# Bilayer WSe2 as a natural platform for interlayer exciton condensates in the
strong coupling limit

Qianhui Shi  Department of Physics, Columbia University, New York, NY, USA
En-Min Shih  Department of Physics, Columbia University, New York, NY, USA
Daniel Rhodes  Department of Mechanical Engineering, Columbia University, New
York, NY, USA  Bumho Kim  Department of Mechanical Engineering, Columbia
University, New York, NY, USA  Katayun Barmak  Department of Applied Physics
and Applied Mathematics, Columbia University, New York, NY, USA  Kenji
Watanabe  Research Center for Functional Materials, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Zlatko Papić  School
of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK  Dmitry A.
Abanin  Department of Theoretical Physics, University of Geneva,24 quai
Ernest-Ansermet, 1211 Geneva, Switzerland  James Hone  Department of
Mechanical Engineering, Columbia University, New York, NY, USA  Cory R. Dean
Department of Physics, Columbia University, New York, NY, USA

Exciton condensates (EC) are macroscopic coherent states arising from
condensation of electron-hole pairs blatt:1962 . Bilayer heterostructures,
consisting of two-dimensional electron and hole layers separated by a tunnel
barrier, provide a versatile platform to realize and study EC lozovik:1975b ;
snoke:2002 ; eisenstein:2004 . The tunnel barrier suppresses recombination
yielding long-lived excitons butov:2002 ; eisenstein:2014 ; liu:2017 ;
li:2017a ; wang:2019 ; ma:2021 . However, this separation also reduces
interlayer Coulomb interactions, limiting the exciton binding strength. Here,
we report the observation of EC in naturally occurring 2H-stacked bilayer
WSe2. In this system, the intrinsic spin-valley structure suppresses
interlayer tunneling even when the separation is reduced to the atomic limit,
providing access to a previously unattainable regime of strong interlayer
coupling. Using capacitance spectroscopy, we investigate magneto-EC, formed
when partially filled Landau levels (LL) couple between the layers. We find
that the strong-coupling EC show dramatically different behaviour compared
with previous reports, including an unanticipated variation of EC robustness
with the orbital number, and find evidence for a transition between two types
of low-energy charged excitations. Our results provide a demonstration of
tuning EC properties by varying the constituent single-particle wavefunctions.

Narrow-gap semiconductors or semi-metals were originally proposed as natural
hosts for EC due to a spontaneous paring instability between the electrons and
holes, provided that the Coulomb interaction between them exceeds the band gap
halperin:1968 . Signatures of EC have been identified in several such
candidate systems lu:2017 ; kogar:2017 ; zli:2019 ; kim:2020 ; jia:2020 ;
ataeie:2021 , though debates are still ongoing due to challenges to
disentangle lattice-related effects baldini:2020 ; kim:2020 ; watson:2020 ;
mazza:2020 . In another direction, EC have been explored in layered structures
where electrons and holes are spatially confined to separate layers
lozovik:1975 ; butov:2004 ; min:2008 ; fogler:2014 . These so-called spatially
indirect excitons can be generated by either optical excitation or electrical
gating snoke:2002 ; jauregui:2019 ; ma:2021 . The spatial separation of the
electrons and holes inhibits recombination and therefore extends the exciton
lifetime, making it possible to observe the EC under equilibrium conditions
ma:2021 .

Quantum Hall bilayers provide a robust model platform to study the spatially
indirect EC eisenstein:2004 ; eisenstein:2014 ; li:2017a ; liu:2017 . In these
systems, a heterostructure consisting of electrically-isolated parallel layers
is exposed to a perpendicular magnetic field. Interlayer excitons can then
form between partially filled LLs with filled (electron) states in one layer
coupling to vacancy (hole) states in the other layer. A key advantage of this
approach is that, within the flat LLs, kinetic energy is quenched and Coulomb
interactions necessarily play a dominant role. Quantum Hall bilayers
fabricated from GaAs double wells provided the first direct evidence of
exciton superfluidity nandi:2012 ; eisenstein:2014 , appearing when each layer
was tuned to half filling of the lowest Landau level. More recently,
superfluid EC have been studied in graphene heterostructures, consisting of
two graphene layers separated by a boron nitride tunnel barrier, where
increased flexibility in device architecture has expanded the accessible phase
space liu:2017 ; li:2017a .

Figure 1:  Independent layer population in bilayer WSe2. (a) Schematic of WSe2
bilayer crystal structure. (b) Schematic illustration of the four flavors in
the valence band in bilayer WSe2. (c) Schematic illustration of the
penetration capacitance measurements. (d) Penetration capacitance of bilayer
WSe2 measured at $B=0$ and $T=0.3$ K, versus $\nu_{tot}$ and $D$. (e)
Schematic phase diagram of layer population according to features in (c). (f)
Penetration capacitance measured at $B=14.9$ T and $T=0.3$ K, versus
$\nu_{tot}$ and $D$. (g) Schematic illustration of features shown in (e). The
filling factors in each layers are marked as $(\nu_{T},\nu_{B})$ for the
integer gaps.

The EC phase diagram is determined by the interplay between the interlayer
Coulomb attraction, $E_{inter}\propto 1/d$, where $d$ is the layer separation,
and intralayer Coulomb repulsion, $E_{intra}\propto 1/l_{B}$, where
$l_{B}=\sqrt{\hbar/eB}$ is the magnetic length eisenstein:2014 . Whereas
$l_{B}$ can be widely varied with $B$, $d$ is more restricted since it must be
small enough to promote strong coupling between the layers, at the same time
remaining large enough to suppress interlayer tunneling. In GaAs bilayers,
$d>10$ nm, and the EC are measurable only in the so-called weak coupling limit
($d/l_{B}\sim 1$) eisenstein:2014 . Recent efforts in graphene double layers
demonstrated the ability to access the strong coupling regime ($d/l_{B}<0.5$),
by reducing the layer separation to just a few nanometers liu:2020 . Owing to
the inability to further reduce $d$ without introducing appreciable tunneling,
the regime of extreme strong coupling remains almost completely unexplored
kim:2021 .

Figure 2:  EC formation and gap opening at LL crossings. (a-c) Penetration
capacitance versus total filling factor $\nu_{tot}$ and displacement field $D$
at $B=14.9$ T around several LL crossings. The bottom panel are linecut along
the red dashed line in the top panel, which tracks equal topmost LL population
in the two layers. The insets of the bottom panel illustrate the orbital, spin
and valley indices.

Here we identify the natural bilayer WSe2 as a system that provides a unique
opportunity to realize EC in the previously inaccessible regime of extreme
strong interlayer coupling. In this case, interlayer tunneling is naturally
suppressed via a spin-blocking mechanism, due to combination of the spin-
valley locking and stacking order. This eliminates the need to insert a
physical tunnel barrier, thereby allowing the layer separation to be reduced
to the atomic limit. Previously, in the weak coupling regime, EC has only been
observed in the lowest LL ($n=0$). Remarkably, here we observe EC in higher
LLs up to $n=6$. Our data indicates that in the strong coupling regime the
low-energy charged excitations have a different nature in different LLs,
rendering the EC more robust in high LLs than in the lowest LL.

Fig. 1(a) shows a schematic of the crystal structure for 2H-stacked bilayer
WSe2, consisting of two monolayers rotated 180 degrees to each other. In
Fig.1(b) we sketch the low-energy valence bands, with the valley and spin
indices indicated. Strong spin-orbit coupling gives rise to spin-valley
locking within each layer, with the K and K’ valleys oppositely spin-
polarized. Due to the stacking order, carriers residing in the same valley
have opposite spin in the two layers, leading to strongly suppressed
interlayer tunneling (Gong  _et al._ , 2013; zeng:2013, ; xu:2014, ;
fallahazad:2016, ; pisoni:2019, ; Pisoni  _et al._ , 2019) (also see SI.1).

In our experiment, we measure the penetration capacitance $C_{P}$, illustrated
in Fig.1(c), which has proven to be an effective probe that circumvents non-
ideal electrical contact and disorder effects in 2D semiconducting transition
metal dichalcogenides Shi  _et al._ (2019). While $C_{P}$ reflects the inverse
compressibility of a monolayer system, for a multi-layer system it has an
additional contribution from the polarizability. The penetration capacitance
normalized to the geometric capacitance between the top and bottom gates
($C_{G}$) can be written as young:2011 ; zibrov:2017 ,

$\frac{C_{P}}{C_{G}}=\frac{2c\partial\mu/\partial
n}{e^{2}+2c\partial\mu/\partial
n}+\frac{ec}{4\epsilon_{0}c_{0}}\frac{\partial\Delta n}{\partial D}.$ (1)

Here, $c$ is the capacitance (per unit area) between the bilayer WSe2 and the
gates, $c_{0}$ is the interlayer capacitance (per unit area) of the bilayer
WSe2, $\partial\mu/\partial n$ is the inverse electronic compressibility where
$\mu$ is the chemical potential and $n$ is the total carrier density, $\Delta
n=n_{T}-n_{B}$ is the carrier density imbalance of the two layers, and
$D=c(V_{B}-V_{T})/2\epsilon_{0}$ is the displacement electric field on the
bilayer. The two terms in the right hand side of Eq. (1) correspond to
“incompressibility” and “polarizability” contributions, respectively; a large
signal in $C_{P}$ indicates that the system is either incompressible or highly
polarizable.

