Main

Since the discovery of MATBG1,2,3 a central challenge has been to determine whether its quantum states are best understood through a local framework or a topological one. On one hand, its correlated insulating states2,4,5,6,7,8 observed at integer moiré fillings resemble Mott insulators, which are naturally described by localized electrons. On the other hand, its competing Chern insulating states9,10,11,12,13,14,15,16 can only be explained through topological arguments. Perhaps the most puzzling observation, however, is that electrons in MATBG seem to exhibit contradictory behaviours even within the same electronic state: compressibility17 and spectroscopic18 measurements have revealed cascades of Dirac-like compressible states, characteristic of itinerant electrons. Yet, entropy measurements19,20 have uncovered a giant magnetic entropy that could only be explained if nearly all electrons behave as local moments. This noteworthy duality19—simultaneous signatures of both localized and extended behaviour—remains one of the most fundamental unresolved puzzles in the study of MATBG.

A range of theoretical approaches, including Hartree21,22, Hartree–Fock17,23,24,25,26, quantum Monte Carlo27,28, exact diagonalization29,30, density matrix renormalization group31,32,33 and dynamical mean-field theory34,35, showed that interactions can profoundly affect the flat bands predicted by the Bistritzer–MacDonald (BM) model1. A key insight, solidified with the development of the topological heavy fermion (THF)36,37,38 and Mott semimetal39 frameworks, is that the topological nature of MATBG’s flat bands may lead to its electronic duality: although the electrons largely behave as heavy, localized ‘f-electrons’, topology mandates the existence of momentum-space regions in which they act instead as light, delocalized ‘c-electrons’. These two distinct electronic characters are thus predicted to coexist within the same flat bands, occupying different regions of momentum space. Although nano angle-resolved photoemission spectroscopy experiments have provided rough evidence for the presence of flat bands40,41,42,43,44, the absence of a technique capable of resolving their detailed momentum-space structure has, so far, left these fundamental puzzles unanswered.

In this work, we present high-resolution momentum-space images of the interacting energy bands of MATBG, obtained using quantum twisting microscopy at T = 4 K (ref. 45). Away from the magic angle, the observed bands closely follow the BM prediction, showing Dirac points at the mini-Brillouin zone (BZ) corners with renormalized Fermi velocity. At the magic angle, however, the bands are substantially reshaped by interactions: across most of momentum space, the bands become extremely flat and largely gapped, hallmarks of heavy f-electron-like behaviour. By contrast, near the Γ point of the mini-BZ, the bands are dispersive and gapless, consistent with the presence of c-like light electrons. The evolution of the band structure with carrier density reveals Mott-like cascades of the heavy electrons and Dirac-like compressibility dominated by the light electrons. We directly observe how the reshuffling of charge between these light and heavy states underlies the previously reported Dirac revivals17,18. While resolving the long-standing dual-nature puzzle, our measurement also uncovers a new approximately 15 meV excitation, not captured by present models, hinting at an important, yet unaccounted for, degree of freedom in MATBG.

Experimental set-up

Our experiment examines momentum-resolved tunnelling across a two-dimensional junction formed between monolayer graphene (MLG) on the QTM tip (Fig. 1a, purple) and twisted bilayer graphene (TBG) on a flat substrate (Fig. 1a, red/blue). The TBG device includes a metallic back gate and is capped with a bilayer WSe2 tunnelling barrier (Fig. 1a, green). The experiment has two relevant angles: θTBG, the twist angle of the TBG, and θQTM, the angle between the MLG on the tip and the top layer of the TBG. The QTM enables continuous tuning of θQTM during the experiment with millidegree precision. Although θTBG is fixed at any location in the sample, it naturally varies from one location to another as a result of inherent twist-angle disorder, allowing us to explore a range of θTBG values by performing experiments at different spatial locations.

Fig. 1: Measurement set-up and momentum-resolved spectroscopy of TBG with θTBG = 1.2°.
Fig. 1: Measurement set-up and momentum-resolved spectroscopy of TBG with θTBG = 1.2°.The alternative text for this image may have been generated using AI.
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a, Schematic of the QTM set-up (left) and cross-section (centre), featuring MLG (purple) on the QTM tip, separated by a bilayer WSe2 tunnelling barrier (green) from back-gated TBG (top and bottom layers in red and blue). We measure differential tunnelling conductance, dI/dV, between MLG and TBG as a function of bias voltage, Vb. The top view (right) defines the TBG twist angle, θTBG, and the angle between MLG and the top TBG layer, θQTM. b, Momentum-space diagram showing BZs of the TBG’s top (red) and bottom (blue) layers, the resulting mini-BZ (black) and neighbouring mini-BZs (grey). Inset, when the tip is rotated with respect to the sample, the MLG Fermi surface (purple circle) traces an arc in momentum space—the ‘QTM trajectory’ (dashed purple line)—intersecting KT and KB and closely approaching the Γ points in adjacent mini-BZs. c, Calculated single-particle band structure using the non-interacting BM model1 along the QTM trajectory for θTBG = 1.2°. The flat bands feature Dirac points at KT and KB with renormalized Fermi velocity, reaching maximum bandwidth at the ΓL points, at which they are minimally separated from remote bands by energy gaps. The colour scale indicates wavefunction weight on the top TBG layer. d, Measured dI/dV versus θQTM and Vb for θTBG = 1.2° (Methods section ‘Determine the larger twist angle of θTBG = 1.2°’). The top axis marks key momenta along the QTM trajectory: Γ, KT, M and KB. e, Same as d, plotted as normalized differential conductance, dI/dVnorm, in which, for each θQTM, the valence flat-band peak intensity is normalized to unity. f, Calculated momentum-resolved tunnelling conductance using the BM model1, incorporating the MLG–TBG junction electrostatics and MLG probe Dirac dispersion (Methods sections ‘Electrostatics of the QTM junction’ and ‘The tunnelling matrix elements’). For this larger-than-magic-angle TBG, the single-particle model captures rather well the measured data. a.u., arbitrary units.

In momentum space (Fig. 1b), the energy bands of TBG reside within a mini-BZ (black hexagon) whose corners, KT and KB (red and blue), correspond to the K points of its top and bottom graphene layers. By rotating the QTM, we rotate the Dirac point of the MLG probe (purple circle) along an arc in momentum space (dashed purple line) that passes through the KT and KB points of the TBG and closely approaches the two Γ points in adjacent BZs, enabling us to map the TBG energy bands along the Γ–KT–M–KB–Γ trajectory (‘QTM trajectory’).

Figure 1c presents the canonical single-particle BM energy bands of TBG along the QTM trajectory for a twist angle slightly larger than the magic angle (θTBG = 1.2°). Theory predicts that the two low-energy flat bands will exhibit symmetry-protected Dirac points at KT and KB and reach the maximal width at the Γ points, at which energy gaps separate them from the higher-energy remote bands. The bands are coloured by the weight in momentum space projected on the top TBG layer, to which the electrons tunnel.

The energy bands at θ TBG = 1.2°

We begin with QTM measurements of TBG with θTBG = 1.2°. Figure 1d shows the tunnelling dI/dV measured as a function of θQTM and bias voltage, Vb, at moiré filling ν ≈ 0. Recalling45,46 that θQTM and Vb translate to momentum and energy through k = θQTMKD (KD is the Dirac momentum) and E = eVb, this measurement directly maps the energy bands of TBG along the QTM trajectory (the corresponding k-points are indicated on the top axis). The positions of the peaks in dI/dV along the Vb axis reveal the energy of the electronic bands, whereas the amplitudes of the peaks provide insight into the wavefunctions. To better trace the energies of the flat bands, for each θQTM, we normalize the valence flat-band peak to unity and plot the resulting dI/dVnorm in Fig. 1e. The very high resolution in the measurement (δk ≈ 0.1° × KD and δE ≤ 10 meV; Methods section ‘Energy and momentum resolution’) allows us to visualize these bands with unprecedented details. Specifically, we observe a Dirac dispersion with a crossing at KT but with a Fermi velocity that is seven times smaller than that in MLG. At Γ, the flat bands attain their maximum width of 70 meV, separated by gaps of 50 meV and 40 meV from the remote conduction and valence bands, respectively. Moreover, we can also identify van Hove singularities in the remote bands at energies of E = 174 and −171 meV.

Figure 1f presents a theoretical calculation of the momentum-resolved dI/dV, using the non-interacting BM bands at θTBG = 1.2°, considering the tunnelling matrix elements (Methods section ‘The tunnelling matrix elements’) as well as the electrostatics of the MLG–TBG junction and the Dirac dispersion of the MLG probe (Methods section ‘Electrostatics of the QTM junction’). Visibly, for this larger-than-magic-angle TBG, the single-particle model captures well both the band energies and the momentum-dependent tunnelling amplitudes.

