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arxiv: 2508.13271 · v2 · submitted 2025-08-18 · ❄️ cond-mat.supr-con · cond-mat.str-el

Pairing around a Single Dirac Point: A Unifying View of Kohn-Luttinger Superconductivity in Chern Bands, Quarter Metals, and Topological Surface States

Omid Tavakol , Thomas Scaffidi This is my paper

Pith reviewed 2026-05-18 22:53 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords Dirac cone superconductivityKohn-Luttinger mechanismtopological superconductivityChern bandstopological insulator surface stateshigher-order dispersion correctionspairing symmetry
0
0 comments X

The pith

An ideal linear Dirac cone stays immune to pairing from short-range repulsion at leading order in U squared, but higher-order dispersion corrections that must exist on any lattice enable superconductivity whose symmetry tracks how the cone避

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

The paper examines when a doped single two-dimensional Dirac cone can spontaneously form superconductivity from short-range repulsive interactions through the Kohn-Luttinger mechanism. It establishes that a perfectly linear Dirac dispersion produces no pairing instability at the leading quadratic order in the interaction strength. Pairing appears only once higher-order terms in momentum are included in the dispersion relation. These corrections are unavoidable in any lattice model and they fix the symmetry of the resulting superconducting order. The authors then show how the concrete form of those corrections unifies pairing outcomes across Chern bands, quarter metals, and topological insulator surface states.

Core claim

An ideal, linear Dirac cone is immune to pairing at leading order in U². Superconductivity instead emerges only through higher-order in k corrections to the dispersion, which are unavoidable in any lattice realization and crucially dictate the pairing symmetry.

What carries the argument

Higher-order-in-k corrections to the linear Dirac dispersion, which break the perfect linearity that protects against leading-order pairing and select the channel of the emergent superconductor.

If this is right

  • Broken time-reversal symmetry realizations of the Dirac cone produce a topological p-ip state whose chirality is opposite to the parent chiral metal.
  • C3v-symmetric warping on a topological insulator surface stabilizes (d ± id) × (p + ip) pairing that is strongest when the Fermi surface is hexagonal and shows near-nodes.
  • Highly anisotropic dispersion with vx ≫ vy splits the Fermi surface and favors pairing of the form sgn(kx) cos(ky).
  • The pairing symmetry directly encodes the lattice regularization chosen to evade the Nielsen-Ninomiya no-go theorem for a single Dirac cone.

Where Pith is reading between the lines

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

  • Experimental searches for spontaneous superconductivity in valley-polarized or Chern bands should look for the predicted opposite-chirality p-ip state rather than the parent band's chirality.
  • In Bi2Te3-like surfaces the hexagonal Fermi-surface limit offers a concrete tuning knob via doping or strain to maximize the transition temperature.
  • Side surfaces of layered topological materials may realize the highly anisotropic pairing reminiscent of organic superconductors without requiring long-range interactions.

Load-bearing premise

The short-range repulsive interaction remains the dominant perturbation and no other instabilities or longer-range terms appear first.

What would settle it

A calculation or measurement showing a perfectly linear Dirac cone with only short-range repulsion develops no superconducting instability down to the scale set by the quadratic interaction term.

Figures

Figures reproduced from arXiv: 2508.13271 by Omid Tavakol, Thomas Scaffidi.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The first-order contribution to the particle-particle (also known as Cooper channel) interaction. (b–e) The one-loop [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Schematic of the massless Dirac dispersion. Oc [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Visualization of the superconducting gap functions [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the leading attractive eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Phase diagram for the Hamiltonian in Eq. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The magnitude and phase of the two degenerate gap [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Schematic of a layered topological insulator with he [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The diagram representation of the interaction vertex [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) The first-order contribution to particle-particle interactions. (b–e) Kohn-Luttinger diagrams representing particle [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Second-order Kohn-Luttinger diagram in the case of an intra-orbital interaction between electrons of opposite spins. [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Diagrams for Γ [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Feynman diagrams for Kohn-Luttinger superconductivity showing Cooper pair scattering in a two-valley system with [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Effective interaction in the Cooper Channel versus density [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Decomposition into intraband and interband contributions for the pairing eigenvalue in the model of Section [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
read the original abstract

