# Strongly coupled magneto-exciton condensates in large-angle twisted double
bilayer graphene

Qingxin Li1† Yiwei Chen1† LingNan Wei1† Hong Chen1† Yan Huang1 Yujian Zhu1
Wang Zhu1 Dongdong An1 Junwei Song1 Qikang Gan1 Qi Zhang1 Kenji Watanabe2
Takashi Taniguchi3 Xiaoyang Shi4 Kostya S. Novoselov 5 Rui Wang1∗ Geliang Yu1∗
Lei Wang1∗ 1National Laboratory of Solid-State Microstructures, School of
Physics, Nanjing University, Nanjing, 210093, China  2Research Center for
Electronic and Optical Materials, National Institute for Materials Science,
1-1 Namiki, Tsukuba 305-0044, Japan.  3Research Center for Materials
Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  4Department of Earth and Environmental Engineering,
Columbia University, New York, NY 10027, USA  5Institute for Functional
Intelligent Materials, National University of Singapore, Building S9, 4
Science Drive 2, Singapore, 117544  †These authors contributed equally to this
work.  ∗Corresponding authors, Emai<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Excitons, the bosonic quasiparticle emerging from Coulomb interaction between
electrons and holes, will undergo a Bose-Einstein condensation(BEC) and
transition into a superfluid state with global phase coherence at low
temperatures. An important platform to study such excitonic physics is built
on double-layer quantum wells or recent two-dimensional material
heterostructures, where two parallel planes of electrons and holes are
separated by a thin insulating layer. Lowering this separation distance ($d$)
enhances the interlayer Coulomb interaction thereby strengthens the exciton
binding energy. However, an exceedingly small $d$ will lead to the undesired
interlayer tunneling, which results the annihilation of excitons. Here, we
report the observation of a sequences of robust exciton condensates(ECs) in
double bilayer graphenes twisted to $\sim$10$\degree$ with no insulating mid-
layer. The large momentum mismatch between the two graphene layers well
suppress the interlayer tunneling, allowing us to reach the separation lower
limit $\sim$ 0.334 nm and investigate ECs in the extreme coupling regime.
Carrying out transport measurements on the bulk and edge of the devices, we
find incompressible states corresponding to ECs when both layers are half-
filled in the $N=0$ and $N=1$ Landau levels (LLs). The comparison between
these ECs and theoretical calculations suggest that the low-energy charged
excitation of ECs can be meron-antimeron or particle-hole pair, which relies
on both LL index and carrier type. Our results establish large-angle twisted
bilayers as an experimental platform with extreme coupling strength for
studying quantum bosonic phase and its low-energy excitations.

An EC is a Bose–Einstein condensate formed when a large number of electron-
hole pairs occupy the ground state with macroscopic phase coherence [1]. In
bulk materials, condensed excitons can be generated by optical pumping but
with short lifetimes [2]. In small-bandgap semiconductors and semimetals, ECs
are predicted to live for longer time whereby exciton binding energy exceeds
the charge gap [3]. But the structural character of spontaneous symmetry
breaking in these solid-state systems may hamper the possibility to realize
superfluidity [4, 5]. Double-layer system subject to finite magnetic field is
shown as impressive platform for exciton condensation [6, 7]. As the
recombination is blocked by mid-layer insulator, electron-like carriers in a
partially filled LL in one layer couple with hole-like carriers in the other,
forming interlayer magneto-excitons, which then experience a Bose-Einstein
condensation to a coherent superfluid ground state [8, 9, 10, 11].

The energy of the magneto-excitons is determined by the ratio of intralayer
and the interlayer Coulomb interaction: $E_{intra}/E_{inter}\sim d/l_{B}$,
where $l_{B}=\sqrt{\hbar/eB}$ is the magnetic length, $\hbar$ is the reduced
Planck constant, $e$ is electron charge, and $B$ is magnetic field. An
attractive characteristic of such quantum hall bilayer structure is that
$d/l_{B}$ can be tuned by $B$ and $d$, providing an opportunity to adjust the
electron-hole coupling strength and average distance between excitons. In this
way, it facilitates exploring quantum condensate phase changes in bosonic
system, e.g., the crossover of weak-coupling Bardeen-Cooper-Schrieffer (BCS)
pairing to a strong-coupling BEC pairing [12] and the transition of superfluid
coherent phase to translational symmetry breaking supersolid coherent phase
[13]. However, the intriguing region with extremely strong coupling which
needs tiny $d/l_{B}$ remains elusive due to the difficulty in achieving
extremely small $d$ without raising interlayer tunneling.

Recently, progress on reducing the $d$ down to sub-nanometer has been made in
natural bilayer WSe2, where interlayer tunneling was avoided by the intrinsic
spin-valley structure [14]. However, the unipolar nature of such semiconductor
system limits the observation of magneto-excitons only on the hole side.
Another candidate reaching such a small $d$ is the large-angle twisted
graphene system, where a large momentum mismatch between different sheets
suppresses the interlayer tunneling [15], making it possible to realize ECs in
the strong coupling limit with layers widely covering both the electron and
hole sides. Although recent studies have shown some plausible traces of ECs in
such twisted bilayers by observing quantum Hall states(QHSs) at some
incomplete odd-integer total fillings limited on one carrier side [16, 17],
due to contact quality issues and disorders[16], the sequence of ECs is yet to
be observed.

Here, we reported the observation of a complete sequence of ECs emerging at
both electron and hole fillings with extremely strong coupling strength in
high quality large twisted angle twisted double bilayer graphene(TDBG)
devices(FIG. 1a). At finite magnetic field, the interlayer tunneling gap due
to spatial wave functions overlap in two bilayers can be
negligible($10^{-10}$meV) based on our numerical calculation(see Supplementary
Information section 4). By measuring the bulk and edge transport properties,
we unambiguously identified these robust ECs which appear at the half-filled
$N$ = 0 and $N$ = 1 LL. Thermally activated measurements combined with
theoretical models indicate that the low-energy charged excitation of ECs is
topologically nontrivial spin-texture in $N$ = 0 LL[18], whereas for $N$ = 1
LL, it changes from such spin-texture on the hole side to particle-hole pair
on the electron side.

