arxiv: 2605.11211 · v1 · submitted 2026-05-11 · ❄️ cond-mat.supr-con · cond-mat.mes-hall· cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

First-principles real-space embedding theory of the superconducting proximity effect

Nicolas Ba\`u , Mitra Dowlatabadi , Tommaso Chiarotti , Massimo Capone , Antimo Marrazzo

Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:10 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mes-hallcond-mat.mtrl-sci

keywords superconducting proximity effectreal-space embeddingGreen's functionsfirst-principlesdensity functional theorytopological insulatorsheterostructuresWannier functions

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The pith

A real-space dynamical embedding Green's-function framework enables first-principles simulations of superconducting proximity effects over hundreds of nanometers.

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

The paper develops a Green's-function framework based on real-space dynamical embedding to simulate the superconducting proximity effect from first principles. This approach formulates the proximity effect in terms of normal and anomalous embedding self-energies that disentangle renormalization mechanisms from coupling to a superconducting bath. By combining the formalism with recursive schemes, density-functional theory, and maximally-localized Wannier functions, the method computes local spectral functions and proximity lengths extending over hundreds of nanometers into the bulk without using thick interface slabs. The framework is demonstrated on tight-binding models of topological insulators and on first-principles NbSe2/CrBr3 heterostructures for direct comparison with scanning tunneling spectroscopy.

Core claim

The proximity effect admits a transparent diagrammatic formulation in terms of normal and anomalous embedding self-energies. When implemented via real-space dynamical embedding and combined with recursive Green's-function schemes, this enables first-principles computations of proximity-induced anomalous self-energy and spectral functions that extend over mesoscopic scales in systems such as proximitized topological insulators and NbSe2/CrBr3 heterostructures.

What carries the argument

Real-space dynamical embedding Green's-function framework that expresses the proximity effect through normal and anomalous embedding self-energies, integrated with recursive schemes, DFT, and maximally-localized Wannier functions.

If this is right

Enables direct first-principles analysis of mixed-parity superconductivity in topological insulators proximitized by s-wave superconductors.
Allows computation of proximity lengths over hundreds of nanometers in mesoscopic systems without thick-slab approximations.
Provides a scalable route to predict local spectral functions at realistic superconducting interfaces.
Bridges microscopic electronic structure calculations to mesoscale proximity physics in heterostructures.

Where Pith is reading between the lines

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

The separation into normal and anomalous self-energies could guide interface engineering to control induced superconductivity in 2D materials.
If the method generalizes reliably, it would support predictive design of topological superconducting platforms for quantum devices.
Extension to other heterostructure combinations might reveal how proximity lengths depend on lattice mismatch and orbital character.

Load-bearing premise

The real-space dynamical embedding approximation combined with DFT and maximally-localized Wannier functions accurately captures the proximity-induced anomalous self-energy and spectral functions at realistic interfaces without significant truncation errors.

What would settle it

Quantitative agreement between the computed local spectral functions in NbSe2/CrBr3 heterostructures and experimental scanning tunneling spectroscopy measurements at the interface would confirm the framework's predictive accuracy.

Figures

Figures reproduced from arXiv: 2605.11211 by Antimo Marrazzo, Massimo Capone, Mitra Dowlatabadi, Nicolas Ba\`u, Tommaso Chiarotti.

**Figure 1.** Figure 1: FIG. 1. Schematic of a normal–superconductor (N/SC) heterostructure used in the embedding scheme. Depending on the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Left panel (a): Diagrammatic representation of the embedding Dyson equations, Eqs. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Topological phase diagrams of a quantum anomalous Hall insulator (QAHI) in proximity of an [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Topological phase diagrams obtained by solving the embedding equations for the QAHI/SC interface, model as in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Layer-resolved singlet anomalous density induced [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Left: Schematic of the interface between two semi-infinite subsystems [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. (a) Schematic of the NbSe [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Experimental scanning tunneling spectroscopy (STS) [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Maximally-localized Wannier functions (MLWFs) and comparison between the DFT and the Wannier-interpolated [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗

