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REVIEW 4 major objections 5 minor 50 references

Magnetic field-free braiding and nontrivial fusion of Majorana bound states in high-temperature planar Josephson junctions

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A planar Josephson junction coupled to a skyrmion crystal can create, fuse, and exchange Majorana bound states in zero external magnetic field, using only gate voltages to move the states through T-shaped and double-cross circuits.

desk verdict A legitimate numerical extension of the skyrmion-coupled junction platform, but the braiding and fusion claims are asserted, not demonstrated. read the letter →

arxiv 2506.04338 v2 pith:OHPZJFRK submitted 2025-06-04 cond-mat.supr-con cond-mat.mes-hall

Pankaj Sharma , Narayan Mohanta This is my paper
classification cond-mat.supr-concond-mat.mes-hall
keywords MajoranaboundstatesplanarJosephsonjunctionskyrmioncrystalnon-Abelianstatisticstopologicalsuperconductivitybraidingfusiond-wavesuperconductor
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a planar Josephson junction whose electron gas is influenced by a skyrmion crystal can host multiple pairs of Majorana bound states without any external magnetic field, and that gate-defined barriers alone can fuse and exchange those states. Standard planar Josephson junctions need a fixed in-plane magnetic field to create a single pair of Majoranas, and that fixed direction blocks the multi-terminal layouts needed to move several states around. The skyrmion crystal supplies a spatially varying Zeeman field and a gauge field that imitates Rashba spin-orbit coupling, so topological superconductivity appears at zero phase bias, with Majorana pairs at the ends of the non-superconducting channels. Numerical diagonalization of a realistic two-dimensional lattice shows barrier-controlled splitting and fusion of the pairs, a three-gate exchange in a T-shaped junction, and a four-gate braiding sequence in a double-cross junction. If the claim holds, this is a scalable, field-free route toward demonstrating non-Abelian statistics and eventually topological quantum gates, and the reported s-wave and d-wave compatibility points toward higher-temperature operation.

What carries the argument

The load-bearing object is the skyrmion crystal: a triangular lattice of nanoscale magnetic whirls placed beneath the two-dimensional electron gas. Its spin texture acts as a spatially varying Zeeman field and generates a gauge-field contribution that mimics Rashba spin-orbit coupling, replacing the uniform in-plane magnetic field of conventional planar Josephson junctions and removing the need for a fixed field direction. The other essential mechanism is gate-defined control: tunable chemical potentials placed on the non-superconducting channels create barriers that split a single topological segment into two, hybridize or fuse the Majorana pairs, and move states through T-junction and double-cross networks. The evidence is produced by a tight-binding Bogoliubov-de Gennes diagonalization of a realistic two-dimensional lattice, from which local density of states maps reveal zero-energy bound states with the expected Majorana signatures.

What would settle it

Compute the ground-state fermion parity after performing the two braiding orders in the double-cross geometry; if the two outcomes are identical, or if a topological invariant over the claimed parameter window vanishes while zero modes persist, the central claim is falsified. An experiment that always sees the same fusion outcome, rather than a parity-dependent choice between vacuum and an unpaired fermion, would likewise rule out the non-Abelian interpretation.

Watch

Extended reading notes

Core claim

The central discovery is that the interplay of a triangular skyrmion crystal with the two-dimensional electron gas of a planar Josephson junction produces a topological superconducting phase in the absence of an applied magnetic field, with a pair of Majorana bound states bound to the ends of the non-superconducting channel at zero phase difference. Because no global field direction pins the system, several pairs can coexist in branched geometries. The paper shows numerically that a local barrier potential moves the system through isolated, entangled, and fusion regimes; in the fusion regime two Majoranas from different pairs fuse into an unpaired fermion, in the T-shape geometry two Majoranas from the same pair are swapped by a three-gate sequence while remaining at zero energy, and in a double-cross geometry a gate sequence carries out the exchange that would distinguish non-Abelian braiding from Abelian exchange. Both s-wave and d-wave pairing in the leads are reported to generate the zero modes, with d-wave leads raising the possibility of higher-temperature operation.

Load-bearing premise

The load-bearing premise is that the zero-energy states seen in the density maps are genuine Majorana bound states with non-Abelian statistics; the paper identifies them through zero-energy localization spectra, not through a topological invariant, a braiding unitary, or a fermion-parity readout.

