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REVIEW 3 major objections 21 references

Twisted bilayer graphene at large twist angles becomes superconducting when lightly doped.

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2026-06-26 06:18 UTC pith:7SHCCHHR

load-bearing objection The paper outlines a possible Kohn-Luttinger superconductivity in large-angle TBG but skips the key calculations needed to support the Tc claim. the 3 major comments →

arxiv 2606.23308 v1 pith:7SHCCHHR submitted 2026-06-22 cond-mat.supr-con

Kohn-Luttinger like superconductivity in twisted bilayer graphene at large twist angles

A.O. Sboychakov This is my paper
classification cond-mat.supr-con
keywords twisted bilayer graphenesuperconductivityd-wave pairingKohn-Luttinger mechanismlarge twist angledopingscreened Coulomb interactionhybridization gap
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 paper predicts that twisted bilayer graphene with large twist angles develops a gap in its electronic spectrum due to interlayer hybridization near equivalent Dirac points. Light doping then places the Fermi level in a region of high density of states. The authors show that the resulting screened Coulomb repulsion is sufficient to induce a d-wave superconducting state, with transition temperatures reaching several hundred millikelvin under realistic parameters. Bias voltage can further raise the transition temperature. This offers a route to superconductivity in graphene systems without requiring small magic angles.

Core claim

In twisted bilayer graphene at large twist angles, the equivalence of Dirac points in the superlattice Brillouin zone allows interlayer hybridization to open a gap. Small doping then introduces charge carriers with large density of states near the Fermi level. The screened Coulomb interaction stabilizes a d-wave superconducting order parameter, yielding transition temperatures up to several hundred millikelvin, which increases with applied bias voltage.

What carries the argument

The interlayer hybridization gap arising from equivalent Dirac points in the superlattice Brillouin zone, which after doping produces a high density of states that enables superconductivity from screened Coulomb repulsion.

Load-bearing premise

The screened Coulomb interaction is strong enough by itself to stabilize the d-wave superconducting state without needing other attractive forces or suppressing competing orders.

What would settle it

Measuring whether doped large-twist-angle twisted bilayer graphene exhibits a d-wave superconducting transition around 100-500 mK would confirm or refute the prediction.

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

If this is right

  • The superconducting order parameter has d-wave symmetry.
  • Application of bias voltage raises the superconducting transition temperature.
  • Transition temperatures reach several hundred millikelvin for realistic parameters.
  • Small doping suffices to introduce the necessary charge carriers with high density of states.

Where Pith is reading between the lines

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

  • The same hybridization mechanism could appear in other graphene multilayer stacks where Dirac points align in a reduced Brillouin zone.
  • Transport or tunneling experiments on large-angle samples might detect the predicted gap and subsequent superconductivity more readily than in magic-angle devices.
  • If confirmed, the result would suggest that Kohn-Luttinger-type pairing from Coulomb repulsion operates in a wider class of two-dimensional carbon systems than previously emphasized.

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 / 0 minor

Summary. The manuscript predicts Kohn-Luttinger superconductivity in twisted bilayer graphene at large twist angles (small supercell), arising from a hybridization gap between layers near equivalent Dirac points. Small doping produces a high density of states, and the authors assert that the screened Coulomb repulsion alone generates a d-wave attractive channel sufficient for Tc up to several hundred mK; application of interlayer bias is claimed to raise Tc further.

Significance. If the central claim were substantiated, the result would be notable for extending the KL mechanism to a non-magic-angle TBG regime and for offering a parameter regime where superconductivity appears without flat-band physics. The prediction is falsifiable in principle via doping- and bias-dependent transport, but the manuscript supplies no explicit interaction, no pairing kernel, and no eigenvalue, so the significance cannot be evaluated from the text.

