pith. sign in

arxiv: 2606.06450 · v1 · pith:LDZHAUJBnew · submitted 2026-06-04 · ❄️ cond-mat.mes-hall · cond-mat.str-el

1/3 Fractional and Gapless Integer Quantum Anomalous Hall States in Rhombohedral Graphene

Jackson P. Butler , Tonghang Han , Andrew DiFabbio , Zach Hadjri , Emily Aitken , Kenji Watanabe , Takashi Taniguchi , Long Ju
show 1 more author
Raymond C. Ashoori
This is my paper

Pith reviewed 2026-06-27 23:52 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-el
keywords fractional quantum anomalous Hallmoiré superlatticerhombohedral graphenefractional Chern insulatorquantum capacitancetopological phase transitioncharge density wave
0
0 comments X

The pith

Rhombohedral five-layer graphene on boron nitride hosts a 1/3 fractional Chern insulator that transitions to a charge density wave.

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

The paper establishes the existence of the fractional quantum anomalous Hall effect at moiré filling factor one-third in rhombohedral five-layer graphene aligned to hexagonal boron nitride. Quantum capacitance and transport data show a gapped fractional Chern insulator state that can be driven into a trivial charge density wave by changing the displacement field. Inclusion of this state produces particle-hole symmetry in the observed fractional states around half filling, matching the pattern seen in conventional lowest-Landau-level fractional quantum Hall systems. At unit filling the same measurements separate a gapped integer quantum anomalous Hall state from a gapless, highly compressible extended state that transport alone does not distinguish.

Core claim

We report the FQAH effect at moiré filling factor ν = 1/3 in R5G/hBN moiré superlattice devices, through a combination of quantum capacitance and transport measurements. By tuning the displacement field, we observed a topological phase transition from a 1/3 fractional Chern insulator (FCI) to a trivial charge density wave state. With the inclusion of the 1/3 state, the FQAH states in R5G/hBN now exhibit a surprising level of particle-hole symmetry about half-filling, closely resembling the behavior of FQH states in the lowest Landau level. Compressibility measurements at ν=1 reveal a distinct transition from a gapped IQAH state to a gapless and highly compressible EQAH state.

What carries the argument

Moiré superlattice in rhombohedral five-layer graphene aligned to hBN, read out by quantum capacitance for thermodynamic compressibility combined with transport Hall response.

If this is right

  • FQAH states in this system display particle-hole symmetry about half filling.
  • A displacement-field-tuned transition separates the 1/3 fractional Chern insulator from a trivial charge density wave.
  • At ν=1 the extended quantum anomalous Hall regime is thermodynamically gapless while the integer state remains gapped.
  • Direct thermodynamic access to the phase diagram opens routes to anyon braiding studies at zero magnetic field.

Where Pith is reading between the lines

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

  • The observed symmetry suggests the underlying interaction and topology mechanism may be closer to conventional Landau-level physics than previously thought for graphene moiré systems.
  • The gapless character at ν=1 could imply metallic or critical behavior that might host different quasiparticle statistics if further tuned.
  • Similar capacitance-plus-transport protocols could be applied to other moiré platforms to map full thermodynamic phase diagrams.

Load-bearing premise

Quantum capacitance data directly reflect intrinsic thermodynamic compressibility of the fractional state without significant masking by disorder, contact effects, or competing orders.

What would settle it

No dip in compressibility at ν=1/3 or no displacement-field-driven transition from incompressible to compressible behavior at that filling.

Figures

Figures reproduced from arXiv: 2606.06450 by Andrew DiFabbio, Emily Aitken, Jackson P. Butler, Kenji Watanabe, Long Ju, Raymond C. Ashoori, Takashi Taniguchi, Tonghang Han, Zach Hadjri.

