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arxiv: 2605.21618 · v1 · pith:UWOAHKDAnew · submitted 2026-05-20 · ❄️ cond-mat.supr-con · cond-mat.str-el

Chiral superconductors from parent states with non-uniform Berry curvature: Momentum-space vortices, BdG topology, and thermal Hall conductivity

L. David Le Nir , Asimpunya Mitra , Yong Baek Kim This is my paper

Pith reviewed 2026-05-22 08:42 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords chiral superconductivityBerry curvaturemomentum-space vorticesBdG topologythermal Hall conductivityChern numberrhombohedral graphene
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The pith

Non-uniform Berry curvature in a parent band nucleates momentum-space vortices in the chiral superconducting gap, with their number fixed by the parent Chern number.

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

The paper establishes that chiral superconductivity arising from electronic states with spatially varying Berry curvature develops additional structure in the gap function. Using a tunable continuum model, the authors solve the BCS gap equation and show that the parent band's Berry curvature profile induces vortices in momentum space away from high-symmetry points. These vortices are not arbitrary; their possible number is strictly limited by the Chern number of the underlying band. The resulting Bogoliubov-de Gennes topology and its measurable consequences, such as thermal Hall response, follow directly from this constraint. A reader would care because the mechanism supplies a concrete, testable link between measurable band geometry and the internal structure of a chiral superconductor.

Core claim

In the continuum λ_N-model that allows independent control of Berry curvature profiles, solving the full BCS gap equation beyond weak coupling reveals that non-uniform Berry curvature enriches the superconducting order parameter and produces momentum-space vortices in the gap function. The parent band Chern number constrains the number of such vortices that can form, independently of the specific details of the λ_N-model. A gauge-invariant expression for the BdG Berry curvature is derived and shown to be set by the winding of a momentum-space phase current around the occupied vortices, which in turn fixes the BdG Chern number. Thermal Hall conductivity is identified as a probe that can track

What carries the argument

momentum-space vortices in the gap function, whose nucleation is driven by non-uniform Berry curvature of the parent band and whose number is constrained by the parent Chern number.

If this is right

  • Tuning the Berry curvature profile produces distinct regimes of vortex nucleation followed by saturation at a number set by the Chern number.
  • Vortex nucleation lowers the condensation energy relative to the uniform-curvature case.
  • The BdG Chern number is determined by the total winding of the phase current around vortices inside the occupied region.
  • Thermal Hall measurements can detect the presence and number of these momentum-space vortices.

Where Pith is reading between the lines

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

  • The same Berry-curvature-driven vortex mechanism may operate in other Chern bands that host superconductivity, such as certain moiré superlattices.
  • Engineering the parent Berry curvature profile could provide a route to control the number of vortices and thereby the topological character of the superconductor.
  • If the vortices survive disorder or lattice effects, they may produce additional signatures in spectroscopic probes beyond thermal Hall conductivity.

Load-bearing premise

The continuum λ_N-model supplies a faithful, tunable representation of the essential physics of parent states in rhombohedral graphene multilayers when the full BCS gap equation is solved.

What would settle it

In a material whose parent band has a known Chern number, if the number of momentum-space vortices extracted from the gap function or from thermal Hall data exceeds or falls short of that Chern number, the claimed constraint is falsified.

Figures

Figures reproduced from arXiv: 2605.21618 by Asimpunya Mitra, L. David Le Nir, Yong Baek Kim.

Figure 1
Figure 1. Figure 1: FIG. 1. The Berry curvature distribution of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The variation of the magnitude [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) likewise tends to an N-independent value. The condensation energy also increases monotonically with N at fixed B, reflecting the broadening of the Berry curvature plateau: a wider plateau suppresses the gap over a larger region, increasing the condensation energy accordingly. For N = ∞ (or the ICB), increasing B leads to an unbounded growth in the number of nucleated vor￾tices, accompanied by a monoton… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The real part of the Cooper vertex [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Columns (1)–(3) illustrate vortices migrating toward [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The gap structure for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The coherence factor [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The thermal Hall conductance [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) shows qualitatively how the gap function can be approximately captured by this ansatz with NV = 9 vortices. We may use this ansatz to try to recreate the trend in the condensation energy as the radius of the vortex ring contracts. We show this trend in [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Gap structure as a function of increasing interaction [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Top panel shows that vortex number [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. All panels are similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The variation of the gap structure with increasing [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The results with an annular Fermi surface. (a) The [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
read the original abstract

We investigate chiral superconductivity emerging from parent electronic states with non-uniform Berry curvature, motivated by recent experiments in rhombohedral graphene multilayers. Using the continuum $\lambda_N$-model-a tunable platform with independently controllable Berry curvature profiles-we solve the full BCS gap equation on a continuum Chern band beyond the weak-coupling limit. We find that a non-uniform Berry curvature of the parent band enriches the superconducting order parameter, leading to the formation of momentum-space vortices in the gap function away from high-symmetry points. By tuning the Berry curvature profile, we identify distinct regimes associated with vortex nucleation and vortex number saturation, and show that the nucleation of momentum-space vortices tends to lower the condensation energy. We then show analytically that the parent band Chern number constrains the number of momentum-space vortices that can nucleate in the gap-independent of details of the $\lambda_N$-model. We also provide a gauge-invariant formulation for computing the Bogoliubov-de Gennes (BdG) Berry curvature for continuum models, and find that it is determined by a momentum-space phase current. The winding of this current around vortices in the occupied region in turn determines the BdG Chern number. Finally, we discuss how thermal Hall measurements can be used to probe the formation of momentum-space vortices. Our results highlight the crucial role of Berry curvature in shaping chiral superconductivity, and offer guiding principles for its identification in systems such as rhombohedral graphene.

