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REVIEW 3 major objections 5 minor 1 cited by

Flat Band and Many-body Gap in Chirally Twisted Triple Bilayer Graphene

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Chirally twisted triple bilayer graphene shows flat bands at charge neutrality and a finite gap at zero electric field.

desk verdict First capacitance study of CTTBG confirms flat bands, but the zero-field gap's ferromagnetic attribution is under-supported; the paper deserves review. read the letter →

arxiv 2501.06852 v1 pith:7KVVANJV submitted 2025-01-12 cond-mat.mes-hall

Wenlu Lin , Wenxuan Wang , Shimin Cao , Miao Liang , Lili Zhao , Kenji Watanabe , Takashi Taniguchi , Jinhua Gao
show 3 more authors
Jianhao Chen Xiaobo Lu Yang Liu
This is my paper · ORCID
classification cond-mat.mes-hall
keywords chirallytwistedtriplebilayergrapheneflatbandsmoirésuperlatticecapacitancemeasurementcompressibilitymany-bodygapferromagnetismLandaulevelfan
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that chirally twisted triple bilayer graphene (CTTBG), a stack of three helically twisted Bernal bilayers, has flat bands at the charge neutral point and that a gap already exists between those flat bands when no perpendicular electric field is applied. The evidence comes from capacitance measurements read as a compressibility probe: unlike monolayer graphene, whose $V_G/n$ grows linearly with $1/\sqrt{n}$ from Dirac dispersion, CTTBG shows a density-independent $V_G/n$, the signature of a flat band. The authors extract gap sizes from the gate-voltage width of the compressibility minima and find an inter-flat-band gap $\Delta_0 \approx 10.8$ meV at zero displacement field. Because this value does not match the continuum-model single-particle gap, they attribute it to spontaneous ferromagnetism stabilized by Coulomb interaction. If the interpretation holds, CTTBG becomes a quantitative platform for studying interaction-driven phases in moiré flat bands.

What carries the argument

The central object is the differential capacitance $C = e\,\partial n/\partial V_G$ between the top gate and the graphene sample, which acts as a local compressibility probe: a dip in $C$ marks a region of low density of states, and the $V_G/n$ ratio plotted against $1/\sqrt{n}$ distinguishes Dirac dispersion from flat bands. The quantitative gap extraction uses the gate-voltage spacing $\Delta V_{TG}$ between the two conductance peaks that flank the compressibility minimum, converted to energy by $\Delta = \frac{2K}{1+K}\Delta V_{TG}$ when both gates are swept at fixed displacement field. The continuum-model single-particle calculation is the contrast object: agreement at large $D$ validates the model, while the discrepancy at $D=0$ isolates the many-body contribution. These elements together turn a capacitance trace into a band-structure measurement.

What would settle it

Measure the magnetization of the CTTBG flake at zero displacement field: if the ~10.8 meV capacitance gap is a ferromagnetic gap, a spontaneous magnetization should appear below the transition temperature and the gap should close upon warming, whereas a strain- or disorder-induced single-particle gap would persist. A complementary check is to measure the gap via activated transport or scanning tunneling spectroscopy on the same sample; if those probes find no gap while capacitance shows one, the gap is a charging gap in a domain percolation picture, not a bulk ferromagnetic gap.

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Extended reading notes

Core claim

At its core, the paper claims that the flat bands predicted for CTTBG are real and directly measurable, and that the many-body gap between them is finite even at $D = 0$. The constant $V_G/n$ observed across densities is presented as direct evidence of a diverging density of states, i.e., flat bands, and the width of the capacitance/conductance minimum at the charge neutral point yields a gap of about 10.8 meV. The computed single-particle gap curve is the key reference: the experimental gap at $D=0$ clearly deviates from it, which the authors take as evidence that the gap is not single-particle in origin. They assign it to spontaneous ferromagnetism under Coulomb interaction and support this by noting the minimum persists up to 10 T. They also measure the higher-energy moiré gap $\Delta_1$ at $\nu = \pm4$, whose closing with displacement field matches the theoretical model, validating the band parameters used in the single-particle comparison.

