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Sawtooth in compressibility tracks superconductivity in twisted trilayer graphene

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In helically twisted trilayer graphene, the superconducting critical temperature scales with the sawtooth compressibility strength as a function of twist angle, with electron-hole asymmetry, while showing no direct link to correlated insulator gaps.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A believable, well-executed same-device comparison of thermodynamic and transport signatures in twisted trilayer graphene, with the main correlation claim qualitatively convincing but not quantitatively nailed down. the 3 major comments →

arxiv 2509.07977 v2 pith:OPAHHULT submitted 2025-09-09 cond-mat.mes-hall cond-mat.str-elcond-mat.supr-con

Link between thermodynamic correlation signatures and superconductivity in twisted trilayer graphene

Jesse C. Hoke , Yifan Li , Yuwen Hu , Julian May-Mann , Kenji Watanabe , Takashi Taniguchi , Trithep Devakul , Aaron Sharpe
show 1 more author
Benjamin E. Feldman
This is my paper
classification cond-mat.mes-hall cond-mat.str-elcond-mat.supr-con
keywords twisted trilayer graphenesuperconductivitycompressibilitycorrelated insulatorssawtoothlight and heavy fermionselectron-hole asymmetrytwist-angle dependence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that, in a twisted trilayer graphene device with unequal twist angles, the strength of a thermodynamic signature called the sawtooth in compressibility is closely tied to the appearance of superconductivity, while the correlated insulating gaps are not. The authors measure both local compressibility and transport in the same region of a single device, allowing them to compare these signatures directly as the local twist angle changes. If correct, the sawtooth strength serves as a thermodynamic predictor for where superconductivity emerges.

Core claim

The central discovery is that, in the same TTG sample, the critical temperature of superconductivity tracks the strength of the sawtooth in inverse electronic compressibility sigma_{e(h)} as a function of twist angle theta_12, but does not track the correlated insulator gap Delta_{pm 2}. This establishes the sawtooth as the thermodynamic signature most closely linked to superconductivity.

What carries the argument

The key object is the sawtooth in the inverse electronic compressibility dmu/dn, quantified by the standard deviation sigma_{e(h)} over the filling range 0.5 < |nu| < 3.5 after filtering out the nu = pm 2 correlated insulator peaks. This measure captures the strength of thermodynamic correlation fluctuations and is compared with the optimal critical temperature T_c^opt extracted from transport measurements.

Load-bearing premise

The comparison assumes that the sawtooth strength measured at T = 1.6 K and near-zero displacement field is representative of the physics governing superconductivity at its optimal displacement field and base temperature.

What would settle it

Measure the sawtooth strength sigma_{e(h)} at the same displacement field, density, and temperature where T_c is maximal (base temperature), and check whether the twist-angle dependence still tracks T_c^opt; if not, the correlation would be coincidental.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • In this TTG system, the sawtooth in compressibility can serve as a local thermodynamic marker predicting where superconductivity is strongest as twist angle varies.
  • Superconductivity persists even where no correlated insulator gap is present, suggesting the two phenomena are independent or competing rather than directly coupled.
  • Electron- and hole-doped correlated states have distinct magic angles, indicating that conduction and valence bands require different conditions for optimal correlations.
  • The same measurement approach can be applied to other twisted multilayer systems to disentangle thermodynamic and transport signatures.
  • The close correlation between the sawtooth and T_c suggests a common microscopic origin, such as the interplay of light and heavy fermions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The correlation between the sawtooth and T_c may extend beyond TTG to other moire systems, where the sawtooth could serve as a universal predictor of superconductivity.
  • If the sawtooth is a proxy for heavy-fermion physics, then tuning the light-heavy carrier coupling could provide a route to enhancing T_c in twisted multilayer graphene.
  • A direct test would be to measure sigma_{e(h)} at the same displacement fields and temperatures where T_c is optimized; the current comparison uses SET data at different conditions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies a helically twisted trilayer graphene device with unequal twist angles (θ12/θ23 ≈ 1/3). Using a scanning single-electron transistor, the authors map the inverse compressibility as a function of local twist angle, extracting three thermodynamic signatures: the chemical-potential width of the flat bands (Δμ), the correlated-insulator gap at ν=±2 (Δ±2), and the 'sawtooth' compressibility strength (σe(h)). They then pattern the same region into a Hall bar and measure transport, finding superconductivity with an electron-hole asymmetry. The central claim is that the optimal superconducting critical temperature T_c^opt tracks the sawtooth strength σe(h) as a function of twist angle, whereas it does not track the correlated-insulator gaps. The authors interpret this as evidence that superconductivity is not directly tied to the correlated insulators, but shares a common origin or link with the sawtooth thermodynamics.

