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

Excitons, pairs of electrons and holes, undergo a Bose-Einstein condensation at low temperatures. An important platform to study excitons is double-layer two-dimensional electron gases, with two parallel planes of electrons and holes separated by a thin insulating layer. Lowering this separation (d) strengthens the exciton binding energy, however, leads to the undesired interlayer tunneling, resulting in annihilation of excitons. Here, we report the observation of a sequences of robust exciton condensates (ECs) in double bilayer graphene twisted to ~ 10° with no insulating mid-layer. The large momentum mismatch between two graphene layers suppresses interlayer tunneling, reaching a d ~ 0.334 nm. Measuring the bulk and edge transport, we find incompressible states corresponding to ECs when both layers are in half-filled N = 0, 1 Landau levels (LLs). 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 a novel platform with extreme coupling strength for studying quantum bosonic phase.

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 doublelayer 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 ∼10°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 ∼ 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].Doublelayer 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 ∼ d/l B , where l B = ℏ/eB is the magnetic length, ℏ 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-arXiv:2405.11761v1[cond-mat.mes-hall]20 May 2024 nanometer has been made in natural bilayer WSe 2 , 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 largeangle 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°.(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: 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 layerselective 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].(SupplementaryInformation 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 ν tot at B = 14T, where ν tot = ν T + ν B , and ν T , ν B are the LL filling fractions of the top and bottom bilayer respectively.In bilayer graphene(BLG), the zero-energy LL(ZLL) has eightfold degeneracy (spin, valley, and accidental orbital degeneracy N = 1, N = 0) and higher LLs have fourfold 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 × 4' matrices (for |ν tot | > 8) and a unique '8 × 8' matrix (for −8 < ν 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 × 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 |D| 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 × 8' matrix centered at the charge neutrality point.In FIG.2a we plot longitudinal conductance σ xx versus D and ν tot for −9 < ν tot < 9.A clear '8 × 8' structural pattern with a forming mechanism analogous to the '4 × 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 = 0V/nm, we notice a series of exceptional crossing points at ν 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.(seeFIG.S4).Besides, for D ̸ = 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 σ 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 ν tot and D on the same '8 × 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 (ν tot , D) maps corresponds to both bilayers are half-filled and we take linecuts at the crossing points showing R xx−Bulk vs ν 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 singleparticle 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 [10,11] and condense into an incompressible superfluid: exciton condensate.Besides, it's worth pointing out that the crossing points for D ̸ = 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 [13,27] 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 < ν tot < 8 with thermal activation measurements.FIG.3a shows temperature dependence of bulk resistance (R xx−Bulk ) as a function of ν tot at D = 0 V/nm for B = 14 T. The EC gap (∆) shows an unexpected hierarchy and manifests as a nonmonotonic behavior with ν tot (FIG.3c).Remarkably, all ECs appear in −4 < ν tot ≤ 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 ≈ 0, LLs of both bilayers with unambiguous orbital index cross with each other, giving rise to the filling sequence of orbital index throughout −8 ≤ ν tot ≤ 8 as: −4 < ν tot ≤ 0, 4 ≤ ν tot ≤ 8 corresponds to N = 1 LL and −8 ≤ ν tot < −4 , 0 ≤ ν 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 antimerons 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 meronantimeron 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 WSe 2 the spin-texture charged excitation only occurs in LLs n ≤ 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 ∆ − 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 spinvalley flavour from ν 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 ν tot =1 and 5, corresponding to spin-texture and particlehole 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 ν tot = 1 whereas at ν 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 ν 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 ν 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(∝ B), rather than increasing the d/l B (∝ √ 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 ∆ = 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 (ρ s ∼ 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 [31,32].Hence, under higher magnetic fields beyond our experiments (about B > 25T) [31,32], 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 meronantimeron 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.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 we = 1.

SAMPLE FABRICATION AND MEASUREMENT SETUP
The three devices (device A∼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°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 .

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.

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 τ bilayer, where τ = ±1 for the top or bottom bilayer respectively, in the valley ξ are where δ = uℏω c /γ 1 reflects the valley splitting, and the two layers are distinguished by the on-site energies ± u 2 [23].D denotes the displacement field.For the higher Landau levels, i.e., N > 1, the Landau levels read as 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.
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 ̸ = 0, the ẑ 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(± e/2) and can pair to skyrmion or antiskyrmion which both carry unit of topological charge(± 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 ∆ SK = 8πρ s [42,43], where denotes the stiffness of the order parameter and V ef f N is the projected Coulomb potential [42,43].As discussed above, the tunneling between the two bilayers is negligible for θ ∼ 10 • , 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 ) × 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 ∆ SK is obtained as ∆ SK ∼ e 2 /l B ∼ √ B, where l B is the magnetic length.Both of the particle-hole and the spin-texture excitation energys are related to stiffness (ρ s ∼ 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. 2 .FIG. 3 .
FIG. 2. ECs in the zero-energy Landau level matrix.a-b, The longitudinal conductance σxx (a) and bulk resistance R xx−Bulk (b) versus D and νtot at B = 14 T and T = 1.5 K for−9 < νtot < 9. Dotted red circles(a) and orange circles mark the LL crossings manifesting as anomalous conductivity minimum states at D = 0 and D ̸ = 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 < νtot < 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 (νT ,νB).d-f, Top panel displays enlarge images of R xx−Bulk versus νtot 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.

2 .FIG. 4 .
FIG. 3. Energy gap of ECs and the low-energy charged excitations.a, Temperature dependence of R xx−Bulk as a function of νtot 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/lB = 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 we = 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 νtot= 1, 5, which correspond to N = 0 and 1 LL, respectively.Red curve regions mark ECs and grey curves correspond to IQH regime.At νtot= 1 in N = 0 LL, red curve corresponding to meron-antimeron-type ECs with lager slope than gray curve corresponding to IQH, and At νtot= 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.)

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

FIG. S2 .FIG.
FIG. S2.Decoupled behavior at finite magnetic field.a, Longitudinal resistance Rxx 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 Rxx 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.