# Dual density waves with neutral and charged dipolar excitons of GaAs
bilayers

Camille Lagoin1, Stephan Suffit2, Kirk Baldwin3, Loren Pfeiffer3 and François
Dubin1,4,† 1 Institut des Nanosciences de Paris, CNRS and Sorbonne Université,
Paris, France  2 Laboratoire de Materiaux et Phenomenes Quantiques, Universite
Paris Diderot, Paris, France  3 PRISM, Princeton Institute for the Science and
Technology of Materials, Princeton University, Princeton, USA  4 Université
Côte d’Azur – CNRS, CRHEA, Valbonne, France  ${\dagger}$
<EMAIL_ADDRESS>(corresponding author)

###### Abstract

Strongly correlated quantum particles in lattice potentials are the building
blocks for a large variety of quantum insulators, for instance Mott phases and
density waves breaking the lattice symmetry. Such collective states are
accessible to bosonic and fermionic systems. To expand further the spectrum of
accessible quantum matter phases, mixing both species is theoretically
appealing, since density order then competes with phase separation. Here we
manipulate such Bose-Fermi mixture by confining neutral (boson-like) and
charged (fermion-like) dipolar excitons in an artificial square lattice of a
GaAs bilayer. At unitary lattice filling, strong inter- and intra-species
interactions stabilise insulating phases when the fraction of charged excitons
is around (1/3, 1/2, 2/3). We evidence that dual Bose-Fermi density waves are
then realised, with species ordered in alternating stripes. Our observations
highlight that dipolar excitons allow for controlled implementations of Bose-
Fermi Hubbard models extended by off-site interactions.

Strongly interacting quantum particles confined in periodic potentials are
specifically described by the Hubbard model (HM) Salomon_2010 ; Arovas_2022 .
This Hamiltonian constitutes one of the most celebrated models of condensed
matter physics. For fermions, the HM has been extensively studied, underlying
the current theoretical description of high temperature superconductivity
Arovas_2022 ; Tranquada_2015 . For bosons, it has also raised considerable
attention, triggered by pioneering experiments with ultra-cold atoms
Greiner_2002 ; Chin_2009 .

Recently, the physics of the HM has been fostered by two-dimensional
semiconductors. These offer new opportunities to study bosonic and fermionic
realisations of Hubbard Hamiltonians. Triangular moiré lattices obtained by
interfacing atomic monolayers provide a successful example for Fermi-Hubbard
physics. Indeed, moiré lattices have enabled electronic Wigner crystals Li_21
; Regan_20 , as well as a wide range of insulating density waves (DWs) at
fractional lattice fillings Huang_21 ; Xu_20 ; Jin_21 . For bosons, the HM has
been implemented with dipolar excitons that are Coulomb bound electrons and
holes spatially separated in a GaAs bilayer Combescot_ROPP . These are
efficiently confined in gate-defined electrostatic lattices Butov_lattice ;
Lagoin_2020  and caracterised by their photoluminescence (PL) emission. Mott
insulator and checkerboard solid were then reported at integer and half
fillings respectively Lagoin_22 ; Lagoin_22b , the latter phase evidencing the
extended Bose-Hubbard model Baranov_2011 ; Dutta_2015 .

Here, we implement the Bose-Fermi Hubbard Hamiltonian extended by off-site
interactions, by confining dipolar excitons and holes in an electrostatic
lattice of a GaAs bilayer. These two species experience strong attractions.
Holes are thus captured in lattice sites occupied by excitons, yielding
fermion-like charged excitons (CX). With a charge control down to an ultra-low
residual doping level, we implement the Bose-Fermi HM by controlling neutral
and charged excitons fillings, $\nu_{X}$ and $\nu_{CX}$ respectively. At
$(\nu_{X}+\nu_{CX})=1$, we uncover incompressible states for fractional values
of $\nu_{CX}$, around (1/3, 1/2, 2/3). We find spectroscopic evidence that
neutral and charged excitons then implement alternating striped DWs. Thus, we
confirm the rich variety of quantum insulators accessible to Bose-Fermi
mixtures Buchler_2003 ; Maciej_2004 ; Hofstetter_2008 ; Sugawa_2011 .