We fabricated high-quality TDBG devices with ‘cut and pick-up’ transfer method
[19, 20] by picking up and twisting two pieces of bilayer graphene, cut from a
single flake, to an angle about $10\degree$. (see in Supplementary Information
section 1). Depicted in FIG. 1b, the structure of device, which contains top
and bottom graphite gates with voltages $V_{T}$ and $V_{B}$, allows us to
independently tune carrier density: $n=(C_{B}V_{B}+C_{T}V_{T})/e$, and
displacement field: $D=(C_{B}V_{B}-C_{T}V_{T})/2$, where $C_{T}$($C_{B}$) is
top(bottom) gate capacitance and $e$ is the elementary charge. FIG. 1c shows
the longitudinal resistance $R_{xx}$ as a function of $n$ and $D$ in the
absence of a magnetic field. A high resistance state appearing around zero
values of $n$ and $D$ suggests the presence of crystal fields [15], which
occurs due to the imbalance of electron occupancy between the outer two layers
and inner ones. Upon increasing the displacement field, the high resistance
state evolves into two splitting resistive traces, dividing the diagram for
$D>$ 0 into five regions(FIG. 1d), which correspond to the different carrier
population configurations in the two bilayers. In region I and V, both
bilayers are simultaneously populated by either holes or electrons
respectively; in region II and IV, one of the bilayers is gapped while the
other is filled; in region III, the two bilayers are populated by opposite
carrier types and the system contains a mixture of holes and electrons. This
layer-selective population behavior evidences the two bilayers in our large-
angle TDBG are decoupled, allowing the top and bottom gates to control them
separately [21, 22].(Supplementary Information section 2).

Next, we investigate the behavior of system under magnetic fields. FIG. 1f
plots the $R_{xx}$ versus the $D$ and total filling factor $\nu_{tot}$ at $B$
= 14T, where $\nu_{tot}$ = $\nu_{T}$ \+ $\nu_{B}$, and $\nu_{T}$, $\nu_{B}$
are the LL filling fractions of the top and bottom bilayer respectively. In
bilayer graphene(BLG), the zero-energy LL(ZLL) has eight-fold degeneracy
(spin, valley, and accidental orbital degeneracy $N$ = 1, $N$ = 0) and higher
LLs have four-fold degeneracy [23]. As for our system consisting of two
decoupled bilayer graphene layers, LLs have an extra two-fold degeneracy
corresponding to ‘top layer’ and ‘bottom layer’ regulated by $D$. At $B$ =14T,
these degeneracies are fully lifted, showing a sequence of QHSs at all integer
fillings as resistance minima lines paralleling to the $n$ = 0 line.
Strikingly, we observed some repeated ‘4 $\times$ 4’ matrices (for
$\left|\nu_{tot}\right|$ $>$ $8$) and a unique ‘8 $\times$ 8’ matrix (for $-8$
$<$ $\nu_{tot}$ $<$ $8$). These matrices can be qualitatively understood using
the single-particle picture of LL crossings as illustrated in FIG. 1e and FIG.
1g. In a ‘4 $\times$ 4’ matrix, along the white dashed line (FIG. 1f) at $D$ =
0, four QHSs at even total fillings stabilized by the spin and valley
degeneracy lifting. As $\lvert D\rvert$ increases, the layer degeneracy is
lifted and each of these four LLs splits into two, subsequently intersect with
their neighbors, forming a series of crossing points (marked by yellow circles
in FIG. 1g). At these crossing points, the double bilayer system is gapless
since both bilayers are in partially filled LLs (more details in Supplementary
Information section 3).

Now we move on to look into the ‘8 $\times$ 8’ matrix centered at the charge
neutrality point. In FIG. 2a we plot longitudinal conductance $\sigma_{xx}$
versus $D$ and $\nu_{tot}$ for $-9$ $<$ $\nu_{tot}$ $<$ $9$. A clear ‘8
$\times$ 8’ structural pattern with a forming mechanism analogous to the ‘4
$\times$ 4’ matrices is displayed, which corresponds to crossings of the
quantum Hall octet from two decoupled bilayers(schematically illustrated in
FIG. 2c). Unexpectedly, focusing on $D=0$V/nm, we notice a series of
exceptional crossing points at $\nu_{tot}=-7,-3,-1,1,3$ and 5(red dotted
circles in FIG. 2a and FIG. 2c) manifesting as anomalous states with quantized
Hall conductivity and vanishing longitudinal conductivity, which is drastic
contrast to the finite conductivity in those normal LL crossings.(see FIG.
S4). Besides, for $D\neq$ 0, a few similar anomalous states also develop with
the reduced longitudinal conductivity marked by orange dotted circles in FIG.
2a. These anomalous states are inadequate to be understood from the single-
particle LL crossings picture, where the system should show a finite
conductivity as both bilayer LLs are half-filled.

In order to investigate the origin of the vanishing of $\sigma_{xx}$ at these
crossing points, we further measure the bulk transport properties of these
anomalous states using the configuration shown in FIG. 2b inset [24]. FIG. 2b
maps the bulk resistance $R_{xx-Bulk}$ as a function of $\nu_{tot}$ and $D$ on
the same ‘8 $\times$ 8’ matrix. Along $D$ = 0 V/nm, each anomalous state
manifests a high $R_{xx-Bulk}$ peak while other LL crossings show $R_{xx-
Bulk}$ dips. Based on these two different patterns of $R_{xx-Bulk}$ at the
crossing points, we group them into three categories as shown in FIG. 2d, e,
f. In top panels, the center of these $R_{xx-Bulk}(\nu_{tot},D)$ maps
corresponds to both bilayers are half-filled and we take linecuts at the
crossing points showing $R_{xx-Bulk}$ vs $\nu_{tot}$ in bottom panels. For
normal crossing points(FIG. 2d), the resistance dip in $R_{xx-Bulk}$ linecut
manifests that the system is compressible at these states. This confirms the
single-particle LL crossings picture, meanwhile, also indicates the tunneling
is negligible between two bilayers otherwise a LL anti-crossing gap would be
induced by tunneling [25, 26]. On the contrary, for the anomalous crossing
points along $D$ = 0(FIG. 2e), the prominent resistive peak demonstrates the
state is incompressible which is beyond the picture of single-particle LL
crossing. Given that the tunneling is negligible here, this phenomenon implies
the emergence of a correlation energy gap due to many-body interactions. When
both bilayers are half-filled, interlayer interactions prompt electrons in one
bilayer and holes in the other to form magneto-excitons [11, 10] and condense
into an incompressible superfluid: exciton condensate. Besides, it’s worth
pointing out that the crossing points for $D\neq$ 0(FIG. 2f) show
comparatively weaker bulk-resistance peak compared to those along $D$ = 0.
This is presumably due to the imbalanced occupancy of LL orbitals of top and
bottom bilayers [27, 13] or slight spatial wave functions overlap in two
bilayers induced by finite $D$.