Load-bearing premise

The load-bearing premise is that the single-particle continuum calculation used as the reference curve is accurate enough, so the discrepancy at zero perpendicular electric field can only be caused by many-body physics rather than by lattice relaxation, strain, disorder, or an incorrect electrostatic conversion from gate-voltage width to energy.

Editorial extensions

If this is right

  • CTTBG is established as a tunable moiré system with isolated flat bands at charge neutrality, so interaction-driven phases such as correlated insulators or fractional quantum Hall states should be sought at integer and fractional fillings of those bands.
  • A finite gap at zero perpendicular electric field implies the ground state is already symmetry-broken without external bias, so the phase diagram of CTTBG should include a spontaneous ferromagnetic region around the charge neutral point.
  • The capacitance method yields quantitative values for both $\Delta_0$ and $\Delta_1$ that can be compared with continuum-model predictions, making it a general tool for extracting moiré band parameters in twisted multilayer systems.
  • The apparent absence of the zero-field gap in transport, together with its presence in capacitance, indicates that charging gaps need not produce a fully insulating transport response, which matters for interpreting transport silence in other moiré materials.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension would be to sweep an in-plane or out-of-plane magnetic field at $D=0$ and look for hysteresis in the capacitance or conductance: ferromagnetic domains should produce a history-dependent response, while a charge-ordered gap would not.
  • The same $D=0$ comparison between measured and single-particle gaps could be applied to other twistronic systems, such as magic-angle twisted bilayer graphene or helical trilayer graphene, to separate interaction gaps from single-particle gaps without assumptions about scattering rates.
  • If the flat bands are as isolated as the $\Delta_1$ measurements suggest, tuning the twist angle toward smaller values should enhance the interaction-to-kinetic ratio and may drive fractional quantum Hall or Wigner-crystal phases at low filling, which could be probed by the same compressibility features.
  • A temperature-dependent measurement of $V_G/n$ at $D=0$ could locate the ordering temperature of the proposed ferromagnet; above that temperature the gap width should collapse, separating the interaction scale from the moiré band parameters.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports capacitance measurements on a chirally twisted triple bilayer graphene (CTTBG) device and compares them with a monolayer graphene calibration. The authors measure the gate-voltage-to-density ratio VTG/n as a function of density; for CTTBG they find a density-independent VTG/n, which they interpret as evidence of flat bands. They also extract a charge-neutrality gap from the width of a capacitance/conductance minimum and study its dependence on displacement field D. A finite gap of about 10.8 meV is observed at D=0, which the authors attribute to spontaneous ferromagnetism stabilized by Coulomb interactions. In addition, they measure moiré gaps at filling factors ν=±4 and report agreement with a continuum-model calculation. The central claims are the existence of flat bands and the many-body origin of the zero-field gap.

Significance. If established, the observation of a finite gap at zero displacement field in CTTBG would be a notable experimental result, as it would point to interaction-driven physics in a moiré system where the single-particle band structure is not expected to open a gap at neutrality. The capacitance method itself is attractive: the monolayer-graphene calibration is careful, with carrier densities determined from quantum oscillations without fitting parameters, and the resulting Fermi velocities are consistent with literature values. The agreement between the measured moiré gap at ν=±4 and the theoretical values provides a useful internal consistency check of the model parameters. However, the two headline claims are not equally supported. The flat-band inference from a constant VTG/n is ambiguous, and the many-body attribution of the D=0 gap rests on a single unshown single-particle comparison, without independent magnetization or temperature-dependence data. The paper therefore reports interesting data but overinterprets the evidence.