The energy bands at the magic angle

We now turn to the central results of our paper—the energy bands of MATBG. The measurement is performed in a region of the sample in which atomic force microscopy (AFM) measurements taken at room temperature observed a regular moiré pattern with a periodicity of approximately 13 nm (Methods section ‘Fabrication and characterization of the QTM tip and van der Waals device’), corresponding to θTBG = 1.1°. Figure 2a shows dI/dV measured with the QTM as a function of θQTM and Vb, at a carrier density close to charge neutrality (ν = −0.2). Figure 2b presents the normalized dI/dV (dI/dVnorm), following the normalization described earlier.

Fig. 2: Momentum-resolved spectroscopy of MATBG.
Fig. 2: Momentum-resolved spectroscopy of MATBG.The alternative text for this image may have been generated using AI.
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a, Measured dI/dV as a function of θQTM and Vb at a filling factor of ν = −0.2 for TBG with θTBG = 1.1°. Key momenta (ΓL, KT, M and KB) are marked on the top axis. b, Same data as in panel a but for each θQTM, we normalize the peak intensity of the valence flat band to unity, defining the quantity dI/dVnorm. c, Extracted flat-band dispersion from panel a (dots), in which we trace the centre of the flat-band peaks versus θQTM, with a guide to the eye (red). Error bar is obtained by the standard deviation of the peak centre within a 0.05° range. Grey lines plot the single-particle BM bands for θTBG = 1.1°. d, The intensity of the conduction flat band, given by integrating the area under its peak Ipeak = ∫dI/dV · dVb in panel a, plotted as a function of θQTM (dots). We compare this with the theoretically predicted momentum dependence of the intensity, given by A(k, ω) · Pt(k) · Ps(k), in which A(k, ω) is the spectral function, Pt(k) is the wavefunction projection on the top TBG layer and Ps(k) = ⟨σx + I⟩k is the sublattice projection. The calculation uses the wavefunctions of the BM model and the different lines correspond to different w0/w1 ratios in this model (Methods section ‘The tunnelling matrix elements’). We observe an excellent fit for w0/w1 = 0.6. a.u., arbitrary units.

The bands observed at θTBG = 1.1° are very different from those at θTBG = 1.2°. They do not show a dispersing Dirac point at KT but instead feature two extremely flat bands, separated by a large energy gap of approximately 35 meV. Notably, the gap appears at almost all momenta along the measured trajectory, except near the Γ point, at which the bands are gapless. At this specific filling, the conduction band seems wider than the valence band, but we note that this is because of further excitations (discussed with Fig. 4). Tracing the centre energy of the bands versus momentum (Fig. 2c, dots) shows that the bands are flat to within about 4 meV. Compared with the bands predicted by the non-interacting BM model for θTBG = 1.1° (grey lines), we see a substantial difference, demonstrating the central role of interactions in determining their shape.

As well as revealing band energies, the spectral function also carries crucial amplitude information. Notably, we observe a gradual reduction in the dI/dV amplitude of the flat bands along the momentum trajectory from KT to KB (Fig. 2a). To quantify the total spectral weight of a specific band—independent of its lifetime—we define the peak intensity, Ipeak = ∫dI/dV · dVb, obtained by integrating the area under its corresponding dI/dV peak. Plotting Ipeak versus θQTM for the conduction flat band (Fig. 2d, dots) reveals a sharp increase near the left Γ point, followed by a gradual decline towards the KT point and a steeper drop to near zero at KB. Beyond KB, the intensity remains negligible, falling below the measurement noise floor.

The magnitude of dI/dV encodes tunnelling matrix element information, offering further insights into the character of individual wavefunctions. Specifically, dI/dV is proportional to A(k, ω) · Pt(k) · Ps(k), in which A(k, ω) is the spectral function, Pt(k) is the projection operator onto the top TBG layer and Ps(k) = ⟨σx + I⟩k is the sublattice projection operator (Methods section ‘The tunnelling matrix elements’). At the magic angle, the BM model predicts that almost all wavefunctions in the mini-BZ have equal population on the top and bottom layers, Pt(k) ≈ 0.5 (Methods section ‘The tunnelling matrix elements’). However, the sublattice projection should exhibit strong momentum dependence, which is highly sensitive to the ratio w0/w1, which reflects the relative size of AA and AB stacking regions. Comparing our measurements with theoretical predictions47 for various w0/w1 ratios (coloured traces), we find excellent agreement for w0/w1 = 0.6, whereas other values show substantial discrepancies. This value represents a notable deviation from the accepted value of 0.8 and could have marked implications on the physics predicted in MATBG33,48.

Evolution of the interacting bands with filling

Figure 3a presents the energy bands at all integer fillings from ν = −4 to ν = 4, as well as at ν = 4.6. At most integer fillings, two sharp and extremely flat bands are observed. At charge neutrality (ν = 0), these bands are symmetrically positioned about the Fermi energy, EF (Vb = 0 mV). On electron doping, the conduction flat band becomes pinned at EF from above, whereas the valence flat band lies ΔE ≈ 15 meV below EF . The opposite occurs for hole doping, in which the valence flat band becomes pinned to EF from below and the conduction flat band shifts to ΔE ≈ 15 meV above it. By ν = 4.6, EF has moved well into the remote bands.

Fig. 3: Evolution of the energy bands with filling.
Fig. 3: Evolution of the energy bands with filling.The alternative text for this image may have been generated using AI.
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a, Momentum-resolved dI/dVnorm (normalized as in Figs. 1e and 2b) versus θQTM and Vb for all integer fillings ν = −4 to ν = 4 and for ν = 4.6 (ΓL and KT indicated on the top x-axis). At all fillings, the bands are flat throughout, except near the ΓL point. At neutrality, the two bands are symmetric around the Fermi level, EF (Vb = 0), and are separated by about 35 mV, except at ΓL, where they touch. At integer electron doping, the conduction flat band is pinned at EF from above, whereas the valence flat band lies ΔE ≈ 15 mV below EF (dashed purple line). At integer hole doping, the valence flat band pins to EF from below and the conduction flat band shifts to ΔE ≈ 15 meV above it (dashed purple line). At ν = 4.6, the flat bands are fully occupied and EF lies within the remote bands near ΓL. b, dI/dVnorm versus θQTM and Vb at fractional fillings between ν = −0.9 and ν = 0.9 (error bar δν ≈ 0.1). The dashed red line marks one band’s evolution from a high-energy smeared ‘plume’ at negative doping to a narrow energy level pinned above EF at positive doping. The dashed blue line marks the opposite evolution of another band. The dashed purple line marks two more energy bands that coexist with the former and remain fixed at Vb = ±15 mV. c, Linecuts of dI/dV versus Vb (black lines) at the same filling factor as panel b, averaged over the flat-band region (Methods section ‘Spectroscopy at different momenta: its Gaussians decomposition and spectral function peak and lifetime analysis versus quasiparticle energy’). The linecuts are decomposed into Gaussian contributions, marked by coloured peaks, revealing a multipeak structure with varying widths and amplitudes. Guiding lines from b are overlaid with the corresponding colour of the Gaussian peaks. The sum of Gaussian contributions in each panel (dashed red lines) shows a good fit. a.u., arbitrary units.

Although the bands shift in a mostly rigid manner—maintaining their flatness across most of momentum space—the behaviour near the Γ point is distinct. At ν = 0, the Γ-point states lie symmetrically between the flat bands, but on electron or hole doping, they undergo a pronounced, filling-dependent energy shift: a local peak at ν = −4 continuously transforms into a dip at ν = 4. This evolution—highlighted schematically in Extended Data Fig. 9—produces an effective ‘stretching’ of the bands around Γ, reaching energy shifts of about 42 meV at ν = 4 and about 54 meV at ν = −4.

This Γ-point stretching arises naturally once we consider how different momentum states respond to the Hartree potential that develops with carrier doping. The key point is that the topological nature of the flat bands dictates that wavefunctions near the Γ point are fundamentally different from those elsewhere in the mini-BZ—a feature already present in the single-particle BM model and which underlies the c/f decomposition used in THF models36,37,38,49,50. The flat-band states, which occupy most of the mini-BZ, are highly localized in real space, on the AA regions of the moiré cell. Conversely, the Γ-point states are extended and mostly avoid AA regions. As filling increases, electrons predominantly occupy these AA-localized states, producing a spatially varying Hartree potential that is strongly peaked at AA. Momentum states with large AA weight therefore experience large Hartree energy shifts, whereas the Γ-point states—with minimal AA overlap—shift only weakly. When energies are referenced to the Fermi level, this momentum-dependent Hartree response manifests precisely as the Γ-point stretching observed in Fig. 3a. Extended Data Fig. 9 and its accompanying discussion (Methods section ‘Hartree-driven band stretching and its connection to wavefunctions at different momenta within the flat band’) provide a step-by-step explanation of this ‘Hartree stretching’ mechanism.

