Entropic evidence for a Pomeranchuk effect in magic angle graphene

In the 1950's, Pomeranchuk predicted that, counterintuitively, liquid 3He may solidify upon heating, due to a high excess spin entropy in the solid phase. Here, using both local and global electronic entropy and compressibility measurements, we show that an analogous effect occurs in magic angle twisted bilayer graphene. Near a filling of one electron per moir'e unit cell, we observe a dramatic increase in the electronic entropy to about 1kB per unit cell. This large excess entropy is quenched by an in-plane magnetic field, pointing to its magnetic origin. A sharp drop in the compressibility as a function of the electron density, associated with a reset of the Fermi level back to the vicinity of the Dirac point, marks a clear boundary between two phases. We map this jump as a function of electron density, temperature, and magnetic field. This reveals a phase diagram that is consistent with a Pomeranchuk-like temperature- and field-driven transition from a low-entropy electronic liquid to a high-entropy correlated state with nearly-free magnetic moments. The correlated state features an unusual combination of seemingly contradictory properties, some associated with itinerant electrons, such as the absence of a thermodynamic gap, metallicity, and a Dirac-like compressibility, and others associated with localized moments, such as a large entropy and its disappearance with magnetic field. Moreover, the energy scales characterizing these two sets of properties are very different: whereas the compressibility jump onsets at T~30K, the bandwidth of magnetic excitations is ~3K or smaller. The hybrid nature of the new correlated state and the large separation of energy scales have key implications for the physics of correlated states in twisted bilayer graphene.

Systems of strongly interacting fermions exhibit a competition between localization, minimizing the potential energy, and itineracy, minimizing the kinetic energy. The advent of two-dimensional moiré systems, such as magic angle twisted bilayer graphene [2][3][4][5][6] (MATBG), opens a new route to study this physics by controlling the ratio between the electronic interactions and bandwidth in a highly tunable artificial lattice. In systems where this ratio is large, such as transition metal dichalcogenides hetero-bilayers, electrons tend to localize to the lattice sites, forming Mott insulators 7,8 . In the other extreme, where the electronic bandwidth is large, as in bilayer graphene with a large twist angle, a Fermi liquid state is formed in which electrons are itinerant. MATBG provides a fascinating example of a system at the boundary between these two extremes. This system shows a host of electronic phases, including correlated insulators 3,9,10 , Chern insulators [11][12][13] , superconductors 4,9,10 , and ferromagnets 14,15 . Scanning tunneling spectroscopy [16][17][18][19] and electronic compressibility measurements 20,21 indicate that in this system Coulomb interactions and kinetic energies are indeed comparable. In this regime, there is an inherent tension between localized and itinerant descriptions of the physics.
Moreover, the growing understanding that the nearly-flat bands in MATBG have a topological character 22-24 implies that a simple "atomic" description, in which electrons are localized to individual moiré lattice sites, may not be appropriate. Instead, a picture analogous to that of quantum Hall ferromagnetism has been proposed as an alternative starting point [25][26][27] . Understanding this interplay between itineracy and localization, and the new physics that emerges from it, remains a major challenge.
In this work we find that, surprisingly, the correlated state in MATBG above a filling of one electron per moiré site has a hybrid nature, with some of its properties resembling those of an itinerant system, and others which are usually associated with localized electrons. Measurements of the electronic entropy at temperatures as low as a few Kelvin reveal that this state has an unusually large excess entropy, which is rapidly suppressed by a moderate in-plane magnetic field. This suggests that even at such low temperatures, there are strongly fluctuating magnetic moments in the system, a behavior that is typically 3 associated with local moments. On the other hand, our measurements find that this state is metallic and has no thermodynamic gap nearby, which is naturally understood within an itinerant picture.
The presence of fluctuating moments at temperatures that are much smaller than the electronic bandwidth indicates the existence of a new, anomalously small energy scale associated with the bandwidth of magnetic excitations, which is an order of magnitude smaller than the energy scale where a jump appears in the compressibility 21,28 .