To characterize the ECs, we examined the excitation energy gap at all odd-
integer filling for $-8$ $<$ $\nu_{tot}$ $<$ $8$ with thermal activation
measurements. FIG. 3a shows temperature dependence of bulk resistance ($R_{xx-
Bulk}$) as a function of $\nu_{tot}$ at $D$ = 0 V/nm for $B$ = 14 T. The EC
gap ($\Delta$) shows an unexpected hierarchy and manifests as a nonmonotonic
behavior with $\nu_{tot}$(FIG. 3c). Remarkably, all ECs appear in
$-4<\nu_{tot}\leq 4$ hold obviously larger gap value than those appear in
other odd-integer fillings. In BLG, orbital character of the ZLL has been
mapped out as the function of filling factors and electric fields [27], and
the holes or electrons are fully polarized in a single orbital component
($N=0$ or $1$) covering the whole of accessible parameter space. Based on this
picture, we displayed the distribution of orbital index of these two decoupled
bilayers with filling factors under strong magnetic field in FIG. 3b. Near
$D\approx 0$, LLs of both bilayers with unambiguous orbital index cross with
each other, giving rise to the filling sequence of orbital index throughout
$-8\leq\nu_{tot}\leq 8$ as: $-4<\nu_{tot}\leq 0$, $4\leq\nu_{tot}\leq 8$
corresponds to $N=1$ LL and $-8\leq\nu_{tot}<-4$ , $0\leq\nu_{tot}<4$ are in
accord with $N=0$ LL. As a result, we find that the EC robustness is tightly
associated with LL index and carrier type(electron–hole asymmetry),
specifically, ECs appearing within the $N=1$ orbital of hole side are more
stable than those within the $N=0$ orbital whereas it is opposite on the
electron side.

The orbital wavefunction plays an important role for formation of correlated
states. For example, in BLG, compared to the conventional $N=0$ orbital with
sharper composite-fermion interactions, the $N=1$ orbital has softer
interactions which are beneficial for pairing due to an additional node in the
single-particle wavefunction [28], and that lead to the observation of the
even-denominator fractional QHSs exclusively within LL $N=1$, not $N=0$. [29,
30, 31, 32]. For our system consisting of two decoupled bilayers, different
orbitals may host distinct low energy excitations, which affects the
robustness of ECs. This influence can be better understood by taking
pseudospin magnetism picture into consideration in which pseudospin up (down)
corresponds to an electron or a hole in the upper(lower) bilayer, and
spontaneous-interlayer-coherence broken symmetry occurs as easy-plane pseudo-
ferromagnetism [18]. In this case, considering the finite interlayer spacing
$d$, topologically stable charged vortices known as meron can emerge.[18, 33]
Then the merons and anti-merons pairing, leads to the topologically nontrivial
spin configurations known as skyrmions. Theoretical works have suggested that
the energy of this excitation increases with orbital index $n$ [34]. The
meron-antimeron spin-texture is not the sole low-energy excitation in the
double-layer system. With increasing of LL orbital index, conventional
particle-hole pairs may host lower energy due to shorter-ranged interactions
caused by excessive screening, and overtakes meron-antimeron pairs. A recent
study suggests that in p-type bilayer WSe2 the spin-texture charged excitation
only occurs in LLs $n\leq 2$ [14], while higher Landau levels host the
particle-hole excitation whose energy decreases with orbital index $n$.

In our TDBG system, theoretical calculations suggest that the low-energy
charged excitation of ECs is not only related to LL index but also associated
with carrier doping type. FIG. 3d shows the theoretical calculation of the
$\Delta-N$(LL index) dependence of these two different charged excitations. It
reveals that the spin-texture charged excitations for ECs are favoured in the
lower LLs on both electron and hole sides, and switch to particle-hole type at
higher LL index. However, the transition points between two types of charged
excitations are different on electron and hole sides, with the EC-gap maximum
occurring between $N$ = 1 and $N$ = 2 for hole side and near $N$ = 0 for
electron side. This discrepancy can be well understood by considering the
different screening strength between the hole and electron sides. Being in ZLL
of TDBG at $D$ =0, with the total filling increasing, carriers populate QHSs
with different spin-valley flavour from $\nu_{tot}=-8$ to 8 sequentially. At
the same time, increasing particle density strengthens the screening, driving
a shorter-ranged interaction. This, in turn, reduces the excitation energy of
particle-hole pair originating from exchange interactions, whereas leaving the
spin-texture excitation energy unaffected. This theoretical result well agrees
with our observed EC-gap trend. On the hole side, ECs in $N$ = 1 and $N$ = 0
hold spin-texture excitations, thereby excitation energies monotonically
increase with the LL index. Meanwhile, on the electron side, ECs in $N$ = 1
with reduced particle-hole pair excitation energy, have smaller gaps compared
to the ECs in $N$ = 0 with spin-texture excitations.

The evolution of the bulk resistance with $D$ provides further insight into
the identification of the two types of charged excitation. FIG. 3e plots the
bulk resistance as a function of $D$ for ECs at $\nu_{tot}$=1 and 5,
corresponding to spin-texture and particle-hole excitation type respectively.
The red curve regions around $D=0$ mark EC regime and the system transition
into integer quantum hall(IQH) phase(grey curve) with the increasing of $D$.
Previous numerical study find that the excitation energy of meron–antimeron
pairs show a sharp increase in gap with layer imbalance, while particle–hole
excitation energy is independent of the layer imbalance until the zeeman
energy exceeds the EC gap [35]. In our system, layer imbalance is regulated by
$D$. In this scenario, we find a sharp $R_{xx-Bulk}$ decrease with $D$ in red
curve region for $\nu_{tot}$ = 1 whereas at $\nu_{tot}$ = 5, $R_{xx-Bulk}$
hold a mild response with $D$ (Fig .3e). This suggests that ECs on electron
side in $N$ = 0 LL host the spin-texture charged excitation while ECs in $N$ =
1 LL have a particle–hole excitation(At $\nu_{tot}$= 1 in $N$ = 0 LL, red
curve corresponding to meron–antimeron-type ECs with lager slope than gray
curve corresponding to IQH, and At $\nu_{tot}$= 1 in $N$ = 1 LL, red curve
corresponding to particle–hole-type ECs with smaller slope than gray curve
corresponding to IQH.). On the other hand, on the hole side, ECs in both $N$ =
0 and 1 exhibit sharp $R_{xx-Bulk}$ changes, corresponding a spin-texture
charged excitation(FIG. S8). This result is consistent with our theoretical
calculation that the low-energy charged excitation of ECs is different on
electron and hole sides.