Figure 1
Figure 1. Figure 1: b shows the penetration capacitance as a func￾tion of density and displacement field in the moir´e-distant limit. At small filling factor, ν <∼ 1/2 and lower displace￾ment field we observe a large penetration capacitance over an extended region of density and displacement field. We attribute this region of the map to a Wigner solid (WS) state, an electronic solid with short range order and high resistivity… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the extracted thermodynamic gap size of the four fractional states at 39 mK. The 1/3 state hosts the largest gap among the FQAH states, ∆µ ≈ 0.6 meV or ∆µ ≈ 7 K. This is the largest reported gap for a graphene fractional Chern insulator. The 2/3 gap is slightly smaller, peaking at around 0.5 meV while the 2/5 and 3/5 gaps are much smaller. The ν = 1 gap peaks at D = 770 mV nm−1 with a value of ∼ 2.7 … view at source ↗
Figure 4
Figure 4. Figure 4: shows both the compressibility and longitudi￾nal resistance, R∗ xx (see Supplementary Information XII for how R∗ xx differs slightly from the usual Rxx), between filling ν ≈ 1/2 and ν ≈ 1. Previous very low temperature results show a vanishing longitudinal resistance along with a quantized h/e2 Hall resistance over a broad re￾gion of n− D phase space between ν = 1 and ν = 1/2[4]. At full-filling, ν = 1, we… view at source ↗
read the original abstract

The fractional quantum anomalous Hall (FQAH) effect occurs in moir\'e superlattices in both twisted bilayer MoTe$_2$ and rhombohedral $n$-layer graphene aligned to hexagonal boron nitride (R$n$G/hBN) as a novel quantum phase driven by intertwined electron correlation and topology. Although several fractional states in the Jain sequence have been identified, the $1/3$ state, the most robust and fundamental state in conventional fractional quantum Hall (FQH) systems, was missing in either FQAH system. Determining whether it exists would have a major impact on understanding the mechanism of FQAH, especially in the theoretically still-debated R$n$G/hBN system. Here we report the FQAH effect at moir\'e filling factor $\nu = 1/3$ in R$5$G/hBN moir\'e superlattice devices, through a combination of quantum capacitance and transport measurements. By tuning the displacement field, we observed a topological phase transition from a $1/3$ fractional Chern insulator (FCI) to a trivial charge density wave state. With the inclusion of the $1/3$ state, the FQAH states in R$5$G/hBN now exhibit a surprising level of particle-hole symmetry about half-filling, closely resembling the behavior of FQH states in the lowest Landau level. Additionally, we perform compressibility and transport measurements at a filling of one electron per moir\'e unit cell, $\nu =1$, and also for $\nu \lesssim 1$, where previous transport measurements displayed the extended quantum anomalous Hall (EQAH) effect. While our transport measurements show no change between the integer quantum anomalous Hall state (IQAH) and the EQAH region, compressibility measurements reveal a distinct transition from a gapped IQAH state to a gapless and highly compressible EQAH state. Our direct thermodynamic characterization of the rich phase diagram paves the way to engineering of anyon braiding and non-Abelian quasiparticles at zero magnetic field.

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

2 major / 1 minor

Summary. The manuscript reports the experimental observation of the 1/3 fractional quantum anomalous Hall (FQAH) state at moiré filling ν=1/3 in rhombohedral 5-layer graphene aligned to hBN, using quantum capacitance and transport measurements. It describes a displacement-field-driven topological transition from a fractional Chern insulator to a trivial charge density wave, notes an apparent particle-hole symmetry in the FQAH sequence around half-filling, and identifies at ν=1 a transition from a gapped integer quantum anomalous Hall (IQAH) state to a gapless, highly compressible extended QAH (EQAH) state via compressibility data.