Editorial extensions

If this is right

  • Multi-terminal topological Josephson junctions become feasible, since the skyrmion crystal removes the directional constraint imposed by an external in-plane field.
  • A single gate-tunable barrier can split one Majorana pair into two pairs, then fuse two of the zero modes, with the charge buildup in the fusion regime serving as a readout of fermion creation.
  • The T-shape three-gate protocol exchanges two Majoranas at zero energy, which is a basic move for topological qubit operations.
  • The double-cross geometry gives a concrete gate sequence for braiding three Majoranas, and comparing the two braiding orders would test their non-Abelian statistics.
  • Using d-wave superconducting leads allows the same fusion and exchange operations at higher temperatures than s-wave leads.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the identification rests on zero-energy spectra and spatial localization, a direct numerical computation of a topological invariant over the same parameter window would be the cleanest way to confirm that the states are Majorana rather than trivial Andreev bound states; the paper does not report such an invariant.
  • The skyrmion radius is described as a control knob, so a testable extension is to sweep it dynamically to switch topological segments on and off, effectively turning the barrier-gate sequence into a time-dependent braiding driver.
  • For d-wave leads, the pairing amplitude is momentum-dependent, so a practical next calculation is to test the braiding sequences under a misaligned d-wave gap or nodal orientation; that would show whether the high-temperature claim survives realistic d-wave proximity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a planar Josephson junction coupled to a skyrmion crystal as a magnetic-field-free platform for Majorana bound states. Using tight-binding Bogoliubov-de Gennes calculations with realistic InSb parameters, the authors report zero-energy states localized at the channel ends for d-wave superconducting leads at zero phase difference. They then study three gate-controlled operations: a barrier-induced fusion of two MBS pairs, a T-junction exchange of two MBS from the same pair, and a double-cross gate protocol intended to demonstrate non-Abelian braiding. The abstract and introduction claim that nontrivial fusion and non-Abelian braiding operations can be performed in this platform, with d-wave leads enabling operation at higher temperatures.

Significance. If fully established, the proposed platform would be significant because it removes the fixed in-plane magnetic field constraint of conventional planar Josephson junctions, allows multi-terminal networks, and potentially permits higher-temperature operation through d-wave superconductors. The numerical model is standard, the parameters are realistic, and the use of Kwant makes the calculations reproducible in principle. However, the central claims of nontrivial fusion and non-Abelian braiding are not supported by the diagnostics presented. The manuscript provides zero-energy spectra and LDOS localization, but no topological invariant, no parity readout of a fusion outcome, no braid unitary or Berry phase, and no comparison of opposite braid orders. These missing elements are exactly what distinguish Majorana bound states from trivial Andreev bound states, so the headline conclusions go beyond what the calculations demonstrate.

major comments (4)
  1. [Sec. II, Figs. 1 and A1] The identification of the zero-energy states as Majorana bound states is not established. The evidence consists of zero-energy eigenvalues and LDOS/CDOS localization profiles, which are also produced by trivial Andreev bound states and disorder-localized states, as the Introduction itself notes near Ref. [12]. No topological invariant (e.g., Majorana number/Pfaffian, local Chern marker, or scattering invariant) is computed for the skyrmion-crystal-coupled junction at phase difference phi=0. The long-junction localization in Appendix A does not remedy this because it again relies only on LDOS profiles. Since the entire fusion and braiding claim depends on this identification, a topological diagnostic is load-bearing and currently missing.
  2. [Sec. III, Fig. 2] The claimed nontrivial fusion of two MBS is not demonstrated. The text invokes the fusion rule gamma x gamma = I + psi, but no fermion parity of the fused state is computed. The only evidence is the disappearance of two localized LDOS peaks and a statement that the total charge increases by more than an order of magnitude. An increase in CDOS cannot distinguish the vacuum (I) outcome from the unpaired-fermion (psi) outcome, and no projection onto even or odd parity sectors is reported. Without a parity readout or an equivalent observable, the calculation shows hybridization of low-energy states, not nontrivial fusion.
  3. [Sec. IV, Fig. 3] The T-junction protocol exchanges two Majorana bound states belonging to the same pair. As the paper itself labels this a 'trivial exchange', it is not a braid and does not provide information about non-Abelian statistics. The abstract's broad statement that braiding operations can be performed therefore cannot be supported by this section. A non-Abelian exchange requires moving Majoranas from different pairs and detecting a nontrivial transformation of the degenerate ground-state manifold, neither of which is presented.
  4. [Sec. V, Fig. 4] The non-Abelian braiding demonstration is absent. The text correctly states that showing sigma1 sigma2 and sigma2 sigma1 lead to different outcomes is required, but only the sigma1 sigma2 sequence is simulated. No final-state readout, Berry phase, or braid unitary is computed, and no comparison with the reversed order is provided. The section therefore demonstrates a sequence of LDOS movements compatible with trivial zero modes, not a non-Abelian braid. This is the central claim of the paper, and the required calculation is missing rather than merely needing clarification.
minor comments (5)
  1. [Abstract and Sec. II] The abstract claims both s-wave and d-wave leads generate MBS, but the manuscript only presents d-wave calculations; the s-wave case is delegated to Ref. [20] without reproduction. Please either show the s-wave result or qualify the claim.
  2. [Sec. III] The term 'entangled regime' overinterprets wavefunction hybridization. Simultaneous localization of one eigenstate at four positions does not by itself establish quantum entanglement of Majorana pairs; a more neutral term such as 'hybridized regime' would be appropriate.
  3. [Sec. V] The section title refers to three MBS, but Fig. 4 tracks four Majoranas and shows two exchange operations. The role of the fourth Majorana should be clarified, or the wording revised.
  4. [Throughout] Figure references are inconsistent, alternating between 'FIG.' and 'Fig.' and 'FIGs.' and 'Figs.'; these should be unified.
  5. [Sec. II] The quantity rho_c is called 'charge density of states', but it is the charge density of a single quasiparticle eigenstate, not a density of states. This terminology is misleading and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the d-wave BdG calculations are self-contained, and the only self-citation ([20] for s-wave leads) is auxiliary and not load-bearing.