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

1 major / 1 minor

Summary. The paper investigates chiral superconductivity emerging from parent electronic states with non-uniform Berry curvature, motivated by experiments in rhombohedral graphene multilayers. Using the continuum λ_N-model as a tunable platform, the authors solve the full BCS gap equation beyond weak coupling and report that non-uniform Berry curvature enriches the superconducting order parameter, producing momentum-space vortices in the gap function away from high-symmetry points. They identify regimes of vortex nucleation and saturation, show that vortex formation lowers condensation energy, and present an analytical argument that the parent band's Chern number constrains the number of vortices independently of λ_N-model details. The work also introduces a gauge-invariant formulation for the BdG Berry curvature based on a momentum-space phase current whose winding determines the BdG Chern number, and discusses thermal Hall conductivity as a probe for these vortices.

Significance. If the central results hold, the manuscript offers valuable insights into how Berry curvature shapes chiral superconducting order parameters in continuum Chern bands. Strengths include the solution of the full BCS gap equation, the identification of momentum-space vortices and their energetic favorability, the gauge-invariant BdG formulation, and the proposed link to thermal Hall measurements. These elements could provide guiding principles for interpreting experiments in systems such as rhombohedral graphene multilayers.

major comments (1)
  1. [Analytical argument following the gap-equation solution] The analytical claim that the parent band Chern number constrains the number of momentum-space vortices independently of λ_N-model details (abstract and associated derivation) rests on a global winding or index relation for Δ(k). Because the continuum model controls large-|k| decay of both dispersion and pairing kernel via the tunable λ_N parameters, the counting argument may inherit continuum-specific assumptions. Explicit verification that the same vortex-number constraint survives lattice discretization (compact Brillouin zone, periodic pairing) or an alternative momentum cutoff would strengthen the independence statement.
minor comments (1)
  1. The abstract refers to 'distinct regimes associated with vortex nucleation and vortex number saturation'; labeling these regimes explicitly in the main text or a figure caption with the corresponding λ_N parameter ranges would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive suggestions. We are pleased that the referee recognizes the significance of our results on momentum-space vortices in chiral superconductors and the gauge-invariant BdG formulation. Below, we provide a point-by-point response to the major comment.

read point-by-point responses

  1. Referee: [Analytical argument following the gap-equation solution] The analytical claim that the parent band Chern number constrains the number of momentum-space vortices independently of λ_N-model details (abstract and associated derivation) rests on a global winding or index relation for Δ(k). Because the continuum model controls large-|k| decay of both dispersion and pairing kernel via the tunable λ_N parameters, the counting argument may inherit continuum-specific assumptions. Explicit verification that the same vortex-number constraint survives lattice discretization (compact Brillouin zone, periodic pairing) or an alternative momentum cutoff would strengthen the independence statement.

    Authors: We appreciate the referee's careful scrutiny of our analytical argument. The derivation indeed employs a global winding relation for the gap function Δ(k), which is tied to the parent band's Chern number through the integral of the Berry curvature. This relation arises from the topological properties of the band and the structure of the BCS gap equation in momentum space. The λ_N parameters control the decay rates to ensure the integrals converge and the Chern number is well-defined, but the constraint on the vortex number is a consequence of the index theorem-like relation and holds for any Berry curvature profile that yields the same Chern number, independent of the specific functional details within the λ_N family. Regarding potential continuum-specific assumptions, we note that the argument does not rely on the absence of a lattice but on the existence of a momentum-space topology and sufficient decay at large momenta, conditions that can be met in discretized models with appropriate regularization. To strengthen the presentation, we have revised the manuscript to include a short paragraph elaborating on the generality of the argument and its applicability beyond the specific continuum setup. We believe this addresses the concern without requiring a full lattice calculation, which would constitute a separate project. revision: partial

Circularity Check

0 steps flagged

No significant circularity; topological constraint derived independently of model details

full rationale

The paper solves the full BCS gap equation numerically within the tunable continuum λ_N-model to identify vortex nucleation regimes, then separately provides an analytical argument that the parent band's Chern number (integral of Berry curvature) constrains the total number of momentum-space vortices in the gap function. This constraint is explicitly stated to hold independent of λ_N details, indicating a general topological relation rather than a reduction to fitted parameters, self-definition, or model-specific ansatz. The BdG Berry curvature formulation is derived from a momentum-space phase current whose winding determines the BdG Chern number, but this follows from gauge-invariant definitions without circular redefinition of inputs. No load-bearing self-citations or uniqueness theorems imported from prior author work are invoked for the central claims. The model serves as an exploratory platform whose outputs are then constrained by an external topological identity, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The λ_N-model is introduced as a tunable platform with controllable Berry curvature profiles; these profiles function as free parameters fitted or chosen to match target systems. The parent band Chern number is treated as an input from prior band topology. No invented entities such as new particles are introduced. The full list of assumptions cannot be audited without the manuscript body.

free parameters (1)
  • Berry curvature profile parameters in λ_N-model
    Tunable parameters that independently control the non-uniform Berry curvature shape in the continuum model.
axioms (1)
  • domain assumption The continuum λ_N-model accurately represents the essential electronic structure of rhombohedral graphene multilayers for superconductivity calculations.
    Invoked to motivate the choice of model for solving the BCS gap equation.

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Reference graph

Works this paper leans on

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