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arxiv: 2606.20849 · v1 · pith:2UKNELG4new · submitted 2026-06-18 · ❄️ cond-mat.str-el · cond-mat.supr-con

Displacement-Field-Driven Semimetal-Superconductor Transition in Magic-Angle Twisted Trilayer Graphene

Bokai Liang , Jing-Yu Zhao , Ya-Hui Zhang This is my paper

Pith reviewed 2026-06-26 15:09 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-con
keywords twisted trilayer graphenedisplacement fieldsemimetal-superconductor transitionself-dopingDirac coneMott gapsuperconductivityparticle-hole asymmetry
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The pith

The displacement field drives a semimetal-to-superconductor transition in magic-angle twisted trilayer graphene at fillings ν=±2 through Dirac-cone energy shift and self-doping.

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

The paper shows that the displacement field primarily shifts the energy of the dispersive Dirac cone, which self-dopes carriers into the flat bands of the twisted bilayer graphene sector. This self-doping effect outweighs any increase in hybridization and produces a transition from semimetal to superconductor at filling ν=±2, as calculated in a slave-particle approach. A sympathetic reader cares because the mechanism supplies a concrete way electric fields can select between semimetal, Mott, and superconducting regimes while respecting the momentum-selective Mott gap at the Gamma point.

Core claim

The dominant effect of the displacement field is an energy shift of the Dirac cone and self-doping into the TBG sector. At filling ν=±2, increasing D drives a transition from a semimetal into a superconducting state. The calculation also finds enhancement of superconductivity by D near ν=±2 together with particle-hole asymmetry of the phase diagram, yielding a unified account of electric-field-tunable superconductivity, Mottness, and heavy-fermion-like behavior.

What carries the argument

The displacement-field-induced energy shift of the Dirac cone, which produces self-doping into the TBG sector.

If this is right

  • Superconductivity near ν=±2 strengthens with rising displacement field.
  • The overall phase diagram acquires particle-hole asymmetry.
  • Electric fields can tune among semimetal, Mott, and superconducting states through the same self-doping channel.

Where Pith is reading between the lines

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

  • The same Dirac-cone shift mechanism may operate in other twisted multilayer stacks that combine flat bands with a dispersive cone.
  • Spectroscopic tracking of the Dirac-cone position versus D at fixed filling would directly test whether self-doping occurs as predicted.
  • Transport signatures of the transition should appear as a function of D at fixed ν=±2 and could be compared with the calculated phase boundaries.

Load-bearing premise

The dominant effect of the displacement field is an energy shift of the Dirac cone and self-doping into the TBG sector rather than enhanced hybridization.

What would settle it

A band-structure measurement or slave-particle calculation that shows hybridization effects dominating the Dirac-cone shift, with no semimetal-superconductor transition appearing at ν=±2 as D rises, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.20849 by Bokai Liang, Jing-Yu Zhao, Ya-Hui Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a)Schematic interpretation of the central mechanism [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Order parameters as a function of filling for TBG and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Color map of the STM signal as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The bare band result of TTG calculated from the BM model at different displacement field [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Mott band calculated at [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Mott band calculated at [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Mott band calculated at [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Mott band calculated at [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Mott band calculated at [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Mott band calculated at [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Results of the toy-model calculation. In the numerical calculation, the constant [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Superconducting order parameters away from [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Evolution of the system properties driven by the energy offset [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Evolution of the system properties driven by the displacement field [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Distribution of different components. The parameters are [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Comparison between the original results and the results after the coefficients in the linear response theory are [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Particle-hole-asymmetric band structure in the presence of the middle-layer shift term. The onsite energy shift is [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Mean-field Mott band structures including the Dirac-cone offset at positive fillings: (a) [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Mean-field Mott band structures including the Dirac-cone offset at nonpositive and negative fillings: (a) [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
read the original abstract