Significance. If the correlation between σe(h) and T_c^opt is genuine, the sawtooth compressibility would serve as a thermodynamic marker for where superconductivity appears in TTG, and the paper would establish a new empirical connection between a many-body thermodynamic signature and a superconducting instability. The experimental design is a notable strength: the same region of the same device is characterized by local compressibility and then by transport, avoiding cross-device variability. The authors also provide multiple independent compressibility datasets (line cuts 2–4 in Fig. ED3, the separate line cut in Fig. 2/ED9, and the 1×1 μm2 grid in ED10) that all show the same qualitative σ–T_c relationship. The negative result that T_c^opt does not correlate with Δ±2 is a useful falsifiable observation. In addition, the methods section explicitly reports robustness checks for the T_c extraction (10% vs 20% definition, and V_t=0 line cuts), which strengthens the internal consistency of the transport analysis. However, the central claim currently rests on visual overlay of data, not on a quantitative correlation analysis, and the comparison is made between measurements taken under different th

major comments (3)
  1. [Fig. 4; 'Comparing compressibility and transport'] The central claim that T_c^opt 'correlates closely' with σe(h) is assessed only by eye. No correlation coefficient, rank correlation, or significance test is provided for any of the datasets in Fig. 4, ED9, or ED10. With only 4–5 independent contact pairs, a monotonic trend in both quantities could arise by chance. Please add a quantitative measure (e.g., Spearman rank correlation with confidence intervals, or a permutation test) for the electron and hole datasets, and ideally for each independent compressibility map. Without this, the abstract's 'correlates closely' is not supported.
  2. [Fig. 4; Methods ('Scanning SET measurements', 'Extraction of the superconducting critical temperature')] σe(h) is measured at T=1.6 K and near-zero displacement field D, while T_c^opt is the maximum over (n,D) at base temperature. The comparison in Fig. 4 therefore assumes that the twist-angle dependence of σ at T=1.6 K, D≈0 is representative of the physics that governs superconductivity at its optimal D. The V_t=0 line-cut check in ED8 partially mitigates this, but V_t=0 is not D=0, and the temperature difference remains. Please either measure σ at or near the optimal D and base temperature, or explicitly demonstrate that the relative θ12 ordering of σ is invariant under D and T changes. Absent this, the correlation could be coincidental.
  3. [Methods: 'Extraction of sawtooth compressibility strength'] σe(h) is a filling-averaged standard deviation over 0.5<|ν|<3.5 with the ν=±2 peaks filtered out, whereas T_c^opt occurs at a specific filling that may vary with θ12. The broad average may dilute or distort the pairing-relevant compressibility. Please test whether the correlation persists when σ is computed in a filling window matched to the superconducting pocket at each θ12, or at the specific ν where T_c is maximal. If the correlation only holds for the broad average, the claim of 'near-identical scaling' needs qualification.
minor comments (5)
  1. [Abstract] The phrase 'e-hasymmetry' should read 'e–h asymmetry' with a space or en-dash; it currently appears as a typographical run-on.
  2. [Superconductivity section] In the sentence 'the electron- and hope-doped superconducting phases', 'hope-doped' is a typo for 'hole-doped'.
  3. [Methods: 'Extraction of the superconducting critical temperature'] The sentence 'the qualitative shape and θ12 dependence of Tc is remains consistent' is grammatically incorrect; 'is remains' should be 'remains' or 'is consistent'.
  4. [Fig. ED8 caption] The notation 'V_t = 0' is used but not defined in the caption; please define the top-gate voltage and clarify that this is a line cut in Fig. 3c, not necessarily D=0. This is relevant to the main text's robustness claim.
  5. [Throughout] The critical temperature is variously written as 'T_c', 'T_c^opt', and 'T^{opt}_c'. A single notation (e.g., T_c^{opt}) should be used consistently in text and figures.