We first demonstrate that we can achieve layer-selective population in bilayer
WSe2 through gate control. In Fig.1(d) we plot the penetration capacitance
versus the displacement field and total density. For a fixed carrier density,
the electron polarization can be tuned through three regions, corresponding to
the bottom, top or both layer populated. The transition from full layer
polarization to partial polarization is visible as a step in the
$C_{P}/C_{G}$, owing to a finite polarizability contribution in the latter.
This step follows a curved shape that defines the polarization boundaries in
the space of $D$ vs $n$ (see SI.3).

The layer-selective population becomes more apparent as a perpendicular
magnetic field is applied, and the energy spectrum splits into discrete LLs.
In Fig.1(e) we plot $C_{P}/C_{G}$ as a function of $D$ and total filling
factor $\nu_{tot}=\nu_{T}+\nu_{B}$, where $\nu_{T}$ and $\nu_{B}$ are the top
and bottom layer LL filling fractions, respectively. Horizontal features are
observed at integer values of $\nu_{tot}$. These peaks in $C_{P}$ indicate
incompressibility when the Fermi level is within an integer quantum Hall (IQH)
gap. In the bilayer populated region, each feature is interrupted by exactly
$\nu_{tot}$ vertical features, as schematically illustrated in Fig.1(g). This
is consistent with discrete filling of the LLs in each layer: with increasing
$D$, $\nu_{T}$ increases step-wise from $0$ to $\nu_{tot}$, and $\nu_{B}$
decreases from $\nu_{tot}$ to $0$. Interlayer charge transfer is allowed only
when the displacement field induces crossings of LLs in the two layers. At the
crossing points, the bilayer system manifests increased polarizability at both
integer and non-integer fillings, giving rise to the vertical features in
Fig.1(e) (see SI.4 for more details).

Figure 3:  Conditions for exciton condensate formation. (a) Penetration
capacitance versus total filling factor $\nu_{tot}$ and displacement field $D$
at $B=14.9$ T. Solid (dotted) circles mark the crossings where the two LLs
have the same orbital number $n=n_{T}=n_{B}>0$ ($n=n_{T}=n_{B}=0$). (b)
Schematic LLs diagram. As $D$ is varied, LLs in the bottom layer increases in
energy and those of the top layer increases, giving rise to LL crossings. Each
layer hosts LLs from two spin branches, with a large spin-splitting energy
several times of the cyclotron gap. LLs from 4 different flavors are marked by
different colors and the orbital number is also marked. (c) Penetration
capacitance at the layer balanced condition $\nu_{T}=\nu_{B}$, as a function
of the filling factor and magnetic field. (d) Schematic illustration according
to (c), with yellow lines standing for the IQH gap, and the green lines
standing for the EC gap. Orbital numbers are marked for different regions for
$\nu_{T}=\nu_{B}=5.5$. (e)Cartoon illustrating the LL structure at low and
high densities respectively. The black arrow points to the Fermi level at
$\nu=5.5$, where active LLs have orbital number $n=5$ and $n=0$, respectively.

Next, we examine the penetration capacitance at the LL crossings. Fig. 2 shows
representative examples of the three types of behaviour we observe. At the
center of each $\nu_{tot}-D$ map, a light blue diamond area demarcates the
region where both layers host fractional filling factors and the system has
finite polarizability. Fig.2(a) and (b) show two examples where the gap
closes, leading to a disappearance of the sharp horizontal feature at integer
$\nu_{tot}$. This behavior is seen for most LL crossings, and is consistent
with the single-particle picture of LL crossings - the bilayer system is
compressible since both layers are partially filled. The gap closing in (a)
and (b) also confirms the suppression of interlayer tunneling, which would
otherwise induce LL anti-crossings and preserve the gap. Fig.2(c), on the
contrary, demonstrates an example where the gap persists throughout the
fractional filling region. The two scenarios regarding the presence or absence
of a gap at the level crossings are clearly distinguishable in linecuts taken
through the center of the fractional-filling region (bottom panels): a sharp
spike in $C_{P}$ is seen at integer $\nu_{tot}$ for (c), but is absent in (a)
and (b). Having established that the two layers are tunnel-decoupled, we
interpret the incompressible states while both layers are partially filled in
(c) as interlayer correlated states, i.e. exciton condensates.

In Fig.3(a) we plot $C_{P}$ as a function of $\nu_{tot}$ and $D$ over a broad
range, and mark the location of EC by solid circles using the criteria
established in Fig.2. Remarkably, EC appear for a large range of LLs - far
beyond what has been previously observed - both where the layers are balanced
(around $D=0$) and where they are very imbalanced (at large $D$). Fig.3(b)
shows the corresponding schematic LL diagram, which identifies the LL orbital
and spin/valley indices (see SI.2). We note first that all of the EC
correspond to states with matched orbital number $n$. The series of EC at
$D=0$ (with Fig.2(c) as an example) corresponds to the crossings of two LLs
with the same orbital number and spin indices but different valley index;
while those at finite $D$ correspond to matched orbital numbers and valley
indices but different spins (see SI.5).

Figure 4:  Different types of charged excitations for different LLs. (a)
Measured excitation gap when the layers are balanced ($D$ = 0 and LLs in both
layer are half-filled), at $B$ = 14.9 T which corresponds to $d/l_{B}$ = 0.1.
Error bar for $n=0$ reflects the uncertainty in distinguishing the EC gap at
density balance from that at density imbalance or the IQH gap. Error bars for
the rest represent the uncertainty in separating the stray capacitance or the
polarizability contribution. Error bars are not shown when they are smaller
than the symbol. (b) Theoretical estimate of charge gap values for layer
balanced condition, for two types of excitations, at $d/l_{B}=0.1$. The
excitation energy of layer-pseudospin texture increases with $n$, while those
for particle-hole excitations decreases with $n$. The filled markers stand for
the lowest-energy excitations, which changes from one type to the other as $n$
varies. (c) $C_{P}/C_{G}$ as a function of $D$, at constant total filling
factor $\nu_{tot}$ = 1, 3, 5, 7, as marked, while the topmost LLs has orbital
numbers $n$ = 0, 1, 2, 3, respectively. A small background due to the
polarizability contribution has been subtracted according to $C_{P}$ at
adjacent non-integer values. The shades highlight the $D$-range where the both
layers are partially filled and charge is being transferred interlayer, while
in the non-shaded regions both layers are at integer fillings as marked and
the system exhibits an IQH gap. (d) Numerically calculated upper limit
$d/l_{B}$ for EC formation. Experimentally accessible ranges of $d/l_{B}$ for
different systems are marked on the right.

Our observations indicate that EC formation requires the orbital wavefunctions
to match, but is independent of the spin/valley degree of freedom. Likewise,
our Hartree-Fock and numerical calculations suggest that formation of EC,
characterized by a good overlap with the Halperin (111) wavefunction and a
gapless neutral mode, only appears when the LLs in the two layers have matched
orbital numbers (see SI.10 and SI.11). The consistency with theory further
confirms the interpretation of the observed incompressible states as EC. In
addition, our observation is consistent with previous experiments in graphene
double-layers li:2017a ; li:2019  which demonstrated that spin and valley
degrees of freedom are largely irrelevant in the formation of EC in the lowest
LL. Intriguingly, the specific orbital number also plays a critical role in EC
formation: EC formation is observed for $n=1-6$, but is suppressed for $n=0$,
as shown by the dashed circles in Fig.3 (a) and (b) (see SI.5 and SI.6). This
is precisely opposite to previous studies in double-layer systems with larger
interlayer spacing eisenstein:2014 ; li:2017a ; liu:2017 , where the EC have
only been observed for $n=0$.

The critical role of the orbital number is further corroborated by the
evolution of specific EC with magnetic field. This is displayed in Fig.3(c),
which plots $C_{P}$ versus the magnetic field and filling factor at the
balanced condition $D=0$. For $B<10.5$ T, EC are clearly observed for
$\nu_{T}=\nu_{B}=4.5$, $5.5$, and $6.5$, but the state at $\nu_{B,T}=5.5$
disappears abruptly above $10.5$ T. This disappearance of EC coincides with a
change in the orbital number from $n=5$ to $n=0$, as illustrated in Fig.3(d).
The orbital number change is due to the rearrangement of the LLs with
different spin indices, as illustrated in Fig.3(e). For a fixed filling
factor, the carrier density scales linearly with the magnetic field, and at
low densities, the $g$-factor which decides the Zeeman splitting between two
spin branches are strongly enhanced Shi  _et al._ (2019). Therefore, as $B$
increases and the carrier density is raised above a threshold value, the
active LLs at a fixed filling factor can switch from one spin to another and
abruptly change orbital number to $n=0$, leading to the disappearance of EC
(see SI.7).

