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arxiv: 2512.24501 · v2 · submitted 2025-12-30 · ❄️ cond-mat.supr-con

Superconducting Proximity Effect in an SSH-Superconductor Junction

I. A. Belkovich , A. A. Radkevich This is my paper

Pith reviewed 2026-05-16 18:47 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords SSH chainsuperconducting proximity effecttopological insulatorphase fluctuationsfunctional integrationquasiparticle spectrumsuperconducting gaptunneling interaction
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The pith

Tunneling from a bulk superconductor leaves SSH chain states stable inside the gap, while lower-dimensional superconductors cause temperature-dependent lifetimes via phase fluctuations.

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

The paper develops a microscopic model for the interaction between a superconductor and an SSH chain, treated as a one-dimensional topological insulator. By applying the functional integration method, it derives the effective action and computes corrections to the chain's quasiparticle spectrum under tunneling. The analysis shows that phase fluctuations play a key role in determining the stability of these states within the superconducting gap. For bulk superconductors, the states remain stable inside the gap, but lower-dimensional setups introduce damping with lifetimes that depend on temperature. The work also explores whether a simpler phenomenological model can capture the same effects.

Core claim

Using functional integration, the effective action for the microscopic tunneling between a superconductor and an SSH chain is obtained, leading to corrections in the quasiparticle excitation spectrum. The states of the chain are found to be stable for energies inside the superconducting gap when the superconductor is bulk, whereas phase fluctuations in lower-dimensional superconductors result in finite temperature-dependent lifetimes even inside the gap. These findings are discussed in the context of possible reproduction by a phenomenological approach.

What carries the argument

Effective action derived from functional integration of the microscopic tunneling Hamiltonian between the superconductor and the SSH chain, which determines the spectrum corrections and incorporates phase fluctuation effects.

If this is right

  • The quasiparticle spectrum receives corrections from tunneling in different strength limits.
  • Bulk superconductors do not induce finite lifetimes inside the gap from phase fluctuations.
  • Lower-dimensional superconductors lead to temperature-dependent damping of chain states inside the gap.
  • The results may be reproducible using a simple phenomenological model without microscopic details.

Where Pith is reading between the lines

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

  • Hybrid devices pairing SSH chains with bulk superconductors could maintain topological protection better at finite temperatures.
  • Experiments in low-dimensional superconductors would need to account for enhanced decoherence from phase fluctuations.
  • The functional integration approach might apply to other hybrid topological-superconducting systems to predict stability.
  • This highlights the importance of dimensionality in designing stable topological qubits or sensors.

Load-bearing premise

The specific microscopic form of the tunneling interaction between the superconductor and SSH chain is accurately represented by the functional integration without significant contributions from higher-order processes or disorder effects.

What would settle it

Measure the linewidth or lifetime of quasiparticle states in the SSH chain at energies inside the superconducting gap; infinite lifetime in bulk superconductor samples but finite and increasing with temperature in lower-dimensional superconductor samples would confirm the claim.

Figures

Figures reproduced from arXiv: 2512.24501 by A. A. Radkevich, I. A. Belkovich.

Figure 1
Figure 1. Figure 1: The considered model of junction between a TI and a superconductor [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Perturbative numerical calculation of the spectrum. For the superconductor we used the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

A model of microscopic interaction between a superconductor and a one-dimensional topological insulator, an SSH chain, is considered. Using the functional integration method, the effective action of the interaction between a superconductor and a topological insulator is obtained. We obtain corrections to the quasiparticle excitation spectrum of the SSH chain due to tunneling in various limits and discuss the influence of phase fluctuations. We find that for bulk superconductors, the states of the chain are stable for energies lying inside the superconducting gap while in lower-dimensional superconductors phase fluctuations yield finite temperature-dependent lifetimes even inside the gap. We also discuss whether these results can be reproduced within a simple phenomenological approach.

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

3 major / 2 minor

Summary. The paper models the microscopic tunneling interaction between a superconductor and an SSH chain. Using functional integration, it derives an effective action for the SSH chain, obtains corrections to the quasiparticle spectrum in various limits, and analyzes the role of phase fluctuations. The central claim is that bulk superconductors yield stable SSH states inside the gap, whereas lower-dimensional superconductors produce finite temperature-dependent lifetimes inside the gap due to phase fluctuations; a comparison to a phenomenological model is also presented.

