physics behind this interplay could guide us in designing next-generation strongly-correlated topological quantum materials.

Magic-angle twisted bilayer graphene (MATBG) serves as a unique platform to investigate interaction driven phenomena in a highly tunable flat-band system. When two layers of monolayer graphene (MLG) are stacked with a small twist angle of 𝜃 ∼ 1.1°, the interlayer hybridization in the resulting moiré superlattice renormalizes the Fermi velocity and creates flat bands at low energies 16,17 . In this regime, a plethora of exotic correlated and topological phenomena have been experimentally demonstrated, including correlated insulator states, superconductivity, and the quantum anomalous Hall effect 1,2,[4][5][6][7] . Scanning tunneling and singleelectron transistor experiments have directly shown the significance of Coulomb-induced phase transitions that break the spin/valley symmetry 9,10,[18][19][20][21] . Despite significant experimental and theoretical progress, the microscopic picture that underlies the broken-symmetry states and their possible connections to the correlated phases and superconductivity still requires investigation.

Here we study the interplay between interaction-driven symmetry breaking and nontrivial topology in MATBG by directly measuring the combined thermodynamic and transport properties of its many-body ground state. We use a unique technique 22 to sense the chemical potential of MATBG. The MATBG is separated from a MLG layer by an ultrathin layer of hBN (∼ 1 nm, Fig. 1a). We use the top gate voltage 𝑉 𝑡𝑔 and back gate voltage 𝑉 𝑏𝑔 to control the densities in MLG and MATBG, and measure the transport properties of the two layers simultaneously. Direct probing of the chemical potential 𝜇 of one layer is achieved by sensing the screening of the electric field from the gates by the other layer 22 (Fig. 1b and Supplementary Information). In particular, when one layer is at the charge neutrality point (CNP), e.g. 𝑛 𝑀𝐿𝐺 = 0, the chemical potential of the other layer (𝜇 𝑀𝐴𝑇𝐵𝐺 ) is given by 𝜇 𝑀𝐴𝑇𝐵𝐺 = -( 𝑒𝐶 𝑡𝑔 𝐶 𝑖 ) 𝑉 𝑡𝑔 , where 𝐶 𝑡𝑔 and 𝐶 𝑖 are the geometric capacitances per unit area of the top and middle hBN dielectrics, respectively.

The MLG layer used in our experiments has very low disorder < 3 × 10 9 cm -2 (Fig. 1c). The MATBG layer has a twist angle of 𝜃 = 1.07 ± 0.03°, and exhibits correlated states at integer filling factors 𝜈 𝑀𝐴𝑇𝐵𝐺 = 4𝑛 𝑀𝐴𝑇𝐵𝐺 𝑛 𝑠 = +1, ±2, +3 of the flat bands (𝑛 𝑠 = 8𝜃 2 /√3𝑎 2 is the superlattice density of TBG and 𝑎=0.246 nm is the graphene lattice constant), as well as superconducting states at both 𝜈 = -2 -𝛿 and +2 + 𝛿, where 𝛿 is a small change in filling (Fig. 1d). The superconducting transition temperature 𝑇 𝑐 reaches as high as 2.7 K for 𝜈 = -2 -𝛿 (Extended Data Figure 1). Figure 1e andf show the resistance of MATBG and MLG as a function of 𝑉 𝑡𝑔 and 𝑉 𝑏𝑔 at 𝐵 ⊥ =0 T and 𝐵 ⊥ =1 T, respectively. As a proof of principle, 𝜇 𝑀𝐿𝐺 as a function of 𝑛 𝑀𝐿𝐺 is obtained by tracking the CNP of MATBG (Fig. 1e inset and Supplementary Information), from which we determine the MLG Fermi velocity to be 𝑣 𝐹 = 1.12 × 10 6 m/s by fitting to |𝜇 𝑀𝐿𝐺 | = ℏ𝑣 𝐹 √𝜋|𝑛 𝑀𝐿𝐺 |. In a magnetic field 𝐵 ⊥ = 1 T, the spectrum of MLG is quantized into discrete Landau levels at energies ±𝑣 𝐹 √2𝑒ℏ𝐵|𝑁| as expected. Our technique can thus determine the chemical potential of either layer with a sensitivity of ≲ 1 meV.