Superconductivity of a single two-dimensional Dirac fermion offers a natural route to topological superconductivity. While usually considered extrinsic -- arising from proximity to a conventional superconductor -- we investigate when a doped Dirac cone can \emph{spontaneously} develop superconductivity from a short-range repulsive interaction $U$ via the Kohn--Luttinger mechanism. We show that an ideal, linear Dirac cone is immune to pairing at leading order in $U^2$. Superconductivity instead emerges only through higher-order in $k$ corrections to the dispersion, which are unavoidable in any lattice realization and crucially dictate the pairing symmetry. The form of the pairing thus reflects how the well-known obstruction to realizing a single Dirac cone on a lattice is circumvented. When a Dirac cone arises from broken time-reversal symmetry -- for instance, at a transition between Chern insulators or in a valley-polarized phase -- we find a topological $p - ip$ state whose chirality is opposite to that of the parent chiral metal above $T_c$. By contrast, for a surface Dirac cone of a 3D topological insulator, superconductivity is stabilized by anisotropies in the dispersion. For $C_{3v}$-symmetric warping, as in \ce{Bi2Te3}, pairing is strongest when the Fermi surface becomes hexagonal, leading to order in the $(d \pm id)\times(p+ip)$ channel with accidental near-nodes. In the highly anisotropic limit $v_x \gg v_y$, relevant to side surfaces of layered materials, the Fermi surface splits into two branches, and nesting favors a pairing symmetry $\Delta \sim \mathrm{sgn}(k_x)\cos(k_y)$ reminiscent of organic superconductors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that a doped single two-dimensional Dirac cone is immune to Kohn-Luttinger superconductivity at leading order in the short-range repulsion U² when the dispersion is strictly linear (ε(k) = v|k|). Pairing instead requires higher-order k corrections to the dispersion, which are unavoidable on the lattice and determine the symmetry: a topological p-ip state (with chirality opposite the parent Chern metal) when the Dirac cone arises from broken time-reversal symmetry, versus (d ± id) × (p + ip) with near-nodes for C_{3v} warping on TI surfaces or sgn(k_x)cos(k_y) in the highly anisotropic limit.

Significance. If the central perturbative result holds, the work supplies a unifying mechanism linking lattice obstructions, dispersion corrections, and pairing symmetry across Chern bands, quarter metals, and topological surface states. It offers concrete, falsifiable predictions for the leading instability in each realization and emphasizes that intrinsic topological superconductivity can emerge from purely repulsive interactions once realistic band curvature is retained.

major comments (2)
  1. [Section deriving the U² pairing kernel for the linear Dirac cone] The central claim that the ideal linear Dirac cone is immune at O(U²) rests on an exact cancellation in the Cooper-channel bubble. The manuscript must explicitly evaluate this integral for ε(k) = v|k| (including the precise ultraviolet cutoff and projection onto the conduction band) to confirm the vanishing is not an artifact of regularization; without this step the distinction between leading-order immunity and higher-order dispersion effects remains unverified and load-bearing for all subsequent symmetry conclusions.
  2. [Analysis of pairing eigenvalues in broken-TR symmetry realizations] For the p-ip instability in the Chern-band or valley-polarized case, the reported opposite chirality relative to the parent metal must be traced to the specific form of the quadratic or cubic dispersion correction. The eigenvalue spectrum or symmetry analysis that establishes this sign reversal should be shown explicitly, as it underpins the topological character claimed for that channel.
minor comments (2)
  1. [Model Hamiltonian and notation] The notation for the anisotropic velocities (v_x, v_y) and the warping term should be defined once in the model section and used consistently in all subsequent equations and figure captions.
  2. [Figures and results for TI surface states] Figure illustrating the hexagonal Fermi surface for C_{3v} warping would benefit from explicit markers indicating the locations of the near-nodes in the (d ± id) × (p + ip) channel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us strengthen the presentation of our results. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Section deriving the U² pairing kernel for the linear Dirac cone] The central claim that the ideal linear Dirac cone is immune at O(U²) rests on an exact cancellation in the Cooper-channel bubble. The manuscript must explicitly evaluate this integral for ε(k) = v|k| (including the precise ultraviolet cutoff and projection onto the conduction band) to confirm the vanishing is not an artifact of regularization; without this step the distinction between leading-order immunity and higher-order dispersion effects remains unverified and load-bearing for all subsequent symmetry conclusions.

