarrow}_{N},z^{\downarrow}_{1},\dots,z^{\downarrow}_{N})\\\
=&\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})\,\int\prod_{i}^{N}dw^{\downarrow}_{i}d(w^{\downarrow}_{i})^{\ast}\,\prod_{i<j}(\partial_{(w^{\downarrow}_{i})^{\ast}}-\partial_{(w^{\downarrow}_{j})^{\ast}})(w^{\downarrow}_{i}-w^{\downarrow}_{j})^{\ast}\\\
&\sum_{\sigma}\,\text{sign}(\sigma)\prod_{i}e^{z^{\uparrow}_{\sigma_{i}}(w^{\downarrow}_{i})^{\ast}/4\ell^{2}}\prod_{i<j}(w^{\downarrow}_{i}-w^{\downarrow}_{j})\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i,j}(z^{\downarrow}_{i}-w^{\downarrow}_{j})\\\
=&\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})\prod_{i<j}(\partial_{z^{\uparrow}_{i}}-\partial_{z^{\uparrow}_{j}})\prod_{i<j}(z^{\uparrow}_{i}-z^{\uparrow}_{j})^{2}\prod_{i<j}(z^{\downarrow}_{i}-z^{\downarrow}_{j})\prod_{i,j}(z^{\downarrow}_{i}-2^{-1}z^{\uparrow}_{j}),\end{split}$
(47)

where the additional factor of $2^{-1}$ appears in the interlayer terms
because we had to use the modified Bargmann identity

$\int dwdw^{\ast}e^{zw^{\ast}/4\ell^{2}}f(w)=f(z/2).$ (48)

In Fig. C.5, we compare the phase structures of the PH transformed versions of
the CB XC wave functions in Ref. Lian and Zhang (2018) and the one proposed
here. They again look almost identical. Only the position of the rightmost
zero differs a little.

Figure C.5: Phases of the wave functions (left)
$(\Theta_{\text{PH}}^{\downarrow})^{-1}[\psi_{\text{CB
XC}}^{(1)}](z^{\uparrow}_{1},z^{\uparrow}_{2}=2,z^{\downarrow}_{1}=-1,z^{\downarrow}_{2}=-2)$
and (right) $(\Theta_{\text{PH}}^{\downarrow})^{-1}[\psi_{\text{CB
XC}}^{(2)}](z^{\uparrow}_{1},z^{\uparrow}_{2}=2,z^{\downarrow}_{1}=-1,z^{\downarrow}_{2}=-2)$