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Near the relativistic Mott transition in twisted double-bilayer WSe2 and twisted bilayer graphene, quantum-critical fluctuations of gap-opening collective modes can bind strongly incoherent electrons into superconductivity, with the allowed

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Near the relativistic Mott transition, quantum-critical fluctuations of time-reversal-even collective modes induce superconductivity in twisted WSe2 and TBG, with a rich spectrum of degenerate pairing states.

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load-bearing objection A solid group-theoretic classification of pairing channels in twisted WSe2 and TBG, with an honest but conditional superconductivity existence claim resting on an uncomputed anomalous dimension. the 3 major comments →

arxiv 2510.06313 v1 pith:NYUKFOUZ submitted 2025-10-07 cond-mat.str-el cond-mat.supr-con

Superconductivity of Incoherent Electrons near the Relativistic Mott Transition in Twisted Dirac Materials

Veronika C. Stangier , Mathias S. Scheurer , Daniel E. Sheehy , J\"org Schmalian This is my paper
classification cond-mat.str-el cond-mat.supr-con
keywords twisted bilayer graphenetwisted double-bilayer WSe2Gross-Neveu transitionquantum criticalitysuperconductivity without quasiparticlesskyrmionsDirac semimetallarge-N random-coupling theory
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.

The reading

This paper argues that superconductivity can appear at the exact twist angle where twisted Dirac materials turn into insulators, driven by the same critical fluctuations that make electrons incoherent. It identifies which collective modes promote pairing in two concrete systems. In twisted double-bilayer WSe2, only time-reversal-even modes do, yielding four nearly degenerate pairing states. In twisted bilayer graphene, fluctuations of inter-valley-coherent order produce six degenerate or nearly degenerate superconducting states, some with finite momentum. The paper also shows that these same conditions are precisely those for charged skyrmions in the insulator, linking the superconducting and topological sectors.

Core claim

The paper establishes that, for a generic two-dimensional Dirac system at its Gross-Neveu (relativistic Mott) critical point, the superconducting instability is governed entirely by the algebraic relations between the pairing matrices and the collective-mode coupling matrices: [Γ_J, α_i]=0 and Γ_J Υ_a = τ_φ Υ_a Γ_J. Applying this to AB-BA stacked twisted double-bilayer WSe2 at filling ν=2, it finds that all time-reversal-even, gap-opening bosons promote pairing — producing four degenerate states, most typically an s-wave state — while time-reversal-odd modes (Haldane, spin-Hall, Kekulé loop-current variants) do not. In a Dirac model of twisted bilayer graphene with the mini-valley flavor inc

What carries the argument

The paper's strong-coupling framework is a generalized large-N random-coupling theory in which melonic diagrams dominate and give universal power laws for the critical fermion and boson propagators, characterized by the fermion anomalous dimension η_ψ. The Eliashberg equation for the anomalous self-energy has a kernel that is universal at the Gross-Neveu critical point, with a dimensionless pairing strength λ_p that grows with η_ψ; superconductivity requires η_ψ ≥ η_c^ψ ≈ 0.146. The central algebraic selection rule is Eq. (5): a pairing matrix Γ_J must commute with the Dirac matrices and satisfy Γ_J Υ_a = τ_φ Υ_a Γ_J, which together with Fermi statistics fixes which pairing states each colle

Load-bearing premise

The superconductivity result rests on adopting the companion paper's Eliashberg kernel and on the assumption that the Gross-Neveu transitions in twisted WSe2 and TBG are in the generalized large-N limit with a fermion anomalous dimension above the universal threshold (about 0.146); the paper does not compute this anomalous dimension from a microscopic model, and the authors explicitly say the precise threshold value is not their key prediction.

What would settle it

A numerical determination of the fermion anomalous dimension η_ψ at the Gross-Neveu critical point in a microscopic lattice model of twisted double-bilayer WSe2 or twisted bilayer graphene that yields η_ψ below ≈0.146 would falsify the claim that these materials superconduct through this mechanism, while leaving the algebraic classification of allowed pairing states intact.