The chemical potential of MATBG is shown in Fig. 2a. Hereafter we will simply use 𝑛 (𝜈) and 𝜇 to denote 𝑛 𝑀𝐴𝑇𝐵𝐺 (𝜈 𝑀𝐴𝑇𝐵𝐺 ) and 𝜇 𝑀𝐴𝑇𝐵𝐺 . The 𝑉 𝑡𝑔 axis is directly proportional to 𝜇 when tracking the CNP of MLG (shown as the green curve). The longitudinal resistance 𝑅 𝑥𝑥 of MLG (purple) and MATBG (orange) is overlaid for qualitative comparison, and the gray dash lines indicate the integer filling factors 𝜈 = 0, ±1, ±2, ±3. Around the MATBG CNP (𝜈 = 0), 𝜇 rises quickly with 𝜈, consistent with a minimal DOS at the Dirac point. However, once we start filling electrons into the flat band, its slope decreases quickly and 𝜇 reaches a local maximum around 𝜈 = 0.6. Surprisingly, it then decreases, exhibiting a negative inverse compressibility 𝜒 -1 = 𝑑𝜇/𝑑𝑛, 23 and gets pinned at a local minimum around 𝜈 = 1. Subsequently, 𝜇 rises again until it reaches the next maximum. This intriguing pinning behaviour repeats at each integer filling factor, including 𝜈 = 4 (Fig. 2a inset). On the hole-doped side (𝜈 < 0), the pinning behaviour is opposite and weaker (i.e. creates weak maxima in 𝜇). The total bandwidth estimated from 𝜇 is around ∼40 meV. We also investigated 𝜇 versus temperature from 2 K to 70 K (Fig. 2d). The observed pinning behaviour persists prominently up to 20 K. We point out that the pinning of 𝜇 should not be interpreted as a measure of the gaps of the insulator states because its energy scale (visible up to 70 K) is much greater than the typical energy scale of the insulator states (typically below 10 K). Instead, the insulator state, and possibly also the superconducting state, might be thought of as low-energy states that emerge from the broken flavour symmetry 'parent' states. We also note that the pinning on the hole-doped side occurs at slightly more negative values of 𝜈 (Supplementary Information).

The pinning of 𝜇 at all integer 𝜈 is reminiscent of the stabilization of half-filled or full-filled electronic shells in atoms, which is known as Hund's rule for maximum spin multiplicity and stems from the Coulomb exchange interaction between the electrons. In MATBG, the pinning behaviour of the chemical potential is naturally explained when both the on-site inter-flavour Coulomb repulsion energy U and inter-site intra-flavour exchange energy J are considered. We focus on 𝜈 > 0 in the following description. Figure 2b shows 𝜇 calculated with a mean-field model for different values of U and J (Supplementary Information), which qualitatively reproduces the experimentally measured 𝜇 only when both U and J are nonzero and of similar magnitude (purple solid curve), beyond the currently established understanding 9,10,21 . A possible mechanism for such stabilization of 𝜇 at 𝜈 = 1 is illustrated in Fig. 2c and elaborated in the Methods. We note that the mean-field treatment of the Coulomb interactions correctly captures the many-body compressibility to leading order 24 , but might not give the same ground state as the exact solution. Other mechanisms, such as the formation of a Wigner crystal 25 , might also be relevant to the observation of negative compressibility.