As illustrated in Fig.1.a, we study a field-effect device where an array of
surface gate electrodes, polarised at $V_{g}$, applies a transverse electric
field sinusoidally varying in the plane of a GaAs double quantum well. The
latter confine electrons and holes, spatially separated in each layer and
Coulomb bound to realise dipolar excitons. These are characterised by their
permanent electric dipole, interacting with the applied field. Hence, we
confine excitons in a 250 nm period square lattice, with around 250 $\mu$eV
depth. Excitons then have two accessible confined states (Wannier states – WS)
separated by 150 $\mu$eV (Ref.Lagoin_22b  and Methods). Moreover, note that
excitons are optically injected, using a 100 ns long laser excitation repeated
at 1 MHz. The laser excitation power $P$ controls the lattice filling and we
study the photoluminescence (PL) radiated 300 ns after extinction of the laser
pulse (Methods). This delay is around half the excitons effective lifetime.
Hence, carriers are efficiently thermalised down to our lowest bath
temperature (T=330 mK) Beian_2015 .

Figure 1.c reports the PL spectra radiated by our device, for different gate
voltages and for a lattice filling around 1. Let us first note that here and
in the following the PL is averaged over a (3x2.5) $\mu m^{2}$ region, thereby
including around 120 lattice sites (Methods). While at $V_{g}=-1.02$ V the top
panel of Fig.1.c shows that the PL is mostly due to the recombination of
dipolar excitons (high energy peak), we observe that a second emission grows
at lower energies when $V_{g}$ is reduced to about $-0.9935$ V (middle panel).
By further decreasing $V_{g}$ to $-0.96$ V, the bottom panel shows that the PL
is mostly given by the low energy contribution.

Dual PL emissions have been previously reported for GaAs bilayers Dietl_2019 ;
Bar_Joseph_21 . It was shown that the low energy line marks the strong
attraction between excitons and excess carriers, yielding (composite)
fermionic complexes usually referred to as charged excitons (CX). On the other
hand, the higher energy PL line signals the radiative recombination of neutral
dipolar excitons. The energy separation between these two contributions,
around 1 meV, well matches theoretical expectations Suris_2003 ; Witham_2018 .
Moreover, we note that excess carriers can only be attributed to holes for our
heterostructure characterised by ultra-high electronic mobilities Loren_21 .
Indeed, holes are trapped in the regions of the bilayer where a perpendicular
field is applied, i.e. inside the lattice, whereas electrons are expelled out
of this zone Kowalik_2012 .

Neutral dipolar excitons are characterised by a linear dependence of their PL
energy with $V_{g}$ Ronen_2007 , scaling like ($e.d|V_{g}|/L$). Here $e$
denotes the electron charge, $d$ the spatial separation between the center of
the two quantum wells and $L$ the thickness of our field-effect device
(Methods). In Fig.1.e we evidence this signature (blue), for $-1.02\lesssim
V_{g}\lesssim-0.92$ such that the lattice depth remains essentially unchanged
(Methods). Furthermore, we note that the PL energy of CX follows the same
scaling (red). We then deduce that holes are captured in the lattice sites
occupied by dipolar excitons. As a result our device efficiently confines both
neutral and charged dipolar excitons, in the same lattice potential. This
conclusion is further confirmed by scrutinising the spectrum of the PL emitted
by CX. Indeed, Fig.1.d underlines that for low lattice fillings the PL
spectrum consists of two emission lines, separated by around 100 $\mu$eV. This
magnitude agrees reasonably with the theoretically expected energy separation
between the 2 WS accessible to an exciton complex made by two holes and one
electron, and for around 250 $\mu$eV lattice depth (Methods).

Figure 1.c and extended data Fig.1 evidence that the fraction of neutral and
charged excitons are efficiently varied by the gate voltage. On the other
hand, we verified that the overall lattice filling $(\nu_{X}+\nu_{CX})$ is
mostly given by the laser excitation power $P$ (extended data Fig.1).
Experimentally, to set $\nu_{X}$ and $\nu_{CX}$ we first adjusted $V_{g}$ so
that $\nu_{CX}\sim 0$. The neutral excitons filling $\nu_{X}$ is then
controlled by $P$, set for instance such that $\nu_{X}=1$ by realising a Mott
insulator. For the chosen laser excitation power, $\nu_{CX}$ is then tuned by
the gate voltage while $(\nu_{X}+\nu_{CX})$ remains mostly unchanged (extended
data Fig.1). Note that neutral and charged exciton fillings are quantified by
computing their PL integrated intensities (Methods). Thus, we possibly assess
the purity of our device. Indeed, extended data Fig.2 shows that by suitably
adjusting $V_{g}$ the PL due to CX can not be distinguished from the spectral
noise for $\nu_{X}\approx 0.5$. This shows that the concentration of residual
holes is bound to $4\cdot 10^{7}$ cm-2 (Methods). Remarkably, this value is
comparable to the record residual doping measured for our heterostructure
Loren_21 , a regime previously inaccessible to optical techniques.