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

If this is right

  • At the critical twist angle where twisted double-bilayer WSe2 becomes insulating, superconductivity or at least strong pairing fluctuations are predicted; the dominant instability is one of four degenerate pairing states, with an s-wave (A1) state the most common outcome.
  • In twisted bilayer graphene, both KIVC and TIVC fluctuations at the onset of inter-valley-coherent order promote pairing, yielding six distinct degenerate or nearly degenerate superconducting states, some with finite center-of-mass momentum.
  • The mechanism only works when electrons are strongly incoherent — well-defined quasiparticles would not pair this way — and the richer the Dirac structure (16-component in TBG vs 8-component in WSe2), the more readily pairs form.
  • The algebraic conditions that select pairing states coincide with the conditions for a Wess-Zumino-Witten term, so insulators with charged skyrmions are a subset of the pairing states; skyrmions carry charge 2e in WSe2 and 4e in TBG.
  • With a Dirac cutoff of about 20 meV, the optimal transition temperature is about 0.4 K.

Where Pith is reading between the lines

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

  • If the threshold η_c is a general feature, the same mechanism could apply to other twisted Dirac systems, suggesting that superconductivity near correlation-driven gap openings may be more widespread than currently suspected.
  • The degeneracy among the four WSe2 states is an artifact of the continuum Dirac theory; lattice corrections will lift it. A natural next step is to compute which specific state survives — the paper leaves that hierarchy open.
  • The coincidence with the skyrmion/WZW conditions suggests the predicted superconducting states may have topological signatures (e.g., chiral or helical Cooper pair wavefunctions) even though the paper does not establish topology explicitly.
  • A microscopic calculation or measurement of η_ψ at the transition in each material would turn the qualitative prediction into a quantitative one; experiments could look for a dome of T_c peaking at the critical twist angle.
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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

3 major / 4 minor

Summary. The paper applies a generalized SYK framework, developed in the companion paper Ref. [27], to study superconductivity near Gross-Neveu (relativistic Mott) critical points in twisted double-bilayer WSe2 and twisted bilayer graphene (TBG). The central input is a critical boson–fermion model (Eq. (1)) whose pairing instability is governed by the linearized Eliashberg equation (Eq. (2)) with dimensionless pairing strength λ_p (Eq. (3)) determined solely by the fermion anomalous dimension η_ψ. The authors then derive algebraic conditions (Eq. (5)) that select the allowed pairing channels for a given collective bosonic mode: [Γ_J, α_i]=0 and Γ_J Υ_a = τ_φ Υ_a Γ_J, together with Fermi statistics. Applying these conditions to the Dirac theories of WSe2 and TBG, they find: for WSe2, all time-reversal-even gap-opening fluctuations induce pairing, but time-reversal-odd ones do not, yielding four degenerate states (Eqs. (9)–(11)); for TBG, KIVC and TIVC fluctuations each give four degenerate states (Eqs. (15), (16)), with six distinct states in total. They further show that a subset of these pairing states satisfies the additional trace condition (Eq. (18)) for a Wess-Zumino-Witten (WZW) term, linking the pairing to charge-carrying skyrmions in the proximate insulator. Using the maximal T_c from Ref. [27], they estimate T_c ≈ 0.4 K for both materials. The paper is transparent in footnote [45] that the threshold value η_c^ψ ≈ 0.146 is model-dependent, but the existence of a threshold is p

Significance. If the framework is valid, this work provides a concrete and experimentally testable scenario for superconductivity near relativistic Mott transitions in twisted Dirac materials, with a rich, symmetry-dictated spectrum of pairing states and a topological mechanism for pairing via skyrmions. The algebraic classification in Eq. (5) and the tables are internally consistent, checkable, and likely correct; they constitute a useful group-theoretic catalog independent of the dynamical details. The connection between the pairing conditions and the WZW-term criteria (Eq. (B1)) is elegant and is a genuinely new observation. The paper is also honest about the main caveat—that the threshold η_ψ ≥ η_c^ψ is not computed for the specific materials. However, the central existence claim (T_c ≈ 0.4 K, 'one should expect either superconductivity or strong pairing fluctuations') rests on two unverified imported ingredients from Ref. [27]: the Eliashberg kernel (Eq. (2)) and the threshold η_c^ψ, plus an uncontrolled extrapolation from the large-N melonic limit to the physical n_γ=8 or 16 systems. The classification would survive even if η_ψ falls below threshold, but the predicted superconductivity wou