To better understand the dependence of EC robustness on orbital number, we
measure the excitation gap $\Delta$ by integrating $\partial\mu/\partial n$
over the gap at integer filling factors (see SI.8 for details). Fig.4(a) plots
$\Delta$ vs. the orbital number $n$ at $D=0$, for $B=14.9$ T: it exhibits a
non-monotonic behavior with the maximum at $n=2$. Tuning the orbital number
modifies the single-particle electron wavefunction: as $n$ increases, the
wavefunctions spread out and have more nodes. In single-layer systems, at
fractional fillings, this evolution often leads to different correlated ground
states for different values of $n$. Here, although the ground state remains
EC, the non-monotonic dependence of $\Delta$ on $n$ stems from the change of
the nature of low-energy charged excitations. The character of these
excitations is more conveniently understood in the pseudospin picture. In this
picture, electron states in the top (bottom) layer are viewed as pseudospin up
(down), while the phase-coherent EC manifest macroscopically aligned
pseudospin pointing in-plane Moon  _et al._ (1995). On top of such a
pseudospin ferromagnet ground state, one type of charged excitation is
associated with a spatially extended pseudospin texture, known as meron-
antimeron pair in the case of bilayer quantum Hall systems Moon  _et al._
(1995) (similar to skyrmions Sondhi  _et al._ (1993) in quantum Hall
ferromagnets in a single-layer). As $n$ increases, the energy of such
excitations increases Wu and Sondhi (1995). On the other hand, a conventional
type of excitation, which represents a localised particle or hole, has an
energy that decreases with $n$. As a result, a change in the nature of the
lowest energy excitations is expected as $n$ is varied. Fig.4(b) shows the
theoretical estimate of the gap for the two types of excitations in a simple
bilayer model without screening effects. The transition between two types of
excitations occurs between $n=0$ and $n=1$, giving a non-monotonic dependence.
This is qualitatively consistent with our data, except that in the model the
gap maximum appears at $n=1$, whereas the maximum appears at $n=2$ in our
measurement, suggesting that the pseudospin textured excitations also dominate
at $n=1$. The slight discrepancy with experimental data may be due to
screening or disorder effects, which may lower the energy of pseudospin-
texture excitations and shift the transition to higher $n$.

The different characteristics of the two kinds of excitations are also
suggested by the density imbalance dependence, as shown in Fig.4(c). Here, we
plot the penetration capacitance as a function of $D$, at a constant total
filling factor $\nu_{tot}$ = 1, 3, 5, 7, which are the four bottom circles in
Fig.3(a) and (b), and correspond to LL orbital number $n$ = 0, 1, 2, 3
respectively. The color shades in the right three panels illustrate the
$D$-range where the individual layers host fractional filling factors, as
suggested by the polarizability contribution at adjacent non-integer fillings.
A larger $C_{P}$ suggests a smaller electronic compressibility and a larger
gap for charged excitations. Our data suggests that, for $n=1$, the EC gap
increases with $D$, and transitions smoothly into the IQH gap. In contrast,
for $n$ = 2 and 3, the EC gap decreases with $D$, and has a minimum before
transitioning into the IQH state.

The distinct behavior of the two types of excitations is supported by our
numerical calculations (see SI.12), which find that the meron-antimeron pairs
have a sharp increase in gap with layer imbalance, while the particle-hole
excitations shows a flat response. While the pseudospin points in-plane for a
density-balanced bilayer system, imbalancing the bilayer is equivalent to
tilting the pseudospin out-of-plane, therefore the in-plane component of the
spin-stiffness decreases Moon  _et al._ (1995). For meron-antimeron pairs, the
excitation energy has the main contribution from the meron and anti-meron
self-energy, the sum of which increases with density imbalance. On the other
hand, the particle-hole excitation energy is just the exchange energy, which
is independent of the charge density imbalance. The decrease of the excitation
gap with imbalance for $n=2$ and 3 is not captured by our calculations and
remains to be understood.

Finally, we remark that, in contrast to our bilayer WSe2, in other quantum
Hall double layer systems the EC has only been observed within the LL of
orbital number $n=0$. We find that, although numerical calculations suggests
that EC could form in LLs of high orbital numbers, a smaller interlayer
spacing is required for higher orbital numbers. As shown in Fig.4(d), the
calculated critical interlayer spacing in units of the magnetic length,
$d_{c}/l_{B}$, decreases sharply as a function of the orbital number $n$. The
critical spacing $d_{c}$ is estimated as the distance where the Goldstone mode
goes soft at finite momentum, signaling the EC becoming unstable (see SI.11).
On the right axis of Fig.4(d), we mark the experimentally accessible ranges of
$d/l_{B}$ for three different systems – GaAs double quantum wells, graphene
double layers with BN barrier, and bilayer WSe2. Our calculation suggests that
the absence of EC in higher LLs in the first two systems is likely due to the
large thickness of the intentional interlayer physical barrier (graphene also
has the complication as the LL wavefunctions are mixture of simple harmonic
oscillator wavefunctions). In contrast, the interlayer spacing in bilayer WSe2
is set by the lattice constant in the vertical direction, which is only about
0.7 nm, and thus EC are expected to occur in higher LLs.

To conclude, our study demonstrates bilayer WSe2 as a unique platform to
access EC in the strong coupling limit, by exploiting the intrinsic spin-
orbital coupling and stacking order in the host material. This designing
principle could be further used in creating other quantum phases from layered
van der Waals materials, including the EC in zero field. Moreover, our study
demonstrates EC composed from tunable single-particle orbital wavefunctions,
and how such tunability could vary the properties of EC and the excitations on
top of the ground state.

We thank William Coniglio and Bobby Pullum for help with experiments at the
National High Magnetic Field Lab. This research is primarily supported by US
Department of Energy (DE-SC0016703). Synthesis of WSe2 (D.R.,B.K.,K.B.) was
supported by the Columbia University Materials Science and Engineering
Research Center (MRSEC), through NSF grants DMR-1420634 and DMR-2011738. A
portion of this work was performed at the National High Magnetic Field
Laboratory, which is supported by National Science Foundation Cooperative
Agreement No. DMR-1157490 and the State of Florida. D.A. acknowledges support
by the Swiss National Science Foundation and by the European Research Council
(grant agreement No. 864597). Z.P. acknowledges support by the Leverhulme
Trust Research Leadership Award RL-2019-015. K.W. and T.T. acknowledge support
from the Elemental Strategy Initiative conducted by the MEXT, Japan (Grant
Number JPMXP0112101001) and JSPS KAKENHI (Grant Numbers JP19H05790 and
JP20H00354).

##  I Method

The heterostructure is assembled using the van der Waals dry transfer
technique, and pre-patterned Pt electrodes are used for electrical contacts to
bilayer WSe2. Hexagonal boron nitride are used as the dielectric, and graphite
or metal as top and bottom gates for bilayer WSe2. The top gate was
lithographically shaped so that its overlap with bottom gate covers only
bilayer WSe2, and the overlap defines the device area. In order to achieve
good electrical contact, we use an additional contact gate on top to heavily
dope the contact area, which is isolated from the top gate by Al2O3
dielectric. Data from a different device are shown in SI.9. Penetration
capacitance was measured with an FHX35X high electron mobility transistor
serving as a low-temperature amplifier, in a similar setup as in Ref. Shi  _et
al._ , 2019. Measurements were performed at $T=0.3$ K unless otherwise
specified.

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Supplementary Information for “Bilayer WSe2 as a natural platform for
interlayer exciton condensates in the strong coupling limit”

##  II S1. Discussion on interlayer tunneling

Here we discuss the suppression of interlayer tunneling in the context of
conventional double quantum wells. In conventional GaAs double quantum wells,
the interlayer tunneling is usually characterized by $\Delta_{\text{SAS}}$.
The two single-particle eigenstates can be written as,

$\displaystyle\psi_{+}$
$\displaystyle=\sin\alpha\;\psi_{t}+\cos\alpha\;\psi_{b}\,,$ (2)
$\displaystyle\psi_{-}$
$\displaystyle=\cos\alpha\;\psi_{t}-\sin\alpha\;\psi_{b}\,.$ (3)

Here, $\phi_{t}$ and $\phi_{b}$ are the wavefunctions localized in the top and
bottom quantum wells, respectively;

$\cos 2\alpha=eV/\sqrt{(eV)^{2}+\Delta_{\text{SAS}}^{2}}$ (4)

characterize the degree of wavefunction localization, where $V$ is the
interlayer electrical potential difference.

For any finite tunneling gap $\Delta_{\text{SAS}}$, $\cos 2\alpha=0$ at zero
bias and the single-particle wavefunctions are just the bounding and anti-
bounding states that is an superposition of wavefunctions in both layers with
equal weight ($\cos\alpha=\sin\alpha$). The single-particle ground state has a
gap $\Delta_{\text{SAS}}$ to the excited state.