major comments (3)
  1. [Section 3, Fig. 3(a)] The constant VTG/n in CTTBG is presented as evidence of flat bands, but the text itself acknowledges that a constant VTG/n gives a constant μ/n, which is equally consistent with a parabolic dispersion. The later statement that 'the constant VG/|n| strongly evidence[s] the existence of a group of flat bands' is therefore an overstatement. To support the flat-band claim, the authors should show that the data are inconsistent with a parabolic band over the measured density range, for example by fitting a parabolic dispersion and demonstrating a statistically significant deviation, or by presenting a complementary observable that directly reflects a diverging density of states at the CNP.
  2. [Section 4, Fig. 4(b)] The central attribution of the D=0 gap (~10.8 meV) to spontaneous ferromagnetism rests on the disagreement between the measured gap and the 'cal' single-particle curve in Fig. 4(b). The calculation underlying this curve is not presented: no Hamiltonian, parameter set, or treatment of lattice relaxation is given. At θ=1.7° the moiré period is about 8 nm, a regime where lattice relaxation is known to modify band structures and can open gaps at neutrality in twisted graphene systems. Without showing the single-particle calculation and testing whether relaxation or strain changes its prediction, the possibility that the observed gap is single-particle in origin is not excluded. The observed constancy of the gap width with magnetic field up to 10 T does not discriminate, because a single-particle gap would also be approximately field-independent. Please provide the calculation details and a relaxed-band calculation, or temper the many-body claim accordingly.
  3. [Section 3, gap extraction and device statistics] The conversion from the gate-voltage width ΔVTG to the energy gap Δ is stated with two different formulas (Δ = 2K/(1+K) ΔVTG and Δ = K/(1+K) ΔVTG) depending on the sweep mode, but no derivation is given. Because the gap magnitude is a central quantitative result, the electrostatic model (including the role of the quantum capacitance inside the gap and the two-gate lever arms) should be derived or explicitly referenced. In addition, all conclusions rest on a single CTTBG device, and no error bars or reproducibility data are provided for the reported gap values; the manuscript should at least state the measurement uncertainty and the number of devices studied.
minor comments (5)
  1. [Throughout] The notation is inconsistent: VTG, VFG, and VT G are used interchangeably, and 'VFG' is never defined. Please unify the notation.
  2. [Reference [25]] Reference [25] contains an unresolved placeholder '[ ? ]' and should be completed.
  3. [Fig. 2(b) inset and main text] The text states that the parasitic capacitance Cp is 'typically about 30 fF', but the inset of Fig. 2(b) reports Cp ≈ 68 fF. Please clarify which value applies to the MLG device and which to the CTTBG device.
  4. [Fig. 3(a) caption] The blue dashed line in Fig. 3(a) is described as the expected curve for a single-particle gap, but neither the caption nor the text gives its functional form or the parameters used. Please provide this information.
  5. [Section 4, transport absence of gap] The statement that no gap feature is observed in transport near D=0, attributed to a charging gap versus percolation, would benefit from a more quantitative explanation; as written, the reconciliation is hand-waving and leaves the physical status of the D=0 gap unclear.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the D=0 gap is an observed deviation from a single-particle benchmark, not a fitted or self-referential input.

full rationale

The derivation chain is not circular. The central experimental quantities—the constant VG/n for CTTBG and the finite gap at D=0—are obtained from capacitance and conductance measurements with carrier densities fixed by quantum oscillations via n=eBν/h, with no fit to the band-structure model. The single-particle benchmarks (blue dashed line in Fig. 3(a), red line in Fig. 4(b)) are external model predictions from the continuum approximation (ref [17], which is co-authored by M. Liang and J.-H. Gao); the paper's key observation is that the D=0 data deviate from that benchmark ('a clear finite gap is seen at D = 0'), so the many-body attribution is a falsifiable comparison, not a quantity manufactured from the model's inputs. No parameter of the theoretical curve is fitted to the CTTBG capacitance data, and the MLG vF analysis is an independent calibration. Self-citations (refs [12], [17], [20]) provide prior predictions, transport context, and the capacitance-bridge method, but the load-bearing claim—a finite gap where the single-particle model predicts none—does not reduce to those citations. The interpretation that the gap is ferromagnetic is underdetermined (no magnetization measurement; a relaxed single-particle band structure might open a comparable gap), but underdetermination is a correctness risk, not circularity. Therefore no prediction or first-principles result in the paper is equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central interpretation rests on the continuum model from refs [13,14,17] (two with present co-authors) for band structure and feature assignment, and on the assumption that the capacitance minimum width reflects a thermodynamic gap. No new free parameters are fit to the CTTBG data; the only fits are the MLG Fermi velocities used for calibration.