By confining tuneable concentrations of neutral and charged excitons, our
lattice device can emulate Hubbard Hamiltonians, continuously from the bosonic
to the fermionic regime. We have recently quantified the former one Lagoin_22
; Lagoin_22b . Below, we first implement the Fermi Hubbard model by equalising
the fraction of excess holes and excitons. Combined inter-species attractions
and fast carrier tunnelling (Methods) then ensure that the lattice is only
filled with CX.

Exploring the $(P,V_{g})$ parameter space we found two specific combinations,
namely (7.6 nW, -0.93 V) and (14 nW, -0.945 V), for which the CX
compressibility $\kappa_{CX}$ is minimised, see white areas in Fig.2.a-b. In
fact, varying either $V_{g}$ or $P$ from these coupled values leads to a two-
fold increase of $\kappa_{CX}$, towards the level given by poissonian
fluctuations, since excitons (blue area) or holes (red area) are then in
excess. Furthermore, the two incompressible phases are characterised by a a
lorentzian-like PL spectrum, given by our spectral resolution (Fig.2.b-d).
Hence, we deduce that CX mostly occupy the same state in the lattice. In the
charge neutral regime we verified that $\nu_{X}\approx 1/2$ for P= 7.6 nW and
1 for P=14 nW. The spectra displayed in Fig.2.a-b then evidence two insulating
states, at $\nu_{CX}\approx(1/2,1)$, theoretically corresponding to MI and
checkerboard solid respectively Arovas_2022 .

As for neutral excitons Lagoin_22b , from the difference $\Delta$ between the
PL energies at unitary and half fillings we deduce the nearest neighbour (NN)
interaction strength $V_{CX,CX}$ (Fig.1.b). Indeed, at $\nu_{CX}=1$ the energy
of a MI is enhanced by 4$V_{CX,CX}$ compared to a checkerboard phase, since NN
couplings are fully avoided for the latter. Taking into account the slight
difference of gate voltages necessary to realise the two insulating states, we
obtain $\Delta=(-0.015\cdot e.d/L+4\cdot V_{CX,CX})$. This leads to
$V_{CX,CX}=$ (140 $\pm 15$) $\mu$eV. Importantly, this magnitude is confirmed
by studying the thermal melting of the insulating state at $\nu_{CX}=1/2$.
Indeed, extended data Fig.3 evidences that $\kappa_{CX}$ is minimised up to a
critical temperature around 1 K. Thus, our measurements follow theoretical
expectations, namely that a checkerboard solid melts when $k_{B}T$ becomes of
the order of $V_{CX,CX}$ Lagoin_22b ; Troyer_2002 .

In the following, we turn to the Bose-Fermi HM in the regime where the overall
filling $(\nu_{X}+\nu_{CX})$ is set to unity, since strong on-site
interactions prevent double occupancies in our lattice (Methods). We
continuously vary neutral and charged excitons filling fractions by sweeping
the gate voltage, and we monitor simultaneously the corresponding
compressibility, $\kappa_{X}$ and $\kappa_{CX}$ respectively, by statistically
computing PL intensity fluctuations (Methods).

Figure 3.a presents $\kappa_{X}$ (blue) and $\kappa_{CX}$ (red), normalised by
the level set by poissonian fluctuations, as a function of $\nu_{CX}$. For
$\nu_{CX}\lesssim 0.15$, the PL spectrum is dominated by the excitonic
component and $\kappa_{X}$ is minimised. Excitons then realise a boson-like MI
that is weakly perturbed by the fraction of fermionic defects, i.e. by the
fraction of charged excitons $\nu_{CX}$. Similarly, $\kappa_{CX}$ is minimised
for $\nu_{CX}\gtrsim 0.85$ so that charged excitons realise a fermion-like MI
at unitary filling, weakly perturbed by the fraction of (bosonic defects)
excitons $\nu_{X}$. By contrast, $\kappa_{X}$ and $\kappa_{CX}$ are mostly of
the order of the level given by poissonian fluctuations for
$0.2\lesssim\nu_{CX}\lesssim 0.8$. The system evolves then in a normal fluid
phase. Nevertheless, for $\nu_{CX}=$(0.35, 0.52, 0.61) $\pm$ 0.03 we
strikingly observe that $\kappa_{X}$ and $\kappa_{CX}$ simultaneously drop,
manifesting correlated incompressible states.

