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Moiré Hofstadter bands keep Josephson phase coherence alive to 6 T in graphene junctions.

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2026-07-14 03:37 UTC pith:Y36X4AI6

load-bearing objection Real multi-device data push bulk Josephson interference to ~6 T in graphene/hBN moiré junctions by restoring group velocity in Hofstadter bands; the EH_Th conjecture is secondary to the observation. the 2 major comments →

arxiv 2607.11721 v1 pith:Y36X4AI6 submitted 2026-07-13 cond-mat.mes-hall cond-mat.supr-conquant-ph

High-field Josephson effect enabled by a moir\'e Hofstadter spectrum

A. D\'iez-Carl\'on , M. C\'ardenes Wuttig , N. Wei , D. Ivanov , P. Altpeter , P. Hakonen , K. Watanabe , T. Taniguchi
show 2 more authors
L. I. Glazman D. K. Efetov
This is my paper
classification cond-mat.mes-hall cond-mat.supr-conquant-ph
keywords Josephson effectmoiré superlatticeHofstadter butterflygraphene/hBNAndreev bound stateshigh-field superconductivitymagnetic Bloch bandsFabry-Pérot interference
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Magnetic fields usually kill coherent supercurrents by bending electron trajectories and flattening Landau levels, so Josephson interferometry has been stuck at relatively low fields. This paper shows that a graphene/hBN moiré weak link changes the picture. Ballistic junctions still show Fabry–Pérot and Fraunhofer signatures across both the main Dirac cone and the reconstructed minibands. Once the Fermi level enters the minibands, phase-coherent Josephson interference survives up to about 6 T—well past the cyclotron-orbit limit that caps ordinary ballistic graphene junctions at roughly 2.5 T. Hofstadter-spectrum calculations locate the surviving supercurrent where the moiré potential turns flat Landau levels into dispersive magnetic Bloch bands that restore finite quasiparticle velocity, letting electron–hole Andreev pairs cross the junction. The result turns the fractal Hofstadter spectrum into a practical route for high-field superconducting interferometry.

Core claim

In ballistic graphene/hBN moiré Josephson junctions, phase-coherent supercurrent persists up to ~6 T inside the Hofstadter-butterfly regime precisely where the moiré potential converts quenched Landau levels into overlapping dispersive magnetic Bloch bands that restore finite group velocity and enable extended Andreev trajectories across the weak link.

What carries the argument

Dispersive Hofstadter (magnetic Bloch) minibands: the moiré potential broadens Landau levels into bands whose average squared velocity ⟨v²⟩/vF² remains large enough that the associated Thouless energy stays comparable to the lead gap, preventing exponential suppression of bulk Andreev transport.

Load-bearing premise

That a finite Hofstadter group velocity large enough to keep the Thouless energy near or above the superconducting gap is what actually rescues bulk supercurrent, rather than some other high-field Andreev path.

What would settle it

Measure the same junctions while deliberately suppressing band velocity (for example by changing twist angle or density so that ⟨v²⟩/vF² drops well below ~0.1 at the densities where supercurrent is now seen) and check whether the high-field Josephson pockets disappear even though Landau levels remain absent.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript reports ballistic graphene/hBN moiré Josephson junctions that sustain phase-coherent Andreev transport and Josephson interference up to ~6 T in the fractal Hofstadter regime, well beyond the semiclassical 2rc = L limit that bounds conventional ballistic graphene junctions (~2.5 T). After establishing Fabry–Pérot oscillations in both Ic and RN across the primary Dirac cone and moiré minibands, and standard Fraunhofer interference at low field, the authors map high-field superconducting pockets whose Ic remains large (tens of nA, approaching the single-mode quantum limit) and oscillates irregularly with B and density. A non-moiré control device and bias-quenched Landau fans show that these pockets sit where Landau quantization is absent. Hofstadter-spectrum calculations (Moon–Koshino continuum model) then correlate the superconducting regions with domains of finite average squared band velocity ⟨v²⟩, interpreted as dispersive magnetic Bloch bands that restore a finite Thouless energy and enable extended electron–hole Andreev trajectories.