assumptions (5)
  • domain assumption Continuum model of CTTBG accurately captures the flat bands and gaps (refs [13,14,17]).
    Used to identify ν=±4 features as Δ1 and to compute the predicted Δ0(D) and Δ1(D) lines in Figs. 4(b) and 4(e).
  • standard math The carrier density is exactly n=eBν/h from Landau level minima.
    Stated in the text as having no fitting parameters; standard relation for a two-dimensional electron gas.
  • domain assumption Capacitance minima at CNP correspond to incompressible states and their width gives the thermodynamic gap via Δ=2K/(1+K)ΔVTG.
    Assumes ideal parallel-plate electrostatics and no inhomogeneous broadening; discussed in the measurement principle section.
  • ad hoc to paper The zero-field gap arises from spontaneous ferromagnetism rather than single-particle or disorder effects.
    Inferred from the mismatch with the single-particle line; the authors note transport shows no insulating behavior and Wigner crystal is called 'courageous' in footnote [30].
  • domain assumption The two twist angles in the CTTBG device are both 1.7 degrees.
    Stated in the text; the band structure and flat-band interpretation depend on this angle.

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Cite this review

Pith. "Pith review of Flat Band and Many-body Gap in Chirally Twisted Triple Bilayer Graphene." pith.science (2026). https://pith.science/paper/7KVVANJV

@misc{pith2026250106852,
  author       = {Pith},
  title        = {Pith review of: Flat Band and Many-body Gap in Chirally Twisted Triple Bilayer Graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KVVANJV}},
  note         = {Machine review of arXiv:2501.06852}
}
read the original abstract

We experimentally investigate the band structures of chirally twisted triple bilayer graphene. The new kind of moir\'e structure, formed by three pieces of helically stacked Bernal bilayer graphene, has flat bands at charge neutral point based on the continuum approximation. We experimentally confirm the existence of flat bands and directly acquire the gap in-between flat bands as well as between the flat bands and dispersive bands from the capacitance measurements. We discover a finite gap even at zero perpendicular electric field, possibly induced by the Coulomb interaction and ferromagnetism. Our quantitative study not only provides solid evidence for the flat-band and interesting physics, but also introduces a quantitative approach to explore phenomena of similar moir\'e systems.

Figures

Figures reproduced from arXiv: 2501.06852 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic diagram of our setup, which measures [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Capacitance data taken from MLG sample at B [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a)(c-d) The [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

Cited by 1 Pith paper

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

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Reference graph

Works this paper leans on

32 extracted references · 27 canonical work pages · cited by 1 Pith paper

  1. [25]

    Fortunately, this effect does not change our conclusion

    The C and G are highly entangled with each other [ ? ]. Fortunately, this effect does not change our conclusion

    work page

  2. [22]

    Emergent dirac gullies and gully-symmetry-breaking quantum hall states in aba trilayer graphene,

    work page

  3. [1]

    J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Com- pressibility of the two-dimensional electron gas: Measure - ments of the zero-field exchange energy and fractional quantum hall gap, Phys. Rev. B 50, 1760 (1994)

    work page 1994

  4. [2]

