Unveiling In-Gap States and Majorana Zero Modes in Superconductor-Topological Insulator Bilayer model

Jukka Vayrynen; Rafal Rechcinski; Satoshi Okamoto; Tatiana de Picoli; Umesh Kumar

arxiv: 2507.23775 · v1 · submitted 2025-07-31 · ❄️ cond-mat.supr-con

Unveiling In-Gap States and Majorana Zero Modes in Superconductor-Topological Insulator Bilayer model

Umesh Kumar , Rafal Rechcinski , Tatiana de Picoli , Jukka Vayrynen , Satoshi Okamoto This is my paper

Pith reviewed 2026-05-19 01:48 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con

keywords Majorana zero modestopological insulatorsuperconductor proximity effectin-gap statesCaroli-de Gennes-Matricon modesbilayer modelhybridization strengthantidot vortex

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The pith

Increasing interlayer tunneling in a superconductor-topological insulator bilayer model enhances the isolation of Majorana zero modes from CdGM states even as the proximity gap decreases.

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

The paper introduces a bilayer model of the surface states of a three-dimensional topological insulator coupled to an s-wave superconductor through a tunable interlayer tunneling strength. It examines how this hybridization affects the proximity-induced gap and the in-gap states when a magnetic vortex is introduced via an antidot. The central finding is that stronger tunneling increases the energy separation between Majorana zero modes and Caroli-de Gennes-Matricon modes, improving MZM isolation. A reader would care because this points to a way to tune interfaces for more stable Majorana modes in potential topological quantum devices.

Core claim

With increasing hybridization strength in the bilayer model, the energy separation between MZMs and CdGM states increases, enhancing the isolation of MZMs. In the strong hybridization limit, the leading CdGM separation remains large in spite of the decrease in the PrI gap. This occurs alongside shifts in the proximity-induced gap minima away from the Gamma point and the emergence of momentum-selective interference patterns and p-wave-like features.

What carries the argument

The tunable interlayer tunneling strength t_perp, which controls the hybridization between the topological insulator surface states and the superconductor, driving shifts in gap minima and state separations.

If this is right

Shifting the proximity-induced gap minima away from the Gamma point produces momentum-selective interference patterns visible as spatial oscillations in in-gap states.
Spin- and spatial-resolved analysis reveals angular momentum asymmetries in the wavefunctions that are absent in conventional s-wave superconductors.
The bilayer geometry exhibits distinct p-wave-like features compared to a standalone s-wave superconductor.
These results offer predictions for experimentally tuning MZM stability and distinguishing them from CdGM modes in SC-3DTI heterostructures.

Where Pith is reading between the lines

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

Engineers could adjust interface properties like layer thickness to achieve stronger hybridization and thus more isolated Majorana modes in fabricated devices.
The approach might extend to other proximity-induced systems where controlling tunneling helps separate topological modes from trivial ones.
Further modeling could explore how these effects scale with system size or disorder to guide real-world experiments.

Load-bearing premise

The bilayer model with tunable interlayer tunneling accurately captures the proximity-induced gap, interference patterns, and in-gap state physics present in actual superconductor-topological insulator heterostructures.

What would settle it

Measuring in real SC-TI bilayers that increasing the interface coupling strength fails to increase the MZM-CdGM energy separation or causes it to decrease along with the proximity gap.

read the original abstract

Interfaces between topological insulators and superconductors are promising platforms for realizing Majorana zero modes (MZMs) via the superconducting proximity effect. We introduce a bilayer model consisting of the surface states of a three-dimensional topological insulator (3DTI) coupled to an $s$-wave superconductor and systematically study the role of interlayer tunneling strength ($t_\perp$). We find that increasing $t_\perp$ shifts the proximity-induced (PrI) gap minima away from the $\Gamma$-point, giving rise to momentum-selective interference patterns that manifest as spatial oscillations in the in-gap states. By introducing an antidot with a magnetic vortex in the SC layer, we investigate the nature of in-gap states including MZMs and Caroli-de Gennes-Matricon (CdGM) modes. With increasing hybridization strength, the energy separation between MZMs and CdGM states increases, enhancing the isolation of MZMs. Importantly, in the strong hybridization limit, the leading CdGM separation remains large inspite of the decrease in the PrI gap. Spin- and spatial-resolved wavefunction analysis reveals angular momentum asymmetries absent in conventional $s$-wave systems. A direct comparison with a standalone $s$-wave superconductor confirms the emergence of distinct $p$-wave-like features in the bilayer geometry. Our results provide experimentally relevant predictions for tuning the stability of MZMs and their differentiation from the CdGM modes in SC-3DTI heterostructures and offer a theoretical framework for probing unconventional superconductivity in engineered topological systems.