This jump marks the boundary between the new state at filling factor > +1 and the more familiar state at lower densities. By tracking the dependence of this boundary on temperature and magnetic field, we find that it exhibits an electronic analogue 29-32 of the famous Pomeranchuk effect 1 in 3 He. In that system, a transition from a Fermi liquid to a solid occurs upon increasing temperature, driven by the high spin entropy of the localized atoms in the solid. Similarly, we find that the new state above = +1 is favored relative to the metallic state at < +1 upon raising the temperature, due the former's high magnetic entropy. The transition near = +1 can also be driven by an in-plane magnetic field, due to the energy gain associated with polarizing the free moments. (A related effect near = −1 was proposed very recently, on the basis of transport measurements 33 .) The existence of the hybrid state we observe here, with its itinerant electrons coexisting with strongly fluctuating magnetic moments, calls for a new understanding of electron correlations in MATBG.
The data reported here is measured using two independent techniques on two conceptually different devices. The bulk of the results are obtained from local measurements of the electronic entropy 34,35 and compressibility using a scanning nanotube single-electron transistor (SET) on hBN-encapsulated twisted bilayer device (Device 1, Fig. 1a). We focus on a spatial region whose twist angle is close to the theoretical magic angle, and is homogenous over a large area (5μm × 4μm) to within the third digit, = 1.130 ± 0.005. Similar results are obtained from global entropy 4 measurements using a monolayer graphene sensor to detect the chemical potential of MATBG (Device 2, Fig. 3a). Both methods have been described elsewhere 21,36 .
The inverse compressibility, / , measured in Device 1 at = 15K as a function of the filling factor, = /( " /4) (where ns corresponds to four electrons per moiré unit cell), is shown in Fig. 1b. As reported previously 21 , sharp jumps in / are observed close to integer 's, where the system rapidly evolves from high to low compressibility, reflecting an abrupt reconstruction of the Fermi surface. These were termed Dirac revivals since they were interpreted as resets of partially filled energy bands back to near charge neutrality, leading to the decreased compressibility. As seen in the figure, the cascade of revivals is already very prominent at these relatively high temperatures (i.e. above typical correlated insulator and superconducting critical temperature scales). Measurements of $ %% vs. at various temperatures ( Fig. 1c) show insulating behavior at = 2 and semimetallic behavior at = 0. As previously-noted 37 , $ %% shows a step-like increase across ≈ 1 at high temperatures, and this feature gradually disappears with lowering the temperature (although a small peak in the resistivity close to = 1 remains). This behavior is different from the insulating behavior observed at other integer filling factors.
The unusual physics of the electronic state around = 1 is revealed by measuring the dependence of the inverse compressibility, / , on temperature, , and parallel magnetic field, ( ∥ . In Fig. 2a we examine the temperature dependence of / near = 1 at ( ∥ = 0T. At low temperature, a jump 21 in / occurs at a filling factor slightly larger than 1. Increasing the temperature moves the jump towards a lower filling factor, and surprisingly, increases the magnitude of the jump, rather than smearing it. A similar measurement with ( ∥ = 12T is shown in Fig. 2b. Compared to ( ∥ = 0T, at low the jump is much larger and closer to = 1. Increasing the temperature at ( ∥ = 12T maintains the jump close to = 1 and, oppositely to the ( ∥ = 0T case, reduces its amplitude and increases its width. The chemical potential, ( , ) (measured relative to that at the charge neutrality point), can be obtained by integrating / over the density at different temperatures ( Fig. 2c,d). We see that has a strong temperature dependence for a certain range of ′s.
This is clearly seen when we plot vs. at two representative ′s (Fig. 2c, inset). At = 0.2, is practically independent of (blue). In contrast, at = 0.9 (red) we see that is nearly constant until ∼ 4K, after which it starts to decrease approximately linearly with . At > 1.15, is again nearly temperature independent. Comparing at ( ∥ = 0T (Fig.   2c) and ( ∥ = 12T (Fig. 2d) reveals a clear contrast: whereas for ( ∥ = 0T, is a decreasing function of temperature for 0.4 < < 1.15, for ( ∥ = 12T, decreases with for < 0.9 and increases for > 0.9. Measurements of the temperature dependence of allow us to directly determine the entropy of the system, by integrating Maxwell's relation: Near = 2 the entropy has a sharp drop, and it starts decreasing towards zero after = 3. We note that the dependence of the entropy on is qualitatively different from that of the compressibility. Specifically, whereas the compressibility drops sharply near = 1 ( Fig. 2a), the entropy does not drop, but rather remains at a high value.