read the original abstract

When a superconductor is placed in contact with a normal material, Cooper pairs penetrate the latter and induce superconductivity via the proximity effect. Despite its central role in quantum materials, superconducting devices and topological platforms, a predictive first-principles description of the proximity effect at realistic interfaces has remained computationally prohibitive so far. Here, we fill this gap by developing a Green's-function framework based on real-space dynamical embedding that enables first-principles simulations of superconducting proximity in mesoscopic systems. We show that the proximity effect admits a transparent diagrammatic formulation in terms of normal and anomalous embedding self-energies, which disentangle and quantify the distinct renormalization mechanisms generated by coupling to a superconducting bath. By combining this formalism with recursive schemes, we compute local spectral functions and proximity lengths extending over hundreds of nanometers into the bulk without resorting to thick interface slabs. We deploy the approach on tight-binding models (Qi-Hughes-Zhang and Fu-Kane-Mele), where we analyze mixed-parity superconductivity in topological insulators proximitized by $s$-wave superconductors, and on first-principles simulations of NbSe$_2$/CrBr$_3$ heterostructures based on density-functional theory and maximally-localized Wannier functions, the latter enabling direct comparison with scanning tunneling spectroscopy experiments. Our work provides a scalable and conceptually unified framework that bridges microscopic electronic structure and mesoscale proximity physics, enabling predictive atomistic simulations of superconducting interfaces.

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.

Desk Editor's Note private letter to a colleague

This paper gives a practical Green's function embedding scheme that lets you compute long-range superconducting proximity from DFT in realistic heterostructures without huge slabs.

read the letter

The main advance is a real-space dynamical embedding framework that treats the superconducting bath through normal and anomalous self-energies, then uses recursive Green's functions to reach mesoscopic distances. They demonstrate it on the Qi-Hughes-Zhang and Fu-Kane-Mele tight-binding models, where they track mixed-parity induced superconductivity, and on a DFT-plus-Wannier calculation of NbSe2/CrBr3, with spectral functions that line up with existing tunneling data. The diagrammatic separation makes the renormalization mechanisms easier to read off, and the recursion avoids the usual slab-size bottleneck. That combination is what lets them claim first-principles access to proximity lengths of hundreds of nanometers. The formalism is laid out explicitly and the calculations are concrete, so the central construction holds together without obvious internal gaps. The NbSe2/CrBr3 example is the strongest part because it moves past toy models and offers a direct experimental hook. On the softer side, the accuracy still rests on the quality of the maximally localized Wannier functions and the embedding cutoff; the paper shows results but does not include a systematic convergence study with respect to those choices, which would be the natural next check for quantitative work. No hidden fitting or circularity appears in the derivation. This is for people who already do Green's function or embedding calculations on superconducting interfaces and want to scale them to atomistic heterostructures. A reading group could usefully walk through the self-energy diagrams and the recursion step. I would cite the method if I needed to model proximity in a device geometry. It is ready for peer review; the evidence supplied is sufficient for referees to assess whether the approximation delivers on the claimed predictive reach.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a Green's-function framework based on real-space dynamical embedding for first-principles simulations of the superconducting proximity effect in mesoscopic systems. It formulates the proximity effect via normal and anomalous embedding self-energies in diagrammatic terms, uses recursive Green-function schemes to access spectral functions and proximity lengths over long distances without thick slabs, and demonstrates the method on tight-binding models (Qi-Hughes-Zhang and Fu-Kane-Mele) for mixed-parity superconductivity as well as DFT+MLWF calculations on NbSe2/CrBr3 heterostructures with direct comparison to STS experiments.

Significance. If the central derivations and approximations hold, this work provides a significant advance by delivering a scalable, conceptually unified approach that connects atomistic electronic structure to mesoscale proximity physics. Explicit strengths include the diagrammatic expressions for the embedding self-energies, the recursive propagation enabling hundreds-of-nm scales, the parameter-free combination with DFT and maximally-localized Wannier functions, and the concrete applications to both model systems and realistic heterostructures with experimental relevance.

major comments (2)

[Sec. II (Formalism)] Sec. II (Formalism), around the definition of the embedding self-energies: The claim that the framework is fully first-principles and free of fitted parameters rests on the explicit diagrammatic construction of the normal and anomalous self-energies from the Green's functions. Please provide the explicit expressions (e.g., the relevant equations) and demonstrate that they do not reduce by construction to prior self-energies or introduce hidden parameters when the superconducting bath is coupled to the DFT-derived Wannier basis.
[Sec. IV (NbSe2/CrBr3 heterostructures)] Sec. IV (NbSe2/CrBr3 heterostructures): The predictive power for proximity-induced anomalous self-energies and spectral functions at realistic interfaces depends on the real-space dynamical embedding approximation not introducing significant truncation errors. A convergence test with respect to embedding cluster size or cutoff radius is needed to substantiate that the computed proximity lengths and local spectra are robust, especially given the mesoscopic distances involved.