full rationale

The paper's central numerical derivation is self-contained: the BdG Hamiltonian in Eq. (3) is diagonalized with Kwant using fixed material parameters, and the zero-energy spectra, LDOS, and CDOS shown in Figs. 1-4 are direct outputs of those calculations rather than quantities fitted to the claimed conclusion. The field-free MBS at zero phase difference is computed for d-wave leads, not assumed from a citation. The only self-citation that stands out is [20], used to state that s-wave leads also produce a robust topological phase; this claim is secondary to the fusion and braiding simulations, is not used to define any parameter, and does not make the d-wave results reduce to its inputs. The paper's weaknesses—absence of a topological invariant, lack of fermion-parity readout for fusion, and failure to compare σ1σ2 with σ2σ1—are evidentiary or correctness concerns about whether the zero modes are truly non-Abelian MBS, not circular-reasoning defects. No equation is defined in terms of the target result, and no prediction is equivalent by construction to an input, so no significant circularity is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the skyrmion texture supplying the required Zeeman and spin-orbit-like interactions, on the mean-field BdG treatment, and on the interpretation of zero-energy LDOS peaks as Majorana modes. The listed parameters are physical inputs, not fitted observables. No new particles or mediators are introduced.

free parameters (2)
  • Skyrmion radius R_s = 60 nm
    Chosen to model nanoscale skyrmions; the topological phase window and gate protocols are shown only for this radius. It is an input, not fitted to a target observable.
  • Magnetic spin amplitude S = 1 T
    Chosen from Co/Pt multilayer values; sets the Zeeman energy scale that creates the MBS phase. Not fitted to the MBS results.
assumptions (4)
  • domain assumption Mean-field BdG description of the proximitized 2DEG is accurate for the proposed device.
    Used in Sec. II; ignores self-consistency of the order parameter and fluctuations.
  • domain assumption The idealized triangular skyrmion texture in Eqs. (1)-(2) faithfully represents a real skyrmion crystal in a multilayer.
    Sec. II; real textures have disorder, thermal fluctuations, and finite size.
  • ad hoc to paper Zero-energy states localized at channel ends are Majorana bound states.
    Identified solely from LDOS profiles in Figs. 1-4 with no topological invariant or braid observable.
  • domain assumption The gate sweeps are adiabatic and preserve the ground-state degeneracy of the Majorana system.
    Assumed in Secs. III-V; only spectra and LDOS are shown, no adiabaticity check or parity tracking.

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Cite this review

Pith. "Pith review of Magnetic field-free braiding and nontrivial fusion of Majorana bound states in high-temperature planar Josephson junctions." pith.science (2026). https://pith.science/paper/OHPZJFRK

@misc{pith2026250604338,
  author       = {Pith},
  title        = {Pith review of: Magnetic field-free braiding and nontrivial fusion of Majorana bound states in high-temperature planar Josephson junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OHPZJFRK}},
  note         = {Machine review of arXiv:2506.04338}
}
abstract

Demonstration of non-Abelian statistics of Majorana bound states (MBS) is crucial for the realization of fault-tolerant topological quantum computation. Two-dimensional platforms such as planar Josephson junctions require an in-plane magnetic field to generate a pair of MBS at its non-superconducting channel ends; however, the fixed direction of the in-plane magnetic field puts a constraint on the realization of a multi-terminal topological planar junction, and hence its ability to physically move multiple MBS -- which is necessary for performing the fusion and braiding operations. Here we show that in a planar Josephson junction coupled to a skyrmion crystal, which can generate multiple pairs of MBS in the absence of any external magnetic field, the non-trivial fusion and braiding operations can be performed. Our numerical calculations, designed for realistic two-dimensional quantum systems, certify the feasibility of experimental realization of the proposed device schemes. We find that both $s$-wave and $d$-wave superconducting leads can generate the MBS; indicating that the MBS movement operations can be performed at higher temperatures using $d$-wave superconducting leads. Our results establish that the skyrmion crystal-coupled planar Josephson junction is a viable platform for the generation and controlled movement of the MBS.

Figures

Figures reproduced from arXiv: 2506.04338 by the authors.

Figure 1
Figure 1. FIG. 1. (a) A planar Josephson junction with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) A planar Josephson junction with a potential barrier (in yellow) in the middle of the non-superconducting channel, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) A T-shape planar Josephson junction considered for exchanging positions of two Majorana bound states. The gates [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) A double-cross shaped planar Josephson junction, that can be used for non-Abelian braiding of three Majorana [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.