An important insight into the origin of this large entropy can be gained by examining its magnetic field dependence. As seen in Fig To test the robustness of our results, we measured the entropy in a completely different setup, in which a sheet of monolayer graphene is used to sense the chemical potential of the MATBG, averaged over the entire device 36 (Fig. 3a). experiments. Of course, we do not expect the free-electron picture to apply at low temperatures, since there interactions change the physics dramatically. We note that the 7 measured 5( ) in the valence band ( Fig. 3b) is approximately a mirror image of 5( ) in the conduction band, although it is somewhat smaller and has less pronounced features. This is consistent with the fact that the observed cascade of revivals in the inverse compressibility is weaker in the valence band relative to that in the conduction band 21,36 .
Our results so far show that a dramatic change occurs in the compressibility and the entropy near = 1. The compressibility experiences an abrupt drop at the revival transition ( Fig. 2a,b), and at approximately the same filling, the magnetic-field-dependent part of the entropy sharply rises (Fig. 2e, inset). This rapid change may be due to a continuous buildup of electronic correlations. Alternatively, it can be interpreted as an underlying first-order phase transition between two distinct phases. Naively, one would then expect a discontinuous jump in thermodynamic properties and hysteretic behavior across the transition, which are not observed. However, we note that a true first-order phase transition can never occur in two dimensions in the presence of disorder or longrange Coulomb interactions 38 , as these will always broaden the transition into a mesoscale coexistence region. Experimentally, since the observed compressibility revival feature is relatively sharp, it can be precisely tracked, mapping a phase diagram as the function of temperature and magnetic field. Below, we show that interpreting this revival feature as a proxy for a first-order transition naturally explains much of the underlying physics.
We define the filling factor @ of the revival feature as the midpoint of the sharp rise in / (tracking the beginning or the end of the rise leads to similar conclusions, see Supplementary Info. SI5). As we have seen in Fig. 2a, raising the temperature leads to an observable change in @ . A similar measurement of / vs. ν at different magnetic fields ( Fig. 4a) shows that @ also shifts with ( ∥ . The measured locations of the revival feature as a function of ( ∥ and form a surface in the ( , ( ∥ , ) space, shown in Fig. 4b.
Examining the magnetic field dependence of @ (Fig. 4c), we see that at = 2.8K, @ is weakly dependent on ( ∥ at low fields, but starts to decrease linearly with field above ( ∥ ≈ 4T. At higher temperatures, @ is similarly insensitive to magnetic field at low ( ∥ , and decreases with increasing ( ∥ at higher fields. The crossover field between the two behaviors increases as the temperature increases.
Another interesting aspect of the evolution of @ is highlighted by looking at its temperature dependence at the different magnetic fields (Fig. 4d). At ( ∥ = 0T (blue) @ is linear in at low temperatures, and curves up at higher temperatures. As the magnetic field increases, the entire curve shifts towards smaller 's, and simultaneously its slope at low temperatures changes sign. At the highest field, ( ∥ = 12T, @ first increases with temperature, reaches a maximum at ≈ 9K, and then decreases at a higher .
The phenomenology seen in Figs. 4b-d can be understood within a simple interpretation, in terms of a first-order phase transition at = @ between two phases: a Fermi liquid phase below @ , and a 'free moment' phase above it. The latter phase has a high concentration of free moments (of the order of one per moiré site), coexisting with a low density of itinerant electrons. Within this framework, the movement of the transition point @ as a function of ( ∥ and reflects the magnetization and entropy differences between the two neighboring phases.
