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arxiv: 2605.06828 · v1 · submitted 2026-05-07 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

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Majorana bound states in chiral ferromagnet-superconductor heterostructures revisited

A. S. Slobodskoi , S. S. Apostoloff , I. S. Burmistrov
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Pith reviewed 2026-05-11 01:13 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords majoranastatesanalyticalheterostructuresboundchirallocalizedmagnetic
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The pith

Analytical expressions for Majorana bound states in skyrmion-vortex pairs demonstrate that spin-orbit coupling is essential for their stabilization, with results matching numerical simulations.

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

The work examines a hybrid system where a superconducting film with a vortex sits next to a chiral ferromagnet that hosts a skyrmion, a stable swirling magnetic texture. Researchers use the Bogoliubov-de Gennes equations, a standard tool for superconductors with magnetic fields, to derive approximate analytical forms for the wavefunctions of low-energy states localized near the vortex. These include potential Majorana zero modes, which are special states that could enable error-protected quantum bits. The analysis shows spin-orbit coupling, an interaction that ties an electron's spin to its orbital motion, plays a key role in creating and stabilizing these modes. Without it, the Majorana states do not appear in the model. The derived formulas agree closely with computer-based numerical calculations of the same system. The authors also examine how real-world factors, such as the vector potential from magnetic fields and small distortions in the magnetic texture, affect the states. This provides a more complete picture than earlier studies of the same setup.

Core claim

Our analytical results explicitly demonstrate the critical role of spin-orbit coupling for the stabilization of Majorana modes and provides approximate analytical expressions for low-lying states localized at the vortex, both with and without an accompanying skyrmion. The derived analytical results show excellent agreement with numerical simulations.

Load-bearing premise

The Bogoliubov-de Gennes model with idealized skyrmion-vortex pairing and dominant spin-orbit coupling accurately captures the low-energy physics, without critical interference from unmodeled effects like disorder or higher-order interactions.

Figures

Figures reproduced from arXiv: 2605.06828 by A. S. Slobodskoi, I. S. Burmistrov, S. S. Apostoloff.

Figure 1
Figure 1. Figure 1: FIG. 1. A schematic illustration of a heterostructure com [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Majorana state localized at the vortex in the absence of a skyrmion: comparison of numerical and analytical solutions [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Majorana state localized at the vortex in the presence of a skyrmion: comparison of numerical and analytical solutions. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Radial probability density of the Majorana state localized at a vortex. Numerical results are presented for homogeneous [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Majorana state localized at the edge: comparison of numerical and analytical solutions. The component [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The spectrum of the localized states as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Wave functions for the states localized at a vortex in [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The dependence of energies [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
read the original abstract

Majorana zero modes are central to the pursuit of fault-tolerant topological quantum computation. While traditionally sought in one-dimensional hybrid nanowires, a robust alternative platform involves heterostructures combining superconductors with noncollinear magnets. This work focuses on a particularly promising system: a chiral ferromagnet hosting a magnetic skyrmion coupled to a superconducting film containing a superconducting vortex. Such skyrmion-vortex pairs have recently been realized experimentally and are theorized to harbor localized Majorana states, offering a potential pathway for braiding operations. We present a comprehensive theoretical analysis of the low-energy quasiparticle bound states in these heterostructures. Extending previous studies, we develop an analytical framework for the Majorana wavefunctions as well as the wavefunctions and spectrum of other lowlying states within a Bogoliubov-de Gennes approach. Our analytical results explicitly demonstrate the critical role of spin-orbit coupling for the stabilization of Majorana modes and provides approximate analytical expressions for low-lying states localized at the vortex, both with and without an accompanying skyrmion. The derived analytical results show excellent agreement with numerical simulations. We further elucidate the role of realistic effects, including vector potentials and texture perturbations from stray magnetic fields, to assess their impact.

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

0 major / 0 minor

Summary. The manuscript revisits Majorana bound states in chiral ferromagnet-superconductor heterostructures with skyrmion-vortex pairs. It develops an analytical framework within the Bogoliubov-de Gennes formalism to derive approximate expressions for the wavefunctions and spectra of low-lying states, including Majorana zero modes, both with and without skyrmions. The work highlights the essential role of spin-orbit coupling in stabilizing these modes and reports excellent quantitative agreement between the analytical results and numerical simulations. Additional considerations include the effects of vector potentials and stray magnetic fields from the texture.

Significance. If the analytical derivations hold, this provides a valuable tool for understanding Majorana modes in an experimentally realized platform. The explicit demonstration of the necessity of spin-orbit coupling, the approximate analytical expressions for vortex-localized states, and the reported quantitative agreement with numerics are strengths that could aid design of topological quantum computing elements. Inclusion of realistic corrections like vector potentials and stray fields enhances applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending acceptance. The referee's summary correctly identifies the central contributions: the analytical Bogoliubov-de Gennes framework for low-energy states in skyrmion-vortex pairs, the explicit demonstration that spin-orbit coupling is required to stabilize Majorana zero modes, the approximate analytical expressions for vortex-localized states, and the quantitative agreement with numerics. We also appreciate the referee's note that inclusion of vector potentials and stray-field effects strengthens the applicability to experiment.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper develops analytical expressions for low-lying quasiparticle states (including Majorana modes) directly from the Bogoliubov-de Gennes Hamiltonian for the chiral ferromagnet-superconductor heterostructure, with explicit dependence on spin-orbit coupling. These are compared to independent numerical simulations for validation, and the framework incorporates vector-potential and stray-field corrections without reducing to parameter fits or self-referential definitions. No load-bearing step equates a claimed prediction to its own input by construction, and external benchmarks (numerics) provide falsifiable checks outside any fitted values.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the applicability of the Bogoliubov-de Gennes formalism to the skyrmion-vortex system and the assumption that spin-orbit coupling is the dominant stabilization mechanism. No explicit free parameters, invented entities, or additional axioms are identifiable without the full text.

axioms (1)
  • domain assumption Bogoliubov-de Gennes equations describe the quasiparticle spectrum in the presence of magnetic textures and superconductivity
    Standard framework invoked for hybrid systems; stated implicitly through the choice of approach.

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Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages

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    The Majorana state localized at the vortex Now we discuss the region of existence of Majorana solutions of Eq. (17). We assume that the vector poten- tialA φ(r) and the magnetic texture angleθ(r) decay fast enough asr→ ∞. We start from the solution localized at the origin (the center of the superconducting vortex). Detailed analysis of Eq. (18) (see Appen...

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    The Majorana state localized at the edge Now, we will discuss the conditions for the existence of the Majorana state localized near the edge of the sys- tem. We assume that the system is in the form of a disk with a radiusL, which is larger than all other relevant scales. This means that we setA φ(r) = 0,θ(r) = 0, and ∆(r) = ∆∞ near the edge. To formulate...

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    Uniform magnetizationθ≡ ±π For the most part of the paper we assume that mag- netizationmis directed in thez-direction far from the vortex,m→e z atr→ ∞. However, it is instructive to discuss shortly the case with homogeneousm=−e z, or θ(r)≡ ±π. The speculations as given in the beginning of the sub- section III A hold true here. Again we consider Eq. (17) ...

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    Spectrum of the edge-localized states The spectrum of the states localized near the edge of the system can be easily described perturba- tively. Indeed, let’s write the Hamiltonian (14) as H(l) ≃H (0) +V l, where Vl = l(1−σ zτz) +l 2τz 2mr2 + αlσxτz r ,(52) Here, we neglect the effect of the vector potential as in the previous sections. At larger, which a...

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