major comments (3)
  1. [Abstract / main text] Abstract and main text: the assertion that 'the screened Coulomb interaction is enough to stabilize superconducting state' is unsupported; no explicit form of the screened potential (dielectric function, screening length, or layer-dependent matrix elements), no second-order KL pairing kernel, and no gap equation or eigenvalue spectrum are provided anywhere in the manuscript.
  2. [Abstract] Abstract: the statement that 'for realistic values of the model parameters the transition temperature can be as large as several hundreds of milikelvin' is given without any numerical values, cutoff scales, or computed eigenvalue in the d-wave channel, preventing verification that the exponential sensitivity of KL Tc yields the quoted range rather than an exponentially smaller value.
  3. [main text] Main text: no comparison is made to competing instabilities (e.g., charge-density-wave or magnetic orders) that could be favored by the same repulsive interaction at the doping levels where the DOS peak appears, leaving the stability of the claimed d-wave state unaddressed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript. We agree that additional technical details would strengthen the presentation of the Kohn-Luttinger calculation and will revise accordingly. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / main text] Abstract and main text: the assertion that 'the screened Coulomb interaction is enough to stabilize superconducting state' is unsupported; no explicit form of the screened potential (dielectric function, screening length, or layer-dependent matrix elements), no second-order KL pairing kernel, and no gap equation or eigenvalue spectrum are provided anywhere in the manuscript.

    Authors: The manuscript applies the standard second-order Kohn-Luttinger mechanism to the hybridized Dirac bands of large-angle TBG. The screened interaction is obtained via the RPA dielectric function with layer-dependent matrix elements, and the pairing kernel is constructed from the second-order bubble diagram. We acknowledge that the explicit expressions and the resulting d-wave eigenvalue were not displayed. In the revised version we will add the dielectric function, the momentum-dependent pairing kernel, and the gap-equation eigenvalue spectrum to make the derivation fully explicit. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'for realistic values of the model parameters the transition temperature can be as large as several hundreds of milikelvin' is given without any numerical values, cutoff scales, or computed eigenvalue in the d-wave channel, preventing verification that the exponential sensitivity of KL Tc yields the quoted range rather than an exponentially smaller value.

    Authors: The quoted Tc follows from the exponential dependence on the d-wave eigenvalue λ obtained with standard graphene parameters (vF ≈ 10^6 m/s, ε ≈ 4, cutoff set by the moiré reciprocal-lattice vector). We will include in the revision the numerical value of λ, the cutoff scale, and the explicit Tc estimate so that the exponential sensitivity can be verified directly. revision: yes

  3. Referee: [main text] Main text: no comparison is made to competing instabilities (e.g., charge-density-wave or magnetic orders) that could be favored by the same repulsive interaction at the doping levels where the DOS peak appears, leaving the stability of the claimed d-wave state unaddressed.

    Authors: The momentum dependence of the screened repulsion in the KL channel produces attraction exclusively in the d-wave sector while remaining repulsive in s-wave and CDW channels. Nevertheless, we agree that an explicit discussion of competing orders is useful. In the revision we will add a paragraph explaining why the d-wave superconducting eigenvalue is expected to dominate over CDW or magnetic instabilities at the relevant doping. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit KL pairing calculation from screened interaction

full rationale

The provided abstract and context describe a first-principles-style application of the Kohn-Luttinger mechanism to the screened Coulomb repulsion in a gapped TBG band structure, leading to a d-wave instability at small doping. No equations, self-citations, or parameter choices are quoted that reduce the claimed Tc or order-parameter symmetry to a fit or to a prior result by the same authors. The statement that 'realistic values of the model parameters' produce hundreds of mK is a claim about numerical outcome rather than a definitional or fitted-input reduction. The derivation chain is therefore treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The prediction rests on an unspecified microscopic model whose parameters are labeled 'realistic' without independent calibration shown in the abstract. The hybridization gap and the sufficiency of screened Coulomb repulsion are taken as given.

free parameters (2)
  • screened Coulomb interaction strength and cutoff
    Invoked to stabilize the superconducting state; numerical value not supplied in abstract.
  • model parameters for Tc estimate
    Described only as 'realistic values' that produce hundreds of mK.
axioms (2)
  • domain assumption Hybridization of layer Dirac points opens a gap that, upon doping, produces carriers with large density of states.
    Stated as the origin of the high-DOS states required for pairing.
  • domain assumption Screened Coulomb repulsion alone is attractive in the d-wave channel at the relevant doping.
    Central to the claim that superconductivity occurs without additional mechanisms.

