Controlling particle-hole symmetry of fractional quantum hall states in trilayer graphene

Aveek bid; Harsimran Singh; Kenji Watanabe; Manish Jain; Rajdeep Sensarma; Simrandeep Kaur; Takashi Taniguchi; Unmesh Ghorai

REVIEW 2 major objections 5 minor 54 references

Controlling particle-hole symmetry of fractional quantum hall states in trilayer graphene

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In ABA trilayer graphene, a displacement field near a Landau-level crossing breaks particle-hole symmetry of fractional quantum Hall states by enhancing Landau-level mixing and three-body interactions.

desk verdict A careful transport study showing controlled, displacement-field-induced particle-hole symmetry breaking in ABA trilayer graphene FQH states near a Landau-level crossing; the empirical correlation is solid, the causal mechanism attribution is the soft spot. read the letter →

arxiv 2411.18910 v2 pith:FJSPVRPP submitted 2024-11-28 cond-mat.mes-hall cond-mat.str-el

Simrandeep Kaur , Harsimran Singh , Kenji Watanabe , Takashi Taniguchi , Unmesh Ghorai , Manish Jain , Rajdeep Sensarma , Aveek Bid This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.str-el

keywords fractionalquantumHalleffectparticle-holesymmetryABAtrilayergrapheneLandaulevelmixingdisplacementfieldthree-bodyinteractionscompositefermionsactivationenergygap

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 transport measurements on Bernal-stacked trilayer graphene showing that particle-hole symmetry of fractional quantum Hall states around half filling is exact when the lowest monolayer-like Landau level is protected from mixing by the lattice mirror symmetry, and that this symmetry can be switched off by an applied displacement field. At small displacement field, the odd-denominator fractional states and their hole conjugates have equal activation gaps, effective masses, Landé $g$-factors, and disorder broadening. In the narrow displacement-field window where the monolayer-like and bilayer-like Landau levels cross, the measured integer quantum Hall gap falls by a factor of 2.5, the Landau-level mixing parameter $\eta$ rises to a peak, and one conjugate fractional state collapses while the other survives. The authors conclude that inter-band Landau-level mixing enhances $\eta$ and activates three-body interactions, which explicitly break particle-hole symmetry; if correct, this makes trilayer graphene the first platform where particle-hole symmetry breaking in the fractional quantum Hall regime can be controlled externally.

What carries the argument

The load-bearing object is the field-tunable crossing between the monolayer-like Landau level $\mathrm{LL}^{0+}_M$ and the bilayer-like Landau level $\mathrm{LL}^{2+}_B$ in ABA trilayer graphene. In the pristine mirror-symmetric lattice the two bands cannot mix, so the FQH states in $\mathrm{LL}^{0+}_M$ are effectively single-Landau-level and particle-hole symmetric. Applying a displacement field breaks the mirror symmetry and, in the window $0.82 < |D| < 0.86$ V/nm, brings the two levels close enough to hybridize. The paper quantifies the resulting mixing through the Landau-level mixing parameter $\eta = E_c/\Delta E_{3,5}$, estimated from the measured integer activation gap; the growth of $\eta$ at the crossing is the proposed trigger that activates three-body interactions and breaks particle-hole symmetry.

What would settle it

Measure the single-particle spacing between $\mathrm{LL}^{0+}_M$ and $\mathrm{LL}^{2+}_B$ by a method that does not rely on the transport activation gap, such as magneto-capacitance or inter-Landau-level tunneling, across the displacement-field window $|D| = 0.82$–$0.86$ V/nm. If the true spacing does not dip by the factor of about 2.5 seen in $\Delta E_{3,5}$, or if the dip is an interaction-induced avoided crossing, then the enhanced-$\eta$ mechanism would not be the cause of the observed collapse of the hole-conjugate fractional state.