To probe the magnetic properties of the correlated states, we measured 𝜇 as a function of inplane magnetic field up to 11 T (Fig. 2e,f for 𝜈 = +1 and 𝜈 = +2 and Extended Data Figure 2 for 𝜈 = -1, -2, +3). At 𝜈 = ±1, the pinning of 𝜇 is clearly strengthened by 𝐵 ∥ , as is the peak in 𝑅 𝑥𝑥 𝑀𝐴𝑇𝐵𝐺 (Methods and Extended Data Figure 3), suggesting that the 𝜈 = ±1 states develop a spin-polarization in response to the magnetic field. To confirm this, we directly obtained the magnetization by integrating the Maxwell's relation 10 (Fig. 2e inset). We indeed find that the magnetization reaches a value on the order of one 𝜇 𝐵 at 𝜈 = ±1, consistent with a spin-polarized state at finite field, which would indicate either a very soft paramagnetic state or a ferromagnetic state at zero field. The 𝜈 = ±2 states, on the other hand, have been speculated to be spinunpolarized insulating states 1,4,26 . However, we find that, while the transport peak there is indeed suppressed by 𝐵 ∥ (see Fig. 2f and Extended Data Figure 3), 𝜇 measured at 𝜈 = ±2 does not show significant dependence on the in-plane magnetic field (Fig. 2f). Furthermore, 𝑀 ∥ does not return to zero when 𝜈 is tuned from ±1 to ±2 (Fig. 2e inset). While the lack of dependence of 𝜇 can be partially captured by our theoretical model (Supplementary Information), the persistence of magnetization near 𝜈 = ±2 is at odds with the finite-field spin-unpolarized ground state inferred from transport. These observations suggest that in an in-plane field the 𝜈 = ±2 gaps might select a ground state with nontrivial spin and/or valley texture, beyond simply occupying two flavours with opposite spins.

Our experiments also constrain the possible mechanism of superconductivity in MATBG. The superconducting dome lies in the region where 𝜒 -1 is high (Extended Data Figure 4b), with maximum 𝑇 𝑐 corresponding to a maximum in 𝜒 -1 . Since a Bardeen-Cooper-Schrieffer (BCS) type superconductivity in the weak-coupling limit would be enhanced when the DOS is high (and thus χ -1 low), our observation of an opposite trend indicates that it is hard to reconcile the superconductivity in MATBG with a weakly-coupled BCS theory. Future theories attempting to model the superconductivity in MATBG will likely need to consider the importance of Coulomb interactions, including both repulsion and Hund's coupling, and the consequent phase transitions.

We now turn to the topological properties of MATBG. By measuring 𝜇 in a perpendicular magnetic field, we can observe the energy gaps that result from the interplay between the Hofstadter spectrum and the Coulomb interactions [26][27][28][29][30] . The helical nature of the Dirac electrons in graphene endows each flat band of MATBG a Chern number of 𝐶 = ±1, which is usually explicitly manifested when the composite 𝐶 2 𝒯 symmetry is broken, either by alignment to the hBN substrate (breaks 𝐶 2 ) or by applying a magnetic field (breaks 𝒯). Figure 3c shows the Hofstadter butterfly spectrum of TBG, where the topologically nontrivial gaps (𝐶 = ±1) and the trivial gaps (𝐶 = 0) are shown. The former gaps are smoothly connected to the Landau level gaps at 𝜈 𝐿𝐿 = 𝜈/(𝜙/𝜙 0 ) = ±4 at low fields, where 𝜙 is the magnetic flux per unit cell and 𝜙 0 = ℎ/𝑒 is the flux quantum. Without interactions, the only possible total Chern number in this picture is 𝐶 𝑡𝑜𝑡 = 0, ±4, since all flavours are in the same gap. The Coulomb interactions cause their Chern numbers to be different, and give rise to new hierarchies of Chern gaps. These topological gaps are directly observed in Fig. 3a. Near charge neutrality, we observe the gaps as steps in 𝜇 at the Landau level filling factors 𝜈 𝐿𝐿 = 0, ±2, ±4, whose positions evolve according to the Streda formula 31 𝑑𝑛/𝑑𝐵 = 𝜈 𝐿𝐿 /𝜙 0 . In the meantime, the extrema of 𝜇 at 𝜈 = 1, 2, 3 at 𝐵 ⊥ = 0 evolve into topological gaps at 𝐵 ⊥ = 6 T. Their evolution follows the same Streda formula 𝑑𝑛/𝑑𝐵 = 𝐶/𝜙 0 that indicates the total Chern number of 𝐶 = 3,2,1 associated with the states originally at 𝜈 = 1,2,3, respectively.