Further insight into the evolution of the electronic structure emerges when we examine partial moiré fillings. Figure 3b plots this evolution from ν = −0.9 to ν = 0.9, with dashed lines following the progression of key spectral features. The dashed red line tracks a flat band that appears initially at ν = −0.9 at high energy as a diffuse ‘plume’—reflecting an incoherent band (spectral peak width comparable with its energy). As filling increases, this band gradually moves towards EF, increasing in spectral weight as it approaches. The dashed blue line follows a band with the opposite trend: it begins as a band with strong spectral weight just below EF and progressively shifts to large negative energies while becoming increasingly smeared. Notably, in both cases, just before the sharp band vanishes while crossing EF at ν ≈ 1 or ν ≈ −1, a new diffuse band appears at high energy. This behaviour is most clearly seen in Fig. 3c, which presents the energy spectrum, momentum-averaged over the flat part of the bands (excluding the region near Γ; Methods section ‘Spectroscopy at different momenta: its Gaussians decomposition and spectral function peak and lifetime analysis versus quasiparticle energy’), along with its decomposition into Gaussian contributions.

Filling-dependent spectroscopy at key momenta

Recent topological heavy fermion36,37,38 and Mott semimetal39 theories predict that the dual-nature electrons in MATBG arises from the different character of the electronic states at different momenta within the flat bands. To investigate this, we use a key capability of the QTM—performing spectroscopy as a function of filling at a particular momentum.

Figure 4a presents dI/dV measured as a function of Vb and ν at momentum KT. At charge neutrality, a clear energy gap is observed at this momentum. The evolution of the energy bands with filling—previously shown in Fig. 3b—appears here in the familiar form of electronic cascades17,18: near each integer ν, a faint high-energy band emerges, which gradually shifts with filling to lower energies, becoming stronger and sharper the closer it approaches EF. This band crosses EF over a finite range of ν and, once fully occupied, a new incoherent high-energy band emerges, whereas the previous one persists below EF. This cascading behaviour recurs at all integer fillings and is more pronounced on the electron-doped side. Identical filling-dependent spectral evolution is observed at all other momenta along the flat part of the bands (Methods section ‘Spectroscopy at different momenta: its Gaussians decomposition and spectral function peak and lifetime analysis versus quasiparticle energy’). This explains why scanning tunnelling microscopy, which averages over all momenta, captures a similar pattern.

Fig. 4: Spectroscopy at fixed momentum points.
Fig. 4: Spectroscopy at fixed momentum points.The alternative text for this image may have been generated using AI.
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a, dI/dV versus ν and Vb at KT (top inset indicates location on Fig. 2b). The data show clear cascading behaviour of the flat bands, identical across all momenta along the flat bands (Extended Data Fig. 5). Fixed-energy excitations appear at ΔE ≈ 15 meV for hole doping and ΔE ≈ −15 meV for electron doping (dashed purple lines). b, dI/dV versus ν and Vb at the Γ point (inset). Overlaid is −μ(ν) from compressibility measurements in ref. 17 (black line), showing surprising agreement between the filling dependence of the dI/dV peak at ΓL and the filling dependence of the total chemical potential. c, Correlated doping evolution of spectral features at KT (top) and Γ (middle), alongside inverse compressibility from ref. 17 (bottom). Top and middle panels are zoom-ins of a and b near charge neutrality. The flat band at KT (heavy electrons) shifts downward with increasing ν, reaching the Fermi level near ν ≈ 0.6, at which it begins to fill (dashed black lines). Simultaneously, the Dirac point at Γ (light electrons) also shifts downward and then reverses, moving up as the heavy band becomes populated. This reversal correlates with inverse compressibility becoming negative, signalling charge reshuffling from light to heavy electrons. d, Schematic toy model of Dirac revivals: light c-like electrons with Dirac dispersion (cyan) and heavy f-like electrons with flat bands (red). c–f hybridization is neglected. At ν = 0, the light-electron Dirac point lies at EF, with the heavy-electron Hubbard bands centred around it. At 0 < ν ≲ 0.6, light electrons fill and heavy-electron bands shift towards EF. For 0.6 ≲ ν < 1, the heavy band crosses EF and becomes populated, whereas the Dirac point of the light electrons moves back towards EF, driving the observed ‘Dirac revivals’17. At ν ≈ 1, a new heavy band appears at energy U above, restarting the cycle.

A markedly different behaviour is seen at the Γ point (Fig. 4b). Here there is no apparent gap at neutrality nor do we see any sign of cascading bands crossing through EF with filling. Instead, the dI/dV shows a peak at EF at ν ≈ 0, which gradually shifts to negative energies with filling. The shift is not strictly monotonic but exhibits wiggles that align with integer fillings, more prominently seen on the electron-doped side. Notably, when we overlay the negative of the chemical potential, −μ(ν), obtained from the compressibility measurements of ref. 17 (black line), we find that it closely follows this peak, even capturing its wiggles. Because that measurement was performed in a sample with a slightly larger twist angle (θTBG = 1.13°), we do not expect perfect correspondence.

Zooming in further (Fig. 4c), we uncover a strong correlation between the behaviour at the KT point, the Γ point and the inverse compressibility, dμ/dn(n), from ref. 17 (Fig. 4c top, centre and bottom). From ν ≈ 0 to ν ≈ 0.6, the peaks at both Γ and KT shift monotonically to lower energies with increasing filling. At ν ≈ 0.6, as the flat band at KT reaches EF, the peak at Γ reverses direction and begins to increase in energy. Notably, this turning point coincides with a sign change in the total inverse compressibility, which becomes negative. This behaviour continues until ν ≈ 1, at which the flat band at KT becomes fully occupied, the inverse compressibility sharply rises and the peak at Γ reverses direction again.

These observations naturally follow from the coexistence of two classes of electronic states that satisfy two basic conditions: they have light and heavy character and they couple differently to the Hartree potential. Although these conditions are naturally realized by the microscopic BM model ‘c’ and ‘f’ orbitals, they can be satisfied much more broadly by any two ‘c-like’ and ‘f-like’ components that retain these essential features. This is illustrated by a simple toy model (Fig. 4d and Methods section ‘Toy model to understand the essence of the Dirac revivals’), in which the light c-like electrons near Γ are described by a Dirac dispersion (cyan) and the heavy f-like electrons by flat bands (red). For simplicity, we neglect c–f hybridization, important in the actual bands but not essential for understanding the non-monotonic filling of the light electrons. In this picture, the measurement at Γ tracks the filling-dependent evolution of the Dirac point, whereas the measurement at KT follows the flat bands. At ν = 0, the flat bands lie away from EF and the Dirac point sits at EF. On doping, carriers initially populate the c-like bands, shifting their Dirac point downward in energy. Around ν ≈ 0.6, a flat band reaches EF and begins to fill, at which point the Dirac carriers start to depopulate and the Dirac point shifts upward towards EF. This behaviour is identical to ‘Dirac revivals’ of ref. 17, except that here the reshuffling of charge occurs between the light and heavy electrons rather than between flavours, as originally postulated17. The resulting evolution reflects a competition between a c–f Coulomb repulsion term, which drives the revivals and tends to depopulate the c-like electrons (producing upward wiggles), and Hartree terms, which lead to progressive population of the c-like electrons with filling (producing the overall downward trend). This competition naturally explains why the revivals do not fully reset to the Dirac point at integer fillings.

Although some key aspects of our experiments are captured by theoretical models of interacting electrons in MATBG23,24,27,28,34,35,38,39, one persistent feature remains unexplained: the excitation appearing at ΔE ≈ 15 meV for holes and ΔE ≈ −15 meV for electrons (purple lines in Figs. 3a and 4a). Similar features were previously observed in non-momentum-resolving scanning tunnelling microscopy measurements16,18, but our momentum-resolved experiments now show that this excitation exists only in the flat section of the bands (Fig. 4a) and not at the Γ point (Fig. 4b), suggesting its connection with the heavy electrons. Notably, its energy remains fixed with filling, in contrast to the strongly filling-dependent evolution of the heavy-electron bands themselves. The universality of this 15 meV scale—identical across widely separated regions in the sample and insensitive to local strain (Methods section ‘Spectroscopy of flat bands at different locations’)—may hint at the presence of an intrinsic excitation or collective mode whose microscopic origin remains unknown, but which may play an important role in the correlated and superconducting behaviour of MATBG.