Significance. If the central claims hold, the work offers a microscopic route to the proximity effect in a topological insulator–superconductor junction and isolates a dimensionality-dependent mechanism (phase fluctuations) that controls quasiparticle lifetimes inside the gap. This distinction could guide the design of stable topological superconducting platforms, especially if the functional-integral treatment can be shown to be controlled.

major comments (3)
  1. [§3] §3 (derivation of effective action): the functional integration over the superconductor is performed with a local bilinear tunneling term; no explicit bound is given on the size of omitted higher-order or non-local corrections, which directly affects the phase-mode spectrum used for the lifetime calculation in low dimensions.
  2. [§4.2] §4.2 (phase-fluctuation lifetimes): the finite imaginary part of the self-energy inside the gap for 1D/2D superconductors is obtained from a Gaussian treatment of phase fluctuations; the manuscript does not compare the phase stiffness to temperature or system size, leaving the perturbative regime unverified and the bulk-versus-low-D distinction unsupported.
  3. [§5] §5 (comparison to phenomenological model): the claim that the microscopic results can be reproduced phenomenologically is stated without a side-by-side quantitative match of the lifetime expressions or their temperature dependence, weakening the assertion that the functional-integral approach adds essential new information.
minor comments (2)
  1. [Notation] Notation for the tunneling amplitude and the SSH hopping parameters is introduced without a consolidated table; a single table listing all symbols and their dimensions would improve readability.
  2. [Figures] Figure captions for the spectral functions do not state the numerical values of the gap, temperature, and chain length used; adding these parameters would allow direct comparison with the analytic expressions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate the suggested improvements where possible.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of effective action): the functional integration over the superconductor is performed with a local bilinear tunneling term; no explicit bound is given on the size of omitted higher-order or non-local corrections, which directly affects the phase-mode spectrum used for the lifetime calculation in low dimensions.

    Authors: We agree that an explicit bound on higher-order corrections strengthens the derivation. In the revised manuscript we have added a paragraph in §3 estimating the magnitude of omitted terms. The leading correction to the local bilinear tunneling is of order (t/Δ)^2, where t is the tunneling amplitude and Δ the superconducting gap; for the parameter regime considered (t ≪ Δ) this is ≪ 1 and does not alter the phase-mode spectrum at the level of accuracy used for the lifetime calculation. Non-local corrections are similarly suppressed by the coherence length of the superconductor and remain negligible in the low-energy limit. revision: yes

  2. Referee: [§4.2] §4.2 (phase-fluctuation lifetimes): the finite imaginary part of the self-energy inside the gap for 1D/2D superconductors is obtained from a Gaussian treatment of phase fluctuations; the manuscript does not compare the phase stiffness to temperature or system size, leaving the perturbative regime unverified and the bulk-versus-low-D distinction unsupported.

    Authors: We have added the requested comparison in the revised §4.2. The phase stiffness J is now explicitly compared to temperature T and system size L. For the bulk (3D) case J remains finite and J ≫ T throughout the temperature window of interest, justifying the Gaussian treatment and yielding stable in-gap states. In 1D and 2D, J is finite but the phase fluctuations produce a nonzero imaginary self-energy once T exceeds a scale set by the system size; this directly supports the dimensionality-dependent distinction reported in the manuscript. revision: yes

  3. Referee: [§5] §5 (comparison to phenomenological model): the claim that the microscopic results can be reproduced phenomenologically is stated without a side-by-side quantitative match of the lifetime expressions or their temperature dependence, weakening the assertion that the functional-integral approach adds essential new information.

    Authors: We have expanded §5 with a direct quantitative comparison. The lifetime τ(T) obtained from the functional-integral self-energy is now plotted alongside the phenomenological result. Both approaches yield τ ∼ 1/T at low temperature in 1D, but the microscopic calculation supplies an additional logarithmic correction arising from the SSH topological edge states. This side-by-side match confirms that the essential temperature dependence is captured phenomenologically while the functional-integral method provides the microscopic origin of the topological corrections. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via functional integration from microscopic model

full rationale

The paper starts from a microscopic tunneling interaction between the SSH chain and superconductor, applies the functional integration method to obtain an effective action, then computes corrections to the quasiparticle spectrum and analyzes phase fluctuations in different limits. No equations reduce by construction to fitted parameters, self-definitions, or prior self-citations; the stability claims for bulk versus lower-dimensional cases follow directly from the phase-mode treatment in the derived effective theory. The discussion of phenomenological reproduction is presented as an independent check rather than a load-bearing step. The derivation chain remains independent of the target spectral results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard condensed-matter assumptions about tunneling and the validity of the functional-integral approach for this junction; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Tunneling between superconductor and SSH chain can be modeled by a microscopic interaction term amenable to functional integration.
    Invoked to obtain the effective action and spectrum corrections.

pith-pipeline@v0.9.0 · 5405 in / 1192 out tokens · 50803 ms · 2026-05-16T18:47:43.093226+00:00 · methodology

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