The broken-symmetry Landau levels and topological Chern gaps can be analyzed in a unified way using a correlated Hofstadter spectrum model 32 . We consider the single-particle DOS to be representative of the Hofstadter spectrum shown in Fig. 3c, and add the mean-field U and J in a similar manner as above. Using this model, we calculate the Chern number 𝐶 as a function of 𝜈 and reproduce the experimentally observed sequence of 0, ±2, ±4 near CNP, and 3, 2, and 1 at densities 𝜈 = 1 + 3𝜙/𝜙 0 , 2 + 2𝜙/𝜙 0 , and 3 + 𝜙/𝜙 0 , respectively (Fig. 3d). By performing a similar calculation (Supplementary Information), we can simulate the evolution of the chemical potential with the magnetic field (Fig. 3b). The remarkable similarity with the experimental data clearly indicates that this model captures the main features of the correlated spectrum with and without a magnetic field.

A more quantitative analysis is performed on the chemical potential measured at 𝐵 ⊥ = 6 T (Fig. 3e). From the steps in 𝜇, we extract the values of the Landau level gaps at 𝜈 𝐿𝐿 = -4, -2, 0, 2, and 4 to be 5.9, 3.3, 5.9, 2.3, and 4.9 meV, respectively. The small values of the gaps at 𝜈 𝐿𝐿 = ±4 translate to a Fermi velocity of approximately 𝑣 𝐹 ∼ 6 × 10 4 m/s, consistent with previous experiments 1,27 . The Chern gaps at 𝜈 = 1 + 3𝜙/𝜙 0 , 2 + 2𝜙/𝜙 0 , and 3 + 𝜙/𝜙 0 are extracted to be 2.2, 5.0 and 1.9 meV respectively. The larger gap at 2 + 2𝜙/𝜙 0 is consistent with the fact that this state is more readily resolved in electronic transport experiments 1,2,4,5,26,29 . Its difference with the gaps at 𝜈 = 1 + 3𝜙/𝜙 0 and 3 + 𝜙/𝜙 0 might be attributed to different magnetic ground states. These gaps have a weak dependence on 𝐵 ⊥ (Extended Data Figure 5), consistent with the Hofstadter spectrum (Fig. 3c).

In correlated metals with multiple bands near the Fermi energy, the atomic Hund's coupling is known to play an important role in their many-body physics, including the strange metal regime 33 . In MATBG, recent experiments have reported evidence for strange metal behaviour 11 , manifested as resistivity linear with T from very low T. As shown in Fig. 4a andc, the resistivity in our MATBG sample is largely linear with T over a range of densities around the correlated states, and with a slope that is weakly dependent on n 11,34 . It has been hypothesized that the strange metal behaviour can be universally described by a 'Planckian' scattering rate bound Γ ∼ 𝑘 𝐵 𝑇/ℏ in the framework of incoherent non-quasiparticle transport 35 . However, the construction of a microscopic picture for this bound is still in progress 36,37 .

A universal framework to investigate the strange metal regime is the Nernst-Einstein relation, which connects the resistivity 𝜌, compressibility 𝜒, and charge diffusivity 𝐷 of a generic conductor by 𝜌 -1 = 𝑒 2 𝜒𝐷. A linear in T resistivity could thus originate from: (i) 𝜒 -1 ∝ 𝑇, which could arise from thermodynamic contributions 38,39 when 𝑘 𝐵 𝑇 ≳ 𝑊; (ii) 𝐷 -1 ∝ 𝑇, which represent a linear scattering rate; or (iii) a combined T-dependence of both. Differentiating between these possibilities could help constrain theoretical models for strange metal behaviour 38,40,41 . However, to the best of our knowledge, there are no reported measurements of the compressibility or charge diffusivity for strange metals, and only recent experiments have begun to explore this physics in ultracold atoms 12 .