Finally, to fully identify the nature of ECs, we demonstrate the evolution of
ECs with magnetic field. FIG. 4a shows $B$ dependence of $R_{xx-Bulk}$ for all
the ECs at $D$ = 0 V/nm. The $R_{xx-Bulk}$ of all ECs decreases with
diminishing of $B$ indicating that our system is in strong coupling regime. In
this regime, the main effect of increasing the magnetic field is to raise the
excitons density($\propto B$), rather than increasing the
$d/l_{B}$($\propto\sqrt{B}$) to soften the exciton pairing strength, which is
preferred in weak coupling system [12]. We further find that all ECs gaps are
positively correlated with magnetic field and well fit by $\Delta$ =
$E_{c}/E_{c}$(14T) (red dashed line in FIG. 4b) which is in line with our
numerical calculations (Supplementary Information section 5). Both of the
particle-hole and the spin-texture excitation energies are related to
stiffness ($\rho_{s}\sim 1/l_{B}$), causing the excitation energy is
proportional to the Coulomb energy ($E_{c}$ = $e^{2}/l_{B}$) (Supplementary
Information section 5). Furthermore, it is worth noting that the $N$ = 1
orbital in BLG differs from conventional $n$ = 1 orbital. In BLG, $N$ = 1
orbital contains a combination of both conventional LL orbital $n$ = 0 and $n$
= 1 wavefunctions distributed on different atomic sites of BLG, with the
relative weight of $n$ = 0 wavefunction increasing with $B$ [32, 31]. Hence,
under higher magnetic fields beyond our experiments (about $B>$ 25T) [32, 31],
the extensive participation of $n$ = 0 wavefunction in BLG $N$ = 1 orbital
renders ECs in decoupled bilayers prone to hold excitation energy deviating
from the trend of $E_{c}$.

In summary, we have experimentally observed remarkable magneto-excitons and
their EC phase in ZLL region of large-angle twisted TDBG. Interlayer tunneling
is suppressed by large momentum mismatch and we demonstrate the ECs in the
strong coupling limit with sub-nanometer atomic separation between the two
bilayers. The different carrier screening strengths in electron and hole sides
lead to distinct stability of ECs in both carrier types and the evolution of
ECs with LL index unveiled a change of the low-energy charged excitation from
meron-antimeron pair to particle-hole pair on different carrier doping types.
The variations in pairing behavior concerning magnetic field, doping, and
temperature are summarized in FIG. 4c. Using electrostatic gating, thus we
achieved unprecedentedly modulating the topology of low-energy charged
excitation of ECs, providing further opportunities on application of skyrmion-
type devices in magnetic data storage and topological quantum computing.
Moreover, the signature of incompressible states in finite displacement field
may lead to an unconventional route to explore non-equilibrium ECs[36] or
multi-polar excitonic [37] physics.

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## Acknowledgments

L.W. acknowledges the National Key Projects for Research and Development of
China (Grant Nos. 2022YFA120470, 2021YFA1400400), National Natural Science
Foundation of China (Grant No. 12074173) and Program for Innovative Talents
and Entrepreneur in Jiangsu(Grant No.JSSCTD202101). K.W. and T.T. acknowledge
support from the JSPS KAKENHI (Grant Numbers 20H00354 and 23H02052) and A3
Foresight by JSPS. K.S.N. is grateful to the Ministry of Education, Singapore
(Research Centre of Excellence award to the Institute for Functional
Intelligent Materials, I-FIM, project No. EDUNC-33-18-279-V12) and to the
Royal Society (UK, grant number RSRP R 190000) for support.

## Author Contributions

L.W., Q.L. conceived the experiment. Q.L., Y.C., L.N.W., H.Y. fabricated the
samples. Q.L., Y.C., L.W. performed transport measurement and data analysis.
R.W., H.C. performed theoretical calculations. K.W. and T.T. supplied hBN
crystals. Q.L., Y.C., H.C. and L.W. wrote the manuscript with input from all
co-authors.

## Competing interest declaration

The authors declare no competing interest.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon request.