major comments (3)
  1. [Eq. (2), footnote [45], and 'To estimate T_c'] The claim that superconductivity actually occurs in twisted WSe2 and TBG near the Mott transition depends entirely on whether the physical anomalous dimension η_ψ exceeds η_c^ψ ≈ 0.146. The paper never computes η_ψ for these systems; footnote [45] explicitly states that the precise threshold 'will depend on details beyond the scope of our analysis.' For 2+1D Gross-Neveu universality classes, η_ψ is typically O(0.1) and is not guaranteed to exceed 0.146; it depends on the number of fermion flavors and on which order parameter condenses. As stated, the T_c ≈ 0.4 K estimate and the abstract's claim that 'superconductivity can emerge' are therefore conditional on a parameter that is neither computed nor measured. This is load-bearing: if η_ψ < η_c^ψ, the algebraic classification (Eq. (5), Tables I–II) remains correct but the central physical prediction fails. The authors should either comput
  2. [SYK large-N limit] The kernel (Eq. (2)) and λ_p (Eq. (3)) are derived in a generalized SYK model with random couplings among N fermion flavors, in a melonic large-N limit. The physical systems have only n_γ=8 (WSe2) or n_γ=16 (TBG) components. The paper does not discuss whether 1/N corrections are small for these values, nor does it test the robustness of the result to the random-coupling assumption. This is an uncontrolled step between the controlled large-N limit and the physical Dirac theories. The authors should provide at least a rough estimate of the corrections for N=8,16, or otherwise justify why the melonic result applies to these small representations.
  3. [WZW partner-state counting] In the main text, the authors state 'we find in total 56 such partner states' for TBG, but Appendix B counts 336 triples before restricting to order parameters that all transform the same under time reversal; only 56 survive that restriction. The phrase 'in total' is misleading and the reader may infer fewer states than actually satisfy the WZW condition. Please clarify the counting and the restriction in the main text.
minor comments (4)
  1. [Title] Typo in title: 'T wisted' should be 'Twisted'.
  2. [Various] There are missing spaces in phrases such as 'atfilling𝜈=2' and 'theDirac point is right at the Fermi level' (intro). Please run a spell/format check.
  3. [Figure 2 caption] The caption's explanation of the +− and ++ labels for 𝜇_z and 𝜇_0 is cryptic; please spell out the convention more clearly.
  4. [Appendix B / main text] The phrase 'in total 56 such partner states' should be clarified to indicate that this is the count after restricting to triples whose components transform the same under time reversal, not the total number of triples satisfying Eq. (18).

Circularity Check

1 steps flagged

SC existence relies on companion-paper threshold η_c^ψ from Ref. [27]; pairing-state classification is independent algebra.

specific steps
  1. self citation load bearing [Section 'Twisted double-bilayer WSe2' / Eqs. (2)–(3), footnote [45]]
    "The superconducting instability only occurs for sufficiently large dimensionless pairing strength λp such that the normal-state anomalous dimension exceeds a critical value, i.e., ηψ≥ηcψ [45]. ... While ηcψ≈0.14628 within our theory [27], we regard the existence of such a minimum anomalous fermion dimension (rather than its precise value, which will depend on details beyond the scope of our analysis) as the key prediction of our theory."

    The existence claim—that quantum-critical fluctuations actually drive superconductivity in twisted WSe2 and TBG—rests on the threshold ηψ≥ηcψ≈0.14628, which is imported from companion paper Ref. [27] by the same authors. The Eliashberg kernel (Eq. 2) and pairing strength λp (Eq. 3) are also taken from [27]; the present paper does not re-derive them and never computes ηψ for either material. Footnote [45] explicitly concedes that the precise threshold value depends on details beyond the paper's analysis. Thus the central physical prediction reduces to a self-cited, not independently derived, result. The algebraic classification of allowed pairing states (Eq. 5, Tables I–II) is nevertheless an independent calculation and is not circular.