In contrast, the bilayer WSe2 manifest a large degree of wavefunction
localization in individual layers even at zero bias, and this is a result of
the strong spin-orbit coupling. It was shown that the spin doublet single-
particle eigenstates in the valence band can be written as Gong  _et al._
(2013),

$\displaystyle|\text{K},\uparrow\rangle$
$\displaystyle=(\sin\beta\;\psi_{t}^{\text{K}}+\cos\beta\;\psi_{b}^{\text{K}})\otimes|\uparrow\rangle\,,$
(5) $\displaystyle|\text{K},\downarrow\rangle$
$\displaystyle=(\cos\beta\;\psi_{t}^{\text{K}}+\sin\beta\;\psi_{b}^{\text{K}})\otimes|\downarrow\rangle\,,$
(6) $\displaystyle|\text{K}^{\prime},\downarrow\rangle$
$\displaystyle=(\sin\beta\;\psi_{t}^{\text{K}^{\prime}}+\cos\beta\;\psi_{b}^{\text{K}^{\prime}})\otimes|\downarrow\rangle\,,$
(7) $\displaystyle|\text{K}^{\prime},\uparrow\rangle$
$\displaystyle=(\cos\beta\;\psi_{t}^{\text{K}^{\prime}}+\sin\beta\;\psi_{b}^{\text{K}^{\prime}})\otimes|\uparrow\rangle\,,$
(8)

where

$\displaystyle\psi_{t}$
$\displaystyle=\frac{1}{\sqrt{2}}(|d_{x^{2}-y^{2}}^{t}\rangle-i\tau|d_{xy}^{t}\rangle)\,$
(9) $\displaystyle\psi_{b}$
$\displaystyle=\frac{1}{\sqrt{2}}(|d_{x^{2}-y^{2}}^{b}\rangle+i\tau|d_{xy}^{b}\rangle)$
(10)

is the wavefunction localized in top and bottom layers and $\tau$ is the
valley index (1 for K and -1 for K’),

$\cos 2\beta=\lambda/\sqrt{\lambda^{2}+t^{2}}\,,$ (11)

$\lambda$ is the spin-orbit coupling strength and $t$ is the interlayer
hopping amplitude.

For holes in bilayer WSe2, $\cos 2\beta\approx 0.96$ according to ab initio
calculations Gong  _et al._ (2013). Therefore, the single-particle eigenstate
wavefunctions are always highly localized in individual layers. The spin-
orbital coupling strength $\lambda$ is effectively an intrinsic bias and only
the low energy branches are accessible in our experiment. All four flavors
described in equations above are degenerate in energy in the absence of
external fields, and no single-particle gap is expected. Such energetically
degenerate yet spatially separated single-particle wavefunctions set the basis
for exciton condensate physics.

Figure 5:  (a) Capacitive and (b) dissipative contributions in the penetration
capacitance measurements as a function of top and bottom gate voltages $V_{T}$
and $V_{B}$.

##  III S2. Landau level structure and wavefunctions in bilayer WSe2

The above discussion suggests a high level of layer localization which
persists to the LL regime. We therefore treat the LL structures in the two
layers independently. In a monolayer WSe2, the LL structure has two isospin
flavors, separated by a large spin-splitting energy. The spin susceptibility,
characterized by $E_{Z}/E_{C}$, where $E_{Z}$ is the spin-splitting energy and
$E_{C}$ the cyclotron energy, increases as carrier density decreases and could
be as large as 9 Gustafsson  _et al._ (2018); Shi  _et al._ (2019). In bilayer
WSe2, the same physics applies, but due to a different dielectric environment
with larger dielectric constant, the spin susceptibility is slightly smaller.
Further, the susceptibility in one layer mainly depends on the carrier density
of this layer, and slightly on the density of the other layer. In our
experiment, the spin of a certain LL is decided from similar analysis as in
Shi  _et al._ (2019), and assisted with transport measurements where different
spin flavors exhibits very different conductivity Lin  _et al._ (2019). At 15
T, five spin-up LLs are populated before the first spin-down LL is populated
in each layer.

It is well known that for the valence band of WSe2, the LL with orbital index
$N=0$ only exists in one valley; in the other valley, the lowest LL has
orbital index $N=1$ Rose  _et al._ (2013). However, it is worth noting that
the LL wavefunctions is not related to the orbital index in the simple way as
in non-Dirac systems, but a mixture of the harmonic oscillator wavefunctions.
Indeed, in the valley with LL orbital index $N=0$, the wavefunction is pure
$|0\rangle$, while the next LL with $N=1$ is mainly $|1\rangle$ with a few
percent of mixing of $|0\rangle$, where $|n\rangle$ stands for the harmonic
oscillator wavefunction of index $n$. On the other hand, in the valley without
a $N=0$ LL, its lowest LL with $N=1$ actually has the wavefunction dominated
by $|0\rangle$ and only with a few percent of mixing from $|1\rangle$.
Therefore, to the first order, the lowest LLs in the two valleys have similar
wavefunctions despite of different LL orbital index. This is important to
establish interlayer EC in the case of mismatched spins. In the main text, the
orbital number $n$ stands for the index associated with the dominant component
$|n\rangle$ of the single-particle wavefunction.

Figure 6:  (a) Penetration capacitance versus total filling factor $\nu_{tot}$
and displacement field $D$ at $B=14.9$ T, focusing on the low $\nu_{tot}$
regime. (b) Schematic LLs diagram. As $D$ is varied, LLs in the bottom layer
increases in energy and those of the top layer decreases, giving rise to LL
crossings. (c) $C_{P}$ vs. $D$ at $\nu_{tot}=3.7$ along the green line in (a).
Blue and red shades mark the region where the Fermi level is in the top and
bottom layer, respectively, with the filling factor marked as
$(\nu_{T},\nu_{B})$. (d) $C_{P}$ vs. $D$ at $\nu_{tot}=4$. The dips mark where
the gap closes as a result of LL crossings. (e) $C_{P}$ vs. filling factor
along $\nu^{*}_{B}=\nu^{*}_{T}$. The left and right panel are along the blue
and red lines in (a), respectively. In the left panel the valence LLs are has
orbital numbers $n_{B}=3$ and $n_{T}=1$, while in the right panel
$n_{B}=n_{T}=2$. In the latter case, a peak is observed at when
$\nu^{*}_{T}+\nu^{*}_{B}=1$, signaling gap opening. (f) $C_{P}$ vs. $D$ along
$\nu_{tot}=5$.The grey shaded region is where both layers host partial filling
factors.

##  IV S3. Negative compressibility

The strong correlation effects in TMDs result in a negative compressibility
which is stronger at lower densities Pisoni  _et al._ (2019). This effect
manifests in several aspects of our data. In Fig.1 (d) and (f) of the main
text we outline the polarization of the bilayer at zero field. Here the
boundary of the fully polarized region is curved. In a simple electrostatic
model,

$E=(D-e\Delta n/2)/\epsilon\,,$ (12)

where $\epsilon$ is the dielectric constant and $E$ is the unscreened
interlayer electrical field. Without interaction effects considered, $E$ would
not have any dependence on the total density, and the displacement field
required to fully polarize the bilayer should increase linearly with the total
density. Due to the negative compressibility, $E$ is of opposite sign to $D$,
and its amplitude increases with decreasing $n$ \- the bilayer electrons tend
to build a larger density imbalance to“overscreen” the displacement field.
Therefore, the displacement field required to fully polarize the bilayer
increases superlinearly with the total density, giving rise to the curve
shaped boundary in Fig.1 of main text.

Similarly, the negative compressibility effect also manifest in the data under
a perpendicular magnetic field. Under a magnetic field, a single-particle
picture would predict that the LL crossings happen at the same $D$ for a
certain filling factor difference $\Delta\nu=\nu_{B}-\nu_{T}$, as shown in
Fig.3(b) in the main text. In contrast, as shown in Fig.3(a) in the main text,
for a fixed $\Delta\nu$, the LL crossings happen at a smaller $D$ for smaller
$n$, and the polarizability segments at partial fillings also assume a slope.

##  V S4. Distinguishing polarizability and incompressibility contribution

The penetration capacitance is a sum of both “incompressibility” and
“polarizability” contributions. First, we confirm that the polarizability
contribution is only in the capacitance channel, and is not dissipation
related. In Fig. 5 we plot the capacitive and dissipative signal in the
measured penetration capacitance in (a) and (b), respectively. While most
integer quantum Hall gaps show up in both, the features cutting the integer
gap only appear in the capacitive channel, which are consistent with the
polarizability contribution.