Significance. If the central observations hold after additional controls, the work would complete the Jain-sequence FQAH states in the R n G/hBN platform, establish a direct analogy to lowest-Landau-level fractional quantum Hall physics, and supply thermodynamic evidence distinguishing gapped versus gapless phases at integer filling. This would strengthen the case for correlation-driven topology in zero-field moiré systems and support future efforts toward anyon manipulation.

major comments (2)
  1. [Abstract and Results (capacitance data)] Abstract and main text description of capacitance measurements at ν=1/3: the interpretation that capacitance minima directly demonstrate a thermodynamic gap of a fractional Chern insulator (rather than broadened features from inhomogeneity or contacts) is load-bearing for the central claim, yet the manuscript provides no explicit discussion of background subtraction, contact calibration, or electrostatic modeling to exclude these confounds.
  2. [Results (ν=1 compressibility and transport)] Results on ν=1 and ν≲1: transport data are stated to show no change across the IQAH-to-EQAH boundary while compressibility reveals a distinct transition, but without reported error bars, sample-to-sample statistics, or quantitative gap values the distinction between gapped and gapless states remains difficult to assess quantitatively.
minor comments (1)
  1. Notation for device labeling (R5G/hBN versus rhombohedral 5-layer graphene) should be standardized in the main text and figure captions for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding data analysis and quantitative assessment. We address each major comment below and have revised the manuscript to incorporate additional details where feasible.

read point-by-point responses

  1. Referee: [Abstract and Results (capacitance data)] Abstract and main text description of capacitance measurements at ν=1/3: the interpretation that capacitance minima directly demonstrate a thermodynamic gap of a fractional Chern insulator (rather than broadened features from inhomogeneity or contacts) is load-bearing for the central claim, yet the manuscript provides no explicit discussion of background subtraction, contact calibration, or electrostatic modeling to exclude these confounds.

    Authors: We agree that an explicit discussion of these analysis procedures would strengthen the presentation. In the revised manuscript we will add a dedicated paragraph in the main text (and expanded details in the supplement) describing the background subtraction method applied to the quantum capacitance traces, the contact resistance calibration procedure, and a simple electrostatic model used to estimate the scale of inhomogeneity broadening. These additions will directly address the possibility of artifacts and support the interpretation of the ν=1/3 minima as thermodynamic gaps of the fractional Chern insulator. revision: yes

  2. Referee: [Results (ν=1 compressibility and transport)] Results on ν=1 and ν≲1: transport data are stated to show no change across the IQAH-to-EQAH boundary while compressibility reveals a distinct transition, but without reported error bars, sample-to-sample statistics, or quantitative gap values the distinction between gapped and gapless states remains difficult to assess quantitatively.

    Authors: We acknowledge the value of quantitative error bars and statistics. The revised manuscript will include error bars on the compressibility data (derived from measurement noise and repeated sweeps) and will report consistency across the devices measured in this study. However, extracting precise numerical gap values from compressibility requires additional modeling assumptions about the density of states; we will clarify these limitations in the text. Full temperature-dependent transport gap extraction across the transition lies outside the present dataset and would require new measurements. revision: partial

Circularity Check

0 steps flagged

Purely experimental observations; no derivation chain present

full rationale

The paper reports direct experimental measurements of quantum capacitance and transport in R5G/hBN moiré devices at fillings ν=1/3 and ν=1. No equations, fitted parameters, or predictions are derived from prior results within the paper. Claims rest on observed capacitance minima, phase transitions under displacement field, and transport features, without any self-referential reduction or ansatz smuggling. The reader's assessment of 0.0 circularity is confirmed; this is a standard experimental report with no load-bearing derivations to inspect.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Experimental paper; relies on standard interpretations of quantum capacitance and transport in 2D electron systems rather than new postulates or fitted parameters.

axioms (2)
  • domain assumption Quantum capacitance measurements faithfully report the thermodynamic density of states and compressibility of the moiré superlattice.
    Central to identifying gapped versus gapless states at ν=1/3 and ν=1.
  • standard math Standard quantum mechanics and band topology apply to the R5G/hBN system under displacement field tuning.
    Background for interpreting the topological phase transition and particle-hole symmetry.

pith-pipeline@v0.9.1-grok · 5950 in / 1435 out tokens · 29681 ms · 2026-06-27T23:52:54.219484+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Valley Valves at Domain Walls in Symmetry-Broken Rhombohedral Graphene

    cond-mat.mes-hall 2026-06 unverdicted novelty 7.0

    Valley domain walls act as impenetrable barriers to transport in metallic rhombohedral graphene unless intervalley interactions mediate transmission, and intervalley mixing is required for appreciable supercurrent in ...

