Majorana Zero Modes in a Heterogenous Structure of Topological and Trivial Domains in FeSe$_{1-x}$Te$_x$

Dirk K. Morr; Eric Mascot; Jasmin Bedow; Prashant Gupta

REVIEW 3 major objections 10 minor 32 references

Majorana Zero Modes in a Heterogenous Structure of Topological and Trivial Domains in FeSe$_{1-x}$Te$_x$

T0 review · 3 major / 10 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proposes that vortices with and without Majorana zero modes in FeSe1-xTex reside in separate topological and trivial superconducting domains, with a Majorana edge mode necessarily localized at each intervening domain wall.

desk verdict A solid, testable proposal for the mixed vortex-MZM puzzle in FeSeTe, but its material premise is assumed, not shown. read the letter →

arxiv 2505.15745 v1 pith:5G7TXPXL submitted 2025-05-21 cond-mat.mes-hall cond-mat.supr-con

Prashant Gupta , Jasmin Bedow , Eric Mascot , Dirk K. Morr This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.supr-con

keywords MajoranazeromodesFeSe1-xTextopologicalsuperconductivitymagneticvorticesdomainwallsbulk-boundarycorrespondencescanningtunnelingmicroscopyironchalcogenidesuperconductor

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 proposes an explanation for a longstanding puzzle in the iron-chalcogenide superconductor FeSe1-xTex: some magnetic vortices host a zero-energy Majorana mode while others, in the same material, do not. The explanation is that the material contains a heterogeneous mixture of strong topological and trivial superconducting domains, and only vortices in the topological domains carry a Majorana zero mode. Because the Chern number changes at the boundary between a topological and a trivial domain, bulk-boundary correspondence requires a Majorana edge mode at every such domain wall. The paper works out the spectroscopic fingerprints of this picture and predicts that a scanning-tunneling linecut between a vortex with and without a zero mode will show a third zero-bias peak at the domain wall. It also simulates, with a non-equilibrium method, a vortex being dragged from a trivial into a topological domain, showing the domain-wall Majorana mode transferred into the vortex as a localized Majorana zero mode.

What carries the argument

The central machinery is a two-dimensional lattice model of FeSe1-xTex in which topological superconductivity emerges from the interplay of s-wave pairing, Rashba spin-orbit coupling, and ferromagnetic exchange. Spatial variations of the chemical potential or the magnetic exchange energy split the system into strong topological and trivial domains separated by a domain wall of width W. Vortices are introduced through Peierls phases and the superconducting order parameter is computed self-consistently. The load-bearing identity is the bulk-boundary correspondence: the Chern number changes across a topological-to-trivial domain wall, forcing a Majorana edge mode localized at the wall. The time-dependent part of the argument uses a non-equilibrium Green's-function formalism to follow the motion of a vortex and the transfer of this edge mode into a zero mode in the vortex core.

What would settle it

Perform an STM zero-bias dI/dV linecut between a vortex that shows a zero-energy mode and one that does not. The scenario predicts a zero-energy peak at the domain wall between them; if no such peak appears while the two vortices still differ in their zero-mode content, the heterogeneous-domain explanation is ruled out. Conversely, imaging a domain wall between two zero-mode-hosting vortices should show no edge-mode peak.

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Extended reading notes

Core claim

We demonstrate that a heterogeneous structure of topological and trivial superconducting domains in FeSe1-xTex explains why only some vortices host Majorana zero modes. We show that vortices in topological domains host an MZM while those in trivial domains do not, and that the two types of domains are necessarily separated by a domain wall carrying a Majorana edge mode. We predict that an STM zero-bias linecut between a topological and a trivial vortex will reveal zero-energy peaks at the topological vortex and at the domain wall, and none at the trivial vortex. Using a non-equilibrium formalism, we show that moving a vortex in real time from a trivial into a topological domain transfers a Majorana edge mode from the domain wall to the vortex as a localized Majorana zero mode.