Figure 1:  Large-angle TDBG Landau level structure and decoupled carrier
population. a, Illustration of magneto-excitons in TDBG. The interlayer
tunneling is suppressed by large momentum mismatch, and at finite magnetic
field, strong interaction induces electron-like carriers in a half-filled LL
in one layer coupling with hole-like carriers in the other, forming interlayer
magneto-excitons. b, Schematic of the TDBG device structure. c-d, Longitudinal
resistance $R_{xx}$ of TDBG at $B$ = 0 T and $T$ = 1.5 K, versus $n$ and $D$.
The map exhibits the different population of charge carriers in two bilayers
and the ‘layer-targeted’ population divides the diagram into different regions
highlighted in (d). e, Calculated LL spectra as a function of energy $E$ and
$D$ at $B$ = 14 T. Inset: Zooming-in on the LL crossings in the 8 $\times$ 8
matrix (top)(accidental degeneracy for $N$ = 0 and 1 ) and a typical 4
$\times$4(bottom) matrix (for $N\geq 2$), respectively. f, Longitudinal
resistance $R_{xx}$ of TDBG at $B$ = 14 T and $T$= 1.5 K, versus $\nu_{tot}$
and $D$. White dashed line marks the four QHSs stabilized by the spin and
valley degeneracy lifting at $D$ = 0 V/nm in non-zero LL. g, Schematic
illustration of LL crossings driven by displacement field under a constant
magnetic field. ‘T’: top bilayer; ‘B’: bottom bilayer.  Figure 2:  ECs in the
zero-energy Landau level matrix. a-b, The longitudinal conductance
$\sigma_{xx}$ (a) and bulk resistance $R_{xx-Bulk}$ (b) versus $D$ and
$\nu_{tot}$ at $B$ = 14 T and $T$ = 1.5 K for$-9<\nu_{tot}<9$. Dotted red
circles(a) and orange circles mark the LL crossings manifesting as anomalous
conductivity minimum states at $D$ = 0 and $D\neq$ 0, respectively. The insets
show the measurement configurations. To measure $R_{xx-Bulk}$, two contacts
are grounded to make sure the signal comes from the bulk instead of edge
resistance of the sample. c, Schematic LL diagram for$-8<\nu_{tot}<8$. These
LL crossings originate from the cross of zero-energy LLs octet of two
decoupled bilayers which are marked by different colours. Yellow shades mark
two typical IQH regions with the filling factor marked as
($\nu_{T}$,$\nu_{B}$). d-f, Top panel displays enlarge images of $R_{xx-Bulk}$
versus $\nu_{tot}$ and $D$ at $B$ = 14 T around the three categories LL
crossings illustrated in maintext. In each bottom panel, the black curve is
the bulk resistance linecut along the red dashed line in the top panel, while
the blue curve is the longitudinal conductance linecut in (a). The yellow,
red, and orange arrows point to normal LL crossings, ECs, and less-developed
ECs, respectively.  Figure 3:  Energy gap of ECs and the low-energy charged
excitations. a, Temperature dependence of $R_{xx-Bulk}$ as a function of
$\nu_{tot}$ at $D$ = 0 V/nm and $B$ = 14 T. Red dotted lines mark the ECs
corresponding to both bilayers are half-filled. b, Evolution of orbital
occupancy($N$ =0, $N$ = 1) with total filling factors in decoupled TDBG. c,
Excitation energy gaps of ECs at $B$ = 14T and $D$ = 0 V/nm. Purple rectangles
correspond to ECs with the spin-texture charged excitation and green rectangle
corresponds to EC with the particle-hole charged excitation. d, Theoretical
calculations of energy gap for two types of excitations on hole and electron
sides, at $d/l_{B}$ = 0.1 and $D$ = 0 V/nm. Owing to the distinct electron and
hole environments, we use the parameter ’$w$’ to quantify the strength of
screening effects. Since the screening effect of carriers is weak when filling
the zero-energy Landau level starting from hole side, we take $w_{h}=0.1$.
Conversely, the screening effect is already significant when carriers fill the
electron side, so we take the average screening effect for electron side as
$w_{e}=1.2$. The spin-texture excitation energy increases with LL index while
particle–hole excitation energy decreases with LL index. Their crossing points
for hole and electron sides appear at different positions of LL index due to
the screening strength increasing with total filling. The filled markers
represent the low-energy excitations of ECs. e, $R_{xx-Bulk}$ as a function of
$D$ for the ECs at $\nu_{tot}$= 1, 5, which correspond to $N$= 0 and 1 LL,
respectively. Red curve regions mark ECs and grey curves correspond to IQH
regime. At $\nu_{tot}$= 1 in $N$ = 0 LL, red curve corresponding to
meron–antimeron-type ECs with lager slope than gray curve corresponding to
IQH, and At $\nu_{tot}$= 1 in $N$ = 1 LL, red curve corresponding to
particle–hole-type ECs with smaller slope than gray curve corresponding to
IQH.(The rate of change of the bulk resistance with the displacement field
near $D=0$ (the magnitude of the slope) determines the phase boundaries of
different phases.)  Figure 4:  Evolution of ECs with magnetic field. a, The
$R_{xx-Bulk}$ versus $B$ and $\nu_{tot}$ at $D$ = 0 V/nm and $T$ = 1.55 K for
$-9<\nu_{tot}<9$. b, Scaled activation gap $\Delta$/$\Delta$(14T) of ECs at
different total fillings as a function of $\sqrt{B}$ for $9$T $\leq B\leq 14$
T. The red dashed curve corresponds to the scaled Coulomb energy,
$E_{c}$/$E_{c}$(14T) ($\propto\sqrt{B}$). Insets: details of activation gaps
of each EC at different magnetic fields. c, Illustration of the low-energy
charged excitations of ECs and their evolution with tuning temperature($T$),
magnetic field($B$) and carrier density. The bottom panel shows four types of
merons(antimerons) (left) and free electrons and holes(right). A meron and an
antimeron have opposite vorticity, but carry the same electric charge($\pm$
$e$/2) and they can pair to form skyrmion or antiskyrmion which both carry
unit of topological charge($\pm$ $e$) but with opposite charge.(shown in top
left panel). The spin-texture(particle-hole) EC ground states form by pairing
skyrmions and anti-skyrmions (electrons and holes) at low temperature and
finite magnetic field. Increasing doping would make the low-energy charged
excitation of ECs change from spin-texture type to particle-hole type.

Supplementary Information for

Strongly coupled magneto-excitons in large-angle twisted double

bilayer graphene

Qingxin Li et al.

Corresponding authors:Rui Wang<EMAIL_ADDRESS>Geliang Yu,
<EMAIL_ADDRESS>
Lei Wang<EMAIL_ADDRESS>
##  1 Sample fabrication and measurement setup

The three devices (device A$\sim$C) in our text were all fabricated using the
‘cut-and-stack’ technique[19]. The raw materials for the preparation of each
device, hexagonal boron nitride(hBN)(about 30nm), graphite and bilayer
graphene are obtained from mechanically exfoliation onto $Si/SiO_{2}$
substrate. Their thickness and quality were then identified by optical
microscopy and atomic force microscopy. Before stacking, we first cut the
bilayer graphene into two pieces using atomic force microscopy. Then we used
hBN, grphite and precut bilayer graphene pieces to assembled the
graphite/BN/tDBG/BN/grphite stack using dry pick-up technique with a stamp
consisting of polyproylene carbonate(PPC) film and polydimethylsiloxane(PDMS).
Using graphite as the gate above and below the TDBG reduces the disorder and
defects introduced during evaporation compared to metal gates. The stack is
then annealed under high vacuum at $400\degree C$ for 15 minutes. Next we
defined the geometry of the topgate and hall bar by $CHF_{3}/O_{2}$ etching.
Finally, electrode contact was evaporated with Cr/Pd/Au (5/15/100nm) metal by
e-beam evaporation.

Transport measurements were carried out in cryogenic superconducting magnets
with base temperature of 1.5K. The four-terminal resistance were measured
using low-frequency lock-in techniques at 17.777Hz with a current excitation
of $20nA$ .

##  2 Decoupled behavior at finite magnetic field.

Here, we analyze the Shubnikov–de Haas (SdH) oscillations at finite magnetic
field. FIG. S2a shows the plot of longitudinal resistance $\log(R_{xx})$
versus the $D$ and carrier density $n$ at $B$=4T. Noticeable decoupled
behavior, with the same driving mechanism at zero field, is conformed by
evolution of different regions(separated by the black dashed lines) in phase
diagram. For further analyzing decoupled behavior at finite magnetic field, we
present the fan diagram $\log(R_{xx})(n,B)$ at $D$=1.25V/nm(marked as red
rashed line in FIG. S2a) as shown in FIG. S2b. There are five qualitatively
different regions at different $n$ range, which correspond to the different
charge carriers occupation in TDBG, which can be interpreted by single-
particle band structure shown in FIG. S2c-g. In large carrier density like
region I and V, both bilayers (top and bottom) are prone to be populated by
holes and electrons respectively, illustrated in FIG. S2c and g; Reducing
carrier density $n$ to region II or IV, only one piece of bilayer(top or
bottom) is populated while the other bilayer’s fermi energy is in the band
gap, illustrated in FIG. S2d and f; Finally, in the vicinity of CNP(region
III), both bilayers are populated by co-existing hole and electron charge
carriers illustrated in FIG. S2e. The overlapping electron-hole bands are mark
by crossing of red line and blue line, which are on behalf of electron bands
and holes bands,respectively.