  2. Superconductivity and non-Fermi liquid metals in a charge-1/3 anyon fluid

    cond-mat.str-el 2026-06 unverdicted novelty 5.0

    Doping a fractional Chern insulator yields an anyon fluid that can form an SC* superconductor with residual Z2 order or a non-Fermi liquid Z3 orthogonal metal.

Reference graph

Works this paper leans on

70 extracted references · 3 canonical work pages · cited by 2 Pith papers

  1. [1]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Nature622, 74 (2023)

    2023

  2. [2]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y. Zhang, X. Liu, and T. Li, Physical Review X13, 031037 (2023)

    2023

  3. [3]

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Nature626, 759 (2024)

    2024

  4. [4]

    Z. Lu, T. Han, Y. Yao, Z. Hadjri, J. Yang, J. Seo, L. Shi, S. Ye, K. Watanabe, T. Taniguchi, and L. Ju, Nature637, 1090 (2025)

    2025

  5. [5]

    J. Xie, Z. Huo, X. Lu, Z. Feng, Z. Zhang, W. Wang, Q. Yang, K. Watanabe, T. Taniguchi, K. Liu, Z. Song, X. C. Xie, J. Liu, and X. Lu, Nature Materials24, 1042 (2025)

    2025

  6. [6]

    J. K. Jain, Physical Review Letters63, 199 (1989)

    1989

  7. [7]

    R. B. Laughlin, Physical Review Letters50, 1395 (1983)

    1983

  8. [8]

    M. Uzan, W. Zhi, M. Bocarsly, J. Dong, S. Dutta, N. Auerbach, N. S. Kander, M. Labendik, Y. Myasoedov, M. E. Huber, K. Watanabe, T. Taniguchi, D. E. Parker, and E. Zeldov, hBN alignment orientation controls moir´ e strength in rhombohedral graphene (2025), version Num- ber: 1

    2025

  9. [9]

    F. Xu, Z. Sun, J. Li, C. Zheng, C. Xu, J. Gao, T. Jia, K. Watanabe, T. Taniguchi, B. Tong, L. Lu, J. Jia, Z. Shi, S. Jiang, Y. Zhang, Y. Zhang, S. Lei, X. Liu, and T. Li, Signatures of unconventional superconductivity near reen- trant and fractional quantum anomalous Hall insulators (2025), version Number: 1

    2025

  10. [10]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. MacDon- ald, Physical Review Letters122, 086402 (2019)

    2019

  11. [11]

    H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Physical Review Research3, L032070 (2021)

    2021

  12. [12]

    H. Yu, M. Chen, and W. Yao, National Science Review 7, 12 (2020)

    2020

  13. [13]

    Devakul, V

    T. Devakul, V. Cr´ epel, Y. Zhang, and L. Fu, Nature Communications12, 6730 (2021)

    2021

  14. [14]

    T. Tan, P. J. Ledwith, and T. Devakul, The ideal limit of rhombohedral graphene: Interaction-induced layer- skyrmion lattices and their collective excitations (2025), version Number: 1

    2025

  15. [15]

    H. Li, B. A. Bernevig, and N. Regnault, Physical Review B112, 075130 (2025)

    2025

  16. [16]

    J. Yu, J. Herzog-Arbeitman, Y. H. Kwan, N. Regnault, and B. A. Bernevig, Physical Review B112, 075110 (2025)

    2025

  17. [17]

    Huang, S

    K. Huang, S. Das Sarma, and X. Li, Physical Review B 111, 075130 (2025)

    2025

  18. [18]

    B. Zhou, H. Yang, and Y.-H. Zhang, Physical Review Letters133, 10.1103/physrevlett.133.206504 (2024)

    work page doi:10.1103/physrevlett.133.206504 2024

  19. [19]

    Z. Dong, A. S. Patri, and T. Senthil, Physical Review Letters133, 206502 (2024)

    2024

  20. [20]

