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Cascade of Even-Denominator Fractional Quantum Hall States in Mixed-Stacked Multilayer Graphene

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Pith's one-line read Mixed-stack pentalayer graphene hosts five half-filled fractional quantum Hall states, up to ν = -13/2, likely of Moore-Read type.

desk verdict Interesting but not yet proven: the even-denominator cascade in ABCBC pentalayer is a real novelty, yet the central claim relies on Rxx minima when the authors clearly know how to measure gaps. read the letter →

arxiv 2507.20695 v1 pith:UQSFLCOM submitted 2025-07-28 cond-mat.mes-hall cond-mat.str-el

Yating Sha , Kai Liu , Chenxin Jiang , Dan Ye , Shuhan Liu , Zhongxun Guo , Jingjing Gao , Ming Tian
show 10 more authors
Neng Wan Kenji Watanabe Takashi Taniguchi Bingbing Tong Guangtong Liu Li Lu Yuanbo Zhang Zhiwen Shi Zixiang Hu Guorui Chen
This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.str-el
keywords fractionalquantumHalleffecteven-denominatorFQHstatesMoore-ReadstateABCBC-stackedpentalayergraphenenon-AbeliananyonsLandaulevelcrossingscompositefermionsdisplacementfieldtuning
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

This paper reports a cascade of even-denominator fractional quantum Hall states—correlated two-dimensional electron fluids with quantized Hall response—at filling factors $\nu = -5/2$, $-7/2$, $-9/2$, $-11/2$ and $-13/2$ in a pentalayer graphene device with ABCBC stacking, a mixed-stacking arrangement of an ABC trilayer and an AB bilayer that had not been explored in this regime. The half-filled states live in the zeroth Landau level, the lowest Landau level in graphene's spectrum, reaching the highest half-fillings reported there, and are switched on and off by an electric displacement field that drives Landau level crossings. The authors identify two groups of half-filled states, each tied to a distinct orbital branch of the zeroth Landau level, and show that tuning the field moves the system between paired fractional quantum Hall states, magnetic Bloch states, and composite Fermi liquids. Numerical calculations for each state yield a sixfold quasi-degenerate ground state and a chiral graviton spectrum, which the paper takes as evidence that all five states are non-Abelian Moore-Read (Pfaffian or anti-Pfaffian) states. If correct, this makes mixed-stacked multilayer graphene a tunable platform for studying non-Abelian quasiparticles.

What carries the argument

The central object is the ABCBC-stacked pentalayer graphene itself: a non-centrosymmetric stack conceptually composed of an ABC trilayer and an AB bilayer, whose band structure mixes parabolic and cubic dispersions and has a small intrinsic gap. In a magnetic field this yields two distinct orbital branches within the zeroth Landau level, each with significant $N=1$ Landau level character, and their relative energies shift under a displacement field $D$, causing Landau level crossings. Those crossings are the control knob: when a crossing places one branch at half-filling, interaction effects stabilize a paired composite-fermion state; numerically this appears as a sixfold quasi-degenerate ground state and a characteristic chiral graviton spectral function (a collective-excitation spectrum that distinguishes Pfaffian from anti-Pfaffian order), the diagnostics used to assign Moore-Read type order. The same Landau level structure also generates the odd-denominator two-flux and four-flux composite-fermion sequences that flank the half-filled states and the Hofstadter minibands (fractal subbands from a periodic moiré potential) observed near crossings.

What would settle it

Perform cross-sectional scanning transmission electron microscopy on the measured pentalayer flake: if the stacking sequence is not ABCBC, the assignment of the two half-filled-state groups to two specific zeroth-Landau-level branches collapses, and with it the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that ABCBC-stacked pentalayer graphene hosts a cascade of even-denominator fractional quantum Hall states at $\nu = -5/2$, $-7/2$, $-9/2$, $-11/2$, and $-13/2$, and that these states are non-Abelian Moore-Read type states. In the data, the half-filled states appear as minima in the longitudinal resistance together with quantized Hall plateaus, while the surrounding odd-denominator composite-fermion sequences are suppressed. Two groups of states are resolved, one centered near $-5/2$ and one near $-9/2$, and both migrate to higher-index Landau levels as the displacement field is increased; the paper attributes this to two distinct orbital branches within the zeroth Landau level, each acquiring a large $N=1$ orbital component through Landau level mixing. Numerical calculations that include Coulomb interactions, Landau level mixing, and the displacement field reproduce half-filled states with sixfold ground-state degeneracy, the Moore-Read hallmark, and chiral graviton spectral weights that favor anti-Pfaffian order for $-5/2$, $-9/2$, and $-13/2$ and Pfaffian order for $-7/2$ and $-11/2$.

