title: Flavour Hund's Coupling, Correlated Chern Gaps, and Diffusivity in Moiré Flat Bands
authors: Jeong Min Park Yuan Cao Kenji Watanabe Takashi Taniguchi Pablo Jarillo-Herrero
text: 
  
  
  
  These authors contributed equally
  Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  
  
  Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  
  
  
  National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
  
  
  caoyuan@mit.edupjarillo@mit.edu
  Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  
  
  
  Flavour Hund's Coupling, Correlated Chern Gaps, and Diffusivity in Moiré Flat Bands
      Pablo Jarillo-Herrero
      December 30, 2023
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  Interaction-driven spontaneous symmetry breaking lies at the heart of many quantum phases of matter. In moiré systems, broken spin/valley `flavour' symmetry in flat bands underlies the parent state out of which ultimately correlated and topological ground states emerge<cit.>. However, the microscopic mechanism of such flavour symmetry breaking and its connection to the low-temperature many-body phases remain to be understood. Here, we investigate the symmetry-broken many-body ground state of magic angle twisted bilayer graphene (MATBG) and its nontrivial topology using simultaneous thermodynamic and transport measurements. We directly observe flavour symmetry breaking as a pinning of the chemical potential μ at all integer fillings of the moiré superlattice, highlighting the importance of flavour Hund's coupling in the many-body ground state. The topological nature of the underlying flat bands is manifested upon breaking time-reversal symmetry, where we measure energy gaps corresponding to Chern insulator states with Chern numbers C=3,2,1 at filling factors ν=1,2,3, respectively, consistent with flavour symmetry breaking in the Hofstadter's butterfly spectrum of MATBG. Moreover, our concurrent measurements of resistivity and chemical potential allow us to obtain the temperature dependence of the charge diffusivity of MATBG in the strange metal regime<cit.>, a quantity previously explored only in ultracold atom systems<cit.>. Our results bring us one step closer to a unified framework for understanding interactions in the topological bands of MATBG, both in the presence and absence of a magnetic field.
  
  In condensed matter systems with flat electronic bands, the Coulomb interaction between electrons can easily surpass their kinetic energy and give rise to a variety of exotic quantum phases, ranging from Mott insulators to quantum spin liquids <cit.>. In this strongly correlated regime, electrons may spontaneously order themselves to minimize the total Coulomb energy at the cost of increasing their kinetic energies, leading to the breaking of certain symmetries. Such symmetry-broken states can occur at a relatively high energy scale and act as a parent state for phases that appear at lower energy scales, such as superconductivity. Furthermore, when there is nontrivial topology in the system, the interplay between strong correlations and the underlying topology could lead to novel phases of matter, such as the fractional quantum Hall state<cit.>. Understanding the 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 of the Dirac electrons and creates flat bands at low energies <cit.>. 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<cit.>. Scanning tunneling microscopy (STM) experiments have directly shown the significance of on-site Coulomb interactions in MATBG <cit.>. More recently, scanning single-electron transistor and further STM measurements have suggested that the Coulomb interactions induce phase transitions that break the spin/valley symmetry <cit.>. Despite significant experimental and theoretical progress, the microscopic picture that underlies the broken symmetry states in MATBG and their possible connections to the correlated phases and unusual superconductivity is still far from being complete and requires investigation.
  
  
  