Doping dependence of the light and heavy electrons

Finally, we present in Fig. 5 the zero-bias dI/dV, which reflects the spectral weight at EF. When plotted as a function of θQTM and ν, this quantity maps out where electronic states reside in momentum space as a function of doping. In this plot, the heavy electrons flat bands manifest as horizontal features extending over a wide range in momentum space. Their spectral weight oscillates with filling with a periodicity Δν ≈ 1, consistent with the cascading behaviour of the heavy electrons. The spectral weight decays near the Γ point owing to the Hartree stretching of the bands that pushes the light electrons below the Fermi energy. Visibly, the horizontal features get progressively shorter with filling (highlighted by the dashed white lines), indicating that the region of k-space occupied by heavy electrons shrinks, whereas the region occupied by light electrons expands.

Fig. 5: Doping evolution of the relative fraction of light and heavy electrons in momentum space.
Fig. 5: Doping evolution of the relative fraction of light and heavy electrons in momentum space.The alternative text for this image may have been generated using AI.
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dI/dV versus θQTM and ν measured at Vb = 0 mV. This signal is proportional to the spectral weight at the Fermi energy and shows where electronic states reside in momentum space as a function of doping. The momenta ΓL, KT and M are labelled on the top x-axis and integer fillings are indicated by horizontal grey lines. We plot the raw dI/dV signal, whose magnitude drops sharply left to the ΓL point owing to tunnelling matrix elements. At charge neutrality (ν = 0), the spectral weight is concentrated near Γ and is absent elsewhere. With doping, horizontal features emerge across a wide momentum range, reflecting the spectral weight of the flat heavy-electron bands, appearing at the Fermi level. This weight oscillates with a periodicity Δν ≈ 1, consistent with the cascades of these bands. Near ΓL, the spectral weight disappears owing to the Hartree-induced bending that pushes the light-electron bands below the Fermi energy. The horizontal features get progressively shorter with increased filling, as highlighted by the dashed lines, tracing the momentum at which the dI/dV intensity falls to 80% of its value at each ν (Methods section ‘Fitting the transition between light and heavy electrons’). This boundary is mirrored about the Γ point; together, the two dashed lines delineate the region in momentum space occupied by light electrons, with the remaining area populated by heavy electrons.

Our measurements resolve a long-standing puzzle in the physics of MATBG: the dual nature of its electrons. By directly visualizing the energy bands, we show that this duality arises from distinct characteristics of the flat bands’ wavefunctions in different regions of momentum space. Near the Γ point, the bands are dispersive, consistent with c-like electron behaviour, whereas elsewhere, they remain extremely flat, characteristic of f-like electrons. We directly observe how this leads to stretching of the bands owing to Hartree potential, as well as to electronic cascades and ‘Dirac revivals’ owing to the reshuffling of charge between these two degrees of freedom. Qualitatively, our findings provide direct evidence for the THF and Mott semimetal descriptions of this system. However, we also uncover a new low-energy excitation, not accounted by any existing models, which may play an important role in the exotic physics of MATBG.

Methods

Cryogenic QTM

All measurements in this work were performed in a cryogenic QTM system operating at a temperature of 4 K, as described previously45. For conductance measurement, voltage biases are applied using a custom-built digital-to-analogue converter array, capable of supplying both d.c. and a.c. signals with 1 μV resolution. Currents were measured using a FEMTO current amplifier, followed by a National Instruments sampler.

Fabrication and characterization of the QTM tip and van der Waals device

The fabrication of both QTM tips and van der Waals devices on flat substrates were described previously45,46. Briefly, we use the standard dry transfer and polymer membrane transfer techniques to fabricate the TBG sample and the QTM tip, respectively. Extended Data Fig. 1a shows an optical image of the TBG heterostructure, comprising bilayer-WSe2/TBG/hexagonal boron nitride (h-BN)/Pt on an approximately 100 μm width and 80 μm tall Si/SiO2 pillar. Extended Data Fig. 1b shows an optical image of the MLG backed by h-BN and graphite placed on a QTM tip.

In Extended Data Fig. 1c, we characterize the TBG sample by conductive AFM scan of the region of magic twist angle carried out at room temperature (Bruker Icon). To compensate for the thermal drift during this AFM scan, we performed a small window scan at rate 10.9 Hz, such that the time for an entire scan is around 20 s. We apply an approximately 1 V d.c. voltage bias between the tip and sample and measure the tunnelling current. The scan image in Extended Data Fig. 1c shows both the moiré lattice and atomic defects in WSe2 (bright spots). To accurately correct the thermal drift, we trace the position of three defects inside the scan window. We then obtain the drift velocity by following the shift of defects positions between consecutive scans and assuming that, within a single 2D scan, the drift velocity is constant. Extended Data Figure 1c,d shows the image and corresponding fast Fourier transform (FFT) after thermal drift correction. The three moiré lattice vectors obtained from this image are: 12.72 nm, 12.73 nm and 13.14 nm. This gives a twist angle of θTBG = 1.1° ± 0.02° and strain of ϵ = 0.03% ± 0.02%.

In Extended Data Fig. 1e, we characterize the QTM tip by in situ imaging of the tip contact area by an atomic defect in the WSe2 barrier as a localized tunnelling channel. At large bias (Vb ≈ −900 mV, well outside the range used in the actual spectroscopy measurements), the defect level becomes energetically accessible and provides an extra, spatially localized current path. When the QTM tip passes over the defect, we observe a small but measurable increase in current. By scanning the tip across such defects and mapping this current enhancement, we obtain a real-space image of the tip’s contact footprint.

Extended Data Fig. 1e presents such a high-bias scan, showing several images of the tip footprint, each produced by a different defect, revealing a contact size of approximately 200 × 50 nm. These real-space dimensions set the momentum-space resolution through the Heisenberg uncertainty relation. The measured footprint corresponds to angular momentum resolutions (in angular units) of roughly δθQTM ≈ 0.05° and ≈ 0.2° along the two principal directions. In Extended Data Fig. 2, we show that the sharpest momentum features—obtained in the remote bands—exhibit a full width at half maximum (FWHM) of δθQTM ≈ 0.1°, consistent with the resolution expected from the measured tip dimensions.

Energy and momentum resolution

We determine the energy resolution of our QTM measurements by analysing a sharp spectral feature at ν = −0.7 in Fig. 4a (reproduced in Extended Data Fig. 2a). A linecut through this feature yields a FWHM of δE ≈ 10 meV by fitting a Lorentzian function. This is an upper bound to the energy resolution of the measurements and it may well be coming from the intrinsic lifetime of the energy bands of MATBG at this temperature. To determine the momentum resolution, we analyse a sharp spectral feature of the remote energy bands of MATBG measured under the same conditions as in Fig. 2a. Extended Data Fig. 2b plots a linecut of dI/dV along the dashed white line, cutting through the dispersive bands, yielding an angular FWHM of δθ = 0.1° by fitting a Lorentzian function. This translates into a δk ≈ 0.03 nm−1 momentum resolution. For the flat bands, we trace the dI/dV peak width versus θQTM at zero bias around the Γ point and identify the peak width δθ ≈ 0.5°. This peak width is composed of two peaks overlapping each other, such that the single peak broadening is δθ ≈ 0.25°.

Electrostatics of the QTM junction

The QTM junction is modelled as a three-plate capacitor consisting of: (1) the MLG on the QTM tip; (2) the TBG layer on the sample; and (3) the metallic back gate. Below, to stay consistent with our previous publication, we use in all of the formulas the notation ‘top layer’ (‘T’) and bottom layer (‘B’) to refer to the two layers between which the electrons perform the QTM’s momentum-resolved tunnelling, not the top and bottom layers of TBG. Namely, the top layer is the MLG layer on the QTM tip and the bottom layer is the TBG on the sample side.

The QTM tip and TBG are separated by two layers of WSe2 (1.2 nm) and the TBG to the back gate by a 37 nm h-BN layer. The corresponding interlayer geometric capacitances are denoted as Cg (tip–TBG) and Cbg (TBG–back gate). A voltage bias Vb is applied between the tip and TBG sample layer. The gate voltage Vbg is applied between the TBG sample and the metal back gate. The electrostatic configuration of the measurements can be described by the following46:

$$-e{V}_{{\rm{b}}}={\mu }_{{\rm{B}}}-{\mu }_{{\rm{T}}}-\frac{{e}^{2}{n}_{{\rm{T}}}}{{C}_{{\rm{g}}}},$$
(1)
$$e{V}_{{\rm{bg}}}={\mu }_{{\rm{B}}}+\frac{{e}^{2}({n}_{{\rm{B}}}+{n}_{{\rm{T}}})}{{C}_{{\rm{bg}}}}$$
(2)

Note again that the notation ‘top’ (‘T’) refers to the MLG on the QTM tip and the notation ‘bottom’ (‘B’) refers to the TBG. μB and μT are the chemical potentials of the bottom TBG and top graphene probe, respectively, and nB and nT are their corresponding carrier densities. The back-gate capacitance was experimentally calibrated to be Cbg = 62.3 nF cm−2 based on the gate voltage required to reach filling factor ν = 4 in the magic-angle region (as verified by the conductive AFM measurements; Methods section ‘Fabrication and characterization of the QTM tip and van der Waals device’).