Load-bearing premise

The explanation assumes that FeSe1-xTex actually contains adjacent strong-topological and trivial superconducting domains, a scenario the paper posits through spatial variations in chemical potential or magnetic exchange; it does not present or cite direct experimental evidence that such domains exist in the material.

Editorial extensions

If this is right

If the heterogeneous-domain picture is right, an STM zero-bias linecut between a vortex with a zero mode and one without will see a zero-energy peak at the domain wall between them, a signature absent in previous all-topological domain proposals.
Vortices inside topological domains always carry a Majorana zero mode; vortices inside trivial domains never do, so the fraction of zero-mode vortices directly maps the spatial distribution of topological versus trivial regions.
The observed increase in the number of trivial vortices with increasing magnetic field is explained by the ferromagnetic ordering driving the system from an odd-Chern topological phase into a trivial or even-Chern phase.
Dragging a vortex across a domain wall moves a Majorana edge mode into the vortex core, providing a controlled way to create or erase a localized Majorana zero mode in real time.
No domain-wall Majorana peak should appear between vortices that lie in the same kind of domain, even if their chemical potential or magnetic moment differ.

Reading between the lines

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

If the scenario holds, the predetermined position of domain walls in FeSe1-xTex could be used as a bottom-up template for arranging Majorana zero modes in arrays, since vortices in topological domains automatically inherit the modes.
The prediction that the domain-wall peak accompanies every topological-trivial pair could be tested against existing STM datasets of Fe(Se,Te) samples, which already contain linecuts through zero and nonzero vortex cores.
Because the model attributes the domains to spatial variation in chemical potential or magnetic exchange, local manipulation of these parameters, for example by a gate or by magnetic adatom deposition, should be able to engineer where MZMs appear and disappear.
The same heterogeneous-domain mechanism may operate in other iron-chalcogenide superconductors in which ferromagnetism coexists with superconductivity, not only FeSe1-xTex.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The takeaway: this is a respectable numerical paper that turns the non-ubiquitous vortex MZM puzzle in FeSe1-xTex into a concrete, falsifiable STM prediction. The new bit is not the idea of domains per se (Refs 18-19 already did patches), but the specific claim that trivial (C=0) superconducting domains host MZM-free vortices, and that the Chern-number-changing domain wall between topological and trivial regions necessarily carries a Majorana edge mode visible in dI/dV linecuts. That's a clean, testable consequence.

What the paper does well: the static BdG calculations are self-consistent; the domain-wall MEM follows from bulk-boundary correspondence and shows up exactly where expected; and the control calculations in Fig 3 (both domains topological, both trivial) convincingly show that a MEM appears only when the Chern number changes. The time-dependent transfer of a MEM into a vortex MZM (Fig 4) is a fresh result, though it's a demonstration using their non-equilibrium formalism and a rigid vortex profile, not a quantitative prediction for a specific experiment.

Soft spots, in proportion: the load-bearing premise—that real FeSe1-xTex actually contains adjacent strong-topological and trivial domains realized by spatial variations in chemical potential or exchange—is assumed, not evidenced. The paper doesn't measure or cite direct evidence for such domains at the required length scale; it imposes them by hand. The paper's own Discussion admits that even-Chern phases can also produce MZM-free vortices, so the experimental puzzle doesn't uniquely select this scenario. The time-dependent transfer uses a prescribed, non-self-consistent order parameter; the authors argue robustness with respect to static profile from prior work, but the transfer dynamics themselves could be sensitive to relaxation. The W-robustness claim is stated baldly without shown data—minor, since it's not the core.

Who it's for: people working on iron-chalcogenide superconductors and vortex Majoranas; it gives STM a concrete thing to look for (zero-bias peak at domain walls between vortices with and without MZMs). It deserves a serious referee. The central scenario is plausible and testable, even if the material evidence is missing. I'd send it to review with a request for the authors to either strengthen the case for domain formation or soften the claim of explaining the experiments.