Watch

Extended reading notes

Core claim

The central claim is that in ABA trilayer graphene the particle-hole symmetry of fractional quantum Hall states about half filling is exact at low displacement fields and can be broken controllably by a displacement field that brings the monolayer-like $\mathrm{LL}^{0+}_M$ and bilayer-like $\mathrm{LL}^{2+}_B$ Landau levels together. Pristine TLG hosts FQH states in $\mathrm{LL}^{0+}_M$ whose activation gaps, effective CF mass parameter, effective $g$-factor, and disorder broadening match those of their hole conjugates; this symmetry is protected by the lattice mirror symmetry that forbids Landau-level mixing. For $|D|$ between 0.82 and 0.86 V/nm, the Landau levels cross, the integer gap $\Delta E_{3,5}$ drops by a factor of 2.5, $\eta = E_c/\Delta E_{3,5}$ peaks, and the hole-conjugate state $8/3$ disappears while its partner $7/3$ remains. The paper argues this is extrinsic particle-hole symmetry breaking: virtual scattering between $\mathrm{LL}^{0+}_M$ and $\mathrm{LL}^{2+}_B$ enhances Landau-level mixing and activates three-body interactions, which are known to destabilize conventional FQH states. This is presented as fundamentally different from the intrinsic, interaction-driven symmetry breaking seen in the lowest Landau levels of single-layer and bilayer graphene.

Load-bearing premise

The causal attribution rests on assuming that the measured activation gap of the integer quantum Hall state is a faithful measure of the single-particle spacing between the two Landau levels; if interactions substantially lift the degeneracy at the crossing, the inferred peak in $\eta$ would overstate the actual Landau-level mixing.

Editorial extensions

If this is right

At low displacement field, odd-denominator FQH states in $\mathrm{LL}^{0+}_M$ and their hole conjugates have equal activation gaps, effective masses, $g$-factors, and disorder broadening, directly confirming particle-hole symmetry.
In the crossing window $|D| = 0.82$–$0.86$ V/nm, the integer gap $\Delta E_{3,5}$ drops by a factor of 2.5, $\eta$ peaks, and one conjugate FQH state (e.g., $\nu=8/3$) collapses while its partner ($\nu=7/3$) survives.
The displacement-field range of particle-hole asymmetry tracks the theoretically predicted Landau-level crossing as the magnetic field is varied, indicating a causal link between inter-band mixing and symmetry breaking.
The symmetry breaking is extrinsic, driven by enhanced Landau-level mixing and three-body interactions, and is distinct from the intrinsic, interaction-driven breaking reported in single-layer and bilayer graphene.
Displacement field $D$ therefore serves as a continuous external control knob to switch particle-hole symmetry on and off in the fractional quantum Hall regime.

Reading between the lines

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

If three-body interactions are indeed the active symmetry-breaking channel, states whose stability depends on three-body physics (for example candidate non-Abelian phases) should be most affected near the crossing; searching for their appearance or destruction in the same $D$ window would test this mechanism.
The same field-tunable inter-Landau-level spacing is available in other multiband graphene systems with tunable band structure, so the extrinsic route to particle-hole symmetry breaking may generalize beyond ABA trilayer graphene.
A direct, model-independent measure of $\eta$ (for example through Landau-level spectroscopy) would convert the inferred peak in $\eta$ from a transport proxy into a quantitative input for theories of three-body interaction effects.
Because the collapse is selective (the particle state survives while the hole state vanishes), the asymmetry could be used as a sensitive probe of the sign and magnitude of three-body interaction terms, which are usually hard to isolate.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth your time. The key new thing: in the LL0+_M level of ABA TLG, the 7/3 and 8/3 FQH states have essentially identical activation gaps at low D, and when the displacement field pushes LL0+_M and LL2+_B through a crossing window (0.82–0.86 V/nm), the hole-conjugate 8/3 collapses while 7/3 survives. That is a clean, controlled violation of FQH particle-hole symmetry, and it is not in the earlier SLG/BLG or TLG work. The authors also show the same behavior for 13/3 vs 14/3, and the B-dependence of the crossing field tracks the SWMC tight-binding calculation with literature parameters (Fig. S6). Multiple devices, activated transport, and the Supplementary Table of which states die at which crossings all support the observation. Good paper in that sense.

The soft spots are about mechanism, not observation. Their headline mechanism is that D shrinks the inter-LL spacing, raises eta = Ec/ΔE3,5, and thereby turns on three-body terms that break PHS. But eta is estimated from the measured activation gap of the ν = 3 and 5 integer states, and the authors themselves note that interactions will lift the accidental crossing degeneracy, leaving a reduced but finite gap. So the measured gap dip is partly a many-body effect, and the eta enhancement is to some degree a restatement of that dip. The selective collapse of 8/3 is direct evidence of PHS breaking; the attribution to enhanced LL mixing plus three-body interactions is the most plausible option but is not uniquely pinned down by this proxy. Also, the “completely destabilized” phrasing oversells it: the breakdown is in a narrow D window and one conjugate survives.