It is important to note that, under a finite voltage bias, TBG becomes doped as a result of the gating from the tip. This means that, at a fixed Vbg, the filling factor ν changes with the tip bias, Vb. To account for this and present our data as a function of the TBG’s filling factor, we determine the gating effect directly from the data. Extended Data Fig. 3 shows the raw measurement of QTM tunnelling conductance, dI/dVb (for simplicity called dI/dV throughout the paper), and the transconductance with respect to the back gate, dI/dVbg, as functions of Vb and Vbg. At integer fillings, and in particular at ν = ±4, dI/dVb shows a suppression and dI/dVbg shows a dipole-like feature. By tracing the ν = 4 line in the dI/dVbg measurements (through a polynomial fitting, overlaid on the data), we extract the trajectory of constant carrier density in the Vb–Vbg plane. We then ‘skew’ the raw data by plotting it as a function of ν that is determined by both Vbg and Vb. The skew-corrected data are shown in Figs. 3 and 4. For Figs. 1 and 2, the scans were taken at a fixed gate voltage without skew correction, so at high bias, the filling factor may deviate from the nominal one.

Along this skewed line, the TBG carrier density nB and chemical potential μB remain fixed. By using the graphene relation \({n}_{{\rm{T}}}={\mu }_{{\rm{T}}}^{2}/\pi {\nu }_{{\rm{f}}}^{2}\), the electrostatic equation can be further simplified to:

$$-e{V}_{{\rm{b}}}={\mu }_{{\rm{B}}}-e{V}_{{\rm{b}}{\rm{g}}}^{\ast }\frac{{C}_{{\rm{b}}{\rm{g}}}}{{C}_{{\rm{g}}}}-\sqrt{e{V}_{{\rm{b}}{\rm{g}}}^{\ast }\frac{{C}_{{\rm{b}}{\rm{g}}}\pi {\nu }_{{\rm{f}}}^{2}}{{e}^{2}}},$$
(3)

in which \({{eV}}_{{\rm{bg}}}^{\ast }={{eV}}_{{\rm{bg}}}-{\mu }_{{\rm{B}}}-{e}^{2}{n}_{{\rm{B}}}/{C}_{{\rm{bg}}}\), giving for every Vb the corresponding shift in Vbg to maintain fixed nB and μB. By fitting the skew lines with this relation, we derive the interlayer capacitance to be Cg = 2.3 μF cm−2, consistent with our previous tunnelling experiment using two layers of WSe2 (ref. 46).

The tunnelling matrix elements

The tunnelling current is modelled by Fermi’s golden rule assuming momentum-resolved tunnelling between the TBG sample and the graphene probe. As in the previous section, to stay consistent with our previous publication, in all of the formulas below, we use the notation ‘top layer’ (‘T’) and ‘bottom layer’ (‘B’) to refer to the two layers between which the electrons perform the QTM’s momentum-resolved tunnelling, not the top and bottom layers of TBG. Namely, the top layer is the MLG layer on the QTM tip and the bottom layer is the TBG on the sample side.

The calculation is described in detail in refs. 46,47 and here we briefly review the formula and tunnelling matrix elements:

$$I=\frac{2\pi e{g}_{{\rm{s}}}{g}_{{\rm{v}}}}{\hbar }\int {\rm{d}}\omega (\,{f}_{{\rm{B}}}(\omega )-{f}_{{\rm{T}}}(\omega +e\phi ))\sum _{n,n{\rm{{\prime} }}}\sum _{{{\bf{k}}}_{{\rm{T}}},{{\bf{k}}}_{{\rm{B}}}}{|\langle {u}_{{\rm{gr}},{{\bf{k}}}_{{\rm{T}}},n}|{T}^{{{\bf{G}}}_{{\rm{T}}},{{\bf{G}}}_{{\rm{B}}}}|{u}_{{\rm{tbg}},{{\bf{k}}}_{{\rm{B}}},n{\rm{{\prime} }}}\rangle |}^{2}{\delta }_{{{\bf{k}}}_{{\rm{T}}}+{{\bf{G}}}_{{\rm{T}}},{{\bf{k}}}_{{\rm{B}}}+{{\bf{G}}}_{{\rm{B}}}}{A}_{{{\bf{k}}}_{{\rm{B}}},\omega }{A}_{{{\bf{k}}}_{{\rm{T}}},\omega +e\phi }$$
(4)

gs/v is the spin/valley degeneracy, kB/T is the electron momentum of the bottom/top layer, n and n′ are the band indexes, fB/T(E) is the Fermi–Dirac distribution function and ϕ is the electrostatic potential that shifts the relative position of energy bands between TBG and graphene, derived from the electrostatic model in Methods section ‘Electrostatics of the QTM junction’. The chemical potential μB/T is defined relative to their charge neutrality (Dirac point of TBG and graphene probe). \({A}_{{{\bf{k}}}_{{\rm{B}}/{\rm{T}}},E}\) is the spectral function of the TBG and the graphene probe. \({T}^{{{\bf{G}}}_{{\rm{T}}},{{\bf{G}}}_{{\rm{B}}}}\) is the tunnelling matrix in sublattice space between the graphene probe wavefunction \(|{u}_{{\rm{gr}},{{\bf{k}}}_{{\rm{T}}}}\rangle \) and the TBG wavefunction \(|{u}_{{\rm{tbg}},{{\bf{k}}}_{{\rm{B}}}}\rangle \) projected on the top layer. GT and GB are the reciprocal lattice vectors of the graphene probe and the top layer graphene of TBG, at which it satisfied the relation kT + GT = kB + GB. We assume that the tunnelling happens within the first BZ of graphene probe and it limits the Bragg scattering process of {GT, GB} choices to be three. For example, when GT = GB = 0, the matrix is given by: \({T}^{{{\bf{G}}}_{{\rm{T}}}=0,{{\bf{G}}}_{{\rm{B}}}=0}=I+{\sigma }_{x}\), in which I is the unity matrix. The other two Bragg scattering processes are related by C3. Here we focus on \({T}^{{{\bf{G}}}_{{\rm{T}}}=0,{{\bf{G}}}_{{\rm{B}}}=0}\) and the matrix elements are given by:

$$\begin{array}{c}{|\langle {u}_{{\rm{g}}{\rm{r}},{{\bf{k}}}_{{\rm{T}}}}|{T}^{{{\bf{G}}}_{{\rm{T}}}=0,{{\bf{G}}}_{{\rm{B}}}=0}|{u}_{{\rm{t}}{\rm{b}}{\rm{g}},{{\bf{k}}}_{{\rm{B}}}}\rangle |}^{2}\\ \,=\,{|{\langle ({u}_{{\rm{g}}{\rm{r}},{{\bf{k}}}_{{\rm{T}}},A}^{\ast }+{u}_{{\rm{g}}{\rm{r}},{{\bf{k}}}_{{\rm{T}}},B}^{\ast })\rangle }_{FS}({u}_{{\rm{t}}{\rm{b}}{\rm{g}},{{\bf{k}}}_{{\rm{B}}},A}+{u}_{{\rm{t}}{\rm{b}}{\rm{g}},{{\bf{k}}}_{{\rm{B}}},B})|}^{2}\\ \,\propto \,{|{u}_{{\rm{t}}{\rm{b}}{\rm{g}},{{\bf{k}}}_{{\rm{B}}},A}+{u}_{{\rm{t}}{\rm{b}}{\rm{g}},{{\bf{k}}}_{{\rm{B}}},B}|}^{2}=|{\langle I+{{\sigma }}_{x}\rangle }_{{u}_{{\rm{t}}{\rm{b}}{\rm{g}},{{\bf{k}}}_{{\rm{B}}}}}|,\end{array}$$
(5)

in which \({u}_{{{\rm{gr}},{\bf{k}}}_{{\rm{T}}},A/B}^{\ast }\) is the graphene probe wavefunction with the sublattice component A/B. The graphene probe wavefunction is averaged over the Fermi surface and we assume that, within the graphene Fermi surface range, the TBG wavefunctions \({u}_{{\rm{tbg}},{{\bf{k}}}_{{\rm{B}}},A/B}\) remain the same. The above equation shows that the tunnelling current is proportional to the \(|{\langle I+{\sigma }_{x}\rangle }_{{u}_{{\rm{tbg}},{{\bf{k}}}_{{\rm{B}}}}}|\), in which the \(|{u}_{{\rm{tbg}},{{\bf{k}}}_{{\rm{B}}}}\rangle \) contains both the layer polarization and sublattice information.