    Huang, X

    K. Huang, X. Li, S. Das Sarma, and F. Zhang, Physical Review B110, 10.1103/physrevb.110.115146 (2024)

    work page doi:10.1103/physrevb.110.115146 2024

  21. [21]

    Z. Guo, X. Lu, B. Xie, and J. Liu, Physical Review B 110, 075109 (2024)

    2024

  22. [22]

    Xie and S

    M. Xie and S. Das Sarma, Physical Review B109, 10.1103/physrevb.109.l241115 (2024)

    work page doi:10.1103/physrevb.109.l241115 2024

  23. [23]

    Song, C.-M

    X.-Y. Song, C.-M. Jian, L. Fu, and C. Xu, Physical Re- view B109, 115116 (2024)

    2024

  24. [24]

    J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Physical Review Letters68, 674 (1992)

    1992

  25. [25]

    J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Physical Review B50, 1760 (1994)

    1994

  26. [26]

    R. K. Goodall, R. J. Higgins, and J. P. Harrang, Physical Review B31, 6597 (1985)

    1985

  27. [27]

    S. C. Dultz and H. W. Jiang, Physical Review Letters 84, 4689 (2000)

    2000

  28. [28]

    A. A. Zibrov, C. Kometter, H. Zhou, E. M. Spanton, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Nature549, 360 (2017)

    2017

  29. [29]

    T. Han, J. P. Butler, S. Ye, Z. Hua, S. Dutta, Z. Had- jri, Z. Wu, J. Yang, J. Seo, P. Pattanakanvijit, E. Aitken, K. Watanabe, T. Taniguchi, P. Xiong, E. Zeldov, Z. Lu, R. Ashoori, and L. Ju, Evidence of Metallic Wigner Crys- tal in Rhombohedral Graphene (2026), version Number: 1

    2026

  30. [31]

    Streda, Journal of Physics C: Solid State Physics15, L1299 (1982)

    P. Streda, Journal of Physics C: Solid State Physics15, L1299 (1982). 7

    1982

  31. [32]

    Streda, Journal of Physics C: Solid State Physics15, L717 (1982)

    P. Streda, Journal of Physics C: Solid State Physics15, L717 (1982)

    1982

  32. [33]

    Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester, K. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo- Herrero, and A. Yacoby, Nature600, 439 (2021)

    2021

  33. [34]

    E. M. Spanton, A. A. Zibrov, H. Zhou, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Science 360, 62 (2018)

    2018

  34. [35]

    M. He, J. Cai, Y.-H. Zhang, Y. Liu, Y. Li, T. Taniguchi, K. Watanabe, D. H. Cobden, M. Yankowitz, and X. Xu, Nano Letters23, 11066 (2023)

    2023

  35. [36]

    Saito, J

    Y. Saito, J. Ge, L. Rademaker, K. Watanabe, T. Taniguchi, D. A. Abanin, and A. F. Young, Nature Physics17, 478 (2021)

    2021

  36. [37]

    L. Wang, Y. Gao, B. Wen, Z. Han, T. Taniguchi, K. Watanabe, M. Koshino, J. Hone, and C. R. Dean, Sci- ence350, 1231 (2015)

    2015

  37. [38]

    Waters, A

    D. Waters, A. Okounkova, R. Su, B. Zhou, J. Yao, K. Watanabe, T. Taniguchi, X. Xu, Y.-H. Zhang, J. Folk, and M. Yankowitz, Physical Review X15, 011045 (2025)

    2025

  38. [39]

    Assouline, T

    A. Assouline, T. Wang, H. Zhou, L. A. Cohen, F. Yang, R. Zhang, T. Taniguchi, K. Watanabe, R. S. Mong, M. P. Zaletel, and A. F. Young, Physical Review Letters132, 046603 (2024)

    2024

  39. [40]

    S. I. Dorozhkin, R. J. Haug, K. von Klitzing, and K. Ploog, Physical Review B51, 14729 (1995)

    1995

  40. [41]