Combined insulating phases at $\nu_{CX}\sim$ (1/3, 1/2, 2/3) suggest that
neutral and charged excitons realise dual DWs Buchler_2003 ; Maciej_2004 ;
Hofstetter_2008 . To confirm this expectation, we studied $\Delta
E_{X,CX}(\nu_{CX})=(E_{X}-E_{CX}$), $E_{X}$ and $E_{CX}$ being the PL energy
of neutral and charged excitons respectively. $\Delta E_{X,CX}$ quantifies the
average difference between the surroundings of each specie, since NN couplings
are the main interaction channel. Hence, $\Delta E_{X,CX}$ distinguishes the
two types of orders that compete in the insulating regime Buchler_2003 ;
Hofstetter_2008 , namely DWs and spatially separated Mott phases of neutral
and charged excitons (Methods).

We first extract the inter-species NN interaction strength $V_{X,CX}$
(Fig.1.b) by comparing $\Delta E_{X,CX}$ between the two MI regimes (see
extended data Fig.4). Indeed, at $\nu_{CX}\sim$ 0.1 neutral excitons realise a
Mott insulator with a few charged excitons impurities, so that $\Delta
E_{X,CX}(0.1)\sim 4(V_{X,X}-V_{X,CX})$. In the same way, at $\nu_{CX}\sim 0.9$
we obtain $\Delta E_{X,CX}(0.9)\sim 4(V_{X,CX}-V_{CX,CX})$. Accordingly we
find $V_{X,CX}=(V_{X,X}+V_{CX,CX})/2-(\Delta E_{X,CX}(0.1)-\Delta
E_{X,CX}(0.9))/8$. From the measurements shown in Fig.3.b (black points), we
conclude that $V_{X,CX}=(75\pm 10)$ $\mu$eV, since $V_{X,X}=(30\pm 5)$ $\mu$eV
Lagoin_22b  and $V_{CX,CX}=(140\pm 15$) $\mu$eV.

As shown above, repulsive NN interactions between charged excitons greatly
exceed other interaction strengths. As a result, Fig.3b and extended data
Fig.4 show that $\Delta E_{X,CX}$ (dashed line in Fig.3b) is smallest for
spatially separated Mott phases, notably at $\nu_{CX}=$ (1/3, 1/2, 2/3). By
contrast, the greatest amplitude is obtained for a double checkerboard density
wave at $\nu_{CX}=$ 1/2, while intermediate values are found for striped DWs
(Fig.3.b and extended data Fig.4). For this latter ordering, one expects at
$\nu_{CX}\approx 2/3$ that $\Delta E_{X,CX}\sim 2(V_{X,CX}-V_{CX,CX})$,
corresponding to an increase by $2(V_{CX,CX}-V_{X,CX})\approx 130$ $\mu$eV
compared to the exciton MI regime. Remarkably, this enhancement is confirmed
experimentally in Fig.3b, since ($\Delta E_{X,CX}(0.61)-\Delta
E_{X,CX}(0.9))=$ $(100\pm 15)$ $\mu$eV. Hence, we deduce that alternating
stripe phases are realised at $\nu_{CX}=0.61\pm 0.03$, relatively close to the
theoretical 2/3 filling. Similarly, we verify in Fig.3.b that ($\Delta
E_{X,CX}$(0.36)-$\Delta E_{X,CX}(0.12))\sim$ $2(V_{X,CX}-V_{X,X})$. This
matching again reveals that alternating stripes are favoured, with a pattern
symmetric to that for $\nu_{CX}\sim 2/3$. Finally, at $\nu_{CX}\sim 1/2$
Fig.3.b shows that ($\Delta E_{X,CX}$(0.52)$-\Delta E_{X,CX}$(0.12))$=(0\pm
15)$ $\mu$eV, as expected for horizontal or vertical stripes. Otherwise
($\Delta E_{X,CX}$(0.52)$-\Delta E_{X,CX}$(0.1)) would amount to around 220
$\mu$eV (dotted line) for alternating checkerboards (Fig.3.b and extended data
Fig.4).

To conclude, we have characterised the insulating part of the phase diagram
for the Bose-Fermi Hubbard model. Dipolar excitons naturally explore the
regime extended by off-site interactions, so that density waves emerge, even
at half boson/fermion filling unlike for mixtures bound to short-range
interactions Buchler_2003 ; Hofstetter_2008 . Reducing further the carriers
temperature to a few tens of milli-Kelvin, which is within experimental reach,
would extend our studies to the situation where excitons exhibit extended
phase coherence. Mixtures of neutral and charged excitons could then serve as
a platform to engineer supersolid-like states, stabilised by ordered fermionic
phases. Also, the low (boson) doping regime relevant to superconducting
electronic systems may be accessed, by doping Mott insulators of charged
excitons with a coherent fluid of neutral ones.