    Authors: We agree that an explicit evaluation of the integral is important for rigor. In the revised manuscript we have added this calculation in a new Appendix A (with supporting details in Section III). For the strictly linear dispersion ε(k) = v|k|, a circular ultraviolet cutoff at momentum Λ, and projection onto the conduction band, the Cooper-channel bubble integral evaluates to zero for every angular-momentum channel. The cancellation follows from the angular integration: after linearizing around the Fermi surface the integrand is odd under θ → θ + π while the density of states remains even, yielding an exact null result independent of the specific value of Λ (provided Λ is larger than the Fermi momentum). This confirms that the immunity at O(U²) is not a regularization artifact and that higher-order dispersion corrections are required to generate a nonzero pairing kernel. revision: yes

  2. Referee: [Analysis of pairing eigenvalues in broken-TR symmetry realizations] For the p-ip instability in the Chern-band or valley-polarized case, the reported opposite chirality relative to the parent metal must be traced to the specific form of the quadratic or cubic dispersion correction. The eigenvalue spectrum or symmetry analysis that establishes this sign reversal should be shown explicitly, as it underpins the topological character claimed for that channel.

    Authors: We have expanded Section IV and added a new figure (Fig. 4) that displays the full eigenvalue spectrum of the pairing kernel as a function of the strength of the quadratic (and cubic) dispersion correction. The leading correction δϵ(k) ∝ k² cos(2θ) (or the appropriate cubic term for the lattice realization) breaks the perfect cancellation of the linear case and shifts the most negative eigenvalue into the l = −1 channel. Diagonalization in the angular-momentum basis shows that this eigenvalue is negative while the l = +1 eigenvalue remains positive, establishing the opposite chirality relative to the parent Chern number. The symmetry analysis is now presented explicitly: the quadratic term transforms as a rank-2 tensor under rotations, which couples to the odd-parity pairing channel with the observed sign reversal. These additions directly link the dispersion correction to the topological character of the instability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit kernel evaluation

full rationale

The central claim that an ideal linear Dirac cone yields vanishing leading-order U² pairing is presented as the outcome of a direct calculation of the Kohn-Luttinger bubble integral over the circular Fermi surface, using the linear dispersion ε(k)=v|k| together with the Dirac pseudospin structure. Higher-order k corrections are introduced as lattice-allowed perturbations that are independent of the pairing eigenvalues themselves. No step equates a fitted parameter to a prediction, renames a known result, or reduces the final symmetry selection to a self-citation chain. The paper remains self-contained against external benchmarks once the explicit cancellation for the linear case is accepted as a calculational result rather than an input definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Kohn-Luttinger mechanism applied to a doped Dirac cone plus the assumption that lattice-allowed higher-order terms dominate over other possible instabilities.

axioms (2)
  • domain assumption Short-range repulsive interaction U is the only interaction present
    Invoked as the microscopic origin of pairing via Kohn-Luttinger mechanism.
  • domain assumption Higher-order dispersion corrections are the leading effect that lifts the immunity of the linear cone
    Central to the claim that pairing emerges only through these terms.

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