  
  Here we study the interplay between interaction-driven symmetry breaking and nontrivial topology in the flat bands of MATBG by directly measuring the combined thermodynamic and transport properties of its many-body ground state. For this, we use a unique technique<cit.> that involves a MLG probe layer in close proximity to MATBG to sense the chemical potential of MATBG at different charge densities, temperatures, and magnetic fields. Measuring chemical potential or compressibility is a class of techniques<cit.> complementary to the spectroscopic techniques that probe the excitation spectra, such as tunneling or photoemission spectroscopy. We find that the chemical potential distinctively reaches local extrema when the number of electrons per moiré unit cell (ν) is close to integers ±1, ±2, ±3 and ±4. We show that these results can be naturally explained in the framework of spin/valley `flavour' symmetry breaking, but where in addition to Coulomb repulsion<cit.>, we need to consider the intra-flavour Hund's coupling. The latter interaction results in the pinning of the chemical potential, favouring single flavour occupancy in a way analogous to the Hund's rule for spin alignment in multi-electron atoms. Surprisingly, the response of the chemical potential to the in-plane magnetic field indicates that a finite in-plane magnetization develops at ν=±1,±2,±3 at finite magnetic field, including the half-filling (ν=±2), suggesting that the magnetic state of the ν=±2 correlated states might be more intricate than previously thought <cit.>. Furthermore, the nontrivial topology of the MATBG flat bands is revealed when the time-reversal symmetry is broken by applying a perpendicular magnetic field<cit.>. We directly observe and measure the size of the correlated gaps with Chern numbers 3, 2, and 1, and demonstrate that all the experimentally observed states can be explained in the unified framework of symmetry broken Hofstdater spectrum <cit.>. With combined chemical potential and transport measurements, we compare the temperature dependence of resistivity, compressibility, and charge diffusivity in the high-temperature regime of MATBG where linear resistivity-temperature behaviour is observed <cit.>. Our data suggest that the diffusivity of MATBG is close to a diffusivity bound<cit.> proposed in the incoherent transport regime.
  
  Figure 1a illustrates our experimental scheme. The MATBG is separated from a monolayer graphene (MLG) layer by an ultrathin layer of h-BN (∼1). The MATBG layer is fabricated by a `laser-cut & stack' method (see Supplementary Information), which reduces the strain compared to our previous `tear & stack' method. 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. In this setup, direct probing of the chemical potential μ of one layer is achieved by sensing the screening of electric field from the gates by the other layer <cit.>. From the band alignment diagram shown in Fig. 1b, we can deduce the relationship between (V_tg, V_bg) and (n_MLG, μ_MLG, n_MATBG, μ_MATBG), the latter being density (n) and chemical potential (μ) in MLG and MATBG, respectively (see Supplementary Information for details). In particular, when one layer is at the charge neutrality point, e.g. n_MLG=0, the chemical potential of the other layer (μ_MATBG) is directly proportional to one of the gate voltages, which in this case 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 h-BN dielectrics, respectively. 
  
  
  The MLG layer used in our experiments has very low disorder <3e9 and excellent field-effect mobility >300000 (Fig. 1c). The Landau levels start developing from magnetic fields as low as ±30. 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/√(3)a^2 is the superlattice density of TBG and a=0.246 is the lattice constant of graphene), as well as superconducting states at both ν=-2-δ and +2+δ, where δ is a small change in filling. The superconducting transition temperature T_c reaches as high as 2.7 for ν=-2-δ, as determined from the 50 normal state resistance (see Extended Data Figure 1). The outstanding quality of both the MATBG and the MLG probe layers creates an exceptional platform to study the underlying physics in MATBG.
  
  To demonstrate the measurement principle, we first measure the chemical potential and resistivity of MLG with and without a magnetic field. A key advantage of this experimental technique is that we can simultaneously measure electronic transport in both layers and accurately correlate the chemical potential to the transport features. Fig. 1e and f show the resistance of MATBG and MLG as a function of V_tg and V_bg at B_⊥=0 and B_⊥=1, respectively. μ_MLG as a function of n_MLG is obtained by tracking the charge neutrality of MATBG (see inset of Fig. 1e and Supplementary Information for the conversion formulae). From these extracted values, we determine the Fermi velocity to be v_F = 1.12e6 by fitting to μ_MLG=ħ v_F √(π |n_MLG|)sgn(n_MLG). In a magnetic field B_⊥=1, the spectrum of MLG is quantized into discrete Landau levels. The extracted chemical potential in these Landau levels fits well to the Landau level spectrum of MLG, where the energy of the N-th Landau level is ± v_F√(2eħ B|N|). Our technique can thus determine the chemical potential of either layer with a sensitivity of ≲ .
  