Magic-angle twisted trilayer graphene(MATTG) hosts versatile displacement-field-tuned correlated phenomena. MATTG consists of a dispersive Dirac cone which hybridizes with the flat band from a twisted bilayer graphene (TBG) sector. The hybridization strength increases with the displacement field $D$ and naively one may expect D-driven heavy fermion physics. However, the TBG Hubbard bands have a momentum-selective Mott gap, which is small at the $\Gamma$ point due to the band topology, and a rigid local moment description as in the familiar Kondo lattice model is invalid. Here we show that the dominant effect of the displacement field is to induce an energy shift of the Dirac cone and self-doping into the TBG sector. We illustrate this picture in a concrete calculation using a slave-particle theory at the filling $\nu=\pm 2$. We find that increasing $D$ drives a transition from a semimetal into a superconducting state. We also discuss the enhancement of the superconductivity by $D$ near $\nu=\pm2$ and the particle-hole asymmetry of the phase diagram. Our results provide a unified picture for electric-field-tunable superconductivity, Mottness, and heavy-fermion-like behavior in MATTG.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript claims that in magic-angle twisted trilayer graphene the dominant effect of the displacement field D is an energy shift of the Dirac cone inducing self-doping into the TBG sector, rather than enhanced hybridization. Using a concrete slave-particle calculation at filling ν=±2, it shows that increasing D drives a transition from semimetal to superconductor. The work contrasts this with naive heavy-fermion expectations, notes the momentum-selective Mott gap at Γ invalidates a simple Kondo picture, and discusses D-enhanced superconductivity near ν=±2 together with particle-hole asymmetry of the phase diagram.

Significance. If the result holds, the work supplies a unified theoretical account of displacement-field-tunable superconductivity, Mottness, and heavy-fermion-like behavior in MATTG. The explicit slave-particle calculation demonstrating the self-doping mechanism, together with its falsifiable prediction of the D-driven semimetal-superconductor transition at ν=±2, constitutes a clear strength.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and the recommendation to accept the manuscript. The referee's summary correctly identifies the central claim regarding the dominant role of the displacement field in inducing self-doping rather than enhanced hybridization, as well as the limitations of a simple Kondo picture due to the momentum-selective Mott gap.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit slave-particle model

full rationale

The central claim is illustrated by an explicit slave-particle calculation at ν=±2 that maps the displacement-field-induced Dirac-cone shift onto self-doping and a semimetal-to-superconductor transition. No quoted equations reduce the reported transition to a fitted parameter renamed as a prediction, nor does any load-bearing step rest on a self-citation whose content is itself unverified. The model is presented as a concrete illustration rather than a parameter-free first-principles derivation, yet the steps shown do not collapse by construction to the input assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that self-doping dominates over hybridization and on the applicability of the slave-particle ansatz at ν=±2; no explicit free parameters or invented entities are named in the abstract.

free parameters (1)
  • displacement-field scale
    The strength of D is varied continuously to induce the transition, but no numerical values or fitting procedure are given in the abstract.
axioms (2)
  • domain assumption Dominant effect of D is Dirac-cone energy shift and self-doping rather than hybridization
    Explicitly stated in the abstract as the key physical picture after dismissing the Kondo-lattice description.
  • domain assumption Slave-particle theory is adequate at ν=±2
    The calculation is performed within this framework without further justification supplied in the abstract.

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Works this paper leans on

84 extracted references · 8 canonical work pages · 2 internal anchors

  1. [1]

    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

  2. [2]

    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

  3. [3]

    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, Science363, 1059 (2019)

    2019

  4. [4]

    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

  5. [5]

    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

  6. [6]

    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

  7. [7]

    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

  8. [8]

    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

  9. [9]

    E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nature materials19, 1265 (2020)

    2020

  10. [10]

    E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moir´ e materials, Nature Reviews Materials6, 201 (2021)

    2021

  11. [11]

    K. P. Nuckolls and A. Yazdani, A microscopic perspec- tive on moir´ e materials, Nature Reviews Materials9, 460 (2024)

    2024

  12. [12]

    H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Ori- gin of mott insulating behavior and superconductivity in twisted bilayer graphene, Physical Review X8, 031089 7 (2018)

    2018

  13. [13]

    J. Ahn, S. Park, and B.-J. Yang, Failure of nielsen-ninomiya theorem and fragile topology in two- dimensional systems with space-time inversion symme- try: application to twisted bilayer graphene at magic an- gle, Physical Review X9, 021013 (2019)