At ( ∥ = 0T, the free moment phase has a higher entropy than the Fermi liquid, due to thermal fluctuations of the moments. Hence, the former phase becomes increasingly entropically-favorable the higher the temperature. This explains the observed decrease of @ with increasing at low fields ( Fig. 4d). Raising the temperature at a fixed may therefore drive a transition from the Fermi liquid to the free moments phase, an electronic analogue of the Pomeranchuk effect. As ( ∥ increases and the Zeeman energy exceeds the temperature, the moments become nearly fully polarized and their entropy is quenched (as is observed directly in Fig. 2e). Consequently, one expects that at low temperatures and sufficiently high fields, the Fermi liquid phase would be favored by raising the temperature, but that this trend will reverse once the temperature becomes larger than the Zeeman energy. This explains the non-monotonic behavior of @ as a function of , seen at ( ∥ = 12T in Fig    (measured by spatial mapping of the C 8D that corresponds to " , as in Refs. 21,39 ). As seen previously 21 , a jump of / appears near all integer filling factors. This jump corresponds to a Fermi surface reconstruction, in which some combination of the spin/valley flavors filling is reset back to near the charge neutrality point, and correspondingly / shows a cascade of sawtooth features as a function of density. The trace is measured at = 15K, showing that even at this high temperature this sawtooth cascade is well developed c. Two-probe resistance, I, measured as a function of and temperature. Notice that unlike the inverse compressibility, which measures a local quantity, the resistance gives an averaged result over domains with different twist angle. Therefore, the resistance maxima are slightly shifted from the usual integer values, probably because another domain with a small difference in twist angle dominates the transport characteristics globally.  By balancing the electrochemical potential of the adjacent layers in the device, we can obtain the relationship between the density and chemical potential of MATBG and MLG and the gate voltages applied to the system. In the special case where the density of MLG is zero, i.e. at its charge neutrality point, the chemical potential of MATBG is directly proportional to the voltage applied to the top gate. This technique allows us to reliably extract the chemical potential and entropy of MATBG at temperatures up to 70 K. b. The measured entropy, in units of 7 8 per moiré unit cell, as a function of at three different temperature ranges (top legend). The entropy derivative, 5/ , is obtained from a linear fit to vs. in the corresponding temperature range, and is then integrated over to yield the entropy per moiré unit cell (similar to Fig. 2e). Inset: comparison between the dependences of the entropies, measured at the low temperature range, obtained from local and global measurements. c. The entropy as a function of and calculated for a system of four degenerate non-interacting Dirac bands (whose density of states climbs linearly with energy from the Dirac point to the end of the conduction or the valence band). The colorcoded lines show the curves whose temperatures correspond to the mean of the temperature ranges of the experimental curves. The gray lines represent the entire evolution from zero temperature to high temperature, where the entropy saturates on a value of 8L (2) ≈ 5.5, where the factor 8 reflects the total number of energy bands. A bandwidth of > = 30GMCis chosen such that the calculated value of the entropy at the highest temperature roughly matches the one obtained from the measured curve at the same temperature. We track the filling factor that corresponds to the center the jump in / (labeled @ ). Visibly, the application of ( ∥ pushes @ to lower values. b.
Measured @ as a function of ( ∥ and , plotted as dots in the ( , ( ∥ , ) space (the dots are colored by their temperature). The dashed lines are polynomial fits to the dots at constant ( ∥ or constant . Inset: the same surface calculated from a simple model that assumes a transition between a Fermi liquid and a metallic phase that contains one free moments per moiré site (see text). c. Projection of the data in panel b onto the ( , ( ∥ ) plane, showing the dependence of @ on ( ∥ for various temperatures. At low fields, @ is independent of field but it becomes linear in ( ∥ at high fields, a behavior expected from the field polarization of free moments (see text). Inset: curves calculated from the model. d. Projection onto the (ν, ) plane, showing the dependence of @ on for various magnetic fields. At ( ∥ = 0 , @ is linear in at small 's and then curves up at higher 's. At high magnetic field, the dependence of @ on becomes non-monotonic. Inset: curves calculated from the model.

SI1. Extraction of the entropy
In both the local and global measurements, we determine the entropy using a Maxwell relation, relating the partial derivatives of the entropy with respect to the filling factor to that of the chemical potential with respect to temperature: where is the entropy per moiré unit cell. In the global measurements, we probe the chemical potential of the MATBG directly using a monolayer graphene sensor. The measurement determines the chemical potential relative to that at the charge neutrality point (CNP): Δ ( , , ∥ ) = ( , , ∥ ) − ( , ∥ ).
In the local measurements, we use a nanotube single electron transistor to measure the inverse compressibility and integrate it over the density, to obtain the same quantity: In these measurements, the inverse compressibility is probed at typical frequencies of few hundred Hz, and with an excitation = 40# on the back gate, chosen to be small enough as to not smear essential features.
The entropy then follows from: We removed a constant background in ds/d ( ) to account for the variation of with T at charge neutrality, such that the entropy at = ±4 is zero. For each temperature range, was assumed to be linearly dependent on T at a given . The confidence bound of 95% is shown for this linear fitting process. The entropy obtained after integration is shown in Fig. S1b. The error highlighted bands show the propagated uncertainty in this integration process. In the scanning SET measurements, we get an additional small component of parasitic capacitance between the SET and the back-gate. This results from the fact that our SET scans at a finite height (hundreds on nm's) above the MATBG. This parasitic capacitance adds a background to the measured inverse compressibility of the order of / < 10 ,> . In the estimation of the entropy this gets doubly integrated yielding a term that depends quadratically on . We remove this term by assuming that the entropy at = 0 is also zero (in addition to assuming it is zero at = 4 as discussed above). As seen in the global entropy measurements ( Fig. 3b and S1b), the entropy curve that correspond to the temperature range = 4K − 16K (blue) shows that the entropy at = 0 is smaller than 0.1' . Since local entropy measurements are performed only in this temperature range, the assumption that = 0 at = 0 is justified.