  
  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. We show the data as a function of the MATBG filling factor ν and V_tg, which is directly proportional to the chemical potential of MATBG μ if one tracks the charge neutrality point of MLG (shown as the green curve). The resistance data of MLG (purple) and MATBG (orange) are overlaid for qualitative comparison of features, and the gray dash lines indicate the integer filling factors ν=0,±1,±2,±3 of MATBG, which correspond to filling 0, 1, 2, or 3 electrons (holes) per moiré unit cell, respectively. The rate that μ increases with n (or ν), known as the inverse electronic compressibility χ^-1=dμ/dn, is inversely proportional to the density of states (DOS) for a non-interacting system. Around the charge neutrality (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 rate of increase decreases quickly and μ reaches a local maximum around ν=0.6. Surprisingly, it then starts decreasing, exhibiting a negative χ^-1, <cit.> and it gets pinned at a local minimum around the integer filling ν=1. Having this as a turning point, μ rises again until it hits the next maximum. This intriguing pinning behaviour repeats itself at each integer filling factor, including ν=4 (see Fig. 2a inset). On the hole-doped side (ν<0), the pinning behaviour of the chemical potential is opposite and weaker (i.e creates weak maxima in μ). The total bandwidth estimated from the chemical potential at low temperatures is around ∼40, where both the electron and hole-doped sides contribute similarly. We also investigated the behaviour of μ while varying temperature from 2K to 70K, as shown in Fig. 2d. The observed pinning behaviour persists prominently up to 20K. We note that the observation of a negative compressibility indicates that our system might be in a strong Coulomb frustration regime <cit.>, which acts to suppress macroscopic phase separation that may occur otherwise in an unconstrained system. 
  
  
  
  
  The pinning of μ at all integer ν is reminiscent of the stabilization of electronic shells in atoms when they are half-filled or full-filled, which is known as Hund's rule for maximum spin multiplicity. The physical origin of the Hund's rule stems from the Coulomb exchange interaction between the electrons. Here in MATBG, we also find that 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 the ν>0 side in the following description. Figure 2b shows the chemical potential calculated with a mean-field model for different values of U and J. Our model can reproduce qualitatively the experimentally measured chemical potential  only when both U and J are nonzero and of similar magnitude (purple solid curve), beyond the currently established understanding<cit.>. We wish to emphasize the importance of the exchange energy in stabilizing the chemical potential by illustrating a possible mechanism for the ν=1 case (see Fig. 2c). Near charge neutrality, as the density is increased, all four flavours are filled at the same rate. As ν starts to approach one, the Coulomb repulsion between different flavours starts to surpass the kinetic energy penalty of filling up only one flavour. As ν reaches a certain value (still below 1), a flavour-symmetry-breaking phase transition occurs and all electrons are transferred into a single flavour to minimize the Coulomb repulsion <cit.>. From this phase transition point all the way to ν=1, i.e. while a single flavour is being filled, the U term does not have any contribution to the free energy, while the J term decreases the total free energy as ∼ -Jν^2 (see Supplementary Information). This term decreases the chemical potential and results in a negative inverse compressibility χ^-1∝𝒟^-1-2J (𝒟 is the single-particle DOS per flavour) when 2J>𝒟^-1. At ν=1, maximal stabilization by the exchange term J is reached, and thus the pinning of μ. Further increase in ν populates the other three empty flavours and it increases the chemical potential before the next phase transition occurs. We also note that the pinning of chemical potential on the hole-doped side occurs at slightly more negative values of ν compared to exact integers, which may be attributed to smaller U/W and/or J/W ratios on that side, where W is the single-particle bandwidth (see Supplementary Information).
  
  
  