Load-bearing premise

The load-bearing premise is that the measured pentalayer flake really has ABCBC stacking: the paper notes in Extended Data Fig. 1 that the near-field optical characterization alone leaves the exact structure uncertain, and if the true stacking is different, the Landau level spectrum, the grouping of the half-filled states, and the crossing mechanism would all have to be reinterpreted.

Editorial extensions

If this is right

  • Half-filled non-Abelian states are not confined to the second Landau level of conventional semiconductor heterostructures: the zeroth Landau level of ABCBC pentalayer graphene sustains them up to $\nu = -13/2$, the highest half-filling reported in a zeroth Landau level.
  • A single device can be swept by displacement field through a paired fractional quantum Hall state, a magnetic Bloch state, and a composite Fermi liquid, giving access to continuous quasiparticle phase transitions in one sample.
  • Landau level crossings controlled by the displacement field provide an in-situ switch between odd-denominator composite-fermion sequences and even-denominator Moore-Read states.
  • The weak daughter-state signatures near $\nu = -5/2$ and $-9/2$, consistent with anti-Pfaffian order, offer a possible experimental handle on non-Abelian quasiparticles at these high half-fillings.

Reading between the lines

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

  • A thermal Hall measurement at $\nu = -5/2$ or $-13/2$, which the paper does not report, would distinguish the Pfaffian, anti-Pfaffian, and particle-hole Pfaffian candidates, since these states carry different half-integer thermal Hall conductances.
  • Because ABCBC is one member of a family of mixed-stacking multilayers, the same crossing-based mechanism could place even-denominator non-Abelian states at different fillings in other stacks, such as ABCB tetralayer or ABCBA hexalayer graphene.
  • The observed difference in sharpness between the composite-Fermi-liquid-to-Bloch and paired-FQH-to-Bloch transitions could be tested by finite-temperature scaling collapse; a topological transition should show universal critical scaling distinct from a Fermi-surface reconstruction.
  • If gate leakage can be reduced, continuing the cascade toward $\nu = -15/2$ is a direct experimental target that would test whether the sequence is limited by the sample or by the Landau level structure.
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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

4 major / 5 minor

Summary. The manuscript reports a high-field magnetotransport study of dual-gated ABCBC-stacked pentalayer graphene. At B = 18 T and T = 14 mK, the authors observe Rxx minima at the half fillings ν = -5/2, -7/2, -9/2, -11/2, and -13/2, which they interpret as a cascade of even-denominator fractional quantum Hall (FQH) states emerging from two distinct intra-zeroth-Landau-level manifolds. The same data show a dense sequence of odd-denominator Jain states with quantized Hall conductivity and thermally activated gaps, from which an effective composite-fermion mass mCF = 0.785 me is extracted. By tuning displacement field and magnetic field, the manuscript identifies Landau-level-crossing-driven transitions between the even-denominator states, Hofstadter/magnetic-Bloch features, and composite-Fermi-liquid behavior. Numerical calculations summarized in the text report sixfold ground-state degeneracies and chiral graviton spectral weights for all five half-filled states, interpreted as evidence for Moore-Read-type (Pfaffian or anti-Pfaffian) order.

Significance. If the central claim is established, this would be a notable advance: the highest-filling half-filled FQH states reported in the zeroth Landau level, in a new mixed-stacking multilayer graphene platform with strong displacement-field tunability. The odd-denominator characterization is a clear strength: the Jain sequences show quantized σxy, activated gaps, a linear gap-versus-Beff trend, and a reasonable CF mass, and the reproducibility in a second device and at reversed magnetic field supports the intrinsic nature of the features. The numerical claim of Moore-Read-type order for all five fillings is interesting but is presented only in summary form. The paper is therefore of substantial interest to the quantum Hall and van der Waals heterostructure communities, provided the experimental identification of the even-denominator states as incompressible FQH states is strengthened.