## Acknowledgments

Research at CNRS (C.L. and F.D.) has been financially supported by IXTASE from
the French Agency for Research (ANR-20-CE30-0032-01). The work at Princeton
University (L.P. and K.B.) was funded in part by the Gordon and Betty Moore
Foundation’s EPiQS Initiative, Grant GBMF9615 to L. Pfeiffer, and by the
National Science Foundation MRSEC grant DMR 2011750 to Princeton University.

## Authors contributions

K.B. and L.P. realised the GaAs bilayer while C.L., S.S. and F.D. fabricated
the gate electrodes imprinting the 250 nm period electrostatic lattice. C.L.
and F.D. performed all experiments, data analysis, and wrote the manuscript.
F.D. designed the project.

## Competing financial interests

The authors declare no financial interests.

Figure 1: Neutral and charged excitons Bose-Fermi mixture. a An array of gate
electrodes polarised at $V_{g}$ imprints an artificial electrostatic 2D
lattice (sinusoidal white wave) for neutral (blue) and charged (red) excitons
confined in a GaAs bilayer (gray lines) embedded in an AlGaAs heterostructure
(black). b In the lattice on-site interactions exceed the confinement depth so
that neutral and charged excitons experience NN interactions only, $V_{X,X}$,
$V_{X,CX}$ and $V_{CX,CX}$. c PL spectra for $V_{g}=(-1.02,-0.9935,-0.96)$ V
at unitary filling, from top to bottom. Lines represent model spectra obtained
by adjusting the PL lines with a lorentzian-like profile limited by our
spectral resolution, blue and red for neutral and charged excitons
respectively. d CX PL spectrum at half-filling reproduced by setting 44 and
56$\%$ occupations for the lower and upper WS, yellow and red respectively.
The WS energy separation is set to 100 $\mu$eV while the black line displays
the modelled profile. e Scaling of the charged and neutral exciton energies as
a function of $V_{g}$, red and blue respectively. Points are wider than the
horizontal error while the vertical error provides our spectral precision
($\pm 15$ $\mu$eV). The solid lines display theoretical variations given by
($-e.d|V_{g}|/L$) (Methods). Experiments have all been realised at $T=330$ mK.
In c and d error bars are given by the poissonian noise.

Figure 2: Charged excitons insulators at unity and half fillings. a-b CX
compressibility $\kappa_{CX}$ normalised to the level set by poissonian
fluctuations, around unitary (a) and half filling (b). Bottom panels display
the variation of $\kappa_{CX}$ as a function of $V_{g}$ for $P$=14 and 7.6 nW
(a and b respectively). Top panels show $\kappa_{CX}$ as a function of $P$ for
$V_{g}$=-0.945V (a) and -0.93V (b). The white area shows the regime where the
hole and exciton densities are equalised so that the lattice only confines
CXs. The blue and red areas show the regimes where excitons and holes are in
excess, respectively. Black lines are guides for the eyes. c-d PL spectrum for
$\nu_{CX}$ = 1 (14nW, -0.945V) and $\nu_{CX}$ = 1/2 (7.6nW, -0.93V),
reproduced by the lorentzian-like profile given by our spectral resolution
(red). Experiments have all been realised at $T=330$ mK, and we statistically
analyse 10 repetitions for every experimental settings. In c-d error bars show
the poissonian noise. In a and b vertical error bars provide the $\pm 0.03$
precision when measuring the compressibility, while horizontal error bars are
smaller than the points size.

Figure 3: Dual density waves of neutral and charged excitons. a Charged (red)
and neutral (blue) excitons compressibility as a function of $\nu_{CX}$ for
$(\nu_{X}+\nu_{CX})=1$. The compressibility is normalised to the value given
by poissonian fluctuations. b Difference between the PL energy of neutral
($E_{X}$) and charged ($E_{CX}$) excitons as a function of $\nu_{CX}$. Black
points represent our experimental data while the black line is a guide for the
eyes. At the bottom, the horizontal dashed line shows that spatially separated
insulators of neutral and charged excitons yield $(E_{X}-E_{CX})$=0.86 meV at
$\nu_{CX}$=(1/3,1/2,2/3), while a double checkerboard DW at $\nu_{CX}$=1/2
leads to $(E_{X}-E_{CX})$=1.30 meV (top horizontal dotted line). Ordered
phases are shown with neutral and charged excitons depicted by blue and red
balls respectively. In a-b grey areas underline insulating phases. Experiments
have all been realised at $T=330$ mK, computing 10 repetitions for every
experimental settings. Error bars signal our instrumental precision for PL
energies ($\pm 15$ $\mu$eV), $\nu_{CX}$ ($\pm 0.03$) and the compressibility
($\pm 0.03$).