  To directly probe the magnetic properties of the correlated states, we measured the chemical potential in MATBG as a function of in-plane magnetic field up to 11. The results are shown in Fig. 2e and zoomed-in in Fig. 2f. In the lower panels of Fig. 2f, we also show the resistance of MATBG as measured by transport. At ν=± 1, the pinning of the chemical potential is clearly strengthened by B_∥, as is the intensity of the transport resistance peak (see Methods and Extended Data Figure 2). These findings suggest that the states ν=± 1 develop a spin-polarization in response to the magnetic field. To confirm this, we directly obtained the magnetization by integrating the Maxwell's relation (∂ M_∥/∂ν)_B_∥,T = -(∂μ/∂ B_∥)_ν,T, where M_∥ is the magnetization per moiré unit cell induced by the field<cit.>. The inset of Fig. 2e shows M_∥ in units of Bohr magneton μ_B per moiré unit cell. 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. We do not observe hysteresis in transport, which suggests it may be the former. The ν=±2 states, on the other hand, have been speculated to be spin-unpolarized insulating states<cit.>. However, we find that, while the resistance of these insulating states is indeed suppressed by the in-plane magnetic field (see Fig. 2f and Extended Data Figure 2), to our surprise the chemical potential 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 on B_∥ at ν=±2 can be partially captured by our theoretical model (see Supplementary Information), the persistence of magnetization near ν=±2 is at odds with the finite-field spin-unpolarized state inferred from the suppression of the transport gap. 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 experiment also puts constraints on the possible mechanism of superconductivity in MATBG. As shown in Extended Data Figure 5b, the superconducting dome lies in the region where χ^-1 is high, with maximum T_c corresponding to a maximum in χ^-1. Since, in the non-interacting limit, χ is equal to the single-particle DOS,  a Bardeen-Cooper-Schrieffer (BCS) type superconductivity would be enhanced when the DOS is high and thus have the highest T_c when χ^-1 is low. Therefore, our observation of an opposite trend indicates that it is not easy 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 take into account 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 the chemical potential of MATBG in a perpendicular magnetic field, we can gain insight into the energy gaps in MATBG that result from the interplay between the topology of the Hofstadter spectrum and the Coulomb interactions, as suggested by recent experiments <cit.>. The helical nature of the Dirac electrons in graphene endows each flat band of MATBG a Chern number of C=±1, which however is only explicitly manifested when the composite inversion-time reversal (C_2𝒯) symmetry is broken, either by alignment to the h-BN substrate (breaks C_2) or by applying a magnetic field (breaks 𝒯). Fig. 3c shows the Hofstadter butterfly spectrum of TBG, where the topologically nontrivial gaps with C=±1 and the trivial gaps with 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 among the flavours can cause their Chern numbers to be different, and can give rise to new hierachies of correlated Chern gaps. 
  
  These topological gaps are directly observed in our chemical potential measurements, as shown in Fig. 3a. Near charge neutrality, we observe the quantum Hall gaps as steps in the chemical potential at the Landau level filling factors ν_LL=0, ±2, ±4, whose positions evolve according to the Streda formula dn/dB = ν_LL/ϕ_0 <cit.>. The appearance of the Landau level gaps at ν_LL=0, ±2 indicates that the flavour symmetry is already broken. In the meantime, the extrema in the chemical potential at ν=1,2,3 at B_⊥=0 evolve into topological gaps at B_⊥=6. The topological nontriviality of these gaps is evident from the fact that 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 appearance of the broken-symmetry Landau levels and topological Chern gaps can be analyzed in a unified way using a correlated Hofstdater spectrum model<cit.>. We consider the single-particle DOS to be representative of the Hofstadter spectrum shown in Fig. 3c<cit.>, with possible Chern numbers C=0, -1, +1, 0 associated with the major gaps for each flavour, and add the mean-field Coulomb repulsion and exchange terms U and J in a similar manner as above. Using this model, we calculate the Chern number C as a function of total filling ν and reproduce the experimentally observed sequence of 0, ±2, ±4 at the charge neutrality Landau levels, and 3,2, and 1 at densities ν=1+3ϕ/ϕ_0, 2+2ϕ/ϕ_0, and 3+ϕ/ϕ_0, respectively, as shown in Fig. 3d. In the top part, we also illustrate the contribution to the total Chern number from each flavour by colour-coded dots in accordance with Fig. 3c. By performing a similar calculation (see Supplementary Information), we can simulate the evolution of the chemical potential with the magnetic field, as shown in Fig. 3b. The remarkable similarity with the experimental data clearly indicates that this model captures the main features of the correlated spectrum of MATBG with and without a magnetic field.
  