Circularity Check

0 steps flagged

No significant circularity: the σ–Tc correlation compares independently measured quantities; self-citations are methodological, not load-bearing.

full rationale

The central claim is an empirical correlation between two independently measured observables: the sawtooth compressibility strength σ_e(h), extracted as the standard deviation of dμ/dn over 0.5<|ν|<3.5 (Methods, 'Extraction of sawtooth compressibility strength'), and the optimal superconducting critical temperature T_c^opt, extracted from R_xx(n,D) maps using a 10% normal-state-resistance criterion (Methods, 'Extraction of the superconducting critical temperature'). No equation connects σ to T_c; neither quantity is fitted to the other; there is no model parameter calibrated on the target correlation. The paper explicitly shows robustness of the comparison to analysis choices and to a V_t=0 line cut (Methods and Fig. ED8), which addresses the reviewer's concern about differing D and T conditions as a validity risk rather than a circularity. Self-citations appear only for standard procedures (e.g., Ref. 13 for thermodynamic-gap extraction and Hofstadter spectra) or contextual theory (Ref. 34 for magic-continuum calculations), and none of these is used to force the central σ–T_c correlation. No uniqueness theorem, hidden ansatz, or renamed known result is invoked. The D/T mismatch between SET and transport measurements is a legitimate experimental caveat but does not make the comparison circular.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim rests on the operational definitions of sigma and T_c and on the assumption that the local twist angle is the dominant tuning parameter, with theta23 and strain playing secondary roles. No new entities are introduced.

free parameters (4)
  • theta23 estimate from v_F* = 3.45 degrees (from v_F* = 0.81e6 m/s)
    Estimated via Eq. 5 using t0=0.105 eV and v0_F=1e6 m/s; not directly measured. The uncertainty is +/-0.28 degrees.
  • Sawtooth strength nu-range = 0.5 < |nu| < 3.5
    The standard deviation of dmu/dn is computed over this chosen filling range. Changing the range would alter sigma values, though the qualitative trend is likely robust.
  • Tc resistance threshold = 10% of normal-state resistance
    Critical temperature is defined at the 10% resistance point; the authors state the qualitative shape is unchanged for 20%.
  • Gap background subtraction kappa_B^-1 = small constant (not quantified)
    In Eq. 7, a constant background is subtracted when integrating dmu/dn to obtain thermodynamic gaps; the value is not specified quantitatively.
axioms (5)
  • domain assumption Continuum model for TTG band structure
    Spectral function calculations (Fig. 1b, ED1) use the Bistritzer-MacDonald type continuum model with parameters vF=0.88e6 m/s, t0=0.11 eV, t1=-0.227 eV*A, kappa=0.7, taken from prior literature. Interpretation of flat bands and Dirac cone relies on this model.
  • domain assumption Identification of nu=+/-4 incompressible peaks with full filling of the layers 1-2 moiré cell
    Local twist angle theta12 is determined from the density of the two largest incompressible peaks, assumed to be at nu=+/-4. This is a standard assumption in moiré graphene.
  • domain assumption theta23 estimation from renormalized Fermi velocity
    Eqs. 4-5 assume equal interlayer AA/BB and AB/BA hoppings and ignore tunneling between layers 1 and 3. This approximation is used to infer theta23 from the Dirac cone Landau level dispersion.
  • domain assumption SET signal measures inverse compressibility dmu/dn
    The measurement technique is established in prior work (refs 12-13).
  • domain assumption D and T differences do not change qualitative theta12 dependence
    SET is at T=1.6K and small D; transport Tc is at optimal D and low T. The comparison assumes the theta12 dependence of sigma is representative of the physics at the superconducting optimal conditions. Partially addressed in ED8.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Link between thermodynamic correlation signatures and superconductivity in twisted trilayer graphene." pith.science (2026). https://pith.science/paper/OPAHHULT