Referee Report

3 major / 10 minor

Summary. The paper proposes that the puzzling observation of vortices with and without Majorana zero modes (MZMs) in FeSe1−xTex can be explained by a heterogeneous mixture of strong-topological and trivial superconducting domains. Using a two-dimensional Rashba ferromagnet/s-wave superconductor model with self-consistently computed order parameters, the authors find that vortices in topological domains carry MZMs while those in trivial domains do not, and that a domain wall between the two types of domain hosts a Majorana edge mode (MEM). They predict a falsifiable STM signature: a dI/dV zero-bias linecut between a topological and a trivial vortex shows a peak at the topological vortex, no peak at the trivial vortex, and a peak at the intervening domain wall. In addition, using a non-equilibrium Green's function formalism with a moving vortex whose profile is prescribed by Eqs. (20)–(22), they show that a MEM is transferred to the vortex as an MZM when the vortex moves from a trivial into a topological domain. The manuscript argues that this explains the non-ubiquitous vortex MZMs and distinguishes the scenario from earlier topological-domain proposals.

Significance. If the central premise is correct, the paper provides a concrete, falsifiable experimental prediction — a zero-bias STM linecut with a distinct domain-wall MEM peak — that could settle the origin of non-ubiquitous vortex MZMs in FeSe1−xTex. The static BdG calculations are self-consistent, and the domain-wall MEM follows from the Chern number change at the interface, so that part of the argument is internally sound and not fitted to experiment. The time-dependent transfer simulation is also a clear and explicit demonstration of a conceptually interesting process, namely MEM-to-MZM transfer during vortex motion. The main weakness is that the existence of adjacent topological and trivial domains in the real material is assumed rather than evidenced, and several robustness claims, including the W-independence and the rigid-vortex approximation, are not fully demonstrated. The paper is therefore a valuable theoretical proposal with testable consequences, but its central explanatory claim is conditional on an unverified material premise.

major comments (3)

[Theoretical Methods / Discussion] The central premise of the paper — that FeSe1−xTex realizes adjacent strong-topological and trivial superconducting domains — is imposed by hand through spatial variations of μ_r and J_r in Eq. (1); no experimental measurement or citation is provided that establishes such domains at the required length scale. Moreover, the paper's own Discussion (final paragraph) concedes that vortices without zero modes can also occur in phases with even Chern number, so the experimental observation of non-ubiquitous MZMs does not uniquely require the proposed topological/trivial domain structure. This weakens the abstract's claim that the observation "can be explained" by the heterogeneous-domain scenario. The authors should either present supporting evidence or arguments for domain formation, or explicitly reframe the result as a scenario with falsifiable predictions and discuss how it can be distinguished from the alternative mechanisms in Refs. [18]–[20].
[Theoretical Methods / Fig. 1(d)] The statement in Theoretical Methods that the qualitative results are "unaffected by changes in W" is not substantiated: all static simulations use W = 4a0 (Figs. 1–3), and no data for other domain-wall widths are shown. This matters because W controls the hybridization and energy discretization of the MEM, which directly affects the height and position of the predicted zero-bias domain-wall peak in Fig. 2(d). The authors should provide a W-dependence study, at least for Fig. 2's linecut, and quantify the MEM energy ϵ0 as a function of W and system size.
[Appendix C / Fig. 4] The time-dependent MEM-to-MZM transfer shown in Fig. 4 rests on the rigid-vortex approximation of Eqs. (20)–(22), in which the order-parameter magnitude and phase move with the vortex core without self-consistent relaxation, and on an artificially increased Δ to shorten the coherence length. The paper cites Refs. [27,28] for robustness of MZM existence to the spatial profile, but those references do not address dynamic profile relaxation during vortex motion. Since the transfer process depends on the instantaneous gap profile and on the vortex crossing the domain wall at finite velocity, the authors should either demonstrate that the result persists when the order parameter is allowed to relax self-consistently during the motion, or state as a clear limitation that the transfer is an approximation within the rigid-profile model.