full rationale

The main new content of the paper—the classification of pairing states for WSe2 and TBG via Eq. (5)—is self-contained: it is a direct algebraic consequence of the stated Dirac matrices, time-reversal operator, and coupling matrices, and is tabulated in Tables I–II. The WZW connection is also explicitly computed in Appendix B, where the first two WZW conditions of Ref. [34] are shown to coincide with Eq. (5); this is a mathematical equivalence, not a renaming. The circularity concern is confined to the existence part of the prediction. The Eliashberg kernel, the pairing strength λp, and the anomalous-dimension threshold ηcψ≈0.14628 are all imported from the companion paper Ref. [27] with overlapping authorship (Stangier, Sheehy, Schmalian). The paper never computes ηψ for twisted double-bilayer WSe2 or TBG from a microscopic model, and footnote [45] explicitly states that the precise threshold value depends on details beyond the present analysis. Hence the claim 'superconductivity can emerge' at these specific transitions is a conditional application of a self-cited result rather than an independent derivation. Because the algebraic classification has independent content, the score is 4 rather than 6+.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper's predictions rely on the validity of the generalized SYK large-N description of the GN critical point and on the algebraic conditions carried over from Ref. [27]. No microscopic computation of the anomalous dimension for these materials is provided; the threshold and T_c scale are imported from the companion paper. The WZW connection uses criteria from Ref. [34].

free parameters (3)
  • anomalous fermion dimension η_ψ = assumed > η_c^ψ ≈ 0.14628 (from Ref. [27], not computed for these materials)
    Superconductivity only occurs if η_ψ exceeds a threshold; the paper does not compute η_ψ for WSe2 or TBG, so the prediction is conditional on this parameter.
  • cutoff energy Λ = ~20 meV (from experiments [13,20])
    Used to convert dimensionless T_c from Ref. [27] into kelvin; the value of T_c ≈ 0.4 K scales linearly with Λ.
  • maximal T_c coefficient 2×10^-3 Λ = 2×10^-3 Λ
    Taken from Ref. [27] as the maximal transition temperature; this paper does not derive it.
axioms (4)
  • domain assumption The GN critical point of the twisted Dirac materials is described by the generalized SYK large-N limit with universal power laws and a large anomalous fermion dimension η_ψ.
    The entire framework rests on this mapping; no microscopic derivation for the specific materials is provided.
  • domain assumption The pairing instability condition from the linearized Eliashberg equation (Eq. 2) and dimensionless pairing strength λ_p (Eq. 3) from Ref. [27] are valid for the materials.
    These are imported from the companion paper and are not re-derived here.
  • domain assumption The only relevant gap-opening collective modes are those described by the Υ matrices listed (Eqs. 7-8, 13-14); other modes do not affect the classification.
    The classification is complete only if the selected set of bosons spans all possible gap-opening orders.
  • domain assumption The Dirac theory at the neutrality point captures the low-energy physics; lattice corrections only lift degeneracies.
    The paper acknowledges this limitation, noting the Dirac model ignores the rest of the mini-Brillouin zone and higher bands.

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Cite this review

Pith. "Pith review of Superconductivity of Incoherent Electrons near the Relativistic Mott Transition in Twisted Dirac Materials." pith.science (2026). https://pith.science/paper/NYUKFOUZ

@misc{pith2026251006313,
  author       = {Pith},
  title        = {Pith review of: Superconductivity of Incoherent Electrons near the Relativistic Mott Transition in Twisted Dirac Materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYUKFOUZ}},
  note         = {Machine review of arXiv:2510.06313}
}
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abstract

We demonstrate that superconductivity driven by strong quantum-critical fluctuations can emerge near relativistic Mott transitions in twisted two-dimensional materials, taking on a remarkably rich character. In twisted double-bilayer WSe$_2$, all time-reversal-even, gap-opening collective modes promote pairing, whereas time-reversal-odd modes do not. In a Dirac model of twisted bilayer graphene, the Gross-Neveu transition into inter-valley-coherent insulators gives rise to a spectrum of degenerate and nearly degenerate superconducting states. More generally, we show that the richer the Dirac structure, the more readily pairs can form. A crucial ingredient of the theory is that critical fluctuations render the electronic states strongly incoherent, allowing attractive pairing channels to overcome the bare Dirac semi-metal behavior. Finally, we demonstrate a direct relation between boson-mediated pairing and the formation of charge-carrying skyrmionic excitations in the proximate insulating state.

Figures

Figures reproduced from arXiv: 2510.06313 by Daniel E. Sheehy, J\"org Schmalian, Mathias S. Scheurer, Veronika C. Stangier.

Figure 1
Figure 1. Figure 1: Schematic phase diagram for the behavior near the Gross [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Top Row: The four possible SC instabilities in double [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

discussion (0)

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.