@misc{pith2026250907977,
  author       = {Pith},
  title        = {Pith review of: Link between thermodynamic correlation signatures and superconductivity in twisted trilayer graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPAHHULT}},
  note         = {Machine review of arXiv:2509.07977}
}
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read the original abstract

Twisted graphene multilayers exhibit strong electronic correlations, which manifest in a range of experimental signatures. Yet how these signatures relate to each other and the microscopic ground states-and how twist angle and band structure reshape them-remains poorly understood. Here we study this interplay by correlating local thermodynamic and transport measurements in a twisted trilayer graphene (TTG) sample with unequal angles and flat electronic bands. We use a scanning single-electron transistor to map the impact of electron-electron interactions in a region of the sample where the local twist angle evolves smoothly. We observe gapped correlated insulators and a sawtooth in electronic compressibility, both exhibiting pronounced electron-hole (e-h) asymmetry with distinct magic angles for conduction and valence bands. Subsequent transport measurements in the same region reveal robust superconductivity with a similar e-h asymmetry. Our measurements indicate that superconductivity is not directly tied to the correlated insulators. Instead, its critical temperature correlates closely with the strength of the sawtooth in compressibility, suggesting a common origin or link between the two. By combining a local probe with transport measurements, we uncover connections between superconductivity and thermodynamic correlation signatures that are not apparent from either technique in isolation, highlighting the power of our dual approach and establishing their dependence on interlayer twist angles in TTG.

Figures

Figures reproduced from arXiv: 2509.07977 by Aaron Sharpe, Benjamin E. Feldman, Jesse C. Hoke, Julian May-Mann, Kenji Watanabe, Takashi Taniguchi, Trithep Devakul, Yifan Li, Yuwen Hu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: c-e, respectively. Interaction strength, as quanti￾fied by both ∆±2 and σe(h) , is maximized at different θ12 for conduction and valence bands. The θ12 where ∆±2 and σe(h) are maximal are not coincident with the min￾imum of ∆µe(h) , as might be expected. Note, however, that ∆µ is not equivalent to the non-interacting band￾width because interactions significant modify the band structure as the bands are fil… view at source ↗
Figure 3
Figure 3. Figure 3: c shows the longitudinal resistance Rxx as a function of n and D between contacts 3 and 4, whose θ12 = 1.20◦ . At temperature T = 40 mK, we observe two large superconducting pockets of zero resistance upon both electron and hole doping. The superconductivity is confirmed by standard temperature dependence, non￾linear I-V characteristics, and magnetic field measure￾ments that exhibit evidence of phase coher… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

66 extracted references · 28 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nature Materials19, 1265 (2020)

    2020

  2. [2]

    Balents, C

    L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moir´ e flat bands, Nature Physics16, 725 (2020)

    work page 2020

  3. [3]

    K. F. Mak and J. Shan, Semiconductor moir´ e materials, Nature Nanotechnology17, 686 (2022)

    2022

  4. [4]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moir´ e bands in twisted double-layer graphene, Proceedings of the Na- tional Academy of Sciences108, 12233 (2011)

    2011

  5. [5]

    Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras,et al., Correlated insulator be- haviour at half-filling in magic-angle graphene superlat- tices, Nature556, 80 (2018)

    2018

  6. [6]

    Yankowitz, S

    M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan- abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, 7 Science363, 1059 (2019)

    work page 2019

  7. [7]

    X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al., Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature574, 653 (2019)

    2019

  8. [8]