Two minor things: they admit the 1/3 and 2/3 gaps fit linear-B and sqrt-B equally well, so the g-factor and effective mass extraction for those states is not overinterpreted; and the table contains one apparent typo (–11/3 listed for two different crossings), easy to fix.

Bottom line: the empirical result deserves a serious referee and a cite. I would ask the authors to soften the causal claim, quantify how much of the gap dip could be interaction-induced, and check the typo. The paper is not fatally flawed. For the record, the LL-crossing benchmark is not fitted to the symmetry-breaking data, so the core correlation stands.

Referee Report

2 major / 5 minor

Summary. The manuscript reports transport measurements on ABA-stacked trilayer graphene (TLG) showing that fractional quantum Hall (FQH) states around half-filling of the monolayer-like zeroth Landau level (LL0+_M) are particle-hole symmetric at small displacement field D, and that this symmetry is broken in a narrow window of D where LL0+_M crosses the bilayer-like LL2+_B level. The authors attribute the symmetry breaking to enhanced Landau level mixing (increased eta = E_C/Delta E_{3,5}) and the activation of three-body interactions, and argue that this constitutes an extrinsic, controlled violation of particle-hole symmetry distinct from intrinsic interaction-driven breaking in single-layer and bilayer graphene.

Significance. If the result holds, this is the first demonstration of controlled, extrinsic particle-hole symmetry violation in the fractional quantum Hall regime, which would be a notable advance. The paper brings multiple devices (three), temperature-dependent activation measurements, and an external tight-binding calculation with literature parameters; the agreement between the theoretically predicted and experimentally observed crossing fields (Fig. S6) is a strong independent check. The direct observation that one conjugate FQH state (e.g., 8/3) collapses while the other (7/3) survives in the same D window is compelling evidence of PHS breaking, independent of the specific mechanism. The central causal attribution, however, relies on a proxy whose validity is not fully established.

major comments (2)

[Main text, Fig. 4(e-h)] The estimate eta = E_C/Delta E_{3,5} is obtained by identifying the measured activation gap of the nu = 3 and 5 integer quantum Hall states with the single-particle inter-LL spacing. In the same D window, the non-interacting calculation predicts a crossing, and the authors state that 'in reality, any interaction will lift this accidental degeneracy, leading to a reduced but finite activation gap.' The measured Delta E_{3,5} is therefore a many-body gap, and the peak in eta(D) shown in Fig. 4(g-h) may overstate the enhancement of Landau-level mixing. Because the headline conclusion attributes the PHS breaking to enhanced eta and three-body interactions, the paper should either provide a quantitative estimate of the interaction-induced contribution to Delta E_{3,5} (for example, by comparing the data across multiple B fields where the crossing is avoided) or explicitly present the eta enhancement as a qualitative indicator rather than a quantitative measure, with the causal mechanism framed as a plausible interpretation. This issue is load-bearing for the central causal claim.
[Abstract and Discussion] The paper asserts that the observed PHS violation arises from enhanced Landau-level mixing and the activation of three-body interactions, which 'explicitly break the PHS of FQHs.' While the data establish a correlation between the D window of PHS breaking and the LL-crossing window, the specific role of three-body interactions is not directly evidenced. The selective collapse of 8/3 (while 7/3 survives) is direct evidence of PHS breaking, but it does not by itself discriminate between the two proposed mechanisms (enhanced eta and three-body terms). To support the mechanism, the authors should compare the measured D-dependence of the gaps with a theoretical model that includes three-body interactions, or at least provide a calculation of the expected PHS asymmetry from eta alone. Without this, the statement that both factors are responsible goes beyond what the data demonstrate.

minor comments (5)