##  3 Landau levels crossing structure

We here explore the Landau levels crossing structures in the decoupled TDBG
model. Taking the spin and valley into consideration, the $N=0,1$ Landau
levels for $\tau$ bilayer, where $\tau=\pm 1$ for the top or bottom bilayer
respectively, in the valley $\xi$ are

$\displaystyle\begin{aligned} E_{0}&=\frac{1}{2}\xi u\pm E_{Z}/2+\tau D/2\\\
E_{1}&=\xi\left(\frac{1}{2}u-\delta\right)\pm E_{Z}/2+\tau D/2\end{aligned}$
(S1)

where $\delta=u\hbar\omega_{c}/\gamma_{1}$ reflects the valley splitting, and
the two layers are distinguished by the on-site energies $\pm\frac{u}{2}$
[23]. $D$ denotes the displacement field. For the higher Landau levels, i.e.,
$N>1$, the Landau levels read as

$\displaystyle
E^{\pm}_{N}=\pm\hbar\omega_{c}\sqrt{N(N-1)}-\frac{1}{2}\xi\delta\pm
E_{Z}/2+\tau D/2$ (S2)

When the displacement field varies, the Landau levels of two bilayers tilt in
opposite directions, resulting in a crossing of Landau Levels. The lowest two
Landau levels i.e. $N=0,1$, interlace to form an 8x8 crossing structure shown
in maintext Fig. 1e.

##  4 Numerical estimation of the tunneling gap

There are two possible origins for the gap formed between the electron and
hole Landau levels in the TDBG. One is the excitonic gap due to the
correlation effect, and the other is the hybridization gap due to the electron
tunneling between the two bilayers. The second mechanism can be excluded for
the case with large twisted angle around $10^{\circ}$. This can be achieved by
examining the tunneling strength, i.e., the off-diagonal elements connecting
different bilayers of the TDBG Hamiltonian under magnetic fields.

In the Landau gauge, the eigenstate $\ket{N,k_{y}}$ reads as

$\displaystyle\psi_{Nk_{y}}(x,y)=L_{y}^{-1/2}\exp(ik_{y}y)\phi_{N}(x-l_{B}^{2}k_{y}),$
(S3)

with $n$ denoteing the LL index. At zero magnetic field, the TDBG system
(valley + or -) can be described by a continuum model:

$\displaystyle
H=\sum_{\mathbf{k}}\Psi_{\alpha}^{\dagger}(\mathbf{k})h_{\alpha,\beta}(\mathbf{k})\Psi_{\beta}(\mathbf{k})+\sum_{\mathbf{k},\mathbf{Q}_{j}}\Psi_{\alpha}^{\dagger}(\mathbf{k})T_{\alpha,\beta}(\mathbf{Q}_{j})\Psi_{\beta}(\mathbf{k})$
(S4)

where $\alpha,\beta$ are the combination of layer and sublattice indeices. The
second term of Eq.(S4) describe the moiré hopping from momentum
$\mathbf{k}$(relative to the Dirac point) to momentum
$\mathbf{k}+\mathbf{Q}_{j}$, where $\mathbf{Q}_{1}=(0,\frac{4\pi}{3a_{M}})$,
$\mathbf{Q}_{2}=(-\frac{2\pi}{\sqrt{3}a_{M}},-\frac{2\pi}{3a_{M}})$ and
$\mathbf{Q}_{3}=(\frac{2\pi}{\sqrt{3}a_{M}},-\frac{2\pi}{3a_{M}})$. The TDBG
under magnetic fields is then described by the Landau quantization of
Hamiltonian(S4). From Eq.(S3) and Eq.(S4), we can read off the off-diagonal
matrix elements describing the tunneling between the Landau states from
different bilayers, i.e.,

$\displaystyle U_{N^{\prime}k_{y}^{\prime},Nk_{y}}$
$\displaystyle=w\bra{N^{\prime}k_{y}^{\prime}}\exp(i\mathbf{Q}\cdot\mathbf{r})\ket{N,k_{y}}$
(S5)
$\displaystyle=w\delta_{k_{y}^{\prime},k_{y}+Q_{y}}\sqrt{\frac{\lambda!}{\Lambda!}}\left(\frac{Q_{x}+iQ_{y}}{|\mathbf{Q}|}\right)^{N-N^{\prime}}\left(\frac{i|\mathbf{Q}|l_{B}}{\sqrt{2}}\right)^{|N-N^{\prime}|}$
$\displaystyle\times\exp\left(-\frac{|\mathbf{Q}|^{2}l_{B}^{2}}{4}-\frac{i}{2}l_{B}^{2}Q_{x}(k_{y}^{\prime}+k_{y})\right)L_{\lambda}^{|N-N^{\prime}|}\left(\frac{|\mathbf{Q}|^{2}l_{B}^{2}}{2}\right)$

where $\lambda$ and $\Lambda$ denote the cutoff of the Landau level indices
$N$ and $N^{\prime}$, respectively, and $L_{\lambda}^{N}$ is the associated
Laguerre polynomial [38, 39].

One observes from Eq.(S5) an exponential factor
$\mathrm{exp}(-|\mathbf{Q}|^{2}l^{2}_{B})$, where $|\mathbf{Q}|$ is given by
$4\pi/(3a_{M})$ and $a_{M}=a/(2\sin(\theta/2))$ is the moiré lattice constant.
For large twisted angle $\theta\sim 10^{\circ}$, $|Q|$ is large and the
tunneling strength is exponentially suppressed. We take the experimental
parameters, i.e., $w=110\text{meV}$, $\theta\sim 10^{\circ}$, $B=14\text{T}$,
and $a=0.246\text{nm}$. Then the hybridization gap, which is of the same order
of the tunneling strength, is obtained as $\Delta\sim U\sim
10^{-10}\text{meV}$. This is negligible, and is much smaller than the
experimentally observed gap $\Delta_{ex}$. Since $\Delta\sim U\sim
10^{-10}\text{meV}\ll\Delta_{ex}$, the interbilayer tunneling can be safely
discarded when analyzing the correlation effect, which is dominant and becomes
more manifested in the Landau levels physics under magnetic fields.

##  5 Calculation of the excitation gap

To calculate the excitation gap, according to the results in the last section,
we discard the negligible tunneling gap and consider two graphene bilayers
interacting with each other. The two bilayers are further subjected to a
perpendicular magnetic field, thus exhibit Landau quantization characterized
by the Landau index $N_{1},N_{2}$ respectively. The excitonic gap is only
favored in the balanced case where the electron and hole densities are equal,
thus we focus on the case $N_{1}=N_{2}=N$, i.e., the electron and hole states
are filled to the same Landau level. This leads to a twofold pseudospin
degeneracy of the studied system.