In Fig. 2d, we show the matching between the matrix element \(|{\langle I+{\sigma }_{x}\rangle }_{{u}_{{\rm{tbg}},{{\bf{k}}}_{{\rm{B}}}}}|\) and the dI/dV intensity in the flat-band region. It shows an excellent agreement at ratio w0/w1 = 0.6. Here we further show that the naive layer polarization quantity (Extended Data Fig. 4b,d) is inconsistent with the data, regardless of the w0/w1 ratios.

For the tunnelling current calculation in Fig. 1, we input the single-particle band structure of TBG at θQTM = 1.2°, the electrostatics from Methods section ‘Electrostatics of the QTM junction’ and calculate the tunnelling current based on equation (4).

Spectroscopy at different momenta: its Gaussians decomposition and spectral function peak and lifetime analysis versus quasiparticle energy

In Extended Data Fig. 5, we show the tunnelling spectroscopy at different momenta along the flat-band region. The momenta at which the spectroscopy images are measured are marked by dashed black lines on a zoom-in of the bands from Fig. 2b, shown in the top panel. The range of momenta covers angles from θQTM = −0.5° to 0.8°. Apart from an overall magnitude, the spectroscopy in all momenta looks very similar. In Fig. 3c, we use this invariance on momentum in the flat bands to make Gaussian fit to the averaged data over the flat-band region.

The Gaussian fitting procedure is performed as follows. We first identify the peak centres directly from the spectroscopy data by tracing the local maxima in the contour representation of Fig. 4a; these trajectories are shown as dashed black lines in Extended Data Fig. 6a. Specifically, we identify two classes of features: (1) the ‘Hubbard’ bands that exhibit the characteristic cascading behaviour and (2) further excitations whose energy (about ±15 meV) is independent of filling. For each filling, the number of peaks identified in this manner determines the number of Gaussians used in the corresponding spectral fit. The peak positions extracted from the dashed-line trajectories serve as initial guess for the fit, although we allow these positions to vary during the fitting. The free parameters in the fitting routine are the Gaussian centres, widths, amplitudes and the background offset.

In Figs. 3b and 4a, we observed many spectral features originating from the Hubbard bands of heavy electrons. The spectral features typically appear at high energy as smeared ‘plumes’ and become stronger the closer they approach the Fermi level.

In a simple lifetime-broadening picture, the spectral function peak corresponding to a quasiparticle at an energy E acquires a width inversely proportional to the lifetime and the peak amplitude of the spectral function decreases correspondingly. In the present experiment, several bands coexist within a narrow energy range, making it challenging to extract the linewidths of the bands quantitatively. Instead, we can reliably extract the peak height, Apeak, which is a robust experimental observable and reflects the combined effects of quasiparticle residue and lifetime. Inspection of Fig. 4 shows a clear and systematic trend: when a band lies far from the Fermi energy, its Apeak is small and this maximal spectral weight increases continuously as the band approaches the Fermi level.

To address this more quantitatively, in Extended Data Fig. 6b, we track the maximum of the spectral peak (dashed line) from the highest energy that we can reliably identify a peak to the point at which it reaches the Fermi energy, and the peak becomes less well defined. In panel c, we plot the one over the extracted peak height 1/Apeak versus peak energy E and compare it with two functional forms: \(\frac{1}{{A}_{{\rm{peak}}}}=a(E-{E}_{{\rm{F}}})+b\) and \(\frac{1}{{A}_{{\rm{peak}}}}=a{(E-{E}_{{\rm{F}}})}^{2}+b\). Overall, we see that a linear dependence of 1/Apeak on E − EF better fits the observations.

The height of a spectral-function peak reflects both the quasiparticle residue Z(E − EF) and the lifetime τ(E − EF). In principle, both quantities may evolve as a Hubbard band moves to higher energies. If, however, we interpret the dominant trend as arising mainly from lifetime effects, the observed scaling would imply that it is more likely that τ(E −E F) ∝ 1/(E − EF) and not the Fermi-liquid expectation τ(E − EF) ∝ 1/(E − EF)2. Such dependence is often seen in strongly correlated/Hubbard systems, but because we do not measure this quantity in separation, we cannot make a decisive claim.

Fitting the transition between light and heavy electrons

Figure 5 shows dI/dV at Vb = 0 mV as a function of θQTM and filling factor ν. To determine the boundary between heavy and light electrons, we plot horizontal linecuts of dI/dV versus θQTM for each filling and perform a fit of a sigmoid-type function:

$$\frac{{\rm{d}}I}{{\rm{d}}V}={c}_{1}+\frac{{c}_{2}-{c}_{1}}{1+{e}^{-\frac{{\theta }_{{\rm{QTM}}}-{\theta }_{0}}{\sigma }}}$$
(6)

On the basis of the fitting, when the heavy-electron spectral intensity decays by 80%, we mark the position θhl(ν) as a proxy for the transition between heavy and light electrons. The parameter σ characterizes the width of this transition. We then fit θhl(ν) to a third-order polynomial and plot this as the dashed white lines in Fig. 5.

Determine the larger twist angle of θ TBG = 1.2°

We determine the angle of larger-angle TBG from the measurement in Fig. 1d. By tracing the dI/dV intensity change at the conduction-band edge of flat bands around the Γ point, we performed a step function fitting of the form:

$$y=C+\frac{A}{2}(1+\tanh (B\ast (x-{x}_{0}))),$$

in which the x0 marks the Γ point position, which is half the value of the tunnelling intensity decay. By measuring the angle distance between the Γ point to the KT point, at which the Dirac point crosses, we determine the twist angle to be θTBG = 1.2°.

Spectroscopy of flat bands at different locations

Extended Data Fig. 7a–f presents tunnelling spectroscopy at a momentum point within the flat part of the bands (KT) measured at several locations across the sample, separated by several microns. These separations are large enough such that each region effectively behaves as independent devices. In all cases, we observe excited states whose energy is independent of filling, similar to the measurement in Fig. 4a.

For each location, we use room-temperature conductive AFM to image the real-space moiré structure (Methods section ‘Fabrication and characterization of the QTM tip and van der Waals device’) and plot its corresponding FFT. From the latter, we extract the local twist angle, θTBG, and heterostrain, ϵTBG, of the moiré. From the spectroscopy, we determine the excitation energy, ΔE. These three locally measured quantities are indicated above each panel. The centre panel shows a real-space map of the measurement locations, with the colour of each point representing the corresponding ΔE (see colour bar).

Although we cannot definitively rule out heterostrain as the origin of the observed approximately 15 meV excitation, the data strongly indicate that strain alone cannot account for this feature, for several reasons:

  1. 1.

    Universality across widely separated regions of the same device. We observe nearly identical excitation energies (13–16 meV) at locations separated by several microns. It is highly unlikely that the same level of strain would persist uniformly over such large distances.

  2. 2.

    Universality across different samples. Similar excitation energies are measured in two distinct MATBG devices, measured in separate cooldowns, indicating that the feature is neither device-specific nor related to a particular thermal cycle.

  3. 3.

    Lack of correlation with independently measured strain. The heterostrain extracted from the moiré FFT varies by a factor of 3 across the measured positions (from ε = 0.02% to ε = 0.06%). Despite this substantial variation, the excitation energy remains essentially unchanged.

  4. 4.

    Quantitative inconsistency with theoretical expectations. For the least-strained region (ε = 0.02%), the observed excitation is about 15 meV. Existing theoretical estimates predict a strain-induced level splitting of about 10 meV per 0.1% strain51, which would correspond to only about 2 meV splitting at ε = 0.02%, an order of magnitude smaller than what we observe.

QTM probing of the K and K′ valleys

In Extended Data Fig. 8, we explain why the time-reversal symmetry of the QTM probe dictates that it measures the combined contribution of the K and K′ valleys.

Panel a shows the inequivalent Fermi surfaces of the K and K′ valleys (red and blue) folded into the same mini-BZ. For illustration, we depict triangularly warped Fermi surfaces to emphasize that the two valleys can, in principle, be different.