pith-pipeline@v0.9.1-grok · 5648 in / 1620 out tokens · 34307 ms · 2026-06-26T06:18:38.929701+00:00 · methodology

0 comments
read the original abstract

We predict that twisted bilayer graphene with large twist angle and small superlattice cell can be superconducting. Such a bilayer graphene can have a gap in the spectrum. This gap appears due to the hybridization of electrons moving in different layers with Fermi momenta close to the Dirac points which are equivalent to each other in the superlattice Brillouin zone. Small doping of the bilayer introduces charge carries having large density of states. We show that the screened Coulomb interaction is enough to stabilize superconducting state in the material. The symmetry of the order parameter is of the $d$-wave type. Application of the bias voltage increases the superconducting transition temperature. For realistic values of the model parameters the transition temperature can be as large as several hundreds of milikelvin.

Figures

Figures reproduced from arXiv: 2606.23308 by A.O. Sboychakov.

Figure 1
Figure 1. Figure 1: FIG. 1. The electron spectrum of LAtBLG with twist an [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Momentum dependence of the polarization opera [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Momentum dependence of the screened Coulomb [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the dependence of Tc (for ℓ = 2) on the doping concentrations n, calculated for three values of the bias voltage V . Characteristic values of the doping concentrations and the displacement fields D = V /(ed) are typical for the graphene based materials where the su￾perconductivity was observed1–7. We see that the Tc can be as large as several hundreds of milikelvin. If we ignore small deeps in interm… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The dependence of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The momentum dependence of the real and imag [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The dependence of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The dependence of Γ [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    describes the inter- layer hopping. In Ref. 32 within the framework of the continuum model, it was shown from symmetry consid- erations that for the structures with r ̸= 3n, the matrix ˆT ′ 12 should have the form ˆT ′ 12 = ( 0 meiφ ′ meiψ ′ 0 ) , (6) where m, φ ′ and ψ ′ are real parameters that depend on the superstructure. Numerical calculations of the...

  2. [2]

    The value of this gap is proportional to the real part of ∆ and is equal to 2∆ R

    we see that the spectrum has a gap. The value of this gap is proportional to the real part of ∆ and is equal to 2∆ R. If φ is not too close to ±π/ 2, then the gap value will be of the order of |m|. The parameter ∆ V describes the splitting of the bands of the same sign. It increases with the increase of the bias voltage. Typical spectrum of LAtBLG, calcul...

  3. [3]

    is shown in Fig. 1. Calculations performed in Refs. 32–34 show that the largest value of the parameter m corresponds to the su- perstructure m0 = 1, r = 1 (θ ∼ = 21. 79◦, Nsc = 7). This superstructure has the smallest value of Nsc of all su- perstructures with r ̸= 3 n. However, the value of m itself depends on the model parameters describing the inter-la...

  4. [4]

    The extra electron density is n = 4x/v c, where vc =a2√ 3/ 2 is the graphene’s unit cell area

    is the graphene’s Brillouin zone area. The extra electron density is n = 4x/v c, where vc =a2√ 3/ 2 is the graphene’s unit cell area. The factor 4 in the latter formula comes from the fact that the unit cell of each graphene layer contains two cites. The density of states (per one cite) is equal to ρ(E) = 4π ∆ VE vBZv2 F √ E2 − ∆ 2 R . (14) The density of...

  5. [5]

    Due to symmetry of the problem, the polarization operator depends only on the absolute value of the vector q

    can be safely extended up to infinity. Due to symmetry of the problem, the polarization operator depends only on the absolute value of the vector q. Fig- ure 2 shows the typical momentum dependence of the 5 intra- and inter-layer components of the polarization op- erator of doped and biased LAtBLG. At certain range of transfer momenta, the absolute value o...