To determine the uncertainty in the local measurements of the entropy (Fig. 2e in the main text), we first extract the noise level in our measured / . We then add to our measured compressibility signal randomly distributed noise with the experimental noise variance and see how it changes the resulting entropy curve. Repeating this over a statistically significant instances of random noise gives us the error bars in our determined entropy, which are shown in Fig. S2, for the traces taken at different parallel magnetic fields (as in Fig. 2e in the main text).

SI2. Entropy of non-Interacting Dirac electrons
To get a rough understanding of the overall dependence of the measured entropy at high temperatures, it is useful to compare it to the entropy in a system of non-interacting Dirac bands. The curves in Fig. 3c in the main text were obtained for such a model with the a singleparticle density of states that rises linearly from zero at the charge neutrality point up to the band top and bottom at ±2/2, where 2 is the bandwidth. The density of states @(A) for each spin/valley flavor is given by: where Θ(I) is the Heaviside step function. The entropy per unit cell is then given by: where c = Md ↑, d ↓, d′ ↑, d′ ↓T is a spin/flavor index, = 1,2 labels the conduction and valence with variational parameters ] , and minimize the grand potential of the trial density matrix @ v =  We expect that at low temperatures, this approximation, built on a density matrix corresponding to a non-interacting Hamiltonian with self-consistently determined ] 's, will exhibit an entropy that is essentially = Š . ‹ ∑ @ ( + ] ) ] . Hence, the entropy is proportional to the total density of states at the Fermi level. SI4. The effect of a magnetic field on the entropy in a mean-field model without free spins A Zeeman field can be included in the Hamiltonian (4) by adding the following term: where 3 • = ∥ is the Zeeman energy, and • ] is the spin projection of electrons of flavor c along the magnetic field. To account for the Zeeman field in the mean-field calculation, we replace → + 3 • • ] in Eqs. (6) and (7).
The entropy vs. at = 10K in the presence of different in-plane magnetic fields is shown in Fig. S4. As seen in the figure, the effect of a field of up to ∥ = 12T is quite small, decreasing the entropy by at most 0.1' relative to the ∥ = 0 value near the maxima of the entropy before the integer fillings. The change in the entropy away from the maxima due to the field is even smaller.
Comparing the mean-field results to the experimentally measured entropy (Fig. 2e in the main text), we see that the calculated entropy is in rough qualitative agreement with the experimental one at ∥ = 12T and ≈ 10K, showing a similar peak structure near each integer filling. The overall magnitude of the calculated entropy at ∥ = 12T is also similar to the measured one. However, the calculated entropy at ∥ = 0 is very different from the measured entropy. In particular, unlike in the calculation, the measured entropy does not drop after = 1, but rather remains nearly constant at a high value. Moreover, the measured entropy is strongly field dependent for > 1, whereas the calculated one is weakly field dependent at all . We ascribe this failure of the mean-field model to the appearance of nearly-free magnetic moments (as discussed in detail in the main text). These free moments, that onset near = 1, fluctuate strongly at low magnetic fields, an effect which is not captured in mean-field theory. Upon applying a strong Zeeman field, these fluctuations are quenched (as seen experimentally by the dramatic decrease in the entropy), and mean-field theory may be adequate.

SI5. Tracking using different features of the / jump
In the main text, the transition from high to low compressibility near = 1 was tracked by following the midpoint of the rise in / . Since the rise is fastest around its midpoint, this procedure gives us excellent resolution in defining the filling factor that corresponds to this rise, of about ~0.005. We note, however, that the overall width of the rise in filling factor can be significantly larger, and in extreme cases can even reach Δ ≈ 0.2. It is thus necessary to check whether tracking different features of the transition as a function of magnetic field or temperature will lead to similar conclusions. While there are quantitative difference between the curves obtained by the different methods, we can see that in the overall dependence and the essential features in all the curves agree. For example, we see that at ∥ = 12T, independently of the method used, increases with temperature at low temperatures, reaching a maximum, and then starts decreasing with increasing temperature at high temperatures, where the crossover occurs at ≈ 9K.