major comments (4)
  1. [§3, Fig. 3] The central claim that the Rxx minima at ν = -5/2, -7/2, -9/2, -11/2, and -13/2 are even-denominator fractional quantum Hall states is not yet supported by the presented evidence. For the odd-denominator states the manuscript reports quantized σxy and thermally activated gaps (Fig. 2), but for the even-denominator states it only describes 'pronounced Rxx minima' and suppression of neighboring Jain states, with no statement of quantized σxy at (e^2/h)ν and no activation-gap data. Since these minima sit in a region of Landau-level crossings where the manuscript itself invokes magnetic-Bloch and composite-Fermi-liquid behavior (Fig. 4), an Rxx dip at half-filling is not by itself diagnostic of an incompressible paired state. Please provide quantized Hall plateaus, activated gaps, or equivalent thermodynamic signatures for at least the strongest of these states (for example, ν = -9/2 at D = 0.24 V/nm, where the Rxx minimum approaches zero in Extended Data Fig. 6), or explicitly downgrade the claim to candidate even-denominator states.
  2. [Methods and Extended Data Fig. 1] The stacking assignment as ABCBC-5LG is load-bearing: the two groups of half-filled states are assigned to distinct intra-ZLL manifolds that exist only for this specific mixed-stacking order. However, the manuscript's own Methods and Extended Data Fig. 1 state that the near-field optical identification is tentative ('the exact structure remains uncertain from this characterization alone'). If the stacking order is misidentified, the LL spectrum, the crossing pattern, and the interpretation of the two groups all change. Please add independent corroboration, ideally a quantitative comparison of the measured Landau fan and LL-crossing evolution with the calculated LL spectrum of ABCBC-5LG, or another stacking-sensitive measurement with unambiguous contrast.
  3. [§3, numerical calculations] The numerical evidence for Moore-Read-type order is not checkable from the manuscript as written. The model Hamiltonian, the treatment of LL mixing, the displacement-field values, the system sizes, and the computed sixfold degeneracies and chiral graviton spectra are only summarized and deferred to a supplementary file that is not included in the preprint. Because the experimental data alone do not fix the topological order, the numerical section needs to be self-contained: at minimum, show the low-energy spectrum for each relevant filling and define the chiral graviton spectral function used to distinguish Pfaffian from anti-Pfaffian order.
  4. [§4, Fig. 4] The abstract and text claim 'continuous quasiparticle phase transitions' between paired FQH states, magnetic Bloch states, and composite Fermi liquids, but the evidence in Fig. 4 consists of qualitative Rxx linecuts and a statement that one transition is sharper than another. No Hall conductivity, compressibility, or scaling analysis is presented, so the transitions could be crossovers or weakly first-order transitions rather than continuous quantum phase transitions. Please either provide quantitative evidence for criticality or describe these as field-tuned transitions without the 'continuous' designation.
minor comments (5)
  1. [§3, first paragraph] There is an internal inconsistency in the grouping of the half-filled states: the text first lists 'ν = -5/2, -7/2 and -2/9' (presumably a typo for -9/2) in the purple group, and then lists '-9/2' in both the purple and the red groups. Please correct the fractions and define the two groups non-overlappingly.
  2. [References] Reference 49 (Halperin, Lee, and Read) is cited both for the composite-Fermi-liquid theory and for the 'daughter states' of the half-filled states; the daughter states require a separate citation (for example, Read-Rezayi or Levin-Halperin) so that the reader can locate the relevant prediction.
  3. [Fig. 2e-f] Please specify the fitting range and the number of states used in the linear fit Δ = ℏeBeff/mCF, and state whether the quoted mCF = 0.785 me changes if the two states closest to half-filling are excluded.
  4. [Abstract and §3] The phrase 'two distinct intra-ZLL' is not defined in the main text; please define the zeroth-Landau-level manifold in this system and explain what 'intra-ZLL' index identifies each of the two groups.
  5. [Methods] The equations for the displacement field contain corrupted or undefined symbols (for example, D = (D￿ + D￿)/2 and V￿￿); please typeset them properly and define all offset voltages and dielectric constants.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the even-denominator claim rests on independent transport data and external numerical benchmarks; self-citations are background only.