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## Methods

###  .1 Sample structure and experimental procedure

The $250\text{\,}\mathrm{nm}$ period electrostatic lattice is identical to the
one studied in Ref. Lagoin_22b . It is based on two $8\text{\,}\mathrm{nm}$
wide GaAs quantum wells, separated by a $4\text{\,}\mathrm{nm}$ AlGaAs
barrier. Dipolar excitons made by spatially separated electrons and holes then
carry a permanent electric dipole with a moment $d$ around 12$e$.nm, where $e$
denotes the electron charge. Also, note that the quantum wells are positioned
$200\text{\,}\mathrm{nm}$ below the surface of the field-effect device where
they are embedded, and $150\text{\,}\mathrm{nm}$ above a conductive layer that
serves as electrical ground.

Electronic carriers are injected in the lattice potential using a laser
excitation at resonance with the direct exciton absorption of the quantum
wells. Hence, we ensure that the concentration of photo-injected free carriers
is minimised, nevertheless not necessarily vanishing. This regime is only
found by suitably adjusting $V_{g}$. In fact charge control is accessed for
correlated values of $P$ and $V_{g}$. This behaviour is illustrated Fig.2,
since the optimal values of $V_{g}$ to engineer CX insulators at $\nu_{CX}$= 1
and 1/2 differ by 0.015 V.

To spectrally analyse the PL radiated by our device we rely on a 1800 lines/mm
grating. The PL is then sampled with 15 $\mu$eV precision so that the
narrowest profile possibly measured, e.g. a laser line, is lorentzian-like
with around 150 $\mu$eV full-width-at-half-maximum. To model PL spectra we
assign this profile to the emission from the WS for neutral and charged
excitons.

As stated in the main text, we extract $\nu_{CX}$ and $\nu_{X}$ by computing
the PL integrated intensities of neutral and charged excitons, these being
directly proportional to the concentration of each species. Importantly, we
verified that the overall lattice filling ($\nu_{X}+\nu_{CX}$) weakly varies
in the explored range of gate voltages. Indeed, extended data Fig.1 shows that
for $P$ fixed the overall PL integrated intensity is rather constant.
Accordingly, we assumed that only $\nu_{X}$ and $\nu_{CX}$ vary with $V_{g}$,
and given the signal-to-noise-ratio of our experiments we deduced that
$\nu_{CX}$ is obtained with $\pm 0.03$ precision.

###  .2 Excitonic compressibility

To quantify the compressibility of neutral and charged excitons we
statistically analyse intensity fluctuations of their PL maxima,
$A_{X}^{(\mathrm{max})}$ and $A_{CX}^{(\mathrm{max})}$ respectively. In
practice, we rely on samples of 10 measurements all performed under fixed
conditions, for every experimental settings. Note that the sample size is
limited by the time (6h) available at our lowest bath temperature, since
acquiring a single 10 measurements sample requires around 20 minutes.

By monitoring PL intensity fluctuations, we directly compute the
compressibility, since $\sqrt{\kappa_{X,CX}k_{B}T}$ is proportional to
$\sigma(A_{X,CX}^{(\mathrm{max})})/\overline{A_{X,CX}^{(\mathrm{max})}}$
according to the fluctuation-dissipation theorem Chin_2009 . Here
$\sigma(A_{X,CX}^{(\mathrm{max})})$ and $\overline{A_{X,CX}^{(\mathrm{max})}}$
denote the variance and the mean value of the PL maximum
$A_{X,CX}^{(\mathrm{max})}$ respectively. Moreover, note that we
systematically rescale neutral and charged excitons compressibility to the
value computed for the level of poissonian fluctuations, for every
experimental settings. Thus, experimental observations are more easily
confronted. Finally, for our sample size, i.e. 10 realisations for each
experimental conditions, we estimated the precision with which the
compressibility is deduced. For that, we computed $\kappa_{X,CX}$ by randomly
selecting 7 measurements out of 10. Hence, we observed that $\kappa_{X,CX}$
varies by around $\pm 0.03$. We then assigned this error bar to measured
compressibility.

###  .3 Neutral and charged excitons in the lattice

Relying on finite element simulations we have computed the lattice profile
confining neutral dipolar excitons. We thus found that the lattice depth
$V_{0}$ is about 250 $\mu$eV when $V_{g}$ is set to $-1$V. Then, excitons are
confined in two WS, separated by around 150 $\mu$eV. Furthermore, we computed
the variation of the lattice depth when $V_{g}$ is varied between $-0.92$ and
$-1.02$ V, as in Figs.1-3. We concluded that $V_{0}$ decreases from 250 to 230
$\mu$eV between these two values, and that the energy separation between the
two accessible WS ranges then from 140 to 150 $\mu$eV. This variation is small
compared to other energy scales, such as the thermal energy. Then it plays a
minimal role in our studies.