The kinetic energy is completely suppressed under strong magnetic fields with
well formed Landau levels. We thus need to project the Coulomb interaction
onto the Landau levels. The interactions before projection reads as,

$H_{C}=\frac{1}{2}\int
d^{2}\mathbf{x}d^{2}\mathbf{y}V_{\alpha,\beta}(\mathbf{x}-\mathbf{y})[\rho_{\alpha}(\mathbf{x})-\rho_{\alpha,0}][\rho_{\beta}(\mathbf{y})-\rho_{\beta,0}],$
(S6)

where $\rho_{\alpha,0}$ is the average electron density which is determined by
$\rho_{\alpha,0}=\nu_{\alpha}/{2\pi l_{B}^{2}}$, and the indices
$\alpha,\beta$ are the pseudospin indices denoting the two bilayers.
$V_{\alpha,\beta}(\mathbf{x})$ is the Coulomb potential and it has the
following Fourier components

$V_{tt}(\mathbf{q})=V_{bb}(\mathbf{q})=\frac{2\pi}{|\mathbf{q}|},$ (S7)

$V_{tb}(\mathbf{q})=V_{bt}({\mathbf{q}})=\frac{2\pi}{|\mathbf{q}|}e^{-d|\mathbf{q}|},$
(S8)

where $d$ is the spacing between the two bilayers which is set to
$d/l_{B}=0.1$ according to the experiments. The layer spacing within each
bilayer is insignificant compared to $d$ and is neglected here. The next step
is to project Eq.(S6) onto the subspace of each Landau level, which leads to

$V_{N}^{\alpha,\beta}(\mathbf{q})=V_{\alpha,\beta}(\mathbf{q})F_{N}^{2}(q)$
(S9)

where $F_{N}(\mathbf{q})=L_{n}(l_{B}^{2}q^{2}/2)\exp(-l_{B}^{2}q^{2}/4)$ is
the Landau level form factor, and $L_{N}(x)$ is an Laguerre polynomial [40].

In the absence of a displacement field, the top and bottom bilayers are
symmetric in terms of their band structures. Under magnetic fields, and
electron and hole Landau level can exhibits the same energy, leading to the
crossing point as shown by Fig.2c of the main text. Interestingly, the
correlation effect can lift the degeneracy of the LLs at the crossing points.
For example, the attractive Coulomb force between an electron and a hole then
would favor formation of excitons, leading to excitonic gap after
condensation. Besides, the effect of the intra-bilayer interaction $V_{tt}$
and $V_{bb}$ also need to be carefully analyzed.

To analyze the correlation physics at the LL crossing points, we evaluate the
excitation gaps with fully taking into account the interactions among the LLs.
To this end, it is more convenient to introduce the Landau site basis [40], in
which the Hamiltonian in Eq.(S6) is cast into:

$H_{C}=\sum_{mnij}V_{mnij}^{\alpha,\beta}\left[\rho(n,m)^{\alpha}-\nu^{\alpha}\delta_{m,n}\right]\left[\rho(j,i)^{\beta}-\nu^{\beta}\delta_{i,j}\right],$
(S10)

and

$V_{mnij}^{\alpha,\beta}=\frac{1}{4\pi}\int
d^{2}\mathbf{k}V_{N}^{\alpha,\beta}(\mathbf{k})\bra{m}e^{i\mathbf{X}\mathbf{k}}\ket{n}\bra{i}e^{-i\mathbf{X}\mathbf{k}}\ket{j},$
(S11)

where $V_{N}^{\alpha,\beta}(\mathbf{k})$ is the projected coulomb interaction
which is dependent on LLs, and $\rho^{\alpha}(n,m)$ is the Landau site density
operator, where $n$, $m$ denotes the degenerate states within each Landau
level $N$.

We firstly calculate the energy cost to excite a particle-hole pair on top of
the filled Landau state. For $N_{1}=N_{2}=N$, both bilayers have the $N>0$ LLs
occupied, and we assume that Landau levels $0,1,2,\ldots,N-1$ are completely
filled. Then, one considers the electron-hole pair state given by

$\ket{e_{k};h_{l}}=c_{t}^{\dagger}(k)c_{b}(l)\ket{g},$ (S12)

where the ground state has the form
$\ket{g}=\Pi_{n}c_{b}^{\dagger}(n)\ket{0}$, and $\ket{0}$ denotes the vacuum
state. The excitation gap corresponding to the particle-hole excitation is
then evaluated via

$\displaystyle\Delta E$
$\displaystyle=\bra{e_{k};h_{l}}H_{C}\ket{e_{k};h_{l}}-\bra{g}H_{C}\ket{g}$
(S13) $\displaystyle\approx\frac{1}{4\pi}\int
d^{2}\mathbf{k}(V_{tt}(k)+V_{bb}(k))\left[L_{N}(\frac{l_{B}^{2}k^{2}}{2})\right]^{2}e^{-\frac{l_{B}^{2}k^{2}}{2}}.$

This result is clearly LL dependent. The dependence of gap on the field $B$ is
then obtained as $\Delta E(B)\sim 1/l_{B}\sim\sqrt{B}$, as a direct result of
the Landau form factor.

Certainly, it’s crucial to recognize that, owing to the distinct electron and
hole environments, their interactions differ when dealing with charged
excitations. In this situation, electrons experience stronger screening than
holes. We use the parameter ’$w$’ to quantify the strength of screening
effects. We model the electron side Coulomb interaction as $2\pi
e^{-w_{e}ql_{B}}/|\mathbf{q}|$, while the hole side potential is $2\pi
e^{-w_{h}ql_{B}}/|\mathbf{q}|$, here the parameter $w_{e}$ and $w_{h}$
quantify the strength of screening effects of electron and hole sides. Since
the screening effect of carriers is weak when filling the zero-energy Landau
level starting from hole side, we take $w_{h}=0.1$. Conversely, the screening
effect is already significant when carriers fill the electron side, so we take
the average screening effect for electron side as $w_{e}=1.2$. Hence, similar
to the calculation of Eq.(S13), taking into account the variation in screening
parameter, our calculations reveal that the energy of particle-hole type low-
energy excitation on the electron side is lower than that on the hole side
(shown in the FIG. S9)