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

The bilayer model claims stronger interlayer tunneling improves MZM isolation from CdGM states even as the proximity gap shrinks, but only the abstract is available to check it.

read the letter

The punchline is that this bilayer model finds better isolation of Majorana zero modes from Caroli-de Gennes-Matricon states as interlayer tunneling increases, even while the proximity gap shrinks, plus some distinct angular momentum features compared to regular s-wave systems. The paper does a reasonable job extending prior proximity effect studies by scanning the tunable t_perp in a setup with an antidot and vortex. This leads to claims about momentum shifts causing spatial oscillations and hybridization-dependent mode separation. The comparison to a standalone s-wave superconductor is a clear way to show the p-wave-like aspects introduced by the topological insulator surface states. These elements could be helpful for designing experiments that tune MZM stability in SC-3DTI heterostructures. The soft spots come down to the missing details. With only the abstract available, there is no way to check the actual calculations or see quantitative data on how the separation scales with t_perp. The central claims rest on model calculations that sound plausible but cannot be evaluated for robustness without the methods and results. The assumption that the simple bilayer accurately represents real interfaces is common in the field but always open to scrutiny. This kind of paper is for condensed matter physicists working on topological quantum computation platforms. A reader who knows the basics of Majorana modes and vortex states would find the specific geometry and parameter dependence worth considering. I think it deserves peer review if the full paper includes solid numerical evidence and clear derivations, because the ideas are concrete and relevant to ongoing work in the area.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces a bilayer model consisting of the surface states of a three-dimensional topological insulator coupled to an s-wave superconductor via tunable interlayer tunneling strength t_perp. It systematically studies the role of this hybridization in shifting the proximity-induced gap minima away from the Gamma point and generating momentum-selective interference patterns. By considering an antidot with a magnetic vortex in the superconductor layer, the authors investigate in-gap states, reporting that increasing hybridization enhances the energy separation between Majorana zero modes (MZMs) and Caroli-de Gennes-Matricon (CdGM) states, thereby improving MZM isolation. In the strong hybridization limit, the leading CdGM separation remains large despite a decrease in the proximity-induced gap. The work also presents spin- and spatial-resolved wavefunction analysis showing angular momentum asymmetries and a comparison to standalone s-wave superconductors revealing p-wave-like features.

Significance. If the findings are supported by the detailed model calculations, this study would offer important predictions for experimentally tuning the stability and isolation of Majorana zero modes in superconductor-topological insulator heterostructures. The results highlight how hybridization can be used to differentiate MZMs from CdGM modes and suggest a framework for probing unconventional superconductivity in these engineered systems.

major comments (1)

[Abstract] The key results on the increase of MZM-CdGM energy separation with t_perp and the robustness of CdGM separation in the strong hybridization limit despite shrinking PrI gap are central to the manuscript's contribution. However, without access to the explicit Hamiltonian, the numerical implementation (such as BdG equations or diagonalization details), or quantitative data/figures, it is not possible to verify if these trends are accurately derived or if they hold under the model's assumptions.

minor comments (3)

[Abstract] The word 'inspite' is a typographical error and should be written as 'in spite'.
[Abstract] The abbreviation 'PrI' for proximity-induced is used without being defined on first use, which could affect readability.
[Abstract] The abstract would be strengthened by a brief indication of the computational approach employed to obtain the reported energy separations and wavefunction properties.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and constructive comments on our manuscript. We respond to the major comment point by point below.

read point-by-point responses

Referee: [Abstract] The key results on the increase of MZM-CdGM energy separation with t_perp and the robustness of CdGM separation in the strong hybridization limit despite shrinking PrI gap are central to the manuscript's contribution. However, without access to the explicit Hamiltonian, the numerical implementation (such as BdG equations or diagonalization details), or quantitative data/figures, it is not possible to verify if these trends are accurately derived or if they hold under the model's assumptions.

Authors: The referee correctly observes that the abstract does not contain the full technical details. The complete manuscript presents the explicit bilayer Hamiltonian in Section II, the Bogoliubov-de Gennes equations and numerical diagonalization procedure in Section III, and all quantitative data and figures supporting the reported trends. To improve accessibility and verifiability, we have added a concise overview of the model and numerical methods to the introduction in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on model calculations

full rationale

The abstract introduces a bilayer model of 3DTI surface states coupled to an s-wave superconductor via tunable interlayer tunneling t_perp and reports results from systematically varying this parameter, introducing an antidot with a magnetic vortex, and analyzing in-gap states (MZMs and CdGM modes) via spin- and spatial-resolved wavefunctions. These findings are presented as outcomes of the model's behavior under increasing hybridization strength, including shifts in PrI gap minima and enhanced MZM isolation, without any equations, parameter fits, or self-citations shown in the available text. No derivation step reduces by construction to its inputs, and the central trends are not presupposed by definitions or renamed known results. The paper is self-contained against external benchmarks in the abstract, with claims appearing to derive from independent model investigation rather than circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

Abstract-only access prevents enumeration of explicit free parameters or axioms; the model relies on standard condensed-matter assumptions about proximity effect and vortex states whose details are not provided.

free parameters (1)

interlayer tunneling strength t_perp
Tunable parameter whose variation drives the reported gap shifts and mode separations; no specific fitted numerical value is given.

pith-pipeline@v0.9.0 · 5799 in / 1233 out tokens · 48860 ms · 2026-05-19T01:48:39.859811+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

We consider a bilayer system... H = HTI + HSC + Hhyb... BdG formalism... antidot with a magnetic vortex... in-gap states including MZMs and Caroli-de Gennes-Matricon (CdGM) modes.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

increasing t⊥ shifts the proximity-induced (PrI) gap minima away from the Γ-point... energy separation between MZMs and CdGM states increases

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