We further distinguish the two contributions according to their behavior at
different filling factors. Fig. 6 (b) shows the schematic LL diagram
corresponding to the data in Fig. 6 (a). In Fig. 6(c) we plot the linecut of
$C_{P}$ along constant non-integer filling factor $\nu_{\text{tot}}=3.7$
[horizontal line in Fig. 6(a)]. Here, the Fermi level changes back and forth
between two sets of LLs, as marked by the thick lines in Fig. 6(b). The
bilayer system is always compressible and the peaks arises from the
polarizability contributions . Indeed, whenever the Fermi level changes
between the LLs localized in different layers, interlayer charge transfer
happens, and gives rise to a peak in $C_{P}$. We observe a total of seven
peaks in the whole $D$ range in Fig. 6(c), separating red and blue regions,
which correspond to the cases that the Fermi level is in the bottom (red
lines) and top (blue lines) layers, respectively. Further, a small peak
corresponds to charge transfer amount of 0.3 of the Landau level degeneracy,
while a large peak corresponds to 0.7, marked by the numbers in Fig. 6(c).

In Fig. 6(d) we plot the linecut of $C_{P}$ along constant integer filling
factor $\nu_{\text{tot}}=4$. We observe that $C_{P}$ show four dips in the
whole $D$ range, which corresponds to four LL crossings, separating regions of
different filling factors marked in Fig. 6(d). At the level crossings, in
principle there is “polarizability” contribution as well, but it is much
smaller, as evidenced from different vertical scales for Fig. 6(c) and (d).

To distinguish the two contributions at LL crossings at integer filling
factors, we make a linecut along the polarizability segments, cutting through
integer filling lines, as shown in Fig. 6(e). Such a linecut represents the
condition $\nu_{T}^{*}=\nu_{B}^{*}$, where $\nu_{T}^{*}$ ($\nu_{B}^{*}$) is
the fractional part of the filling factor in the top (bottom) layer. While the
polarizability contribution continuously changes from non-integer
$\nu_{\text{tot}}$ to integer $\nu_{\text{tot}}$, an incompressible
contribution would only be possible at integer $\nu_{\text{tot}}$, and shows
up as a sharp peak at $\nu_{T}^{*}=\nu_{B}^{*}=0.5$. The two panels in Fig.
6(e) demonstrate the two situations, respectively. In the right panel, an EC
gap exist as orbital numbers match in the two layers; in the left panel, an EC
gap does not exist as the orbitals are mismatched.

In the presence of EC, the linecut along constant $\nu_{tot}$ is shown in Fig.
6(f). Here, while $C_{P}$ at other LL crossings demonstrate simply a dip at
the closing of IQH gap, its behavior in the EC (grey shaded) region is
different, exhibiting an “reentrant” increase of $C_{P}$, which corresponds to
the emergence of a correlation gap.

##  VI S5. Exciton condensate at density imbalance

Figure 7:  Penetration capacitance versus total filling factor $\nu_{tot}$ and
displacement field $D$ at $B=14.9$ T around LL crossings of opposite spins.
The bottom panel are linecut along the red dashed line in the top panel, which
tracks equal topmost LL population in the two layers. The inset of the bottom
panel illustrates the orbital, spin and valley indices.

In Fig. 7 we plot $C_{P}/C_{G}$ for LL crossings with the same orbital number
but different spin. For $n=0$, as shown in Fig. 7 (a), no incompressible state
is observed when both layers host partially filled LLs. In contrast, for
$n=2$, as shown in Fig. 7 (b), incompressible state persists when both layers
are partially filled, consistent with EC formation. The suppression of EC at
$n=0$ is consistent with what we observe for the LL crossings between the same
spins, at $D=0$, demonstrated in Fig.3 of the main text.

##  VII S6. Data for $n=0$ at 30 T

Figure 8:  Penetration capacitance as a function versus $\nu_{tot}$ and $D$ at
(a) $B$ = 14.9 T and (b) $B$ = 30 T.

In Fig. 8 (a) and (b) we show the capacitance as a function of top and bottom
gate voltages at 14.9 T and 30 T, focusing at $\nu_{tot}=1$. The charge gap at
$\nu_{tot}=1$ disappears at the layer balanced point $D=0$ at 14.9 T, while it
recovers at 30 T. The difference could be due to a larger interaction energy
scale at 30 T, which enhances the EC gap that overcomes disorder effects.

##  VIII S7. Magnetic field dependence at density balance

Figure 9:  Penetration capacitance at the balanced density condition
$n_{T}=n_{B}=n$, plotted versus $n$ and $B$. Filling factor for individual
layers are marked on the top.

In Fig. 9(a) we plot the same data as in Fig.3(c) in the main text, which
shows $C_{P}$ at density balance as a function of the magnetic field, yet here
versus the density and in a larger filling factor range. The transition at
$\nu_{T}=\nu_{B}=5.5$ is clearly distinct from the behavior at other half-
integer filling factors. This transition originates from a density dependent
$g$-factor, which varies the ratio of spin-splitting energy $E_{z}$ and the
cyclotron energy $E_{c}$ Gustafsson  _et al._ (2018); Shi  _et al._ (2019) and
shuffles LLs of different orbital and spin index as density is varied.

##  IX S8. Extraction of charge gap

For fixed excitation amplitude and frequency, the measured penetration signal
$V_{P}$, to a good approximation, can be written as Dultz and Jiang (2000);
Goodall  _et al._ (1985),

$V_{P}^{complex}\propto
C_{P}=C_{G}\left[1-\frac{C_{q}}{C_{t}+C_{b}+C_{q}}\frac{\tanh(\alpha)}{\alpha}\right]\,,\\\
$ (13)
$\alpha=\sqrt{i\omega\frac{C_{q}(C_{t}+C_{b})}{C_{t}+C_{b}+C_{q}}R_{s}}\,,$
(14)

where the imaginary and real part, denoted $V_{P}$ and $V_{P}^{disp}$ stands
for capacitive and dissipative contribution, respectively. In the equation
above, $C_{t}$ ($C_{b}$) is the geometry capacitance between WSe2 and the top
(bottom) gate, $C_{G}=C_{t}C_{b}/(C_{t}+C_{b})$ is the geometry capacitance
between the two gates, $C_{q}=Ae^{2}(\partial\mu/\partial n)^{-1}$ is the
quantum capacitance of the WSe2, with $A$ being the area of the device, and
$R_{s}$ is the resistance of the WSe2 channel.

In the low frequency and low resistance limit ($\alpha\to 0)$, the real part -
dissipative contribution is zero, and the penetration capacitance $C_{P}$ can
be written as,

$C_{P}=C_{G}\frac{C_{t}+C_{b}}{C_{b}+C_{t}+C_{q}}\,;$ (15)

in the high resistance limit ($\alpha\to\infty)$ or when the WSe2 is gapped,
the penetration capacitance is equal to the geometry capacitance between the
two gates,

$C_{P}=C_{G}\,,$ (16)

and the dissipative contribution is also zero.

In our experiment, the resistance at EC states are large, giving rise to
finite dissipative contribution. To obtain the gap value, we obtain the
quantum capacitance $C_{q}$ and resistance $R_{s}$ by numerically solving Eq.
(13) and Eq. (14). An example of the obtained values $C_{q}$ and $R_{s}$ are
shown in Fig. 10. The chemical potential jump due to the formation of
incompressible states was then found by integrating $\partial\mu/\partial n$
over the peaks in the EC region. Such gap values are plotted in Fig.4(a) of
the main text. In addition, we fit the temperature dependence of $R_{s}$ at
the EC state by an activation behavior $R_{s}\sim\exp(-\Delta/2T)$. Such
obtained gaps are plotted in Fig. 11 as circle markers. The quantitative
differences in the gap values may be due to the contact resistance, which are
not accounted for in our analysis. Nevertheless, they demonstrate the same
trend, with a maximum at $n=2$.

Figure 10:  (a) Imaginary ($V_{P}$) and real ($V_{P}^{disp}$) part of the
penetration signal measured at $B=14.9$ T, as a function of the filling factor
at the density balanced condition. For $\nu_{B}=\nu_{T}\approx 0.5$, the gap
depends extremely sensitive on the total filling factor then filling factor
imbalance, as shown in Fig. 8 (a), so the $n=0$ gap is overestimated in this
measurement due to a small density imbalance. In Fig.4 in the main text, we
obtained the gap for $n=0$ when the density balance condition is more
accurately satisfied. (b) $C_{G}/C_{q}$ and (c) $\omega RC_{G}$ extracted from
data in (a) by solving for Eq. (13) and Eq. (14). The contact resistance is
large at low densities, and may give rise to additional contribution for
$\nu_{B}=\nu_{T}<2$, which is not accounted for in our analysis. The gaps for
$n=0$ and $n=1$ may thus be overestimated, but it doesn’t change the
qualitative trend, i.e., the gap has a non-monotonic dependence on $n$.
Figure 11:  Square markers represent the gap by integrating
$\partial\mu/\partial n$ extracted from the capacitance contribution, the same
as in Fig.4(a) in the main text. Circle markers represent the gap obtained
from the temperature dependence of the dissipative contribution, $R\sim
exp(-\Delta/2T)$. The error bars for $n>0$ stands for the standard deviation
of the fitting, and the error bar at $n=0$ stands for the uncertainty in
distinguishing the EC at density balance from at density imbalance or IQH gap.