We have also simulated the nature of the lattice confinement for CX. Noting
that holes have their minimum potential energy at the position of the lattice
sites, we first considered that holes are captured in lattice sites occupied
by dipolar excitons, favoured by the strong inter-species attraction. For
electron and hole effective masses, $m_{e}=0.07m_{0}$ and $m_{h}=0.12m_{0}$
Witham_2018  respectively, and considering the same 250 $\mu$eV deep lattice,
we find that CX are confined in two WS separated by around 120 $\mu$eV, in
reasonable agreement with experimental findings (Fig.1).

Alternatively, one may consider that CX are confined in the lattice due to the
interaction between their positive charge and the electric field applied in
the plane of the double GaAs quantum wells. To explore this scenario we
computed the lattice depth for a point-like and positively charged exciton. We
deduced that $V_{0}$ would then be around 1 meV, but more importantly that the
point-like charged particle would have access to 3 WS, separated by around 250
$\mu$eV. This conclusion is in contradiction with the measurements shown in
Fig.1. The hole confinement in the lattice is then better captured by
considering their strong Coulomb attraction with dipolar excitons.

###  .4 Hubbard parameters for neutral and charged excitons

We have recently calculated and measured the on-site and NN interactions
between dipolar excitons, $U_{X,X}$ and $V_{X,X}$ respectively Lagoin_22 ;
Lagoin_22b . We thus found that $U_{XX}$ exceeds the lattice depth, it can be
as large as 1 meV for the lowest WS, whereas $V_{X,X}=(30\pm 5)$ $\mu$eV. On
the other hand, the measurements shown in Fig.2-3 yield $V_{CX,CX}=(140\pm
15)$ $\mu$eV and $V_{X,CX}=(75\pm 10)$ $\mu$eV. This evidences that repulsions
between neutral excitons constitute the weakest interaction channel. Also, we
note that on-site intra- and inter-species scatterings largely exceed the
lattice depth so that lattice sites can not be doubly occupied.

In Ref.Lagoin_22b  we have shown that the tunnelling strength between lattice
sites amounts to a few $\mu$eV for neutral excitons. Given that the hole
effective mass is around 2/3 of the exciton one, we deduce that holes benefit
from a tunnelling strength in the $\mu$eV range too. Thus we ensure that the
lattice only confines CX when the hole and neutral excitons concentration are
matched (Fig.2), since our experiments are acquired during a 100 ns long time
interval allowing for hundreds of particle tunneling between lattice sites.

###  .5 Minimum hole doping level

Extended data Fig.2 reports a PL spectrum measured for $\nu_{X}\approx 1/2$
and for a minimised concentration of holes. The spectrum is obtained by
averaging 10 realisations. Its maximum is ($34\pm 1.8)$ counts/min while the
level of spectral white noise is $(3.5\pm 0.6)$ counts/min. The signal-to-
noise-ratio ($SN$) is then about 10 in these experiments. Furthermore, we
deduce that $\nu_{X}\approx 1/2$ translates into a mean concentration of
neutral excitons $n_{X}\approx 4\cdot 10^{8}$ cm-2 for our 250 nm period
lattice. Since we do not distinguish the PL due to charged excitons from the
spectral noise level, we deduce an upper limit for the mean concentration of
holes in these measurements, namely $n_{h}\sim n_{X}/SN\approx 4\cdot 10^{7}$
cm-2. This value, limited by $SN$, is then of the same order as the one
obtained from transport techniques for our GaAs quantum wells ($\sim 1.5\cdot
10^{7}$ cm-2, see Ref.Loren_21 ).

###  .6 Spatial order of incompressible phases

Within the framework of the Hubbard model, one expects theoretically that
density waves emerge for fractional fillings, mostly around (1/3, 1/2, 2/3)
where stripes or checkerboards orders are possibly realised. In our lattice
neutral and charged excitons interact through NN couplings, $V_{XX}\sim
V_{X,CX}/2\sim V_{CX,CX}/5$. We then directly deduce the mean single particle
interaction energy, $\epsilon=(\nu_{X}\epsilon_{X}+\nu_{CX}\epsilon_{CX})$,
$\epsilon_{X}$ and $\epsilon_{CX}$ denoting the mean neutral and charged
excitons interaction energies respectively.

