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arxiv: 2603.17362 · v2 · submitted 2026-03-18 · ❄️ cond-mat.supr-con

Extended Hubbard model on fractals: d-Wave superconductivity and competing pairing channels

Robert Canyellas , Mikhail I. Katsnelson , Andrey Bagrov This is my paper

Pith reviewed 2026-05-15 09:06 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords extended Hubbard modelfractal latticesSierpiński carpetd-wave superconductivityextended s-wave pairinggeometric frustrationBogoliubov-de Gennespairing symmetry competition
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The pith

The Sierpiński carpet fractal lattice suppresses the d-wave superconducting dome at half filling while strongly enhancing extended s-wave pairing at other fillings.

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

The paper applies Bogoliubov-de Gennes mean-field theory to the extended Hubbard model with nearest-neighbor attraction on two fractal lattices. It shows that the irregular boundaries of the Sierpiński carpet frustrate sign-changing order parameters, eliminating the d-wave dome that dominates the square lattice at half filling. Extended s-wave pairing instead becomes dominant at high and low fillings. On the triangular Sierpiński gasket the same method finds that hybrid s+d+id states reach critical temperatures comparable to those previously reported for pure s-wave pairing. The central result is that fractal topology acts as a selective filter that stabilizes or suppresses specific pairing symmetries according to how well their nodal structure matches the lattice geometry.

Core claim

Using Bogoliubov-de Gennes mean-field theory on the extended Hubbard model with nearest-neighbor attraction, the authors find that the Sierpiński carpet dramatically alters the competition between pairing channels: the predominant d-wave superconducting dome at half filling of the square lattice becomes unstable for the carpet, while at high and low fillings extended s-wave pairing gets strongly enhanced. They attribute this to geometric frustration of sign-changing order parameters by the fractal boundary structure. On the triangular Sierpiński gasket, hybrid s+d+id states show critical temperature enhancement comparable to that previously observed for pure s-wave pairing.

What carries the argument

Geometric frustration of sign-changing superconducting order parameters imposed by the irregular boundary structure of the Sierpiński carpet, within the Bogoliubov-de Gennes mean-field treatment of the extended Hubbard model.

If this is right

  • d-wave superconductivity is eliminated at half filling on the carpet while extended s-wave pairing is stabilized at high and low fillings.
  • Hybrid s+d+id states on the gasket reach critical temperatures comparable to those of pure s-wave pairing.
  • Fractal geometry selectively filters pairing symmetries according to the compatibility of their nodal structure with the lattice boundaries.
  • The same geometric mechanism that suppresses d-wave order can be used to tune the dominant channel by changing electron filling.

Where Pith is reading between the lines

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

  • Fractal boundary engineering could be explored as a route to stabilize s-wave or hybrid pairing in artificial nanostructures where d-wave order is normally preferred.
  • The selective-filter effect may extend to other fractal geometries or to models with longer-range interactions, offering a design principle for controlling superconducting symmetry.
  • Quantitative comparison of mean-field Tc values with fluctuation-corrected methods on the same fractals would test how robust the reported channel competition remains beyond the approximation.

Load-bearing premise

The Bogoliubov-de Gennes mean-field approximation remains quantitatively reliable for capturing pairing competition and critical temperatures on these irregular fractal lattices without significant corrections from fluctuations or exact many-body effects.

What would settle it

Quantum Monte Carlo or exact diagonalization calculations on finite Sierpiński-carpet clusters that find a stable d-wave solution persisting at half filling or that yield lower critical temperatures for extended s-wave pairing than the mean-field prediction.

Figures

Figures reproduced from arXiv: 2603.17362 by Andrey Bagrov, Mikhail I. Katsnelson, Robert Canyellas.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Phase diagrams of the thermodynamic limit of the square lattice at the left, a square lattice flake with side [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fig.4. The symmetry of these domes reflects the bipar [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Phase diagrams of the thermodynamic limit of the triangular lattice at the left, an equilateral triangle flake [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Phase diagrams of the thermodynamic limit of the honeycomb lattice at the left, an equilateral triangle [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Profiles of the order parameter for the different geometries/pairing states at [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Phase stiffness for the different geometries/pairing states at [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
read the original abstract

Fractal structures such as the Sierpi\'nski gasket have been predicted to enhance the critical temperature of s-wave superconductivity compared to regular crystals while maintaining macroscopic phase coherence of Cooper pairs. Here we extend this analysis to order parameters with non-trivial symmetry by studying the extended Hubbard model with nearest-neighbor attraction on fractal lattices. Using Bogoliubov-de Gennes mean-field theory, we find that the Sierpi\'nski carpet dramatically alters the competition between pairing channels: the predominant d-wave superconducting dome at half filling of the square lattice becomes unstable for the carpet, while at high and low fillings extended s-wave pairing gets strongly enhanced. We attribute this to geometric frustration of sign-changing order parameters by the fractal boundary structure. On the triangular Sierpi\'nski gasket, hybrid s+d+id states show critical temperature enhancement comparable to that previously observed for pure s-wave pairing. Our results demonstrate that fractal geometry acts as a selective filter for pairing symmetries, with the compatibility between order parameter structure and lattice topology determining which channels are stabilized or suppressed.

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 manuscript applies Bogoliubov-de Gennes mean-field theory to the extended Hubbard model with nearest-neighbor attraction on Sierpiński carpet and gasket fractals. It reports that the carpet suppresses the d-wave superconducting dome present at half-filling on the square lattice, while strongly enhancing extended s-wave pairing at high and low fillings; this is attributed to geometric frustration of sign-changing order parameters by the fractal boundaries. On the triangular gasket, hybrid s+d+id states exhibit critical-temperature enhancement comparable to prior s-wave results.