##  X S9. Exciton condensate in another device

Figure 12:  (a) $C_{P}/C_{G}$ as a function of top and bottom gate voltages
$V_{T}$ and $V_{B}$ at $B$ = 15 T in a different device than in the main text.
White circles mark the situation where the LL orbital numbers in the top and
bottom layers are matched, with different spins. (b) Zoom-in of (a) focusing
on EC with $n=4$. The filling factors in adjacent integer quantum Hall states
are marked as ($\nu_{T}$,$\nu_{B}$). (c) LL diagram, illustrating the
situation of matched orbital number in the two layers. Thick lines mark the
relevant LL in panel (b).

EC with similar behavior as discussed in the main text have been observed in
several devices. Here we show the data in a different device. In Fig. 12(a) we
display $C_{P}/C_{G}$ versus the top and bottom gate voltages. In this device,
the top gate also covers the contact region, so $V_{T}$ is kept at a large
negative value in order to highly dope the contact region and maintain good
electrical contact. As a result, the top layer always host a large filling
factor. For $V_{B}\gtrsim-1.5$ V, the bottom layer is empty, and only one set
of LLs gaps from the top layer are observed; for $V_{B}\lesssim-1.5$ V, both
layers are populated, and the polarizability features appear, cutting the
incompressible features along integer $\nu$ to discontinuous segments. It can
be seen that the layer transitions are sharp when the bottom layer filling
$\nu_{B}$ is low; that is, the carriers in the topmost LL transfers from the
top to bottom layer within a small voltage change. The sharp segments smears
out to a diamond shape as $V_{B}$ and the filling factor difference
$\Delta\nu=\nu_{T}-\nu_{B}$ decreases. The white circles mark where the LLs in
the two layers host the same orbital index, which corresponds to n = 2, 3, 4
and 5 respectively. Within these “diamonds”, we observe an incompressible
feature when $\nu$ is integer while $\nu_{B}$ and $\nu_{T}$ are both
fractions. Fig. 12(b) shows a highlight for $n=4$, with the LL structure
sketched in Fig. 12(c). In contrast, the incompressible features are not
present in the cases where the LL orbital indices are mismatched across the
two layers.

In addition, in Fig. 13 we plot $C_{P}$ as a function of the bottom gate
voltage $V_{B}$ and $B$ with the top gate voltage $V_{T}$ fixed. Three fans of
high penetration signal are revealed. First, a main fan tracks constant
integer fillings of $\nu=\nu_{T}+\nu_{B}$ and emerges from the band edge. For
$V_{B}<1.5$ V, as both layers are populated, this fan is cut by layer
transition features and becomes discontinuous. Second, a side fan emerges at
the population onset of bottom layer, $V_{B}=1.5$ V, indicated by dotted
lines. Third, a fan tracks constant $\Delta\nu=\nu_{T}-\nu_{B}$, marked by
dashed lines. While the first two fans are both indications incompressible
states at integer LL filling, the third fan is an indication of charge
transfer between the layers. The black circles mark the position where the top
most LLs in the two layers host the same LL orbital index, and an
incompressible feature appear which signals the emergence of EC.

Figure 13:  Penetration signal as a function of magnetic field $B$ and bottom
gate voltage $V_{B}$ at fixed top gate voltage $V_{T}$ = -8 V. Dotted lines
mark the trajectory of $\nu_{B}$ = 1, 2, 3; dashed lines mark the trajectory
of $\Delta\nu=\nu_{T}-\nu_{B}$ = 0, 2, 4. Black circles mark the situation
where the LL orbital numbers in the top and bottom layers are matched.

##  XI S10. Theoretical model

Our theoretical analysis of bilayer $\mathrm{WSe_{2}}$ is based on a
simplified model that comprises two copies of a two-dimensional electron gas
(2DEG), separated by distance $d$ in the perpendicular $z$-direction. The two
layers are threaded by the magnetic field $B\hat{z}$ and they can have
arbitrary Landau level (LL) indices, $n_{1}$ and $n_{2}$ (for simplicity, we
neglect the spin degree of freedom). The case of $n_{1}=n_{2}=0$ and
$d/\ell_{B}\gtrsim 1$ has been studied extensively in previous work on GaAs
electron bilayers Chakraborty and Pietiläinen (1987); Moon  _et al._ (1995);
Scarola and Jain (2001); Schliemann  _et al._ (2001); Sheng  _et al._ (2003);
Park (2004); Shibata and Yoshioka (2006); Möller  _et al._ (2009); Milovanović
_et al._ (2015); Zhu  _et al._ (2017); Lian and Zhang (2018). Although the
single-particle orbitals in $\mathrm{WSe_{2}}$ are spinors, the admixture of
different LL orbitals is strongly suppressed compared to other materials such
as graphene Rose  _et al._ (2013).

In momentum space, our model is described by the Hamiltonian

$\displaystyle
H=\frac{1}{N_{\phi}}\sum_{\sigma,\sigma^{\prime}=\uparrow,\downarrow}\sum_{\mathbf{q}}V_{\sigma\sigma^{\prime}}(\mathbf{q})\bar{\rho}_{\sigma}(\mathbf{q})\bar{\rho}_{\sigma^{\prime}}(-\mathbf{q}),$
(17)

where $\sigma=\uparrow,\downarrow$ labels the two layers, $N_{\phi}$ is the
number of magnetic flux quanta through the system, $\bar{\rho}_{\sigma}$ is
the projected electron density in a layer $\sigma$.
$V_{\sigma\sigma^{\prime}}$ is the Fourier transform of the Coulomb
interaction which is layer-dependent:

$\displaystyle V_{\uparrow\uparrow}(\mathbf{q})$ $\displaystyle=$
$\displaystyle
V_{\downarrow\downarrow}(\mathbf{q})=\frac{2\pi}{|\mathbf{q}|},$ (18)
$\displaystyle V_{\uparrow\downarrow}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\frac{2\pi}{|\mathbf{q}|}e^{-d|\mathbf{q}|},$ (19)

where $d$ is the distance between the layers which “softens” the repulsion
between $\uparrow$ and $\downarrow$ electrons since $V_{\uparrow\downarrow}$
corresponds to interaction $1/\sqrt{r^{2}+d^{2}}$ in real space. Thus,
$\uparrow$ and $\downarrow$ are sometimes referred to as “pseudospin” degrees
of freedom Moon  _et al._ (1995) to distinguish them from real electron spin.
Furthermore, the effective interaction is also dependent on which LLs the
layers are in, which modifies the projected density operators,

$\displaystyle\bar{\rho}_{\sigma}(\mathbf{q})=e^{-\mathbf{q}^{2}/4}L_{n_{\sigma}}\left(\frac{\mathbf{q}^{2}}{2}\right)\sum_{j=1}^{N}e^{i\mathbf{q}\mathbf{R}_{j\sigma}}.$
(20)

Here $\mathbf{R}_{j\sigma}$ stands for the guiding center coordinate of $j$th
electron with the layer index $\sigma$ Prange and Girvin (1987), and
$L_{n_{\sigma}}$ is the $n$th Laguerre polynomial for the layer $\sigma$
Macdonald (1994).

Below we work in units $e^{2}/\epsilon\ell_{B}=1$. We focus on the total
electron filling factor $\nu=1$, corresponding to the total number of
electrons being equal to the number of flux quanta, $N=N_{\phi}$. If a layer
has LL index $n>0$, we assume that the LLs $0,1,\ldots,n-1$ are fully occupied
and inert, while the $n$th LL is partially occupied. We study the model in Eq.
(17) using exact diagonalization of the Hamiltonian matrix appropriate for
systems of finite size, either on the surface of a sphere Haldane (1983) or
torus Yoshioka  _et al._ (1983). Various symmetries, either spatial (e.g.,
translation, inversion) or internal (e.g., pseudospin U(1) symmetry in the
absence of interlayer tunneling), are used to block-diagonalize the
Hamiltonian and classify its eigenstates. We also interpret our results
against Hartree-Fock theory Côté  _et al._ (1992), which at $\nu=1$ captures
the leading effects as quantum fluctuations are fairly weak given the small
$d/\ell_{B}$.

##  XII S11. Theoretical evidence for exciton condensation

Experimental results in the main text demonstrate the quantum Hall effect at
integer filling factors when the two layers have matched LL indices,
$n_{1}=n_{2}$. Here we provide theoretical evidence that identifies this phase
as exciton condensate, where each exciton consists of an electron and a hole
in the opposite layer (see the recent review Eisenstein (2014)). While the
exciton phase exhibits quantized Hall effect, its distinctive signature is the
Goldstone mode which governs the gapless excitations of the system in the
neutral channel, probed by driving oppositely directed electrical currents
through the two layers Spielman  _et al._ (2000); Tutuc  _et al._ (2004).