    2019

  14. [14]

    Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies- Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, Strange metal in magic-angle graphene with near planckian dissipation, Physical review letters 124, 076801 (2020)

    2020

  15. [15]

    P. J. Ledwith, E. Khalaf, and A. Vishwanath, Strong coupling theory of magic-angle graphene: A pedagogical introduction, Annals of Physics435, 168646 (2021)

    2021

  16. [16]

    Z.-D. Song, B. Lian, N. Regnault, and B. A. Bernevig, Twisted bilayer graphene. ii. stable symmetry anomaly, Physical Review B103, 205412 (2021)

    2021

  17. [17]

    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

  18. [18]

    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

  19. [19]

    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

  20. [20]

    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

  21. [21]

    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)

    2022

  22. [22]

    Mukherjee, S

    A. Mukherjee, S. Layek, S. Sinha, R. Kundu, A. H. Marchawala, M. Hingankar, J. Sarkar, L. V. Sangani, H. Agarwal, S. Ghosh,et al., Superconducting magic- angle twisted trilayer graphene with competing magnetic order and moir´ e inhomogeneities, Nature Materials , 1 (2025)

    2025

  23. [23]

    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

  24. [24]

    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

  25. [25]

    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)

    2025

  26. [26]

    M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, X. Liu, K. Watanabe, T. Taniguchi, and A. Yazdani, Evidence for unconventional superconductivity in twisted bilayer graphene, Nature600, 240 (2021)

    2021

  27. [27]

    J. M. Park, S. Sun, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Experimental evidence for nodal su- perconducting gap in moir´ e graphene, Science391, 79 (2026)

    2026

  28. [28]

    H. Kim, G. Rai, L. Crippa, D. C˘ alug˘ aru, H. Hu, Y. Choi, L. Kong, E. Baum, Y. Zhang, L. Holleis,et al., Resolv- ing intervalley gaps and many-body resonances in moir´ e superconductors, Nature , 1 (2026)

    2026

  29. [29]

    Heavy fermion phase diagram in magic-angle twisted trilayer graphene

    L. Zhang, W. Zhou, X. Fang, Z. Zhan, K. Watan- abe, T. Taniguchi, Y.-f. Yang, and S. Xu, Electri- cally tunable heavy fermion and quantum criticality in magic-angle twisted trilayer graphene, arXiv preprint arXiv:2507.12254 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  30. [30]

    Khalaf, A

    E. Khalaf, A. J. Kruchkov, G. Tarnopolsky, and A. Vish- wanath, Magic angle hierarchy in twisted graphene mul- tilayers, Physical Review B100, 085109 (2019)

    2019

  31. [31]

    C˘ alug˘ aru, F

    D. C˘ alug˘ aru, F. Xie, Z.-D. Song, B. Lian, N. Reg- nault, and B. A. Bernevig, Twisted symmetric trilayer graphene: Single-particle and many-body hamiltonians and hidden nonlocal symmetries of trilayer moir´ e systems with and without displacement field, Physical Review B 103, 195411 (2021)

    2021

  32. [32]

    C. Lei, L. Linhart, W. Qin, F. Libisch, and A. H. Mac- Donald, Mirror symmetry breaking and lateral stacking shifts in twisted trilayer graphene, Physical Review B 104, 035139 (2021)

    2021

  33. [33]

    F. Xie, N. Regnault, D. C˘ alug˘ aru, B. A. Bernevig, and B. Lian, Twisted symmetric trilayer graphene. ii. pro- jected hartree-fock study, Physical Review B104, 115167 (2021)

    2021

  34. [34]

    Christos, S

    M. Christos, S. Sachdev, and M. S. Scheurer, Correlated insulators, semimetals, and superconductivity in twisted trilayer graphene, Physical Review X12, 021018 (2022)

    2022

  35. [35]

    Fischer, Z

    A. Fischer, Z. A. Goodwin, A. A. Mostofi, J. Lischner, D. M. Kennes, and L. Klebl, Unconventional supercon- ductivity in magic-angle twisted trilayer graphene, npj Quantum Materials7, 5 (2022)

    2022

  36. [36]