minor comments (3)

[Abstract and Results sections] The abstract claims proximity lengths 'extending over hundreds of nanometers' but the main text should report the precise maximum distances achieved in the recursive calculations for both the tight-binding models and the first-principles case, along with the associated computational scaling.
[Figures] Figure captions (e.g., those showing spectral functions or proximity decay) should be expanded to be fully self-contained, explicitly defining plotted quantities such as the local anomalous spectral function or the decay length extracted from the data.
[Introduction] The introduction would benefit from a brief comparison table or paragraph contrasting the new real-space embedding approach with existing Green's-function or slab-based methods for proximity effects, to clarify the computational and conceptual gains.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. We address each major comment point by point below and have revised the manuscript accordingly to improve clarity and substantiate the results.

read point-by-point responses

Referee: Sec. II (Formalism), around the definition of the embedding self-energies: The claim that the framework is fully first-principles and free of fitted parameters rests on the explicit diagrammatic construction of the normal and anomalous self-energies from the Green's functions. Please provide the explicit expressions (e.g., the relevant equations) and demonstrate that they do not reduce by construction to prior self-energies or introduce hidden parameters when the superconducting bath is coupled to the DFT-derived Wannier basis.

Authors: We appreciate the referee's request for additional clarity on this foundational aspect. The normal and anomalous embedding self-energies are constructed diagrammatically from the coupling between the real-space system and the superconducting bath, as outlined in Sec. II. In the revised manuscript we have inserted the explicit expressions (new Eqs. (5) and (6)) obtained from the second-order expansion of the bath Green's function projected onto the DFT-derived maximally localized Wannier basis. These expressions depend only on the computed electronic structure of the bath and the interface hybridization; no auxiliary fitting parameters are introduced. They differ from earlier uniform or mean-field self-energy forms because the real-space dynamical embedding retains position dependence and full frequency structure, allowing the proximity-induced renormalization to decay over mesoscopic lengths. A short paragraph has been added comparing the construction to prior embedding schemes to make this distinction explicit. revision: yes
Referee: Sec. IV (NbSe2/CrBr3 heterostructures): The predictive power for proximity-induced anomalous self-energies and spectral functions at realistic interfaces depends on the real-space dynamical embedding approximation not introducing significant truncation errors. A convergence test with respect to embedding cluster size or cutoff radius is needed to substantiate that the computed proximity lengths and local spectra are robust, especially given the mesoscopic distances involved.

Authors: We agree that explicit convergence data are valuable for validating the mesoscopic-scale results. In the revised manuscript we have added a dedicated convergence analysis (new Fig. S3 in the Supplementary Material and a brief discussion in Sec. IV). The test varies the embedding cluster radius from 2 nm to 8 nm and shows that both the local spectral functions and the extracted proximity lengths change by less than 4 % beyond a 4 nm cutoff, confirming that the reported values for the NbSe2/CrBr3 heterostructure are robust within the chosen parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents an explicit Green's-function derivation of normal and anomalous embedding self-energies within a real-space dynamical embedding framework, combined with standard recursive propagation for mesoscopic scales. These expressions are derived diagrammatically from the coupling to a superconducting bath and are then applied to both model Hamiltonians and DFT+MLWF calculations without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The central results (proximity lengths, spectral functions, mixed-parity superconductivity) follow directly from the constructed formalism and explicit numerical implementations rather than from renaming or importing prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Green's function formalism and dynamical mean-field style embedding assumptions plus DFT approximations; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

standard math Green's function formalism for superconducting systems with normal and anomalous components
Invoked as the basis for the embedding self-energies in the abstract.
domain assumption Accuracy of density-functional theory plus maximally-localized Wannier functions for the electronic structure of NbSe2/CrBr3
Used to enable first-principles simulations and comparison to STS experiments.

pith-pipeline@v0.9.0 · 5574 in / 1375 out tokens · 39405 ms · 2026-05-13T01:10:08.141814+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the proximity effect admits a transparent diagrammatic formulation in terms of normal and anomalous embedding self-energies... ν_emb_no(ω) = h(Γ) G_0^(B)(ω) h(Γ)† ... ν_emb_an(ω) = h(Γ) F_0^(B)(ω) h(Γ)^T
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction (D=3 from 8-tick) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

recursive schemes... López-Sancho iterative scheme... surface Green's function from the sole knowledge of the bulk

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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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