SI6. Thermodynamic model for Fermi liquid to free moment phase transition
Here, we describe the simple thermodynamic model we used in the main text to describe the first order phase transition.
The experiment is done under conditions where the temperature , parallel magnetic field ∥ , and gate voltage ¢ 4 are fixed. The appropriate thermodynamic potential to be minimized under these conditions is the grand canonical potential, Ω(¢ 4 , , ∥ ). It is convenient to express the gate voltage in terms of the equivalent filling factor, = † w £ 4 ¢ 4 (£ 4 is the geometric capacitance from the MATBG to the gate per moiré unit cell). For clarity, it is useful to derive the grand canonical potential starting from the free energy •, which is a function of the filling factor , and then obtain Ω by a Legendre transformation.
Our simple model postulates the existence of a first order transition between two phases.
The first phase is a relatively simple metallic phase, which we model as a Fermi liquid. The second phase is characterized by the existence of free moments. This phase is also metallic, although its density of states is lower than that of the first phase. We assume that in the second phase, there is one free spin per unit cell, coexisting with metallic Fermi liquid electrons.
The free energies per moiré unit cell of the two phases are chosen as follows: Here, ¯= 1,2 labels the two phases, A š and š are reference energies and chemical potentials, ¥ š = F We now carry out a Legendre transformation, Ω = • − +¢ 4 , minimize Ω with respect to , and thus eliminate in favor of = † w £ 4 ¢ 4 . Since in our experiment + G /£ 4 is much larger than 1/¥ š , we keep only terms to lowest order in Here, Ãš = A š −°) Gw . š G . In terms of Ω( , , ∥ ), the thermodynamic variables are given by: which is the relation we used in SI7, with * , * , and ∥ * identified as the filling factor ( ), temperature, and magnetic field at the Dirac revival point.
The jump in compressibility seen at is sharp, but not discontinuous, as one may naively expect from a first order phase transition. Indeed, in the presence of long-range Coulomb interactions and disorder in two dimensions, a first order transition is not expected to be sharp. If we assume that the revival transition at = 1 represent a smeared first-order phase transition, we can derive from the shape of the phase boundary the relation between magnetization and entropy.
We demonstrated this relation by analyzing the slope of the phase boundary via the Clausius-Clapeyron equation: Δ#/Δ = −( / ∥ ) º . Here, Δ and Δ# are the differences in the entropy and magnetization per moiré unit cell between the free moment and the Fermi liquid phases, and ( / ∥ ) º is the derivative of the transition temperature with respect to magnetic field at constant . To obtain the ratio Δ#/Δ we reconstruct such equi-contours by fitting a polynomial surface in the ∥ and plane to the measured points, and extract the slope of the contour lines at different points (Fig. S7). Consider point A in Fig. S7: At this point, ( / ∥ ) º ≈ 0. The Clausius-Clapeyron equations then imply that Δ# ≈ 0. In contrast, at point B, the equal contours are nearly vertical, implying that Δ ≈ 0. This clear anti-correlation between Δ and Δ# follows naturally from our simple model, where both Δ and Δ# originate from the same free moments, that are either strongly thermally fluctuating, or polarized along the magnetic field. At point C, the contour has a positive slope, from which we deduce that Δ < 0, Δ# > 0.

SI8. Comparison of transport measurements and compressibility.
Using the multilayer device shown in Fig. 3a, we can simultaneously obtain the transport resistances and the chemical potential of MATBG. Fig. S8a shows the longitudinal resistance » oeoe versus at different temperatures from 1K to 70K. The peaks in resistance near = −1 denoted by the blue dots start appearing at a finite temperature of ∼ 5K, and subsequently move to lower absolute value of filling factor as the temperature increases. The Hall coefficient and density, as shown in Fig. S8b and c, also show a similar trend. The shift of the resistive peak at = −1 has been attributed 3 to a Pomeranchuk-like mechanism, similar to the one near = 1.
The shift of the peak at = +1, on the other hand, is much smaller, as was also observed in Device 1 shown in Fig. 1. Indeed, from our analysis in Fig. 4, the shift of the = +1 state as a function of temperature is on the order of < = 0.06, which might be shadowed in the transport measurement by a moderate twist angle inhomogeneity on the order of ±0.02°.