In the numerics we can access the Goldstone mode directly by computing the
energy dispersion, $E(k)$, as a function of momentum $k$ in the torus
geometry. The torus spectra for a fixed interlayer distance $d/\ell_{B}=0.1$
and both layers having the same LL index $n=0,1,2,3$ are shown in Fig.
14(a)-(d). We compare exact results against the prediction of Hartree-Fock
theory Fertig (1989), shown by the blue line in Fig. 14. We observe, overall,
an excellent agreement between the exact results and the Hartree-Fock
prediction, in particular in view of the fact that Hartree-Fock results do not
contain any adjustable parameters. The agreement could be anticipated given
that we are looking at small $d/\ell_{B}$ and the Hartree-Fock theory captures
the ground state exactly for $d/\ell_{B}=0$. In comparison, much bigger
discrepancy between exact results and Hartree-Fock theory are seen already at
$d/\ell_{B}=0.5$, even in the long-wavelength limit (data not shown).

Figure 14: Energy dispersion $\Delta E(k)$ as a function of 2D momentum,
$k\ell_{B}$, on a torus geometry. Dots represent exact results for various
system sizes ($N=N_{\phi}=10,12,14,16$ electrons) and boundary conditions
(square and hexagon unit cells). In all plots, the two layers have matched LL
indices $(n,n)$, where $n=0,1,2,3$ and we fix $d/\ell_{B}=0.1$. Lines are the
result of the Hartree-Fock theory, which are in excellent agreement with the
exact data in the long-wavelength limit.

Results in Fig. 14 confirm the existence of a gapless mode with a dispersion
in quantitative agreement with the Hartree-Fock theory based on easy-plane XY
ferromagnetism Girvin and Macdonald (1995). Unfortunately, accessing the long-
wavelength of this mode, $k\to 0$, is challenging for exact numerics where the
smallest available (non-zero) $k$ scales as $\propto 1/\sqrt{N_{\phi}}$, thus
we cannot resolve finer features of the mode, such as whether the dispersion
is quadratic (as expected at $d=0$ when the system is SU(2)-invariant) or
linear (for $d>0$). Interestingly, the exact spectra in Fig. 14 suggest that
the Goldstone mode is increasingly better separated from the continuum of the
excitation spectrum as LL index is increased, _at all wavelengths_ – see Fig.
14(d). While this could be due to stronger finite-size effects at larger $n$,
the fully exposed collective mode in bilayer $\mathrm{WSe_{2}}$ may lead to a
different optical response compared to GaAs and graphene materials, where
typical $d/\ell_{B}$ are several times larger and the collective mode is only
exposed below the continuum in the range of momenta $k\ell_{B}\lesssim 1$.

Figure 15: Exciton condensate on the sphere. (a) Overlap between the exact
ground state of the bilayer, $\psi_{\mathrm{exact}}$, and the Halperin 111
state, as a function of interlayer distance $d/\ell_{B}$ for several matched
LL indices $n=0,1,2,3$. (b) Exciton order parameter as a function of matched
LL index $n$ and several values of interlayer distance shown in the legend.
All data is for $N=14$ electrons on a sphere, where “0” denotes the north pole
and $r_{\mathrm{max}}$ is chosen to be the south pole.

Further evidence for the exciton condensate follows from the study of the
system’s ground state properties. At $d=0$, the exact ground state of the
system is given by the first-quantized wave function known as the 111 Halperin
state Halperin (1983):

$\displaystyle\psi_{111}(\\{z_{i}\\},\\{w_{j}\\})$ $\displaystyle=$
$\displaystyle\prod_{i<j}(z_{i}-z_{j})\prod_{i<j}(w_{i}-w_{j})\prod_{i,j}(z_{i}-w_{j})$
(21)
$\displaystyle\times\exp\left(-\sum_{k}\frac{|z_{k}|^{2}}{4\ell_{B}^{2}}-\sum_{k}\frac{|w_{k}|^{2}}{4\ell_{B}^{2}}\right),\quad$

where $z_{i}$, $w_{i}$ are complex 2D coordinates of electrons in top and
bottom layers, respectively. The Jastrow factor, $\prod_{i,j}(z_{i}-w_{j})$,
builds in the exciton correlation between an electron in one layer ($z_{i}$)
and the corresponding hole at the same position $w_{i}$ in the opposite layer.
Although $\psi_{111}$ is only an exact ground state when strictly $d=0$, it
remains a good representative wave function for the entire exciton phase
extending to $d\sim\ell_{B}$ Girvin and Macdonald (1995). Thus, by evaluating
the inner product between the exact ground state of a bilayer and $\psi_{111}$
we can test the existence of the exciton phase in bilayers with $n>0$. These
overlaps are shown in Fig. 15(a) as a function of $d/\ell_{B}$ in the sphere
geometry. We observe that for any value of $n$, as long as $d/\ell_{B}\lesssim
0.3$, the overlap with 111 state is very close to unity. At larger values of
$d$, the overlap drops rapidly, and the decay is faster for larger values of
$n$, indicating the sensitivity of the exciton phase in higher LLs to the
increase in layer separation.

Further, we do not need to rely on a microscopic candidate wave function to
detect the exciton phase. In Fig. 15(b) we have computed the exciton order
parameter Shibata and Yoshioka (2006),

$\displaystyle\langle\psi_{\mathrm{exact}}|\hat{\Psi}_{\mathrm{exc}}^{\dagger}(r_{\mathrm{max}})\hat{\Psi}_{\mathrm{exc}}(0)|\psi_{\mathrm{exact}}\rangle,$
(22)

expressed in terms of an operator that creates an exciton at a given point
$\mathbf{r}$,

$\displaystyle\hat{\Psi}_{\mathrm{exc}}(\mathbf{r})=\hat{\Psi}_{\uparrow}(\mathbf{r})^{\dagger}\hat{\Psi}_{\downarrow}(\mathbf{r}),$
(23)

where $\hat{\Psi}_{\sigma}$ is the standard electron field operator. The
normalization of $\hat{\Psi}_{\mathrm{exc}}$ is chosen such that the
expectation value is 1 when $r\gg\ell_{B}$ in the ideal exciton case (i.e.,
for the exact 111 Halperin state). In contrast, a value close to 0 indicates
no exciton correlations. Specifically, on a sphere geometry, we can pick the
origin to be the north pole and the point far away is the south pole. The
value of the correlator in Eq. (22) is shown in Fig. 15(b); when $d/\ell_{B}$
is small ($\lesssim 0.4$), the exciton order parameter is small in any of the
LLs ($n=0,1,2,3$). For $d/\ell_{B}=0.6$ the correlator decays sharply with
$n$, and the exciton phase is only likely to exist in $n=0$ LL.

Figure 16: (a) A few illustrative examples of exciton dispersion curves $E(k)$
derived in Hartree-Fock theory for $n=2$ and $d/\ell_{B}=0,0.1,0.2,0.4,0.6$.
For any $n$, the dispersion develops a minimum that goes soft at a finite $k$
when $d/\ell_{B}$ is sufficiently large, coinciding with the quantum phase
transition to a density wave phase and the destruction of the exciton
condensate. (b) Hartree-Fock estimate for the critical distance
$d_{c}/\ell_{B}$ as a function of LL index $n$.

By generalising the Hartree-Fock theory of Ref. Fertig (1989), we can
conveniently estimate the critical interlayer distance $d_{c}/\ell_{B}$ when
the exciton condensate is destroyed by a density wave that breaks translation
symmetry. In any LL, we find that the exciton dispersion develops a
“roton”-like minimum as $d/\ell_{B}$ becomes sufficiently large – see an
example of $n=2$ in Fig. 16(a). The minimum becomes more pronounced by
increasing $d/\ell_{B}$ and eventually goes soft, signalling a quantum phase
transition where the ground state changes its momentum from 0 to a finite
value $k^{*}\sim 1/\sqrt{2n+1}$ Brey and Fertig (2000). We numerically perform
the golden section search and determine the critical value of
$d_{c}/\ell_{B}$, which is shown in Fig. 16(b) as a function of $n$. The
critical distance decays sharply for low values of $n$, reaching
$d_{c}/\ell_{B}\sim 0.3$ for $n\lesssim 10$. Note that this estimate does not
include the effect of quantum fluctuations, which become significant near the
transition and potentially further decrease $d_{c}/\ell_{B}$. Nevertheless,
this estimate is consistent with the possibility of the exciton phase in
bilayer $\mathrm{WSe_{2}}$ where we estimate $d/\ell_{B}\approx 0.1$. We note
that in recent works Shi  _et al._ (2008); Zhu  _et al._ (2019) the breakdown
of the exciton phase was studied at GaAs total filling $\nu=5$, which
corresponds to $n=1$ in our case. Refs. Shi  _et al._ (2008); Zhu  _et al._
(2019) estimated $d_{c}/\ell_{B}\approx 0.9-1.2$, which is somewhat larger
than our estimate in Fig. 16(b), possibly due to finite size effects.