    Hu, Y.-C

    Y. Hu, Y.-C. Tsui, M. He, U. Kamber, T. Wang, A. S. Mohammadi, K. Watanabe, T. Taniguchi, Z. Papi´ c, M. P. Zaletel, and A. Yazdani, Nature Physics21, 716 (2025)

    2025

  41. [42]

    Y. Choi, Y. Choi, M. Valentini, C. L. Patterson, L. F. W. Holleis, O. I. Sheekey, H. Stoyanov, X. Cheng, T. Taniguchi, K. Watanabe, and A. F. Young, Nature639, 342 (2025)

    2025

  42. [43]

    Polshyn, H

    H. Polshyn, H. Zhou, E. Spanton, T. Taniguchi, K. Watanabe, and A. Young, Physical Review Letters121, 226801 (2018)

    2018

  43. [44]

    R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Physical Review Letters70, 2944 (1993)

    1993

  44. [45]

    R. Xiao, F. Tasn´ adi, K. Koepernik, J. W. F. Venderbos, M. Richter, and M. Taut, Physical Review B84, 165404 (2011)

    2011

  45. [46]

    H. Park, W. Li, C. Hu, C. Beach, M. Gon¸ calves, J. F. Mendez-Valderrama, J. Herzog-Arbeitman, T. Taniguchi, K. Watanabe, D. Cobden, L. Fu, B. A. Bernevig, N. Reg- nault, J.-H. Chu, D. Xiao, and X. Xu, Observation of High-Temperature Dissipationless Fractional Chern Insu- lator (2025), version Number: 1

    2025

  46. [47]

    Redekop, C

    E. Redekop, C. Zhang, H. Park, J. Cai, E. Anderson, O. Sheekey, T. Arp, G. Babikyan, S. Salters, K. Watan- abe, T. Taniguchi, M. E. Huber, X. Xu, and A. F. Young, Nature635, 584 (2024)

    2024

  47. [48]

    A. S. Patri, Z. Dong, and T. Senthil, Physical Review B 110, 245115 (2024)

    2024

  48. [49]

    J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Physical Review Letters 133, 206503 (2024)

    2024

  49. [50]

    J. Jang, B. M. Hunt, L. N. Pfeiffer, K. W. West, and R. C. Ashoori, Nature Physics13, 340 (2017)

    2017

  50. [51]

    Y. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West, Physical Review Letters 91, 016801 (2003)

    2003

  51. [52]

    Lewis, Y

    R. Lewis, Y. Chen, L. Engel, D. Tsui, P. Ye, L. Pfeiffer, and K. West, Physica E: Low-dimensional Systems and Nanostructures22, 104 (2004)

    2004

  52. [53]

    H. Zhou, H. Polshyn, T. Taniguchi, K. Watanabe, and A. F. Young, Nature Physics16, 154 (2020)

    2020

  53. [54]

    M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Physical Review Letters82, 394 (1999)

    1999

  54. [55]

    M. M. Fogler, A. A. Koulakov, and B. I. Shklovskii, Phys- ical Review B54, 1853 (1996)

    1996

  55. [56]

    A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys- ical Review Letters76, 499 (1996)

    1996

  56. [57]

    J. P. Eisenstein, K. B. Cooper, L. N. Pfeiffer, and K. W. West, Physical Review Letters88, 076801 (2002)

    2002

  57. [58]

    R. M. Lewis, Y. P. Chen, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Physical Review B71, 081301 (2005)

    2005

  58. [59]

    N. Deng, A. Kumar, M. J. Manfra, L. N. Pfeiffer, K. W. West, and G. A. Cs´ athy, Physical Review Letters108, 086803 (2012)

    2012

  59. [60]

    S. Chen, R. Ribeiro-Palau, K. Yang, K. Watanabe, T. Taniguchi, J. Hone, M. O. Goerbig, and C. R. Dean, Physical Review Letters122, 026802 (2019)

    2019

  60. [61]

    Gr¨ uner, A

    G. Gr¨ uner, A. Zettl, W. G. Clark, and A. H. Thompson, Physical Review B23, 6813 (1981)

    1981

  61. [62]