In addition to particle-hole excitations, there is another typical charged
excitations, i.e., the skyrmion and anti-skyrmion excitations. Treating the
bilayer degrees of freedom as pseudospins, the excitations can carry
nontrivial pseudospin textures. In our system where $d/l_{B}\neq 0$, the
$\hat{z}$ component of the order paramgeter is massive and the system has U(l)
symmetry. In this scenario, topologically stable charged objects called meron
and antimeron form, which have opposite vorticity, but carry the same
topological charge($\pm$ $e$/2) and can pair to skyrmion or antiskyrmion which
both carry unit of topological charge($\pm$ $e$) but with opposite charge. The
skyrmion or anti-skyrmion configuration in the pseudospin textures are
energetically favorable[41], which should be compared in energetics with the
electron-hole excitations. Such pseudospin structures bear topological
nontrivial properties, which was investigated in the context of $SU(M)$
quantum Hall ferromagnets. The minimum excitation energy for a Skyrmion pair
is $\Delta_{SK}=8\pi\rho_{s}$ [42, 43], where

$\rho_{S}=\frac{1}{32\pi^{2}}\int_{0}^{\infty}q^{3}V_{N}^{eff}(q)dq$ (S14)

denotes the stiffness of the order parameter and $V_{N}^{eff}$ is the
projected Coulomb potential[42, 43]. As discussed above, the tunneling between
the two bilayers is negligible for $\theta\sim 10^{\circ}$, and thus the TDBG
can be treated as an untwisted double bilayers when calculating the skyrmion
pair excitation energy. In this scenario, the effective model here is
described by the $SU(M)\times SU(M)$ quantum Hall ferromagnets with $M=4$.
Importantly, one observes that the stiffness in Eq.(S14) is independent of
$M$. Thus, from Eq.(S14), the skyrmion pair excitation energy $\Delta_{SK}$ is
obtained as $\Delta_{SK}\sim e^{2}/l_{B}\sim\sqrt{B}$, where $l_{B}$ is the
magnetic length.

Both of the particle-hole and the spin-texture excitation energys are related
to stiffness ($\rho_{s}\sim 1/l_{B}$), consequently, the excitation energy is
proportional to the Coulomb energy ($E_{c}$ = $e^{2}/l_{B}$). We show the
comparison between two types excitation gap as a function of the LL index $n$
on both of hole and electron sides leads to FIG. S9(Fig. 3d of the main text).

FIG. S1:  Optical image of device. Optical image of three TDBG devices near
$10\degree$ in the text. The dashed red and black lines denote two bilayer
graphene sheets.

FIG. S2:  Decoupled behavior at finite magnetic field. a, Longitudinal
resistance $R_{xx}$ of TDBG at $B$=4T and $T$=1.5K, versus $n$ and $D$. Black
lines separate different regions which correspond to different charge carriers
polarization in two bilayers. b, Longitudinal resistance $R_{xx}$ versus $B$
and $n$ at $D$=1.25V/nm and $T$=1.5K. Red and black lines in region III marked
electron bands and holes bands, respectively. c, Schematic of the band
structure near $K$ point. $K1$ and $K2$ represent the $K$ point of bottom and
top bilayer, respectively. Red shaded areas represent the energy range of
corresponding regions.

FIG. S3:  the 8$times$8 structure and temperature dependence of ECs in other
devices. a-c, Longitudinal resistance $R_{xx}$ of TDBG at $B$ = 14 T and $T$ =
1.5 K, versus $\nu_{tot}$ and $D$ in device A(a) and device B(c). b, Bulk
resistance $R_{xx-bulk}$ versus $T$ and $\nu_{tot}$ at $D$=0V/nm in device A.
d, Longitudinal conductance $\sigma_{xx}$ versus $T$ and $\nu_{tot}$ at
$D$=0V/nm in device B.

FIG. S4:  $\sigma_{xy}$ of all ECs. a, $\sigma_{xy}$ versus $D$ and
$\nu_{tot}$ at $B$ = 14 T and $T$ = 1.5 K. b, The linecut of (a) along the
black dashed line at $D$ = 0 V/nm. A series of anomalous states at
$\nu_{tot}=-7,-3,-1,1,3$ and 5 are marked by red crosses with quantized Hall
conductivity and accompanying vanishing longitudinal conductivity(see maintext
Fig. 2a). Given that the tunneling is negligible here, this phenomenon implies
the emergence of a correlation energy gap due to many-body interactions. When
both bilayers are half-filled, interlayer interactions prompt electrons in one
bilayer and holes in the other to form magneto-excitons and condense into an
incompressible superfluid: exciton condensate.

FIG. S5:  All ECs at $D$ = 0V/nm.  Bulk resistance $R_{xx-bulk}$ versus total
filling factor $\nu_{tot}$ and displacement field $D$ at $B$ = 14 T around $D$
= 0V/nm. Each bottom panel is a linecut along the grey dashed line in the top
panel, which tracks the equal topmost LL population in the two layers.  FIG.
S6:  All ECs at $D\neq$ 0. Bulk resistance $R_{xx-bulk}$ versus total filling
factor $\nu_{tot}$ and displacement field $D$ at $B$ = 14 T when $D\neq$ 0\.
Each bottom panel is a linecut along the grey dashed line in the top panel,
which tracks the equal topmost LL population in the two layers.

FIG. S7:  Arrhenius Fits for all ECs at $\nu_{tot}$ =-7, -1, 1, 3, 5.
Logarithm of bulk resistance $\ln(R_{xx})$ versus $1/T$ at $D$ =0V/nm at
$\nu_{tot}$ =-7, -1, 1, 3, 5.  FIG. S8:  Layer imbalance of ECs on hole side.
$R_{xx-Bulk}$ as a function of $D$ for the ECs at $\nu_{tot}$= -7, -3, -1
which correspond to $N$= 0, 1 and 1 LL, respectively. Red curve regions mark
ECs and grey curves correspond to IQH regime.

FIG. S9:  ECs in the ‘8$times$8’ matrix at different magnetic fields The
longitudinal conductivity $\sigma_{xx}$ versus $D$ and $\nu_{tot}$ at $T$ =
1.5 K and $B$ = 14 T, 11 T, 8 T, 6 T. when $B$ = 14 T, ECs appearing at
$\nu_{tot}=-7,-3,-1,1,3$ and 5 manifest as vanishing longitudinal conductivity
states, and as the magnetic field weakens, these ECs gradually diminish,
giving way to finite conductivity in these fillings.

FIG. S10:  The comparison between two types excitation gap with Landau level
index on electron and hole sides. Theoretical calculations of energy gap for
two types of excitations on hole and electron sides, at $d/l_{B}$ = 0.1 and
$D$ = 0 V/nm. The spin-texture excitation energy increases with LL index while
particle–hole excitation energy decreases with LL index. Their crossing points
for hole and electron sides appear at different positions of LL index due to
the screening strength increasing with total filling. The filled markers
represent the lowest-energy excitations of ECs.