##  XIII S12. Charged excitations

Figure 17: Chemical potential discontinuity $\Delta\mu$ (in units
$e^{2}/\epsilon\ell_{B}$) in Eq. (24) as a function of charge imbalance $D$,
which plays the role of a Zeeman field for pseudospin. (a) $n_{1}=n_{2}=0$.
(b) $n_{1}=n_{2}=2$. Data is for $N=14$ electrons on the surface of a sphere,
with $d/\ell_{B}=0.1$. In shaded regions, the imbalance fully polarizes the
pseudospin and the system behaves as a single layer $\nu=1$ state, whose gap
varies linearly with $D$.

We have also studied the response of the exciton phase to adding or removing
charge, which is characterized by the chemical potential discontinuity
Yoshioka  _et al._ (1983); Moon  _et al._ (1995):

$\displaystyle\Delta\mu\equiv
E_{N_{\phi},N+1}+E_{N_{\phi},N-1}-2E_{N_{\phi},N},$ (24)

where $E_{N,N_{\phi}}$ is the ground state energy for the given values of $N$
and $N_{\phi}$. This quantity plays the role of the activation gap of the
system: the non-zero value of $\Delta\mu$, in the limit $N\to\infty$, signals
an incompressible ground state that would give rise to the quantized Hall
effect. Alternatively, one could keep $N$ fixed and vary flux $N_{\phi}\to
N_{\phi}\pm 1$. At $\nu=1$, it is more convenient to keep the geometry of the
system unchanged, i.e., fix $N_{\phi}$ and vary $N$.

As we are adding or removing charge, it is important to include the relevant
electrostatic corrections when evaluating $\Delta\mu$ in finite-size systems.
For example, the Coulomb interaction gives rise to electrostatic charging
energy when particles are shuffled between the layers Moon  _et al._ (1995).
Formally, this term originates from the finite $\mathbf{q}=0$ contribution in
the pseudospin Coulomb interaction,
$(V_{\uparrow\uparrow}(0)-V_{\uparrow\downarrow}(0))/2=\pi d$, and it has a
form $\epsilon_{c}\propto dS_{z}^{2}$, where the total pseudospin
$S_{z}\equiv(N_{\uparrow}-N_{\downarrow})/2$. On a finite sphere, the charging
energy can be shown to be given by

$\displaystyle\epsilon_{c}=\frac{S_{z}^{2}}{N^{2}}\frac{2R^{2}+1}{R}\left(1+\frac{d}{2R}-\sqrt{1+\frac{d^{2}}{4R^{2}}}\right),$
(25)

where the radius of the sphere is $R=\sqrt{N_{\phi}/2}$ (in units
$\ell_{B}=1$).

In Fig. 17 we have computed $\Delta\mu$ as a function of density imbalance,
$-D(N_{\uparrow}-N_{\downarrow})$. Panel (a) shows the charge gap for
$n_{1}=n_{2}=0$, while panel (b) illustrates the case of a higher LL with
$n_{1}=n_{2}=2$. We find the behaviour of the gap of any $n>0$ to be
qualitatively similar to each other and distinct from $n=0$ LL. Although we
have previously established the ground state for any $n_{1}=n_{2}=n$ is the
same exciton condensate, approximately described by the 111 state in Eq. (21),
we now see that the charged excitations are very different in $n=0$ compared
to $n>0$.

In $n=0$ case, the gap rises sharply upon density imbalance, which is
reminiscent of a skyrmion excitation Sondhi  _et al._ (1993); Schmeller  _et
al._ (1995) that involves a large number of pseudospin flips. In the absence
of Zeeman field ($D=0$), the number of flipped pseudospins is extensive in
system size; for a finite value of $D$, the number of flips is optimized to
minimize the total energy due to pseudospin stiffness and the competing Zeeman
term. When $D$ is sufficiently large, all pseudospins orient in the
$z$-direction [shaded regions in Fig. 17(a)], and the gap then trivially grows
linearly with $D$. Adding a small amount of interlayer tunneling,
$-\Delta_{\mathrm{SAS}}S_{x}$, does not qualitatively change the behaviour of
$\Delta\mu$. We note that the increase of the gap in $n=0$ GaAs bilayers has
been observed experimentally in Refs. Spielman  _et al._ (2004) and Tutuc  _et
al._ (2003). In these cases, however, the gap has a much slower parabolic rise
with $D$ due to the larger value of $d/\ell_{B}$.

In contrast, in higher $n>0$ LLs, the gap is largely insensitive to imbalance
until full pseudospin polarization is reached, see Fig. 17(b). This is
consistent with the behavior found in Hartree-Fock approximation which does
not include a possibility of pseudospin-texture excitations Jungwirth  _et
al._ (1998); Jungwirth and MacDonald (2000). In the Hartree-Fock picture, the
ground state pseudospin smoothly rotates as bias potential is varied. The full
pseudospin polarization is reached when the bias potential reaches the value
(in the notation of Ref. Jungwirth  _et al._ (1998))

$\displaystyle U_{\sigma\sigma}^{(n)}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}q}{(2\pi)^{2}}\left(V_{\sigma\sigma}(\mathbf{q}=0)-V_{\sigma\sigma}(\mathbf{q})\right)$
(26) $\displaystyle\times\left(L_{n}(q^{2}/2)\right)^{2}e^{-q^{2}/2},$

where $V_{\sigma\sigma}$ denotes the interaction in the pseudospin channel. It
follows that $U_{\sigma\sigma}^{(n)}\propto\sqrt{2n+1}$, thus it is harder to
polarise the pseudospin in higher LLs, which is consistent with the numerics
as well as the experimental data presented in the main text. In Fig. 17(b)
tiny wiggles can be observed near $D=0$ – those correspond to the ground state
pseudospin changing in integer steps as a single electron is transferred
between the two layers.

Figure 18: (a) Comparison of analytically evaluated gaps for creating a
pseudospin texture vs. ordinary electron-hole excitation, as a function of LL
index. (b) The difference in energy gap of the pseudospin texture versus the
particle-hole (color scale) for $n=1$, plotted as a function of interlayer
distance $d$ and the phenomenological screening parameter $w$ (see text for
details).

The difference in types of charged excitations between $n=0$ and $n>0$ cases
is reminiscent of integer quantum Hall ferromagnets, where it has been
established that skyrmions are energetically competitive with ordinary
electron-hole excitations only in $n=0$ LL Wu and Jain (1994); Wu and Sondhi
(1995). In Fig. 18(a) we analytically compare the energy gap for creating a
spin texture with that for creating an ordinary particle-hole excitation
(i.e., with a single pseudospin flipped). Indeed, in $n=0$ we find the
skyrmion excitation to be significantly lower in energy Sondhi  _et al._
(1993), while in $n=1$ the ordinary particle-hole excitation is already more
stable and continues to be energetically favored in all $n\geq 1$.

Experimental data in the main text shows the gap in bilayer $\mathrm{WSe_{2}}$
varies non-monotonically with $n$ and reaches a maximum in $n=2$. This
suggests that excitations in both $n=0$ (at sufficiently high magnetic fields)
and in $n=1$ LL are pseudospin textures. This discrepancy with respect to
theory could be explained by disorder, which affects the two types of
excitations very differently due to their different spatial profiles, but it
could be also be an interaction effect, e.g., due to screening of the Coulomb
interaction. In Fig. 18(b) we have explored the difference in the gap for
creating the particle-hole excitation vs. a pseudospin texture in $n=1$ LL, by
varying the interlayer distance $d$ and the amount of screening of the
interaction. The latter is phenomenologically modelled by finite thickness
ansatz, i.e., we model the intralayer Coulomb repulsion as
$1/\sqrt{r^{2}+w^{2}}$ , while the interlayer potential is
$1/\sqrt{r^{2}+(w+d)^{2}}$. We observe that either the increase in $d$ or some
screening $w\gtrsim 0.1$ is sufficient to stabilize the skyrmion excitation,
in line with previous results for the integer quantum Hall ferromagnets Cooper
(1997); Wójs and Quinn (2002). Thus, it is likely that these interactions
effects, perhaps in combination with disorder, stabilize the pseudospin
texture in $n=1$ LL of $\mathrm{WSe_{2}}$ over the particle-hole excitation.

Finally, we comment on the case of _unmatched_ LLs, $n_{1}\neq n_{2}$. In this
case, experimental data in the main text suggest an absence of the exciton
phase. In Hartree-Fock theory Jungwirth  _et al._ (1998); Jungwirth and
MacDonald (2000), the system in this case behaves an easy-axis ferromagnet.
When bias potential is introduced, the gap varies linearly with $D$ until a
critical value when the pseudospin flips, $S_{z}=N/2\to-N/2$. The excitation
spectrum on the torus is found to be very different from the Goldstone
collective mode seen in Fig. 14, possibly due to the existence of domain wall
type excitations Rezayi  _et al._ (2003). Similarly, the overlap with the 111
state as well as the exciton order parameter are found to be small and rapidly
diminishing with system size.

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