    D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yaz- dani, Cascade of electronic transitions in magic-angle twisted bilayer graphene, Nature582, 198 (2020)

    2020

  37. [37]

    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

  38. [38]

    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)

    2023

  39. [39]

    K. P. Nuckolls, R. L. Lee, M. Oh, D. Wong, T. Soejima, J. P. Hong, D. C˘ alug˘ aru, J. Herzog-Arbeitman, B. A. Bernevig, K. Watanabe,et al., Quantum textures of the many-body wavefunctions in magic-angle graphene, Na- ture620, 525 (2023)

    2023

  40. [40]

    Herzog-Arbeitman, J

    J. Herzog-Arbeitman, J. Yu, D. C˘ alug˘ aru, H. Hu, N. Reg- nault, C. Liu, O. Vafek, P. Coleman, A. Tsvelik, Z.- d. Song,et al., Topological heavy fermion principle for flat (narrow) bands with concentrated quantum geome- try, arXiv preprint arXiv:2404.07253 (2024)

    work page arXiv 2024

  41. [41]

    G. Rai, L. Crippa, D. C˘ alug˘ aru, H. Hu, F. Paoletti, L. De’Medici, A. Georges, B. A. Bernevig, R. Valent´ ı, G. Sangiovanni,et al., Dynamical correlations and order in magic-angle twisted bilayer graphene, Physical Review X14, 031045 (2024)

    2024

  42. [42]

    S. Youn, B. Goh, G.-D. Zhou, Z.-D. Song, and S.- S. B. Lee, Hundness in twisted bilayer graphene: Corre- lated gaps and pairing, arXiv preprint arXiv:2412.03108 8 (2024)

    work page arXiv 2024

  43. [43]

    J.-Y. Zhao, B. Zhou, and Y.-H. Zhang, Ancilla theory of twisted bilayer graphene i: topological mott localization and pseudogap metal in twisted bilayer graphene, arXiv preprint arXiv:2502.17430 (2025)

    work page arXiv 2025

  44. [44]

    J.-Y. Zhao, B. Zhou, and Y.-H. Zhang, Mixed valence mott insulator and composite excitation in twisted bi- layer graphene, arXiv preprint arXiv:2507.00139 (2025)

    work page arXiv 2025

  45. [45]

    Zhao and Y.-H

    J.-Y. Zhao and Y.-H. Zhang, Resonating-valence-bond superconductor from small fermi surface in twisted bi- layer graphene, arXiv preprint arXiv:2510.26801 (2025)

    work page arXiv 2025

  46. [46]

    J. Xiao, A. Inbar, J. Birkbeck, N. Gershon, Y. Zamir, Y. Vituri, T. Taniguchi, K. Watanabe, E. Berg, and S. Ilani, Imaging the flat bands of magic-angle graphene reshaped by interactions, Nature653, 68 (2026)

    2026

  47. [47]

    Kang and O

    J. Kang and O. Vafek, Symmetry, maximally localized wannier states, and a low-energy model for twisted bi- layer graphene narrow bands, Physical Review X8, 031088 (2018)

    2018

  48. [48]

    Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All magic angles in twisted bilayer graphene are topological, Physical review letters123, 036401 (2019)

    2019

  49. [49]

    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

  50. [50]

    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

  51. [51]

    Ghosh, S

    A. Ghosh, S. Chakraborty, R. Dutta, A. Agarwala, K. Watanabe, T. Taniguchi, S. Banerjee, N. Trivedi, S. Mukerjee, and A. Das, Thermopower probes of emergent local moments in magic-angle twisted bilayer graphene, Nature Physics21, 732 (2025)

    2025

  52. [52]

    Bultinck, E

    N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vish- wanath, and M. P. Zaletel, Ground state and hidden symmetry of magic-angle graphene at even integer fill- ing, Physical Review X10, 031034 (2020)

    2020

  53. [53]

    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)

    2021

  54. [54]

    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

  55. [55]

    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

  56. [56]

    Z. Wu, Z. Zhan, and S. Yuan, Lattice relaxation, mirror symmetry and magnetic field effects on ultraflat bands in twisted trilayer graphene, Science China Physics, Me- chanics & Astronomy64, 267811 (2021)