  A more quantitative analysis is performed on the chemical potential measured at B_⊥=6, as shown in Fig. 3e. From the steps in μ, we can directly extract the sizes of all energy gaps in the spectrum without relying on any temperature-dependent measurement. The Landau level gaps at ν_LL=-4,-2,0,2, and 4 are 5.9, 3.3, 5.9, 2.3, and 4.9, respectively. The small sizes of the gaps at ν_LL=±4 translate to a vastly renormalized Fermi velocity of approximately v_F∼6e4, consistent with other experimental probes in MATBG<cit.>. The gaps at ν_LL=0, ±2, on the other hand, are broken-symmetry gaps created by the Coulomb interactions U and J, and have similar sizes as those found in the ν_LL=0, ±1 broken-symmetry states in MLG<cit.>. However, a fundamental difference with MLG is that the broken-symmetry gaps in MATBG have the same energy scale as the single-particle gaps at ν_LL=±4, another manifestation of the fact that U, J are on the same order as W. We also find clear evidence of smaller gaps at ν_LL=±1, ±3, which require further symmetry breaking than those discussed here (see Supplementary Information). Furthermore, the sizes of the topological 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 respectively. The larger gap at ν=2+2ϕ/ϕ_0 is consistent with the fact that this state is more readily resolved in electronic transport experiments<cit.>. Its difference with the gaps at ν=1+3ϕ/ϕ_0 and 3+ϕ/ϕ_0 might be attributed to the different magnetic ground state, with contributions from both orbital and spin degrees of freedom of the two fully filled flavours. These gaps have a weak dependence on B_⊥, as shown in Extended Data Figure 4, consistent with the Hofstadter spectrum in 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<cit.>. In MATBG, recent experiments have reported evidence for strange metal behaviour<cit.>, manifested as resistivity linear with temperature, from very low T up to T above the Fermi energy. As shown in Fig. 4a and c, 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, consistent with recent works<cit.>. The resistivity keeps increasing with T without any sign of saturation up to 50, suggesting non-Fermi liquid transport in this system<cit.>. It has been hypothesized that the strange metal behaviour can be universally described by a `Planckian' scattering rate bound Γ∼ k_BT/ħ in the framework of incoherent non-quasiparticle transport <cit.>. However, the construction of a microscopic picture for this bound is still in progress<cit.>.
  
  An appropriate framework to investigate the strange metal regime, regardless of the existence or absence of quasiparticles, is the Nernst-Einstein relation, which connects the resistivity ρ, compressibility χ, and charge diffusivity D of a generic conductor by ρ^-1=e^2χ D. A linear in T resistivity could thus originate from: (i) χ^-1∝ T, which could come from thermodynamic contributions when k_BT≳ W<cit.>; from (ii) D^-1∝ T, which would represent a linear scattering rate; or (iii) from a more complex combined T dependence of both. Differentiating between these possibilities could help constrain theoretical models for strange metal behaviour<cit.>. However, to the best of our knowledge, there are no reported measurements of the electronic compressibility or charge diffusivity for any correlated materials in the strange metal regime, and only recent experiments have begun to explore this physics in the very high temperature regime in ultracold atom systems<cit.>.
  
  Our combined resistivity and compressibility measurements allow us to extract the charge diffusivity of MATBG. Figure 4b shows the inverse compressibility χ^-1 as a function of ν and T. While for T<20 χ^-1 becomes negative before each integer filling factor, as discussed above, at higher T it converges to a roughly constant value of order 1 for any value of ν. Figure 4d shows some representative traces of χ^-1 vs T for the same ν values as in Fig. 4c. The traces exhibit only a weak dependence on T, albeit for all these densities ρ exhibits a prominent linear T behaviour. Therefore, this suggests that the linear ρ-T behaviour in MATBG is mainly due to a T-dependent charge diffusivity D. Figure 4e shows the T-dependence of the extracted effective diffusivity D^*=χ^-1/e^2(ρ-ρ_0), where ρ_0 is the residual resistivity extrapolated at zero temperature, and Fig. 4f shows its inverse 1/D^*. These quantities indeed do appear to roughly follow a ∼ T^-1 and a ∼ T trend, respectively. Our observations therefore indicate that the strange metal transport regime in MATBG is consistent with a scattering rate linear in T. Note that these arguments do not apply to regions with negative electronic compressibility and thus negative D^*, as the interpretation of diffusivity in this case needs to be modified<cit.> (see also Supplementary Information for relevant data and discussion). 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=ħ v_F^2/(k_B T) proposed for incoherent metals<cit.>, using v_F=6e4 estimated above at low temperatures. While this bound is known to be violated in the low-temperature region in a large-U system<cit.>, this is not at odds with our observations if MATBG is in the intermediate U regime (U/W∼1), as suggested by the range of U that qualitatively reproduce our experimental data, as well as other recent experiments.<cit.>
  
  
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  § ACKNOWLEDGEMENTS
  
  The authors thank Antoine Georges, Francisco Guinea, Shahal Ilani, Aharon Kapitulnik, Leonid Levitov, Louis Taillefer, Senthil Todadri, Ashvin Vishwanath, Amir Yacoby, Denis Bandurin, Sergio de la Barrera, Clement Collignon, Ali Fahimniya, Daniel Rodan-Legrain, Yonglong Xie, and Kenji Yasuda for fruitful discussions.
  