    J. Yu, B. A. Foutty, Y. H. Kwan, M. E. Bar- ber, K. Watanabe, T. Taniguchi, Z.-X. Shen, S. A. Parameswaran, and B. E. Feldman, Spin skyrmion gaps as signatures of strong-coupling insulators in magic-angle twisted bilayer graphene, Nature Communications14, 6679 (2023)

    work page 2023

  9. [9]

    Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional super- conductivity in magic-angle graphene superlattices, Na- ture556, 43 (2018)

    2018

  10. [10]

    H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi, H. Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu,et al., Superconductivity in metallic twisted bilayer graphene stabilized by wse2, Nature583, 379 (2020)

    2020

  11. [11]

    Y. Cao, D. Rodan-Legrain, J. M. Park, N. F. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. Jarillo-Herrero, Nematicity and competing orders in superconducting magic-angle graphene, science372, 264 (2021)

    2021

  12. [12]

    Zondiner, A

    U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern,et al., Cascade of phase transitions and dirac revivals in magic-angle graphene, Nature582, 203 (2020)

    2020

  13. [13]

    J. Yu, B. A. Foutty, Z. Han, M. E. Barber, Y. Schat- tner, K. Watanabe, T. Taniguchi, P. Phillips, Z.-X. Shen, S. A. Kivelson,et al., Correlated hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene, Nature Physics18, 825 (2022)

    work page 2022

  14. [14]

    Saito, F

    Y. Saito, F. Yang, J. Ge, X. Liu, T. Taniguchi, K. Watan- abe, J. Li, E. Berg, and A. F. Young, Isospin pomer- anchuk effect in twisted bilayer graphene, Nature592, 220 (2021)

    2021

  15. [15]

    Rozen, J

    A. Rozen, J. M. Park, U. Zondiner, Y. Cao, D. Rodan- Legrain, T. Taniguchi, K. Watanabe, Y. Oreg, A. Stern, E. Berg,et al., Entropic evidence for a pomeranchuk ef- fect in magic-angle graphene, Nature592, 214 (2021)

    2021

  16. [16]

    A. T. Pierce, Y. Xie, J. M. Park, E. Khalaf, S. H. Lee, Y. Cao, D. E. Parker, P. R. Forrester, S. Chen, K. Watan- abe,et al., Unconventional sequence of correlated chern insulators in magic-angle twisted bilayer graphene, Na- ture Physics17, 1210 (2021)

    work page 2021

  17. [17]

    Saito, J

    Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young, Independent superconductors and correlated in- sulators in twisted bilayer graphene, Nature Physics16, 926 (2020)

    2020

  18. [18]

    Stepanov, I

    P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K. Efetov, Untying the insulating and supercon- ducting orders in magic-angle graphene, Nature583, 375 (2020)

    2020

  19. [19]

    X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. Li, Tuning electron correlation in magic-angle twisted bilayer graphene using coulomb screening, Sci- ence371, 1261 (2021)

    2021

  20. [20]

    X. Gao, A. Jimeno-Pozo, P. A. Pantaleon, E. Code- cido, D. L. Sharifi, Z. Zhang, Y. Liu, K. Watanabe, T. Taniguchi, M. W. Bockrath,et al., Double-edged role of interactions in superconducting twisted bilayer graphene, arXiv preprint arXiv:2412.01578 (2024)

    Pith/arXiv arXiv 2024

  21. [21]

    Barrier, L

    J. Barrier, L. Peng, S. Xu, V. Fal’ko, K. Watan- abe, T. Tanigushi, A. Geim, S. Adam, and A. I. Berdyugin, Coulomb screening of superconductivity in magic-angle twisted bilayer graphene, arXiv preprint arXiv:2412.01577 (2024)

    Pith/arXiv arXiv 2024

  22. [22]

    Y. Wang, J. Zhu, G. W. Burg, A. Swain, K. Watanabe, T. Taniguchi, Y. Zheng, A. H. MacDonald, and E. Tutuc, Independently tunable flat bands and correlations in a graphene double moir´ e system, Physical Review Letters 134, 096204 (2025)

    work page 2025

  23. [23]

    J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled supercon- ductivity in magic-angle twisted trilayer graphene, Na- ture590, 249 (2021)