Significance. If the result holds, the work supplies a concrete materials route—moiré-engineered Hofstadter minibands—to keep bulk, phase-coherent Josephson transport alive deep into the multi-tesla regime normally associated with quantum Hall physics. That would open high-field superconducting interferometry and phase-sensitive probes of fractal spectra, Brown–Zak transport, and related correlated states. Strengths that support the claim include multi-device reproducibility (GH1–GH4), a non-moiré control, Fabry–Pérot signatures of ballistic transport in the supercurrent itself across minibands, large and irregular high-field Ic oscillations that distinguish bulk mesoscopic trajectories from fragile chiral Andreev edge modes, and transparent theory that uses a standard continuum model with literature parameters rather than free-fitting the spectrum. The experimental observation of phase-coherent supercurrent to ~6 T is already a substantial advance; the Hofstadter-velocity interpretation supplies a physically motivated mechanism.

major comments (2)
  1. Sec. II.D and Fig. 3: The central interpretive claim—that survival of bulk supercurrent is enabled by dispersive Hofstadter bands—rests on the conjecture that EH_Th = ħ vH/L ≳ Δ once ⟨v²⟩/vF² is not much smaller than ~0.1–0.17 (vH ~ 0.42 vF). The visual correlation between 1/R and ⟨v²⟩ maps is compelling and multi-device (SI Figs. S18–S21), but the numerical threshold is chosen post hoc to match the experimental boundary. The paper would be stronger if it either (i) quantified the spatial overlap between high-⟨v²⟩ and superconducting regions (e.g., a simple overlap or ROC-style metric across several flux fractions) or (ii) showed that the observed high-field pockets collapse when ⟨v²⟩ falls well below that window. Without such a check the mechanism remains correlative rather than predictive, even though the raw observation of phase-coherent Ic to ~6 T stands independently.
  2. Sec. II.C and SI Sec. G: The argument that the high-field pockets are bulk mesoscopic Andreev trajectories (not chiral edge modes) relies on three pillars—large Ic (~IQ), irregular period, and absence of Landau levels under a 200 nA bias quench. The first two are solid. The third would be more decisive if the authors also reported the normal-state conductance (or dV/dI) at the same high-B, high-n points with the bias reduced just above Ic, to confirm that the underlying spectrum remains ungapped/non-quantized rather than that a large bias simply heats or depopulates edge channels. A short additional panel or SI note would close this residual ambiguity.
minor comments (5)
  1. Fig. 3c–d and SI theory section: State explicitly how the Fermi-level window η used for ⟨v²⟩ is chosen and whether the maps are robust to reasonable variations of η; a one-sentence sensitivity check would help readers who wish to reproduce the proxy.
  2. Fig. 2a and SI Fig. S13: The orange 2rc = L contour is drawn with the zero-field Dirac velocity; a brief note that magnetic-breakdown-restored orbits (dashed orange Fermi contours in Fig. 2d) still lie below the observed superconducting boundary would make the failure of the semiclassical picture even clearer.
  3. SI Sec. E (flux creep): The procedure of slow sweeps and repeated field cycling is described; adding a quantitative statement of the residual period scatter after this protocol (already partly given as ±0.2–0.4 mT) would help readers assess whether residual flux motion could systematically bias the high-field periods.
  4. Abstract and Introduction: The phrase “well beyond the range expected for conventional ballistic graphene junctions” is accurate; citing the non-moiré control (SI Sec. H) already in the main text would make the comparison self-contained.
  5. Notation: EH_Th and ETh are used for the Hofstadter and zero-field Thouless energies; a single consistent subscript convention would avoid momentary confusion when scanning Sec. II.D.

Circularity Check

0 steps flagged

No significant circularity: experimental high-field Josephson data and independent Hofstadter continuum calculations are compared, not forced by construction.

full rationale

The load-bearing experimental claim (phase-coherent Ic oscillations and zero-resistance pockets up to ~6 T in moiré minibands, multi-device, with Fabry-Pérot, Fraunhofer, bias-quench LL absence, and non-moiré control) stands on measured transport and does not reduce to any theoretical input. The interpretive link uses the standard Moon–Koshino continuum model with literature moiré parameters, magnetic translation operators, and a Fermi-level average ⟨v²⟩ proxy taken from external work (Krishna Kumar et al.; Chang–Niu wave-packet picture). The EH_Th ≳ Δ / ⟨v²⟩/vF² ≳ 0.1–0.17 threshold is an explicit conjecture derived from device scales (Δ, L, vF), not a fit to the high-field superconducting pockets; the paper then shows a correlation between the independent numerical map and the measured 1/R map. Self-citations involving Glazman (magnetic breakdown, SNS edge effects) supply standard theoretical language but are not uniqueness theorems that force the result, nor do they replace the experimental evidence. No equation equals its own input by construction, no fitted parameter is renamed a prediction, and no ansatz is smuggled in as a first-principles derivation. Score 0 is therefore the correct, proportionate finding.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard ballistic SNS theory, the continuum moiré Hamiltonian of Moon & Koshino, and the semiclassical cyclotron-radius criterion 2rc = L, plus one explicit conjecture linking Hofstadter group velocity to the survival of Ic. No new particles or forces are introduced; free parameters are device-specific (twist angle, length) or literature values (vF, Δ).