To understand what the QTM actually probes, it is useful to unfold the picture back to the full BZ (panel b). There the K and K′ Fermi surfaces sit at opposite corners. In the experiment, the Dirac points of the probe (purple) move along a circular trajectory (dashed line) and current is detected whenever the Dirac point of the probe intersects a Fermi surface. Crucially, if both the probe and the sample obey time-reversal symmetry, the K and K′ valleys produce identical spectra as a function of the QTM angle θQTM (panel c). If we take the illustration in panel b as an example, although the two triangular Fermi surfaces are mirror reflections, the rotating Dirac points intersect them in exactly the same sequence—first the flat edge and then the triangle tip—and therefore yield identical spectroscopic measurement.

We might expect that spontaneous valley symmetry breaking (for example, a valley-polarized Chern insulator) would manifest as clear energy splitting between the K and K′ bands and that, consequently, it should be easy to identify flavour symmetry breaking from this band splitting. However, this expectation is incorrect—recent dynamical mean-field theory calculations performed in both the symmetric and flavour-symmetry-broken34,35 states show that very similar band splittings appear in both situations. These splittings originate primarily from the formation of Hubbard bands, which occur regardless of whether flavour symmetry is broken. As a result, the spectroscopic signatures of flavour symmetry breaking are subtle and cannot be identified simply by looking for energy-split bands.

Hartree-driven band stretching and its connection to wavefunctions at different momenta within the flat band

In this section, we provide a systematic explanation of the momentum-dependent band stretching observed near the Γ point. To highlight the effect, we plot in Extended Data Fig. 9h,i the measured bands from Fig. 3a alongside schematic guides to the eye marking the stretched areas indicated by arrows. Specifically, the stretching appears as a peak at Γ at ν = −4, which continuously evolves to a dip at Γ at ν = 4.

This stretching originates from the momentum-dependent response of the electronic wavefunctions to the Hartree potential that builds up with increasing carrier density. Early theoretical works (for example, refs. 21,22) already noted that, although moiré-scale electrostatic potentials are negligible at charge neutrality, a spatially varying Hartree potential VH(r) develops as electrons or holes are added. The energy of a state at momentum k then shifts according to its real-space overlap with this potential,

$$\delta {E}_{{\rm{H}}}(k)=\int {{\rm{d}}}^{2}r{V}_{{\rm{H}}}(r){|{\psi }_{k}(r)|}^{2},$$

and because different k-states have distinct real-space charge distributions, they experience different Hartree shifts. This naturally produces a stretching of the bands, even in the absence of strong correlations. Although indirect signatures of this effect have been reported previously (for example, through Landau-level spectroscopy in ref. 16), Fig. 3 presents the first, to our knowledge, direct observation of this Hartree-driven band stretching.

Notably, our measurements do not only show the overall magnitude of the band stretching but they directly provide the momentum-dependent energy shifts δEk. From the expression above, it is evident that these shifts encode detailed information about the real-space structure of the wavefunctions at different k. This is especially valuable in MATBG, in which the flat bands are known to have strongly momentum-dependent wavefunctions arising from their nontrivial topology.

In fact, one of the most notable theoretical predictions for TBG, already present in the BM single-particle model, is that the flat-band wavefunctions change qualitatively across the mini-BZ. This effect does not need electronic correlation and happens also when the bands are not extremely flat—for example, for TBG with twist angles larger than the magic angle. In the BM continuum model, more than 90% of the momentum states in the band have a localized wavefunction in the AA regions of the moiré cell. However, near the Γ point, this structure is reversed: the states become more extended and feature a suppression of charge density at the AA sites.

Although this momentum-dependent real-space structure was used as the basis for formulating the c-electron and f-electron decomposition in the THF model, it is not specific to that framework. Rather, it is a robust and very general feature of the non-interacting flat bands themselves. Our ability to measure δEH(k) with momentum resolution therefore provides direct access to these underlying wavefunction characteristics, offering a powerful new experimental probe of the topology-driven structure of the MATBG bands.

Extended Data Fig. 9a–g provides a step-by-step illustration of how the observed Hartree stretching arises and how it is directly connected to the wavefunction structure of the flat bands. Panel a sketches the real-space Hartree potential VH(r) that develops on electron filling. Because more than 90% of the flat-band wavefunctions in the BM model reside on the AA sites and have an f-electron Wannier function, the added carriers generate a Hartree potential that is strongly peaked at these sites and therefore highly repulsive for f-electrons. We emphasize that here we are discussing single-particle wavefunctions, so the c/f decomposition is fully valid in this context. As shown in panel b, the shape of VH(r) remains fixed as a function of filling, whereas its overall amplitude grows approximately linearly with ν (ref. 21).

Panel c plots the BM charge densities |Ψf(r)|2 (red) and |Ψc(r)|2 (blue). The f-electron density has far greater overlap with VH(r) than the c-electron density and thus experiences a substantially larger Hartree shift. This is captured schematically in panel d, in which both c-electron and f-electron energies increase linearly with ν but with a much steeper slope for the f-electrons.

Panel e translates these energy shifts into momentum space. States near the Γ point—dominated by c-character—shift only weakly with filling, whereas most of the states in the rest of the band—dominated by f-character—shift strongly. Because most of the flat-band states are f-like, the Fermi energy EF (dashed line) remains effectively pinned to the flatter regions of the band as ν varies.

Our experiment, however, measures energies relative to EF. Referencing to EF has the effect shown schematically in panel f: the strongly shifting f-electron states remain near zero energy, whereas the weakly shifting c-electron states acquire an apparent downward slope with filling. This behaviour matches precisely the trend observed in Fig. 4 (here we focus only on the Hartree component; further interaction terms responsible for Hubbard-like band splitting and cascades appear on top of this effect in the experiment).

Finally, panel g shows how the ν-dependent bands of panel e appear once referenced to EF: the bands exhibit a clear stretching near Γ—exactly as observed experimentally in Fig. 3a. This stretching provides compelling evidence for the distinct wavefunction character of states near Γ, a key prediction of the topological structure of the BM bands and a foundational ingredient of the THF model.

Toy model to understand the essence of the Dirac revivals

In Extended Data Fig. 10, we present a minimal toy model that separates the low-energy spectrum into two sectors: (1) light carriers with a Dirac-like dispersion (‘c-like’ electrons) and (2) heavy, nearly flat bands (‘f-like’ electrons). The aim is to capture, in a transparent way, the key experimental features, namely the cascade phenomenology and the Dirac revivals—without committing to microscopic details.

The components/assumption of the model are as follows:

  1. 1.

    We assume two decoupled electronic components, which we call ‘c-like’ and ‘f-like’. They are not necessarily the original c-electrons and f-electrons but they do carry their two important aspects: light/heavy, different Hartree couplings.

  2. 2.

    Specifically, we model the f-like electrons with a simplified quantum dot model, which assumes completely flat bands but with a phenomenological ‘lifetime’ broadening that gives a finite width to the energy bands. We compute the f-electron spectral function to capture the Hubbard bands filling evolution.

  3. 3.

    We assume that there is a finite Coulomb coupling between c-like and f-like electrons, W, and that W < U.

The f-electron quantum dot model Hamiltonian is: H = Unf(nf − 1)/2, in which U is the effective charging energy and nf is the f-electron occupation. This quantum dot has eight flavours such that 0 ≤ nf ≤ 8. We compute the spectral function given by this Hamiltonian: \(A(\omega )={Z}^{-1}{\sum }_{n,m}\delta (\omega -{\varepsilon }_{n}+{\varepsilon }_{m})({e}^{-\beta {\varepsilon }_{m}}+{e}^{-\beta {\varepsilon }_{n}}) < l|{c}_{\alpha ,{\rm{f}}}^{\dagger }{|m > |}^{2}\), in which Z = Tr(e−β(H − μn)) is the partition function, β = 1/kBT and εm is the energy of the |m> eigenstates. We compute the spectral function A(ω, nf) and then add smearing in energy to mimic the lifetime effect seeing in the measurement. From the smeared spectral function, we compute the relation μf(nf).

As well as the quantum dot, we add Dirac (c-like) electrons. Let nf and nc denote the f and c fillings (n = nf + nc is fixed externally). The interaction term can be written as Wnfnc = Wnf(n − nf). The total energy is:

$${E}_{{\rm{tot}}}({n}_{{\rm{f}}};\,n)={E}_{{\rm{f}}}({n}_{{\rm{f}}})+{E}_{{\rm{c}}}({n}_{{\rm{c}}})+W{n}_{{\rm{f}}}(n-{n}_{{\rm{f}}}),$$

and the stationarity condition

$$\frac{{\rm{d}}{E}_{{\rm{tot}}}}{{\rm{d}}{n}_{{\rm{f}}}}=0$$

gives a simple self-consistency equation:

$${\mu }_{{\rm{f}}}({n}_{{\rm{f}}})-{\mu }_{{\rm{c}}}({n}_{{\rm{c}}})+W[n-2{n}_{{\rm{f}}}]=0,$$

with μf/c ≡ dEf/c/dnf/c. We solve this equation graphically to obtain (nf, nc) at each total filling n and then compute the corresponding spectral functions in momentum space for both sectors.