    2021

  24. [24]

    Z. Hao, A. Zimmerman, P. Ledwith, E. Khalaf, D. H. Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene, Science 371, 1133 (2021)

    2021

  25. [25]

    X. Liu, N. J. Zhang, K. Watanabe, T. Taniguchi, and J. Li, Isospin order in superconducting magic-angle twisted trilayer graphene, Nature Physics18, 522 (2022)

    work page 2022

  26. [26]

    Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant superconductivity in moir´ e graphene, Nature595, 526 (2021)

    2021

  27. [27]

    Zhang, R

    Y. Zhang, R. Polski, C. Lewandowski, A. Thomson, Y. Peng, Y. Choi, H. Kim, K. Watanabe, T. Taniguchi, J. Alicea,et al., Promotion of superconductivity in magic-angle graphene multilayers, Science377, 1538 (2022)

    work page 2022

  28. [28]

    J. M. Park, Y. Cao, L.-Q. Xia, S. Sun, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Robust supercon- ductivity in magic-angle multilayer graphene family, Na- ture Materials21, 877 (2022)

    2022

  29. [29]

    A. Uri, S. C. de la Barrera, M. T. Randeria, D. Rodan- Legrain, T. Devakul, P. J. Crowley, N. Paul, K. Watan- abe, T. Taniguchi, R. Lifshitz,et al., Superconductivity and strong interactions in a tunable moir´ e quasicrystal, Nature620, 762 (2023)

    work page 2023

  30. [30]

    A. T. Pierce, Y. Xie, J. M. Park, Z. Cai, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and A. Yacoby, Tunable interplay between light and heavy electrons in twisted trilayer graphene, Nature Physics , 1 (2025)

    work page 2025

  31. [31]

    G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi, et al., Tunable correlated chern insulator and ferromag- netism in a moir´ e superlattice, Nature579, 56 (2020)

    work page 2020

  32. [32]

    Zhang, K.-T

    X. Zhang, K.-T. Tsai, Z. Zhu, W. Ren, Y. Luo, S. Carr, M. Luskin, E. Kaxiras, and K. Wang, Correlated insulat- ing states and transport signature of superconductivity in twisted trilayer graphene superlattices, Physical review letters127, 166802 (2021)

    work page 2021

  33. [33]

    Z. Zhu, S. Carr, D. Massatt, M. Luskin, and E. Kaxi- ras, Twisted trilayer graphene: A precisely tunable plat- form for correlated electrons, Physical review letters125, 116404 (2020)

    work page 2020

  34. [34]

    C. Yang, J. May-Mann, Z. Zhu, and T. Devakul, Multi- moir´ e trilayer graphene: lattice relaxation, electronic structure, and magic angles, Physical Review B110, 115434 (2024)

    work page 2024

  35. [35]

    D. Foo, Z. Zhan, M. M. Al Ezzi, L. Peng, S. Adam, and 8 F. Guinea, Extended magic phase in twisted graphene multilayers, Physical Review Research6, 013165 (2024)

    work page 2024

  36. [36]

    F. K. Popov and G. Tarnopolsky, Magic angle butterfly in twisted trilayer graphene, Physical Review Research 5, 043079 (2023)

    work page 2023

  37. [37]

    Banerjee, Z

    A. Banerjee, Z. Hao, M. Kreidel, P. Ledwith, I. Phinney, J. M. Park, A. Zimmerman, M. E. Wesson, K. Watanabe, T. Taniguchi,et al., Superfluid stiffness of twisted trilayer graphene superconductors, Nature638, 93 (2025)

    2025

  38. [38]

    Tanaka, J

    M. Tanaka, J. ˆI.-j. Wang, T. H. Dinh, D. Rodan-Legrain, S. Zaman, M. Hays, A. Almanakly, B. Kannan, D. K. Kim, B. M. Niedzielski,et al., Superfluid stiffness of magic-angle twisted bilayer graphene, Nature638, 99 (2025)

    2025

  39. [39]

    Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, J. Wang, P. Ledwith, Z. Cai, K. Watanabe, T. Taniguchi, E. Kha- laf,et al., Strong interactions and isospin symmetry breaking in a supermoir´ e lattice, Science , eadl2544 (2025)

    work page 2025

  40. [40]

    Turkel, J

    S. Turkel, J. Swann, Z. Zhu, M. Christos, K. Watan- abe, T. Taniguchi, S. Sachdev, M. S. Scheurer, E. Kaxi- ras, C. R. Dean,et al., Orderly disorder in magic-angle twisted trilayer graphene, Science376, 193 (2022)

    work page 2022

  41. [41]

    H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Al- icea, and S. Nadj-Perge, Evidence for unconventional su- perconductivity in twisted trilayer graphene, Nature606, 494 (2022)

    2022

  42. [42]

    H. Kim, Y. Choi, ´E. Lantagne-Hurtubise, C. Lewandowski, A. Thomson, L. Kong, H. Zhou, E. Baum, Y. Zhang, L. Holleis,et al., Imaging inter- valley coherent order in magic-angle twisted trilayer graphene, Nature623, 942 (2023)

    2023

  43. [43]

    C. Shen, P. J. Ledwith, K. Watanabe, T. Taniguchi, E. Khalaf, A. Vishwanath, and D. K. Efetov, Dirac spec- troscopy of strongly correlated phases in twisted trilayer graphene, Nature Materials22, 316 (2023)

    2023

  44. [44]

    J. C. Hoke, Y. Li, J. May-Mann, K. Watanabe, T. Taniguchi, B. Bradlyn, T. L. Hughes, and B. E. Feldman, Uncovering the spin ordering in magic-angle graphene via edge state equilibration, Nature communi- cations15, 4321 (2024)

    work page 2024

  45. [45]

    L.-Q. Xia, S. C. de la Barrera, A. Uri, A. Sharpe, Y. H. Kwan, Z. Zhu, K. Watanabe, T. Taniguchi, D. Goldhaber-Gordon, L. Fu,et al., Topological bands and correlated states in helical trilayer graphene, Nature Physics21, 239 (2025)

    work page 2025

  46. [46]

    J. C. Hoke, Y. Li, Y. Hu, J. May-Mann, K. Watanabe, T. Taniguchi, T. Devakul, and B. E. Feldman, Imaging supermoire relaxation and conductive domain walls in he- lical trilayer graphene, arXiv preprint arXiv:2410.16269 (2024)

    arXiv 2024

  47. [47]

    A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science365, 605 (2019)

    2019

  48. [48]

    Y. Choi, H. Kim, C. Lewandowski, Y. Peng, A. Thomson, R. Polski, Y. Zhang, K. Watanabe, T. Taniguchi, J. Al- icea,et al., Interaction-driven band flattening and corre- lated phases in twisted bilayer graphene, Nature Physics 17, 1375 (2021)

    work page 2021

  49. [49]

    J. Xiao, A. Inbar, J. Birkbeck, N. Gershon, Y. Zamir, T. Taniguchi, K. Watanabe, E. Berg, and S. Ilani, The interacting energy bands of magic angle twisted bilayer graphene revealed by the quantum twisting microscope, arXiv preprint arXiv:2506.20738 (2025)

    Pith/arXiv arXiv 2025

  50. [50]

    Q. Hu, S. Liang, X. Li, H. Shi, X. Dai, and Y. Xu, Link between cascade transitions and correlated chern insulators in magic-angle twisted bilayer graphene, arXiv preprint arXiv:2406.08734 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  51. [51]

    Zhang, S

    Z. Zhang, S. Wu, D. C˘ alug˘ aru, H. Hu, T. Taniguchi, K. Wanatabe, A. B. Bernevig, and E. Y. Andrei, Heavy fermions, mass renormalization and local moments in magic-angle twisted bilayer graphene via planar tun- neling spectroscopy, arXiv preprint arXiv:2503.17875 (2025)

    Pith/arXiv arXiv 2025

  52. [52]