[Abstract] The claim that 'conventional FQHs are completely destabilized' is an overgeneralization; the data show that only one member of each conjugate pair (e.g., 8/3 but not 7/3) collapses in the crossing window.
[Supplementary Information, Device characterization] The mobility of device 2 is written as '11,00,000 cm2V-1s-1'; this should be 1,100,000 cm2V-1s-1.
[Supplementary Table 1] The row for nu between -4 and -5 lists the most affected state as -11/3, but the text and Fig. S8(d) identify -14/3 as the affected hole-conjugate state; this appears to be a typo.
[Conclusion] The statement that particle-hole symmetry requires 'the cyclotron energy is significantly greater than the interaction strength' is imprecise: PHS in a single Landau level holds for arbitrary interaction strength as long as LL mixing is negligible. The condition should instead be phrased as the validity of single-LL projection.
[Introduction and Main text, Fig. 4(e-h)] The definition of eta changes from eta = E_C/E_cyc in the Introduction to eta = E_C/Delta E_{3,5} in the results; the relation between the two should be clarified, noting that Delta E_{3,5} serves as a proxy for the cyclotron gap.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PHS-violation window is anchored by an independent tight-binding model with literature parameters, and the η diagnostic is a derived proxy rather than a self-referential prediction.

full rationale

The paper's central derivation chain is not circular. The small-D particle-hole symmetry is established by direct transport measurements of activation gaps, g-factors, effective masses, and disorder broadening; these are independent observables, not quantities fitted to the PHS-breaking claim. The key prediction — that FQH PHS is violated in a specific D window around the LL0+_M / LL2+_B crossing — comes from a Slonczewski-Weiss-McClure tight-binding calculation using fixed literature parameters (γ0 = 3.1 eV, γ1 = 0.39 eV, etc., from Zibrov et al.), with no fitting to the present FQH data. The agreement between Dtheory and Dexperiment across magnetic fields (Supplementary Fig. S6) provides an external benchmark. The η = Ec/ΔE3,5 diagnostic is a derived quantity whose peak in the gray-shaded region is definitionally tied to the measured dip in the integer-QH activation gap; however, the paper uses it only as corroborating evidence, and the PHS breaking itself is separately observed through the collapse of one conjugate FQH state (e.g., 8/3) while the other (7/3) survives. The text explicitly acknowledges that interactions lift the accidental degeneracy, so the measured gap is a many-body gap; this is a measurement-interpretation caveat, not a circular reduction. Load-bearing support is not carried by self-citations: the only same-author reference ([9], Kaur et al.) is cited for the standard composite-fermion mapping, and no uniqueness theorem or fitted ansatz is imported from the authors' prior work. Therefore the central conclusion retains independent content and the circularity score is 0.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard FQH projection theory, a literature SWMC tight-binding model, and the assumption that measured activation gaps faithfully represent single-particle LL spacings at the crossing. The parameters used in the gap models are fitted to the data. No new particles, forces, or conserved quantities are introduced; three-body interactions are an existing theoretical construct.

free parameters (3)

CF effective mass parameter alpha = 0.077 to 0.27, state dependent
Fitted from Delta E versus Beff using Delta = hbar e Beff / m_eff - Gamma; equality for conjugate states is used as evidence of PHS (Tables S3-S4).
Effective Lande g-factor g_eff = 2.65 to 3.93
Fitted from Delta E versus Beff using the Zeeman form Delta = 0.5 mu_B g (2p+1) Beff - Gamma for 1/3 and 2/3 states; values are model-dependent because linear-B and sqrt-B fits both work.
Disorder broadening Gamma = 2.85 to 22 K depending on state and fit form
Extracted as an intercept in gap fits; its equality across conjugate states is part of the PHS evidence, but the value depends on the assumed B-dependence of the gap.

assumptions (7)