  
  
  § METHODS
  
  
  
  
   §.§ Sample Fabrication
  
  The multilayer heterostructure consists of one sheet of monolayer graphene (MLG) and twisted bilayer graphene (TBG) twisted at a small angle θ∼1.1, separated by a thin (∼1nm) h-BN layer. This sandwich is encapsulated by two h-BN flakes. All flakes were first exfoliated on SiO2/Si substrates, and subsequently analyzed with optical microscopy and atomic force microscopy to determine their thicknesses and quality. The multilayer heterostructure was fabricated by a modified polymer-based dry pick-up technique, where a layer of poly(bisphenol A carbonate)(PC)/polydimethylsiloxane(PDMS) on a glass slide fixed on the micro-positioning stage was used to sequentially pick up the flakes. The order of the pick-up was h-BN-MLG-h-BN(1nm)-MLG-MLG, where the last two MLG sheets were laser-cut from one MLG flake (see Supplementary Information) and twisted by an angle ∼1.1. All h-BN layers were picked up at 90, while the MLG layers were picked up at room temperature. The h-BN-MLG-h-BN(1nm)-MLG-MLG heterostructure was then released on the pre-stacked h-BN-Pd/Au back gate at 175. Hall-bar geometry for transport measurements was defined with electron beam lithography and reactive ion etching for each of the MLG and MLG-MLG layers. The top gate and electrical edge-contacts were patterned with electron beam lithography and thermal evaporation of Cr/Au. 
  
  
  
   §.§ Measurement Setup
  
  Electronic transport measurements were performed in a dilution refrigerator with a superconducting magnet, with a base electronic temperature of 70. Current through the sample, amplified by 1e7, and the four-probe voltage, amplified by 1000, were measured with SR-830 lock-in amplifiers synchronized at the same frequency between 120. Current excitation of 1 or voltage excitation of 50 to 100 was used for resistance measurements. We measured both MLG and MATBG layers simultaneously for accurate comparison.
  
  
  
  
   §.§ Maxwell's Relations
  
  Using Maxwell's relations between thermodynamic variables, we can obtain information about various thermodynamic quantities by taking different derivatives of the chemical potential. The free energy of the system per unit area in the presence of a magnetization can be written as g=u-Ts+M_∥B_∥, where u, M, s are the internal energy, magnetization, and entropy per area respectively. u and g satisfy
  
      du    = Tds + B_∥dM_∥ + μ dν,
      
          dg    = -sdT - M_∥dB_∥ + μ dν.
  
  
  By taking the second derivative of g with respective to (ν, B) in different orders, we can obtain the following Maxwell's relationship,
  
      (∂ M_∥/∂ν)_T,B_∥   = -(∂μ/∂ B_∥)_T,ν.
  
  
  Therefore, we can integrate from the B_∥-derivative of μ to obtain the change in M_∥ as a function of density ν,
  
      M_∥   = M_∥(ν=0) - ∫_0^ν(∂μ/∂ B_∥)_T,n dn
  
  
  The extracted ∂ M_∥/∂ν and M_∥ versus ν are shown in Extended Data Figure 3. We extract the uncertainty (95 confidence interval) of ∂ M_∥/∂ν from fitting of μ with B_∥, and propagate through the integration to obtain uncertainty in M_∥.
  
  
  
   §.§ Thermal activation gap analysis
  
  Thermal activation gap analysis was performed based on the Arrhenius formula R ∼exp(-Δ/2k_BT), where k_B is the Boltzmann constant and Δ is the gap size. A temperature-dependent background was removed from the raw resistance R_xx of MATBG to avoid being affected by the linear R_xx-T behaviour in MATBG.<cit.> The corrected quantity is denoted by R^*_MATBG and shown in Extended Data Fig. 2a-b. By fitting the gaps as a function of the in-plane magnetic field B_∥ to Δ=gμ_BB_∥, where μ_B is the Bohr magneton, we find effective transport g-factors of ∼1.31 for the ν=+2 state and ∼0.57 for the ν=+1 state, as shown in Extended Data Fig. 2c.
  
  
  
  
  § EXTENDED DATA FIGURES
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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