    2021

  57. [57]

    C. Chen, K. P. Nuckolls, S. Ding, W. Miao, D. Wong, M. Oh, R. L. Lee, S. He, C. Peng, D. Pei,et al., Strong electron–phonon coupling in magic-angle twisted bilayer graphene, Nature636, 342 (2024)

    2024

  58. [58]

    Wang, G.-D

    Y.-J. Wang, G.-D. Zhou, S.-Y. Peng, B. Lian, and Z.- D. Song, Molecular pairing in twisted bilayer graphene superconductivity, Physical review letters133, 146001 (2024)

    2024

  59. [59]

    Wang, G.-D

    Y.-J. Wang, G.-D. Zhou, B. Lian, and Z.-D. Song, Electron-phonon coupling in the topological heavy fermion model of twisted bilayer graphene, Physical Re- view B111, 035110 (2025)

    2025

  60. [60]

    Zhang and S

    Y.-H. Zhang and S. Sachdev, From the pseudogap metal to the fermi liquid using ancilla qubits, Physical Review Research2, 023172 (2020)

    2020

  61. [61]

    Mascot, A

    E. Mascot, A. Nikolaenko, M. Tikhanovskaya, Y.-H. Zhang, D. K. Morr, and S. Sachdev, Electronic spec- tra with paramagnon fractionalization in the single-band hubbard model, Physical Review B105, 075146 (2022)

    2022

  62. [62]

    Zhou, H.-K

    B. Zhou, H.-K. Jin, and Y.-H. Zhang, Variational wave- function for a mott insulator at finite u using ancilla qubits, Physical Review B112, 115159 (2025)

    2025

  63. [63]

    Zhang and D

    Y.-H. Zhang and D. Mao, Spin liquids and pseudogap metals in the su (4) hubbard model in a moir´ e superlat- tice, Physical Review B101, 035122 (2020)

    2020

  64. [64]

    H. Hu, B. A. Bernevig, and A. M. Tsvelik, Kondo lattice model of magic-angle twisted-bilayer graphene: Hund’s rule, local-moment fluctuations, and low-energy effective theory, Physical review letters131, 026502 (2023)

    2023

  65. [65]

    Chou and S

    Y.-Z. Chou and S. Das Sarma, Kondo lattice model in magic-angle twisted bilayer graphene, Physical Review Letters131, 026501 (2023)

    2023

  66. [66]

    Emergent polaronic correlations in doped spin liquids

    L. Shackleton and S. Zhang, Emergent polaronic correlations in doped spin liquids, arXiv preprint arXiv:2408.02190 (2024)

    work page internal anchor Pith review arXiv 2024

  67. [67]

    M¨ uller, R

    T. M¨ uller, R. Thomale, S. Sachdev, and Y. Iqbal, Po- laronic correlations from optimized ancilla wave func- tions for the fermi–hubbard model, Proceedings of the National Academy of Sciences122, e2504261122 (2025)

    2025

  68. [68]

    Zhao and Y.-H

    J.-Y. Zhao and Y.-H. Zhang, Pseudogap and non-fermi- liquid criticality in double kondo model for bilayer nick- elates, arXiv preprint arXiv:2603.25742 (2026)

    work page arXiv 2026

  69. [69]

    L. L. Lau and P. Coleman, Topological mixed valence model for twisted bilayer graphene, Physical Review X 15, 021028 (2025)

    2025

  70. [70]

    Zhang and S

    Y.-H. Zhang and S. Sachdev, Deconfined criticality and ghost fermi surfaces at the onset of antiferromagnetism in a metal, Physical Review B102, 155124 (2020)

    2020

  71. [71]

    Kang and O

    J. Kang and O. Vafek, Pseudomagnetic fields, particle- hole asymmetry, and microscopic effective continuum hamiltonians of twisted bilayer graphene, Phys. Rev. B 107, 075408 (2023)

    2023

  72. [72]

    Kang and O

    J. Kang and O. Vafek, Analytical solution for the relaxed atomic configuration of twisted bilayer graphene includ- ing heterostrain, Phys. Rev. B112, 125138 (2025). Appendix A: Review: BM model of TTG The typical twisted trilayer graphene(TTG) system is constructed from a AAA-stacking graphen, with the top layer(l= 1) and the bottom layer(l= 3) twisted by...