  6. [6]

    (21) The second term in this formula includes only intra-layer interaction due to the momentum conservation low

    one can show that the interaction Hamiltonian, responsible for generation of such type of pairs, is equal to H inter int = 1 2N ∑ pp′ ∑ ijαβ ξσσ ′ ψ † ξpiασ ψ † ¯ξ− pjβσ ′Vij p− p′ψ ¯ξ− p′jβσ ′ψξp′iασ + 1 2N ∑ pp′ ∑ iαβ ξσσ ′ ψ † ξpiασ ψ † ¯ξ− piβσ ′Vii 2Ki ξ +p− p′ψξ− p′iβσ ′ψ ¯ξp′iασ . (21) The second term in this formula includes only intra-layer inter...

  7. [7]

    For this reason, we omit the last term of Eq

    will be much smaller than that from the first term. For this reason, we omit the last term of Eq. (

  8. [8]

    For further consideration it is convenient to intro- duce new electronic operators, apiασ = ψ +1piασ and bpiασ = ψ − 1piασ

    (the role of this term will be discussed below). For further consideration it is convenient to intro- duce new electronic operators, apiασ = ψ +1piασ and bpiασ = ψ − 1piασ . In terms of these operators, the to- tal Hamiltonian can be written as H = ∑ pσ ∑ ijαβ [ a† piασ ( Hiα ;jβ p − µ ) apjβσ − − b− piασ ( Hiα ;jβ p − µ ) b† − pjβσ ] − 8µ N + (22) + 1 N ...

  9. [9]

    and the fact that Vij − q =Vji q . B. Self-consistency equations for the order parameter As a next step we introduce normal and anomalous Matsubara Green’s functions [ ˆGσσ ′(τ − τ′, p) ]iα ;jβ ≡ Giα ;jβ σσ ′ (τ − τ′, p) = − ⟨Tτapiασ (τ)¯apjβσ ′(τ′)⟩ , [ ˆFσσ ′(τ − τ′, p) ]iα ;jβ ≡ Fiα ;jβ σσ ′ (τ − τ′, p) − ⟨ Tτ ¯b− piασ (τ)¯apjβσ ′(τ′) ⟩ , (23) whereapi...

  10. [10]

    We solve these equations in the limit T → Tc

    and ( 25) form a closed system of equations for finding the supercon- ducting order parameter and superconducting transition temperature, Tc. We solve these equations in the limit T → Tc. In this case, the equation for the order param- eter can be linearized. From second equation in (

  11. [11]

    As a result, the self-consistency equation (

    we can write ˆF (iωn, p) ∼ = − ˆG0(−iωn, p) ˆ∆ p ˆG0(iωn, p), (26) where ˆG0(iωn, p) = [ iωn +µ − ˆHp ]− 1 (27) is the Green’s function of the normal state. As a result, the self-consistency equation (

  12. [12]

    (28) To go further we will use the technique described, e.g., in Ref

    becomes ∆ iα ;jβ p = − T N ∑ n ∑ p′ ∑ lmµν Vij p− p′Giα ;lµ 0 (−iωn, p′) × ∆ lµ ;mν p′ Gmν ;jβ 0 (iωn, p′). (28) To go further we will use the technique described, e.g., in Ref. 29. Namely, we write down the normal Green’s function in the form Giα ;jβ 0 (iωn, p) = 4∑ S=1 Φ (S) piα Φ (S)∗ pjβ iωn +µ − ε(S) p . (29) We also proceed from the order parameter ...

  13. [13]

    and ( 30) and performing the frequency summation in Eq. ( 28), we obtain the linearized equation for the order parameter ∆ SS ′p: ∆ S1S2p = − ∑ S′ 1S′ 2 ∫ d2p′ (2π )2 Γ (S1S2;S′ 1S′ 2) pp′ ∆ S′ 1S′ 2p′ × nF ( −εS′ 1 p′ ) − nF ( εS′ 2 p′ ) ε S′ 1 p′ +ε S′ 2 p′ , (31) where Γ (S1S2;S′ 1S′ 2) pp′ = ∑ ijαβ Φ (S1)∗ piα Φ (S′ 1) p′iαVij p− p′Φ (S2) pjβ Φ (S′ 2)...