    Song and B

    Z.-D. Song and B. A. Bernevig, Magic-angle twisted bi- layer graphene as a topological heavy fermion problem, Physical review letters129, 047601 (2022)

    2022

  53. [53]

    Steglich and S

    F. Steglich and S. Wirth, Foundations of heavy-fermion superconductivity: lattice kondo effect and mott physics, Reports on Progress in Physics79, 084502 (2016)

    2016

  54. [54]

    Tinkham,Introduction to superconductivity(Courier Corporation, 2004)

    M. Tinkham,Introduction to superconductivity(Courier Corporation, 2004)

    2004

  55. [55]

    J. Yu, M. Xie, B. A. Bernevig, and S. Das Sarma, Magic- angle twisted symmetric trilayer graphene as a topo- logical heavy-fermion problem, Physical Review B108, 035129 (2023)

    work page 2023

  56. [56]

    Y. H. Kwan, G. Wagner, T. Soejima, M. P. Zaletel, S. H. Simon, S. A. Parameswaran, and N. Bultinck, Kekul´ e spi- ral order at all nonzero integer fillings in twisted bilayer graphene, Physical Review X11, 041063 (2021)

    2021

  57. [57]

    Wagner, Y

    G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and S. Parameswaran, Global phase diagram of the normal state of twisted bilayer graphene, Physical review letters 128, 156401 (2022)

    2022

  58. [58]

    S. Liu, E. Khalaf, J. Y. Lee, and A. Vishwanath, Ne- matic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality, Physical Review Research3, 013033 (2021)

    2021

  59. [59]

    D. E. Parker, T. Soejima, J. Hauschild, M. P. Zaletel, and N. Bultinck, Strain-induced quantum phase tran- sitions in magic-angle graphene, Physical review letters 127, 027601 (2021)

    work page 2021

  60. [60]

    Z. Bi, N. F. Yuan, and L. Fu, Designing flat bands by strain, Physical Review B100, 035448 (2019)

    work page 2019

  61. [61]

    L.-Q. Xia, A. Uri, J. Yan, A. Sharpe, F. Gaggioli, N. S. Ticea, J. May-Mann, K. Watanabe, T. Taniguchi, L. Fu, et al., Magic continuum in multi-moir\’e twisted trilayer graphene, arXiv preprint arXiv:2509.03583 (2025)

    Pith/arXiv arXiv 2025

  62. [62]

    Amorim and E

    B. Amorim and E. V. Castro, Electronic spectral proper- ties of incommensurate twisted trilayer graphene, arXiv preprint arXiv:1807.11909 (2018)

    Pith/arXiv arXiv 2018

  63. [63]

    Koshino, Interlayer interaction in general incommen- surate atomic layers, New Journal of Physics17, 015014 (2015)

    M. Koshino, Interlayer interaction in general incommen- surate atomic layers, New Journal of Physics17, 015014 (2015)

    2015

  64. [64]

    L. Wang, I. Meric, P. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. Campos, D. Muller,et al., One-dimensional electrical contact to a two-dimensional material, Science342, 614 (2013)

    work page 2013

  65. [65]

    Pizarro, M

    J. Pizarro, M. R¨ osner, R. Thomale, R. Valent ´ ı, and T. Wehling, Internal screening and dielectric engineering in magic-angle twisted bilayer graphene, Physical Review B100, 161102 (2019). 9 METHODS Spectral function calculations In this section, we describe our method for calculat- ing the electronic structure at incommensurate twist an- gles. We use ...

    work page 2019

  66. [66]

    We find that the application of the Savitzky–Golay filter has no impact on the position or width of the peak in σe(h) and simply shiftsσ e(h) down by an overall constant (Fig

    remove contributions from noise in the measurement and 2) remove contributions of the peak atν=±2 to the extracted standard deviation, thereby isolating only con- tributions related to the sawtooth compressibility itself. We find that the application of the Savitzky–Golay filter has no impact on the position or width of the peak in σe(h) and simply shifts...

    work page 2012

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.