domain assumption SWMC tight-binding parameters from Zibrov et al. (gamma0 = 3.1 eV, gamma1 = 0.39 eV, gamma2 = -0.005 eV, gamma3 = 0.275 eV, gamma4 = 0.041 eV, gamma5 = 0.005 eV, delta = 0.0108 eV, Delta2 = 0.003 eV) describe the TLG Landau levels.
Used in Figs. 3(c-d) and S4 to compute the LL crossings that are matched to the PHS-breaking D windows.
domain assumption Pristine ABA TLG has a lattice mirror symmetry that decouples monolayer-like and bilayer-like bands; D breaks this symmetry.
Central to the claim that D-induced hybridization is the extrinsic PHS-breaking mechanism.
domain assumption A single Landau level with small mixing parameter eta yields PH-symmetric FQH Hamiltonians.
Standard FQH theory used to justify PHS at small D; cited from refs. [13,17,18].
domain assumption Three-body interactions explicitly break PH symmetry, and LL mixing enhances their role.
Invoked to connect enhanced eta to PHS violation; taken from refs. [13,14,19,26], not derived for this TLG system here.
domain assumption Activation transport follows Rxx proportional to exp(-Delta/2 k_B T), and activation gaps obey the CF formulas Delta = hbar e Beff / m_eff - Gamma or Delta = 0.5 mu_B g (2p+1) Beff - Gamma.
Used to extract all gaps, effective masses, g-factors, and broadening in Figs. 2 and S12.
domain assumption The LL0+_M wavefunction is single-component with negligible valley-isospin contribution.
Used to rule out valley-isospin transitions as the PHS-breaking mechanism; taken from refs. [33,54].
domain assumption At the LL crossing, interactions lift the accidental degeneracy into an avoided crossing with a finite gap.
The paper uses this to interpret the measured non-zero Delta E3,5 as the denominator for eta at the crossing.

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Pith. "Pith review of Controlling particle-hole symmetry of fractional quantum hall states in trilayer graphene." pith.science (2026). https://pith.science/paper/FJSPVRPP

@misc{pith2026241118910,
  author       = {Pith},
  title        = {Pith review of: Controlling particle-hole symmetry of fractional quantum hall states in trilayer graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJSPVRPP}},
  note         = {Machine review of arXiv:2411.18910}
}

abstract

We present a detailed experimental study of the particle-hole symmetry (PHS) of the fractional quantum Hall (FQH) states about half filling in a multiband system. Specifically, we focus on the lowest Landau level of the monolayer-like band of Bernal stacked trilayer graphene (TLG). In pristine TLG, the excitation energy gaps, Land\'e g-factor, effective mass, and disorder broadening of the odd-denominator FQH states are identical to their hole-conjugate counterpart. This precise PH symmetry stems from the lattice mirror symmetry that precludes Landau-level mixing. Introducing a non-zero displacement field $D$ disrupts this mirror symmetry, facilitating the hybridization between the monolayer-like and bilayer-like Landau levels. This inter-band coupling enhances the Landau level mixing factor $\eta$ and activates three-body interactions -- both of which explicitly break the PHS of FQHs. As a result, conventional FQHs are completely destabilized, offering a route to engineer symmetry breaking of FQHs in a controlled way. We establish that the PHS breaking in TLG is of extrinsic origin and is fundamentally distinct from the intrinsic, interaction-driven symmetry breaking observed in the lowest Landau levels of single-layer and bilayer graphene.

Figures

Figures reproduced from arXiv: 2411.18910 by the authors.

**Figure 1.** Device characterization and Fractional Quantum Hall states in ABA TLG. (a) Optical microscope image of the device1. The scale bar is 5 µm. (b) 2D map of longitudinal conductance Rxx in the B and number density n plane. The rectangular box marks the crossings between the four LL0 M LLs with the BLL LLs. (c) Plots of Rxx (left y-axis) and Gxy (right y-axis) as a function of filling factor ν at B = 12 T and T = 0.02 K.… view at source ↗

**Figure 2.** Particle hole symmetry and activation gaps of FQHs. (a) Plots of Rxx versus filling factor ν at a few representative T for D = 0.1 V/nm between ν = 3 and ν = 4. (b) Symbols are the Arrhenius plots for various FQH states between ν = 3 and 4. The dashed lines are the fit to data points using Arrhenius fit. Plot of activation gaps as a function of Bef f for FQHs between (c) ν = 3 and 4, and (d) ν = 2 and 3. The red das… view at source ↗

**Figure 3.** Effect of Landau level mixing on FQHs in ABA trilayer graphene Contour plots of Gxx as a function of D and ν for (a) electron doping, and (b) hole doping. The data are taken at B = 10 T. The white dotted rectangles mark the regions of PHS violation. Simulated Landau level spectrum as a function of interlayer potential ∆1 and energy E for (b) electron doping and (e) hole doping. The data are plotted over the same ran… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Controlled violation of PHS. Contour plot of Gxx as a function of D and ν around the LL crossing between filling factor (a) ν = 2 and 3, (b) 4 and 5. The white dotted rectangle marks the region of observed PHS violation. Plot of D-dependence of activation gap ∆E for (c…

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