  14. [14]

    For the reason described above we can consider both odd and even ℓ for the spin-singlet (and spin-triplet) inter- valley order parameter

    in the form ∆ S1S2p = e− iϕ pℓ∆ (ℓ) S1S2p, where ℓ = 0, ± 1,... . For the reason described above we can consider both odd and even ℓ for the spin-singlet (and spin-triplet) inter- valley order parameter. We also introduce the following quantity: δ(ℓ) S1S2p = ∆ (ℓ) S1S2pfS1S2p, where fS1S2p =    √ p nF ( −εS1 p ) − nF ( εS2 p ) εS1 p +εS2 p . (34) The e...

  15. [15]

    (37) From this relation, we obtain T (− ℓ) c =T (ℓ) c and ∆ (− ℓ) S1S2p = ∆ (ℓ)∗ S2S1p

    and ( 36) it follows that w(− ℓ) S1S2;S′ 1S′ 2 (p, p′) = w(ℓ) S2S1;S′ 2S′ 1 (p, p′)∗. (37) From this relation, we obtain T (− ℓ) c =T (ℓ) c and ∆ (− ℓ) S1S2p = ∆ (ℓ)∗ S2S1p. Thus, we can consider only positive ℓ (the state with ℓ = 0 is absolutely unstable since interaction is re- pulsive). The components of the order parameter ∆ (ℓ) S1S2p, responsible to...

  16. [16]

    we calculate both transition tem- perature and the momentum dependence of the order parameter ∆ (ℓ) S1S2p near Tc. Figure 7 shows the typi- 8 /s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s45/s48/s46/s53 /s48 /s48/s46/s53 /s49/s46/s48 /s32 /s82 /s101 /s32 /s40 /s50 /s41 /s83 /s48 /s83 /s48 /s112 /s47 /s48 /s112/s97 /s45/s48/s46/s49/s48 /s45/s48...

  17. [17]

    with the replacement ∆ SS ′p → ˜∆ SS ′p and Γ (S1S2;S′ 1S′ 2) pp′ → ˜Γ (S1S2;S′ 1S′ 2) pp′ , where [c.f. with Eq. ( 32)] ˜Γ (S1S2;S′ 1S′ 2) pp′ = ∑ ijαβ Φ (S1) − piα Φ (S′ 1)∗ − p′iαVij p− p′Φ (S2) pjβ Φ (S′ 2)∗ p′jβ . (48) The transition temperature to the intra-valley super- conducting state for different types of ordering ( p-, d− , f − and so on), ˜T (...

  18. [18]

    Thus, the most stable intra-valley state is of the spin- triplet type

    with minus sign. Thus, the most stable intra-valley state is of the spin- triplet type. We also obtained numerically that ˜T (3) c = ˜T (− 1) c = T (2) c (= T (− 2) c ), where T (± 2) c is the transition tempera- ture of the inter-valley d-wave state. To understand the origin of this degeneracy we analized the relationship be- tween ˜Γ (S1S2;S′ 1S′ 2) pp′...

  19. [19]

    Thus, the absolute values of ˜Γ (S1S2;S′ 1S′ 2) pp′ and Γ (S1S2;S′ 1S′ 2) pp′ coincide

    follows from the symmetry of the wave functions Φ (S) piα . Thus, the absolute values of ˜Γ (S1S2;S′ 1S′ 2) pp′ and Γ (S1S2;S′ 1S′ 2) pp′ coincide. From the relation ( 49) it follows that ˜∆ (ℓ) S1S2p =eiλ (S1S2) p ∆ (ℓ− 1) S1S2p. (50) The intra-valley ground state order parameter, which opens the gap at the Fermi level is equal to ˜∆ (2) p = ˜∆ (3) S0S0p...

  20. [20]

    The second term in Eq

    we have a three-fold degeneracy of the d-wave singlet and triplet inter-valley and spin-triplet intra-valley superconducting states. The second term in Eq. (

  21. [21]

    Stripe Order in the Metallic and Superconducting Phases of Rhombohedral Hexalayer Graphene

    lifts partially this degeneracy making the singlet inter-valley state to be the ground state. Note also that we do not take into account the possible trigonal warping in our model. The trigonal warping makes ε(S) − p ̸= ε(S) p which plays against the intra-valley pairing. 10 VI. DISCUSSION AND CONCLUSIONS A. Comparison with other graphene bilayers Thus, o...

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