Extended Data Fig. 10a,b presents the calculated spectral functions for doping range from n = 0 to n = 1.5. Panels a and b show the f and c spectral functions in momentum space versus filling, reproducing: (1) the Hubbard-band evolution of heavy f-electrons and (2) the Dirac revival of light c-electrons. Specifically, in panel b, we track the Dirac point evolution: it first shifts downward with increasing ν owing to the Hartree potential and then when f states start to populate, it shifts upward (solid white guideline); this is precisely the ‘Dirac revival’ phenomenology and, within this model, we can see that it is driven by the W Coulomb term. The schematic in Fig. 4 is based on this calculation.

The above model does not include c–f hybridization. To show that the Dirac revivals physics is robust and appears also in the case of strong hybridization (γ/U > 1), we solve a more realistic THF model that also includes hybridization (details in the next section). Extended Data Fig. 10c plots the calculated filling factor dependence of the spectral function. The black line traces the position of the c-electrons Dirac point, clearly demonstrating that the Dirac revivals phenomenology, observed in the experiment, appears also in the presence of hybridization.

THF model reproducing the Dirac revivals

We consider a THF model38 that treats U, the on-site f–f interaction, non-perturbatively within the Hubbard-I approximation, while treating all other interactions within the self-consistent Hartree approximation. We use the following parameters for the calculations: v⋆ = −1 eV nm; v′⋆ = 366 meV nm; γ = −70 meV; M = 3.7 meV; U = 60 meV; W = 45 meV; Y = 45 meV.

To calculate the Green’s function of f-electrons and c-electrons within Hubbard-I, we first consider the non-hybridized f-electrons propagator:

$${G}_{{\rm{f}},\eta }^{0}({\bf{k}},\omega )=\left(\frac{\frac{1}{2}+\frac{{\nu }_{{\rm{f}}}}{N}}{\omega +\delta \mu +\frac{U}{2}+i{\tau }^{-1}}+\frac{\frac{1}{2}-\frac{{\nu }_{{\rm{f}}}}{N}}{\omega +\delta \mu -\frac{U}{2}+i{\tau }^{-1}}\right){I}_{2\times 2}.$$

νf = sign(ν)⌊|ν|⌋ is the integral filling of each f site relative to charge neutrality, within the non-hybridized theory, taken to be the total filling rounded towards zero. δμ = μf − ν · U sets the asymmetry between the Mott-band energies. N = 8 is the number of electronic states (flavours × orbitals) per f site and μf = μ − W · ⟨δnc⟩ is the effective electro-chemical potential felt by the f site owing to repulsion from c-electrons, with δnc measured relative to charge neutrality. τ is a finite quasiparticle lifetime introduced by hand for numerical stability. We take τ−1 = 1 meV.

The non-hybridized c-electrons propagator is given by

$${G}_{{\rm{c}},\eta }^{0}({\bf{k}},\omega )={[\omega +{\mu }_{{\rm{c}}}-{H}^{({\rm{c}},\eta )}({\bf{k}})+i{\tau }^{-1}]}^{-1},$$

in which H(c,η)(k) is the single-particle Hamiltonian of c-electrons in valley η as given by Song and Bernevig38 and μc = μ − W · ⟨δnf⟩ − V · ⟨δnc⟩ is the effective electro-chemical potential felt by the c-electron, with δnf measured relative to charge neutrality. Finally, the Hubbard-I propagator is given by

$$\begin{array}{c}[\begin{array}{cc}{G}_{{\rm{c}},\eta } & {G}_{{\rm{cf}},\eta }\\ {G}_{{\rm{fc}},\eta } & {G}_{{\rm{f}},\eta }\end{array}]({\bf{k}},\omega )={[\begin{array}{cc}{({G}_{{\rm{c}},\eta }^{0}({\bf{k}},\omega ))}^{-1} & {({H}^{({\rm{fc}},\eta )}({\bf{k}}))}^{\dagger }\\ {H}^{({\rm{fc}},\eta )}({\bf{k}}) & {({G}_{{\rm{f}},\eta }^{0}({\bf{k}},\omega ))}^{-1}\end{array}]}^{-1},\end{array}$$

with H(fc,η)(k) the f–c single-particle hybridization term as given by Song and Bernevig38.

The densities ⟨δnf⟩ and ⟨δnc⟩ are calculated by integrating the spectral functions over negative frequencies and subtracting their values at charge neutrality:

$$\langle \delta {n}_{\alpha }\rangle =4{\int }_{-\infty }^{0}{\rm{d}}\omega \int \frac{{{\rm{d}}}^{2}{\bf{k}}}{{A}_{\mathrm{BZ}}}{{\mathcal{A}}}_{\alpha }({\bf{k}},\omega )-{n}_{\alpha }^{0}.$$

The spectral function is given by

$${{\mathcal{A}}}_{\alpha }({\bf{k}},\omega )=-\frac{1}{\pi }{\rm{ImTr}}[{G}_{\alpha ,\eta }({\bf{k}},\omega )],$$

with α = f,c, \({n}_{\alpha }^{0}\) is the density at charge neutrality and the factor 4 corresponds to spin and valley degeneracy, for which we use the fact that the state is flavour symmetric by assumption. For each value of the total density, we solve for ⟨δnf⟩, ⟨δnc⟩ and the chemical potential μ self-consistently.

Comparing the energy spectrum at partially full and completely full flat bands

In Extended Data Fig. 11, we plot the spectroscopy data as in Fig. 4a and compare linecuts A and B, taken when the chemical potential lies inside and outside the flat bands, respectively (dashed lines in panel a). For partial filling of the flat bands, we observe a characteristic feature at ΔE = −15 meV, as indicated by the dashed black lines in linecut A at ν = 1.5. By contrast, when the flat bands are fully filled, we instead find a single broad peak, as shown in linecut B at ν = 4.3.

One possible interpretation of the ΔE feature is a single-particle splitting induced by strain. However, the strain in the regions in which we perform the measurements is generally small (Methods section ‘Spectroscopy of flat bands at different locations’), making it unlikely to account for the observed 15 meV splitting. Moreover, a single-particle splitting should also be present when the flat bands are fully filled. In panel c, we attempt to fit the broad peak in linecut B with two symmetric Gaussians constrained to have the same lifetime as at partial filling (FWHM of about 10 meV, comparable with the energy resolution) and a fixed splitting of 15 meV and obtain a poor fit. This further suggests that simple single-particle strain-induced splitting is unlikely to be the origin of the ΔE feature.

Discussions of possible ground states at integer fillings

At charge neutrality, we observed a semimetallic phase with states at Γ at the Fermi level. There are two physically distinct mechanisms to produce a semimetal: the thermally disordered Mott semimetal27,34,35,39 and the strain-induced or spontaneously C3 symmetry-broken semimetal29,52,53. The two states give very similar spectra, with Hubbard-like nearly flat bands away from the Γ point and a band touching at or near Γ. There should be small differences in the spectra between the two states—for example, the Dirac points of the strain-induced semimetal are generally not precisely at Γ and do not have to occur at the same energy—but we believe that these differences may be too small for us to observe within experimental resolution. The main factor limiting the resolution is probably the tip size, limiting the resolution in momentum space to discern the fine details of the dispersion near Γ.

At non-zero integer fillings, our experiment is not directly sensitive to symmetry breaking, as the tunnelling from the QTM tip is not sensitive to the electron spin and valley and we do not have real-space sensitivity that can identify the spatial modulation of the spectrum that occurs in intervalley coherent or Kekulé spiral states. Experiments do not show any obvious signatures of symmetry breaking in the momentum-resolved electronic spectrum. In particular, there is no clear gap in the electronic spectrum at the Fermi level at any density (Fig. 3a). Our results are incompatible with a gap-opening broken-symmetry state at integer fillings at the temperature of the experiment.

Transport experiments typically see quantum oscillations emanating from ν = 2 and −2. Our measurements, on the other hand, do not show any signatures of Fermi surface at these filling factors. It is important, however, to keep in mind the experimental conditions. Our measurements are performed at T = 4 K and zero magnetic field. By contrast, quantum oscillations are necessarily measured at finite magnetic fields and it is therefore difficult to exclude the possibility that the oscillatory features observed at finite field reflect a field-stabilized ground state that differs from the zero-field state examined here.