Our combined resistivity and compressibility measurements allow us to extract the charge diffusivity of MATBG (Fig. 4b). While 𝜒 -1 becomes negative before each integer filling factor at T < 20 K, as discussed above, at higher T it converges to a roughly constant value of order 1 eV•nm 2 regardless of 𝜈. Figure 4d shows selected traces of 𝜒 -1 vs T, which exhibit only a weak dependence on T, albeit 𝜌 exhibits a prominent linear-in-T behaviour, suggesting that the linear 𝜌-T behaviour in MATBG is mainly due to a T-dependent charge diffusivity. Figure 4e-f shows the T-dependence of the extracted effective diffusivity 𝐷 * = 𝜒 -1 /𝑒 2 (𝜌 -𝜌 0 ) and its inverse, where 𝜌 0 is the residual resistivity extrapolated at zero temperature. These quantities indeed appear to roughly follow a ∼ 𝑇 -1 and a ∼ 𝑇 trend, respectively. Our observations therefore indicate that the strange metal regime in MATBG is consistent with a scattering rate linear in T. These arguments do not apply to regions with negative electronic compressibility as the interpretation of diffusivity in this case needs to be modified 42 (Supplementary Information). Interestingly, we find the extracted diffusivity D*(T) at all these fillings to be within about a factor of 2 from a diffusivity bound 𝐷 𝑏𝑜𝑢𝑛𝑑 = ℏ𝑣 𝐹 2 /(𝑘 𝐵 𝑇) proposed for incoherent metals 38 . While this bound is known to be violated in the low-temperature region in a large-U system 40 , this is not at odds with our observations if MATBG is in the intermediate U regime (𝑈/𝑊 ∼ 1, deduced from our modeling and other experiments 9,10,18 ). The twist angle of MATBG is 𝜃 = 1.07 ± 0.03°. We find correlated states at filling factors 𝜈 𝑀𝐴𝑇𝐵𝐺 = 1, ±2, 3, as well as superconducting domes (blue) at -2 -𝛿 and +2 + 𝛿, respectively. (e-f) Combined plot of the resistance of MLG and MATBG, represented by purple and orange colour scales, respectively, and overlaid in the same axes. As a proof of principle, we use the charge neutrality point (CNP) of MATBG (orange diagonal feature) to probe the chemical potential of MLG, at (e) B=0 and (f) 𝐵 ⊥ =1 T. The horizontal purple stripes are the resistive features in MLG. From the CNP of MATBG, we extract the chemical potential 𝜇 𝑀𝐿𝐺 versus density 𝑛 𝑀𝐿𝐺 , which is shown in the insets of (e-f). The white line in the inset of (e) is a fit to |𝜇 𝑀𝐿𝐺 | = ℏ𝑣 𝐹 √𝜋|𝑛 𝑀𝐿𝐺 | . The red ticks in the inset of (f) denote the expected Landau level (LL) energies ±𝑣 𝐹 √2𝑒ℏ𝐵|𝑁|, where 𝑣 𝐹 = 1.12 × 10 6 m/s and N is an integer. . Near charge neutrality we find gaps that correspond to Landau level filling factors 𝜈 𝐿𝐿 = 0, ±2, and ±4, while the pinning of 𝜇 at 𝜈 = 1,2,3 shown in Fig. 2 evolves into topological gaps with Chern numbers C=3,2,1, respectively, as evident from their slope in magnetic field 𝑑𝑛/𝑑𝐵 = 𝐶/𝜙 0 , where 𝜙 0 is the flux quantum. (c) The Hofstadter's butterfly spectrum of TBG up to a flux per unit cell of 𝜙 0 /2 (calculation shown for 1.8°, but spectrum is qualitatively similar for the magic angle). The major gaps in the spectrum have Chern numbers of C=0, -1, +1, 0 per flavour, respectively. (d) Calculated total Chern number of TBG using the mean-field model with Coulomb repulsion and exchange interactions for a flux of 𝜙 0 /6. The correct Chern number is reproduced, both in the Landau levels near the charge neutrality (C=-4,-2,0,2,4, indicated by red bars) and in the correlated Chern gaps (C=3,2,1, indicated by the blue bars). The dots above the plot show the configuration of the four flavours in each gap. The colouring scheme of the dots matches the ones shown in (c). Adding the Chern number of each flavour gives the total Chern number. (e) Extraction of energy gaps in the correlated spectrum of MATBG at 𝐵 ⊥ = 6 T. See Extended Data Figure 5 for their dependence on 𝐵 ⊥ .