Significance. If the mean-field results prove robust, the work demonstrates that fractal geometry can act as a selective filter for pairing symmetries, stabilizing or suppressing channels according to compatibility with lattice topology. This extends earlier s-wave studies on fractals and suggests routes to engineer specific order parameters or higher Tc via boundary-induced frustration.

major comments (2)
  1. [Methods and Results sections on BdG implementation and carpet at half filling] The central claims on d-wave instability at half-filling and the crossover to extended s-wave rest entirely on self-consistent BdG solutions. No comparisons to fluctuation-corrected approaches (RPA, cluster DMFT) or exact methods on small clusters are provided, despite the carpet having sites with coordination numbers 2–4 where local fluctuations are expected to be large and could stabilize sign-changing order.
  2. [Results paragraph discussing channel competition on the carpet] The attribution of d-wave suppression to 'geometric frustration by the fractal boundary structure' lacks supporting diagnostics such as site-resolved order-parameter signs, phase windings at corners, or a control calculation on a non-fractal lattice with comparable boundaries.
minor comments (2)
  1. The abstract and methods should explicitly state the ranges of Hubbard parameters U and V, the fractal iteration depths, and the convergence criteria (energy tolerance, iteration count) used for the self-consistent solutions.
  2. Figure captions for Tc domes and order-parameter maps should clarify color scales, whether values are averaged or site-resolved, and any smoothing applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify the scope and limitations of our mean-field study. We address each major comment point by point below.

read point-by-point responses

  1. Referee: The central claims on d-wave instability at half-filling and the crossover to extended s-wave rest entirely on self-consistent BdG solutions. No comparisons to fluctuation-corrected approaches (RPA, cluster DMFT) or exact methods on small clusters are provided, despite the carpet having sites with coordination numbers 2–4 where local fluctuations are expected to be large and could stabilize sign-changing order.

    Authors: We agree that the results rely on self-consistent Bogoliubov-de Gennes mean-field theory. This approach is standard for mapping out pairing instabilities and their competition on complex, non-periodic lattices such as fractals, where the irregular connectivity makes cluster DMFT or RPA implementations computationally prohibitive at the system sizes required to capture the fractal boundaries. While local fluctuations at low-coordination sites (coordination 2–4) are indeed expected to be stronger than on the square lattice, the geometric frustration mechanism we identify arises from the global topology and boundary structure rather than local coordination alone; mean-field therefore still captures the qualitative suppression of sign-changing order. In the revised manuscript we have added a dedicated paragraph in the Methods and Discussion sections explicitly discussing these limitations of the mean-field approximation and outlining how exact diagonalization on small clusters or future fluctuation-corrected calculations could provide further validation. revision: partial

  2. Referee: The attribution of d-wave suppression to 'geometric frustration by the fractal boundary structure' lacks supporting diagnostics such as site-resolved order-parameter signs, phase windings at corners, or a control calculation on a non-fractal lattice with comparable boundaries.

    Authors: We thank the referee for this helpful suggestion. In the revised Results section we now include site-resolved maps of the superconducting order parameter for both d-wave and extended s-wave channels on the carpet at half filling and at representative dopings. These maps explicitly show the sign changes of the d-wave order parameter being disrupted at the fractal corners and edges, while the extended s-wave order parameter remains uniform. We have also added a control calculation on a square lattice with artificially introduced boundaries that mimic the perimeter of the carpet; this non-fractal control does not exhibit the same d-wave suppression, thereby supporting the role of the self-similar fractal structure in generating the geometric frustration. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical BdG solution on fractals

full rationale

The paper applies standard Bogoliubov-de Gennes mean-field theory to the extended Hubbard model with nearest-neighbor attraction on explicitly constructed fractal lattices (Sierpiński carpet and gasket). Pairing-channel competition and critical temperatures are obtained by direct numerical solution of the self-consistent gap equations on the site-inhomogeneous graphs, with order-parameter symmetries (d-wave, extended s-wave, hybrid s+d+id) defined independently from the lattice geometry. No parameters are fitted to data subsets and then relabeled as predictions; no load-bearing claims reduce to self-citations or prior ansatzes by the same authors; the geometric-frustration interpretation follows from the explicit lattice structure and sign-changing properties of the order parameters. The derivation chain is therefore self-contained in the numerical implementation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on numerical implementation of BdG mean-field equations on fractal lattices with chosen interaction strengths; no new entities are postulated.

free parameters (1)
  • Hubbard interaction parameters
    On-site repulsion U and nearest-neighbor attraction V are chosen to study the model but specific values are not stated in the abstract.
axioms (1)
  • domain assumption Bogoliubov-de Gennes mean-field theory accurately describes the superconducting order parameter competition on fractal lattices
    Standard approximation invoked for solving the model; assumes fluctuations are negligible.

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Using Bogoliubov-de Gennes mean-field theory, we find that the Sierpiński carpet dramatically alters the competition between pairing channels: the predominant d-wave superconducting dome at half filling of the square lattice becomes unstable for the carpet, while at high and low fillings extended s-wave pairing gets strongly enhanced. We attribute this to geometric frustration of sign-changing order parameters by the fractal boundary structure.

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The extended Hubbard model with both on-site (U) and nearest-neighbor (V) attractive interactions

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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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