    E. M. Hajaj, O. Shtempluk, V. Kochetkov, A. Razin, and Y. E. Yaish, Chemical potential of inhomogeneous single-layer graphene, Phys. Rev. B 88, 045128 (2013)

    work page 2013

  5. [3]

    J. L. McChesney, A. Bostwick, T. Ohta, T. Seyller, K. Horn, J. Gonz´ alez, and E. Rotenberg, Extended van hove singularity and superconducting instability in doped graphene, Phys. Rev. Lett. 104, 136803 (2010)

    work page 2010

  6. [4]

    S. L. Tomarken, Y. Cao, A. Demir, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and R. C. Ashoori, Elec- tronic compressibility of magic-angle graphene superlat- 5 tices, Phys. Rev. Lett. 123, 046601 (2019)

    work page 2019

  7. [5]

    G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Cas- tro Neto, A. Reina, J. Kong, and E. Y. Andrei, Observa- tion of van hove singularities in twisted graphene layers, Nature Physics 6, 109 (2010)

    work page 2010

  8. [6]

    S. L. Tomarken, Y. Cao, A. Demir, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and R. C. Ashoori, Elec- tronic compressibility of magic-angle graphene superlat- tices, Phys. Rev. Lett. 123, 046601 (2019)

    work page 2019

Show all 32 references
  1. [7]

    Weiss, K

    D. Weiss, K. V. Klitzing, K. Ploog, and G. Weimann, Magnetoresistance oscillations in a two-dimensional elec - tron gas induced by a submicrometer periodic potential, Europhysics Letters 8, 179 (1989)

    1989

  2. [8]

    Drienovsky, J

    M. Drienovsky, J. Joachimsmeyer, A. Sandner, M.- H. Liu, T. Taniguchi, K. Watanabe, K. Richter, D. Weiss, and J. Eroms, Commensurability os- cillations in one-dimensional graphene superlattices, Phys. Rev. Lett. 121, 026806 (2018)

    2018

  3. [9]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moir´ e bands in twisted double-layer graphene, Proceedings of the National Academy of Sciences 108, 12233 (2011)

    2011

  4. [10]

    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, T. Taniguchi, A. Vishwanath, P. Jarillo-Herrero, and A. Yacoby, Unconventional sequence of correlated chern insulators in magic-angle twisted bilayer graphene...

    2021

  5. [11]

    S. D. Sarma and A. Pinczuk, Perspectives in quantum hall effects: Novel quantum liquids in low-dimensional semiconductor structures (1997)

    1997

  6. [12]

    W. Wang, G. Zhou, W. Lin, Z. Feng, Y. Wang, M. Liang, Z. Zhang, M. Wu, L. Liu, K. Watanabe, T. Taniguchi, W. Yang, G. Zhang, K. Liu, J. Gao, Y. Liu, X. C. Xie, Z. Song, and X. Lu, Correlated charge density wave insulators in chirally twisted triple bilayer graphene, Phys. Rev....

    2024

  7. [13]

    Devakul, P

    T. Devakul, P. J. Ledwith, L.-Q. Xia, A. Uri, S. C. de la Barrera, P. Jarillo-Herrero, and L. Fu, Magic-angle helical trilayer graphene, Science Advances 9, eadi6063 (2023)

    2023

  8. [14]

    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, T. Devakul, and P. Jarillo-Herrero, Helical trilayer graphene: a moir´ e platform for strongly-interacting topologi- cal bands, arXiv e-prints , arXiv:...