free parameters (4)
  • twist angle θ = 0.21 ± 0.01° (GH1)
    Extracted from Landau-fan slopes and satellite Dirac peaks for each device (θ ≈ 0.21° for GH1); enters the moiré wavelength and therefore the entire Hofstadter spectrum.
  • junction length L and effective area = L ≈ 200 nm (GH1)
    Designed and verified by Fraunhofer period; sets the 2rc = L boundary and the Thouless energy scale.
  • MoRe gap Δ and Tc = Δ ≈ 1.4 meV
    Taken from measured Tc ≈ 9 K via BCS relation; sets the absolute scale of Ic and the EH_Th ≳ Δ criterion.
  • velocity threshold ⟨v²⟩/vF² ≳ 0.1–0.17 = ~0.17 (corresponding to vH ~ 0.42 vF)
    Chosen so that EH_Th remains comparable to Δ; used to color the theoretical maps that are then compared with experiment.
axioms (4)
  • domain assumption Ballistic long-junction Ambegaokar–Baratoff / Thouless scaling Ic ~ W n^{1/2} (e E_Th / ħ) when E_Th ≲ Δ
    Invoked in Sec. II.D to relate finite group velocity to finite critical current; standard for ballistic SNS junctions.
  • domain assumption Semiclassical cyclotron-orbit criterion 2rc = L marks the loss of bulk Andreev trajectories in the primary Dirac cone
    Used to draw the orange boundary in Fig. 2a; previously established for non-moiré graphene JJs.
  • domain assumption Moon–Koshino continuum Hamiltonian with literature moiré parameters (V0, V1, ψ) correctly describes the graphene/hBN spectrum
    Basis of all Hofstadter and ⟨v²⟩ calculations in the SI.
  • ad hoc to paper Average squared band velocity ⟨v²⟩ is a faithful proxy for the Hofstadter group velocity that enters the Thouless energy
    Explicitly adopted following Ref. [47] because full dispersive spectra at generic flux are impractical; central to the theory–experiment comparison in Fig. 3.

pith-pipeline@v1.1.0-grok45 · 37432 in / 2969 out tokens · 26468 ms · 2026-07-14T03:37:42.657178+00:00 · methodology

0 comments
read the original abstract

Magnetic fields generally suppress phase-coherent Josephson transport, limiting superconducting interferometry to relatively low fields. Here we show that moir\'e-engineered graphene Josephson junctions can overcome this constraint. Using ballistic graphene/hBN junctions, we establish phase-coherent Andreev transport through Fabry-P\'erot oscillations and Fraunhofer interference that persist across both the primary Dirac cone and reconstructed moir\'e minibands. We then demonstrate phase-coherent Josephson interference up to 6 T in the fractal Hofstadter-butterfly regime, well beyond the range expected for conventional ballistic graphene junctions. Comparison with Hofstadter-spectrum calculations reveals that superconductivity survives where the moir\'e potential transforms Landau levels with quenched group velocity into dispersive magnetic Bloch bands with finite quasiparticle group velocity, enabling extended electron-hole Andreev trajectories across the junction. Our results show that Hofstadter minibands can stabilize phase-coherent superconductivity deep into the parameter domain conventionally associated with the quantum Hall regime, establishing a new platform for high-field superconducting interferometry.

Figures

Figures reproduced from arXiv: 2607.11721 by A. D\'iez-Carl\'on, D. Ivanov, D. K. Efetov, K. Watanabe, L. I. Glazman, M. C\'ardenes Wuttig, N. Wei, P. Altpeter, P. Hakonen, T. Taniguchi.

Figure 1
Figure 1. Figure 1: Ballistic superconducting proximity effect in a moiré graphene/hBN Josephson junction. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: High field superconductivity at the moiré bands. a) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Absence of Landau levels from dispersive bands in the superconducting regions. a) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

discussion (0)

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