    2025

  73. [73]

    (A2) wherew 0 = 110 meV andw 1 = 0.8w0 are the AA and AB tunneling

    is the Dirac momentum, and Tj =w0σ0 +w 1 cos 2π 3 (j−1) σx + sin 2π 3 (j−1) σy . (A2) wherew 0 = 110 meV andw 1 = 0.8w0 are the AA and AB tunneling. WhenD= 0, based on the mirror symmetry, equation (A1) can be decomposed into a twisted bilayer grahene and a monolayer graphene. To be specific, we can define the symmetric part ˜ϕ† r,t,ηs = 1√ 2 ϕ† r,3,ηs +ϕ...

  74. [74]

    Since the localizedforbitals are centered on the AA sites and carry orbital, valley, and spin indices, the on-site Hilbert space contains a large multiplet structure

    Spin interaction and local-moment pairing In this subsection we discuss the on-site spin interaction used in our paper, the main discussion follows the Appendix of [45]. Since the localizedforbitals are centered on the AA sites and carry orbital, valley, and spin indices, the on-site Hilbert space contains a large multiplet structure. The role of the spin...

  75. [75]

    The construction follows the parton mean-field theory of Ref

    Details of the mean-field construction In this subsection we give the explicit mean-field decoupling used in the slave-particle calculation. The construction follows the parton mean-field theory of Ref. [45], with the only essential extension that the physical-electron sector contains both the TBG-likecfermions and the monolayer Diracdfermions. As discuss...

  76. [76]

    (D20) follow from flavor counting in the restricted Hilbert space and from the normalization of the recombined triplon in Eq

    The numerical coefficients in Eq. (D20) follow from flavor counting in the restricted Hilbert space and from the normalization of the recombined triplon in Eq. (D15). With these definitions, the hybridization-channel mean-field Hamiltonian is HMF B = X k,G,λ,α c† k+G;λγλα(k+G) Bsψ′ k;α +αB tψ′† −k;¯α + h.c. + X k,p,λ,α d† p;λ eh† λα(p, k) Bsψ′ k;α +αB tψ′...

  77. [77]

    Perturbation theory based on diagonal band basis We first setD= 0 and then diagonalize thef-ψblock, Hf ψ = µ1 2 ϕ1 ϕ1 − µ1 2 .(F7) Its two eigenvalues are ϵ± =±Λ,Λ = r ϕ2 1 + µ2 1 4 = 1 2 q µ2 1 + 4ϕ2 1.(F8) These two bands correspond to the upper and lower Hubbard bands in this minimal model. The corresponding normalized eigenstates of these Hubbard band...

  78. [78]

    Full eigenvalue calculation LetEdenote an eigenvalue of Eq. (F6). The characteristic equation is obtained from det h E−H (0) K− i =E 3 − 2D2 +ϕ 2 1 + µ2 1 4 E−µ 1D2 = 0.(F18) AtD= 0, the three eigenvalues are E0 = 0, E ± =± r ϕ2 1 + µ2 1 4 =± 1 2 q µ2 1 + 4ϕ2 1.(F19) The eigenvalueE 0 = 0 corresponds to the original Dirac point. WhenDis turned on, the mid...

  79. [79]

    Construction of the linear response matrix To systematically investigate the phase transition driven by the energy offsetwand the displacement fieldD, we em- ploy a rigorous linear response formalism. This approach is naturally connected with the method of expanding the free energy, while it directly evaluates how the quantum fluctuations within the paren...

  80. [80]

    Explicit calculation of the susceptibility matrix Evaluating the susceptibility matrixχinvolves calculating the fermionic bubble diagrams using the unperturbed Nambu-Gor’kov Green’s functions of the parent state, where Bs =B t =B ′ c = ∆′ c =B ′ d = ∆′ d = 0,∆̸= 0.(H22) By exactly diagonalizing the unperturbed Hamiltonian HBdG(k)|um(k)⟩=E m(k)|um(k)⟩,(H23...

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