    2023 arXiv

  9. [15]

    R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin, and A. Yacoby, Broken-symmetry states in doubly gated suspended bilayer graphene, Science 330, 812 (2010)

    2010

  10. [16]

    W. Bao, K. Myhro, Z. Zhao, Z. Chen, W. Jang, L. Jing, F. Miao, H. Zhang, C. Dames, and C. N. Lau, In situ observation of electrostatic and ther- mal manipulation of suspended graphene membranes, Nano Letters 12, 5470 (2012)

    2012

  11. [17]

    Liang, M.-M

    M. Liang, M.-M. Xiao, Z. Ma, and J.-H. Gao, Moir´ e band structures of the double twisted few-layer graphene, Phys. Rev. B 105, 195422 (2022)

    2022

  12. [18]

    See Supplemental Material for additional methods, data, and analyses

  13. [19]

    For simplicity, we always define the gate bias VG and the graphene chemical potential µ in reference to the condi- tion when the graphene is tuned to its CNP throughout this manuscript, see the black dash-dotted line in Fig. 1(c)

  14. [20]

    L. Zhao, W. Lin, X. Fan, Y. Song, H. Lu, and Y. Liu, High precision, low excitation capacitance mea- surement methods from 10 mK to room temperature, Review of Scientific Instruments 93, 053910 (2022)

    2022

  15. [21]

    Hazeghi, J

    A. Hazeghi, J. A. Sulpizio, G. Diankov, D. Goldhaber- Gordon, and H. S. P. Wong, An integrated ca- pacitance bridge for high-resolution, wide temper- ature range quantum capacitance measurements, Review of Scientific Instruments 82, 053904 (2011)

    2011

  16. [23]

    Assouline, T

    A. Assouline, T. Wang, H. Zhou, L. A. Cohen, F. Yang, R. Zhang, T. Taniguchi, K. Watanabe, R. S. K. Mong, M. P. Zaletel, and A. F. Young, Energy gap of the even-denominator fractional quantum hall state in bi- layer graphene, Phys. Rev. Lett. 132, 046603 (2024)

    2024

  17. [24]

    B. E. Feldman, A. J. Levin, B. Krauss, D. A. Abanin, B. I. Halperin, J. H. Smet, and A. Yacoby, Fractional quantum hall phase transitions and four-flux states in graphene, Phys. Rev. Lett. 111, 076802 (2013)

    2013

  18. [26]

    S. Y. Zhou, G. H. Gweon, J. Graf, A. V. Fedorov, C. D. Spataru, R. D. Diehl, Y. Kopelevich, D. H. Lee, S. G. Louie, and A. Lanzara, First direct observation of dirac fermions in graphite, Nature Physics 2, 595 (2006)

    2006

  19. [27]

    Li and E

    G. Li and E. Y. Andrei, Observation of landau levels of dirac fermions in graphite, Nature Physics 3, 623 (2007)

    2007

  20. [28]

    G. Li, A. Luican, and E. Y. Andrei, Scanning tunneling spectroscopy of graphene on graphite, Phys. Rev. Lett. 102, 176804 (2009)

    2009

  21. [29]

    Bai, Y.-C

    K.-K. Bai, Y.-C. Wei, J.-B. Qiao, S.-Y. Li, L.-J. Yin, W. Yan, J.-C. Nie, and L. He, Detecting giant electron- hole asymmetry in a graphene monolayer generated by strain and charged-defect scattering via landau level spectroscopy, Phys. Rev. B 92, 121405 (2015)

    2015

  22. [30]

    Also, the localization gap can also be excluded because our trans- port measurement shows no insulating behavior; see Fig

    The collective pinning of electron Wigner crystal may also lead to a gap-feature, however, it is rather coura- geous to suggest this senario in our sample. Also, the localization gap can also be excluded because our trans- port measurement shows no insulating behavior; see Fig. S5

  23. [31]

    In Fig. 3(e), the electric field is kept to be constant, the gap size ∆ is tuned by two electrical gates simul- taneously, so that the ralation between ∆ and VT G is ∆ = 2K 1+K × VT G while if the gap size is only tuned by VT G, the ralation becomes ∆ = K 1+K × VT G

  24. [32]

    E. P. De Poortere, E. Tutuc, and M. Shayegan, Crit- ical resistance in the alas quantum hall ferromagnet, Phys. Rev. Lett. 91, 216802 (2003)

    2003

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