    R. M. Fleming and C. C. Grimes, Physical Review Let- ters42, 1423 (1979)

    1979

  62. [63]

    Xiang, H

    Z. Xiang, H. Li, J. Xiao, M. H. Naik, Z. Ge, Z. He, S. Chen, J. Nie, S. Li, Y. Jiang, R. Sailus, R. Banerjee, T. Taniguchi, K. Watanabe, S. Tongay, S. G. Louie, M. F. Crommie, and F. Wang, Science388, 736 (2025)

    2025

  63. [64]

    Y.-C. Tsui, M. He, Y. Hu, E. Lake, T. Wang, K. Watan- abe, T. Taniguchi, M. P. Zaletel, and A. Yazdani, Nature 628, 287 (2024). 8 VI. METHODS A. Device Fabrication The graphene and hBN flakes were prepared by me- chanical exfoliation onto SiO 2/Si substrates. The rhom- bohedral domains of pentalayer graphene were identified and confirmed using IR camera, ...

    2024

  64. [66]

    S. C. De La Barrera, S. Aronson, Z. Zheng, K. Watanabe, T. Taniguchi, Q. Ma, P. Jarillo-Herrero, and R. Ashoori, Nature Physics18, 771 (2022)

    2022

  65. [67]

    Aronson, The Electronic Compressibility of Rhombohedral Graphene Multilayers, PhD Thesis, Massachusetts Institute of Technology (2025)

    Samuel H. Aronson, The Electronic Compressibility of Rhombohedral Graphene Multilayers, PhD Thesis, Massachusetts Institute of Technology (2025)

    2025

  66. [68]

    Spencer Louis Tomarken, Thermodynamic and Tunneling Measurements of van der Waals Heterostructures, PhD Thesis, Massachusetts Institute of Technology (2019)

    2019

  67. [69]

    #/"n (eV nm 2 ) 4.44.03.63.22.8 n (x10 11 cm -2 ) $ = 1/3 %# = 0.59 ± 0.02 meV 4 3 2 1 0 -1 6.56.05.55.0 n (x10 11 cm -2 ) !

    Raymond C. Ashoori, The Density of States in The Two-Dimensional Electron Gas and Quantum Dots, PhD Thesis, Cornell University (1991). 10 11 Extended Data 8006004002000-200-400-600D (mV/nm) 43210νmoiré 0123n (x1012 cm-2) 1.00.80.60.40.20.0CP/CG FIG. Extended 1: Full phase diagram as measured by penetration capacitance atf= 3.78 MHz andT= 320 mK. We identi...

    1991

  68. [70]

    (meV) 860855850845840835830 D (mV/nm) 10 5 10 6 f (Hz) # = 1/3 1.0 0.8 0.6 0.4 0.2 0.0 !

    Error Analysis of Compressibility and the Thermodynamic Gap Each of the values used to determine the compressibility carries some uncertainty that must be propagated through to the final gap value. First, we derive the uncertainty of ∂µ ∂n . We defineK= χ−χband χgap−χ . σ2 ∂µ ∂n =e 2 1 ctg +c bg " ∂K ∂χband 2 σ2 χband + ∂K ∂χgap 2 σ2 χgap + ∂K ∂χp 2 σ2 χp...

  69. [71]

    S. H. Aronson, T. Han, Z. Lu, Y. Yao, J. P. Butler, K. Watanabe, T. Taniguchi, L. Ju, and R. C. Ashoori, Physical Review X15, 031026 (2025)

    2025

  70. [72]

    Aronson, The Electronic Compressibility of Rhombohedral Graphene Multilayers, PhD Thesis, Massachusetts Institute of Technology (2025)

    Samuel H. Aronson, The Electronic Compressibility of Rhombohedral Graphene Multilayers, PhD Thesis, Massachusetts Institute of Technology (2025). 25 5 μm V FIG. SI 1: Picture of device and configuration of the four-terminal transport measurement on sample D1. The drain contact here is kept grounded in the refrigerator

    2025