full rationale

The paper's load-bearing derivation chain is experimental observation first, numerical interpretation second. The even-denominator states are identified from Rxx minima and D/B-tuned evolution (Fig. 1d, Fig. 3), not from any parameter fitted in the paper. The effective CF mass mCF = 0.785 me is extracted from activation gaps of odd-denominator Jain states (Fig. 2) and is never used to force or predict the half-filled states; it is not an input to the even-denominator identification. The numerical assignment to Moore-Read type order rests on sixfold quasi-degeneracy and chiral graviton spectra, which are external theoretical benchmarks calculated from a model that includes Coulomb interactions, LL mixing, and D; the paper does not fit these calculations to the observed Rxx minima. Self-citations (refs. 36, 38-42) supply background on mixed-stacking transport and related graphene platforms and are not used to justify the central result. The acknowledged stacking-order uncertainty (Extended Data Fig. 1: 'the exact structure remains uncertain from this characterization alone') is an experimental limitation that could affect the LL-level interpretation, but it is not circular reasoning and does not reduce any prediction to its inputs. No equation in the paper defines the observed half-filled states in terms of a fitted quantity or a cited claim; no self-citation is load-bearing. The derivation is self-contained against external benchmarks, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the experimental identification of the material and the transport signatures; no new entities are introduced. The only fitted parameter (mCF) is standard and does not enter the even-denominator analysis. The numerical part relies on established theoretical markers for Moore-Read states.

free parameters (1)
  • Composite fermion effective mass (mCF) = 0.785 me
    Extracted from a linear fit of activation gaps Δ = ℏeBeff/mCF for odd-denominator Jain states; this is a standard characterization parameter and is not used to derive the even-denominator claim.
assumptions (4)
  • domain assumption The sample is ABCBC-stacked pentalayer graphene.
    The stacking is identified by SNOM contrast; the authors acknowledge uncertainty in Extended Data Fig. 1, yet all subsequent Landau level assignments depend on this stacking order.
  • domain assumption The Rxx minima at half-fillings correspond to incompressible fractional quantum Hall states.
    The text reports pronounced Rxx minima and suppressed Jain states, but does not explicitly report quantized Hall plateaus or activation gaps for the even-denominator states.
  • domain assumption Landau level crossings, inferred from enhanced Rxx at integer fillings, are the mechanism that stabilizes the even-denominator states.
    The paper attributes the appearance and disappearance of half-filled states to LL crossings, but this is a post-hoc interpretation of the transport data.
  • standard math The Moore-Read (Pfaffian) state has a sixfold ground-state degeneracy and a characteristic chiral graviton spectrum.
    These are established theoretical fingerprints cited from prior literature; the paper uses them to assign the nature of the observed states.

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Pith. "Pith review of Cascade of Even-Denominator Fractional Quantum Hall States in Mixed-Stacked Multilayer Graphene." pith.science (2026). https://pith.science/paper/UQSFLCOM

@misc{pith2026250720695,
  author       = {Pith},
  title        = {Pith review of: Cascade of Even-Denominator Fractional Quantum Hall States in Mixed-Stacked Multilayer Graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UQSFLCOM}},
  note         = {Machine review of arXiv:2507.20695}
}
abstract

The fractional quantum Hall effect (FQHE), particularly at half-filling of Landau levels, provides a unique window into topological phases hosting non-Abelian excitations. However, experimental platforms simultaneously offering large energy gaps, delicate tunability, and robust non-Abelian signatures remain scarce. Here, we report the observation of a cascade of even-denominator FQH states at filling factors ${\nu}$ = ${-5/2}$, ${-7/2}$, ${-9/2}$, ${-11/2}$, and ${-13/2}$, alongside numerous odd-denominator states in mixed-stacked pentalayer graphene, a previously unexplored system characterized by intertwined quadratic and cubic band dispersions. These even-denominator states, representing the highest filling half-filled states reported so far in the zeroth Landau level (ZLL), emerge from two distinct intra-ZLL and exhibit unprecedented displacement field tunability driven by LL crossings in the hybridized multiband structure. At half fillings, continuous quasiparticle phase transitions between paired FQH states, magnetic Bloch states, and composite Fermi liquids are clearly identified upon tuning external fields. Numerical calculations, revealing characteristic sixfold ground-state degeneracy and chiral graviton spectral analysis, suggest the observed even-denominator FQH states belong to the non-Abelian Moore-Read type. These results establish mixed-stacked multilayer graphene as a rich and versatile crystalline platform for exploring tunable correlated topological phases.

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