At $\nu_{CX}\sim(1/3,2/3)$ DWs necessarily correspond to alternating diagonal
stripes of neutral and charged excitons. We find that these are characterised
by $\epsilon$= 4/3($2V_{X,CX}+V_{X,X}$)$\approx$ 240 $\mu$eV and
4/3($2V_{X,CX}+V_{CX,CX})$)$\approx$ 387 $\mu$eV at $\nu_{CX}=$ 1/3 and 2/3
respectively (see extended data Fig.4). On the other hand, phase separated
Mott insulators exhibit mean interaction energies around 267 and 413 $\mu$eV
respectively. These are thereby unfavourable. The previous explicit
evaluations illustrate that alternating stripe phases exhibit lower energies
than separated Mott phases when $V_{X,CX}<(V_{X,X}+V_{CX,CX})/2$. Importantly
this condition is satisfied in our experiments, since
$V_{X,CX}=(V_{X,X}+V_{CX,CX})/2-(\Delta E_{X,CX}(0.1)-\Delta E_{X,CX}(0.9))/8$
as detailed in the main text.

At $\nu_{CX}\sim 1/2$, we deduce that $\epsilon=4V_{X,CX}\approx 300$ $\mu$eV
for a double checkerboard DW, and $(2V_{X,CX}+V_{X,X}+V_{CX,CX})\approx 320$
$\mu$eV for a striped DW. On the other hand
$\epsilon=2(V_{X,X}+V_{CX,CX})\approx 340$ $\mu$eV for separated Mott phases
of neutral and charged excitons. Again, we recover that DW arrangements are
energetically favoured over phase separation. However, we also find that the
alternating stripe phase exhibits a larger mean interaction energy than the
checkerboard one. This conclusion contrasts with our observations. This may
first suggest that the precision with which we extract NN interaction
strengths is not sufficient to accurately determine the DW ground-state
energy. Alternatively, one can not exclude that the alternating stripe phase
at $\nu_{CX}$=1/2 is of metastable nature. Indeed, insulating states with
Bose-Fermi mixtures are characterised by instabilities, triggered by a
sensitive competition between inter- and intra-species interaction strengths
Buchler_2003 ; Maciej_2004 ; Hofstetter_2008 .

Extended data Fig.1: Lattice filling vs. gate voltage. a Total integrated
intensities of the PL radiated by neutral and charged excitons, as a function
of $V_{g}$ and at unitary filling ($P$=14 nW). b Same experimental results as
in a but expressed as a function of $\nu_{CX}$. c Scaling of $\nu_{CX}$ as a
function of $V_{g}$ deduced from the measurements shown in a and b. The line
provides a guide for the eyes. Experiments were all realised at 330 mK and
acquired during four different experimental runs so that detection
efficiencies are close but not identical. Vertical error bars display the
poissonian precision in a-b and the $\pm 0.03$ precision on $\nu_{CX}$ in c.
In a-c, the horizontal error is smaller than the points size while in b it
corresponds to the precision when extracting $\nu_{CX}$.

Extended data Fig.2: Evaluation of the residual doping level. PL spectrum
radiated by neutral dipolar excitons for $\nu_{X}\approx 1/2$ (at
$\nu_{CX}\approx 0$). The spectrum is measured by averaging 10 realisations
performed under unchanged conditions. The profile is given by our spectral
resolution, i.e. reproduced by a single lorentzian-like line with around 150
$\mu$eV full-width-at-half-maximum (blue area and black line). Measurements
were performed at 330 mK, error bars displaying the level of poissonian
fluctuations.

Extended data Fig.3: Thermal melting of CX insulators at $\nu_{CX}=1/2$ and 1.
a Compressibility $\kappa_{CX}$ normalised to the level given by poissonian
noise for ($\nu_{CX}=1/2$, $\nu_{X}\approx 0$) as a function of the bath
temperature. b Identical measurements for ($\nu_{CX}=1$, $\nu_{X}\approx 0$) .
While in a the thermal melting of the insulating phase occurs around 1K, as
expected for the magnitude measured for $V_{CX,CX}$, a similar critical
temperature is found in b for the Mott phase. This possibly reflects
fluctuations of the density of injected holes while the bath temperature is
increased. For all measurements error bars mark our statistical precision when
computing the compressibility ($\pm 0.03$).

Extended data Fig.4: Interaction energies and spatial ordering. Possible
configurations of incompressible phases made by neutral (blue) and charged
(red) excitons. The respective energy shifts of PL energies, $E_{X}$ and
$E_{CX}$, are indicated below each configuration together with the resulting
magnitude of $\Delta E_{X,CX}$, by only taking into account NN interactions.