minor comments (10)

[Abstract] The word "Majoarana" in the abstract is a typo and should read "Majorana."
[Title] "Heterogenous" should be "Heterogeneous."
[Introduction] The phrase "hards ±-wave superconducting gap" should be "hard s±-wave superconducting gap."
[Eq. (1)] The notation e_{r'−r} for the unit vector in the Rashba term is not defined; please define it explicitly and clarify the sign convention for the cross product.
[Fig. 1 caption / Results] The parameter sets for points 1–4 in Fig. 1(c) are not fully specified in the caption; only the values for Fig. 4 are given. Please list the (μ, J) values for all points used in Figs. 1–3, along with lattice size and boundary conditions.
[Fig. 2(d)] The text acknowledges that the MEM has finite energy discretization ϵ0 due to finite size, but the zero-bias linecut in Fig. 2(d) shows a peak at the domain wall. Please clarify whether ϵ0 is small enough relative to the STM thermal broadening for the peak to be observable at zero bias in a realistic experiment, and specify how ϵ0 scales with the domain-wall length.
[Appendix D, Eq. (24)] The displayed expression for ρ(r, σ, t) appears typeset incorrectly (with tilde symbols over V(t) and V†); please correct the notation or define the matrices involved.
[Discussion] "we previous predicted" should be "we previously predicted."
[References] Ref. [21] is an arXiv preprint from 2025; if a published version has appeared, please update the citation.
[Supplementary Material] Supplementary Movie 1 is referenced but its content and accessibility are not described; please ensure it is provided with the submission or clearly state where it can be obtained.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the new predictions follow from an assumed domain structure plus topology, not from data fitted to the target observation.

full rationale

Walking the derivation chain: the Hamiltonian Eq. (1) is a model adopted from the authors' earlier FeSe1-xTex works (Refs. [15-17]); topological and trivial domains are imposed by hand via spatial variations in mu_r and J_r (Theoretical Methods), and the superconducting order parameter is computed self-consistently. The results that vortices in odd-Chern-number domains bind MZMs while vortices in even-Chern/trivial domains do not are consequences of the computed Chern numbers and the bulk-boundary correspondence, not quantities fitted to the experimental observation that some vortices exhibit zero modes and others do not. The paper's central falsifiable prediction--a zero-energy dI/dV peak at the domain wall between a topological and a trivial vortex (Fig. 2(d))--is derived from the assumed domain structure and the topological invariant change, not from the target data. The non-equilibrium MEM-to-MZM transfer (Fig. 4) is a time-dependent BdG simulation with an explicitly stated rigid-vortex-profile approximation; this is a modelling approximation, not a circular reduction. The main weakness is evidential rather than logical: the paper assumes, without measuring or citing direct evidence, that FeSe1-xTex actually contains adjacent strong-topological and trivial domains at the required length scale. That is a support gap, not a circularity. The same-group citations (Refs. [15-17], [22], [24]) supply the model, the non-equilibrium formalism, and finite-size MEM-energy results, but none is invoked as an external uniqueness theorem or fitted to the present observation; the central claim retains independent, newly computed content. Accordingly, there is no significant circularity; the score of 2 reflects only minor reliance on the authors' prior model and formalism.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on several hand-chosen model parameters and on the assumption that the real material contains the heterogeneous domain structure. All other inputs are standard lattice/BdG machinery. No new particles or forces are introduced.

free parameters (6)

Rashba spin-orbit coupling alpha = 0.2 t_e (Figs. 1-3), 0.8 t_e (Fig. 4)
Chosen from prior model of FeSeTe; controls the topological phase; not fit to the vortex data being explained.
Domain-wall width W = 4 a0
Set by hand; authors assert results are unchanged by W but show no sweep.
Domain parameters (mu, J) for topological and trivial domains = (mu, J) = (-3.8, 2.0) t_e and (-3.8, 0.6) t_e in Fig. 4; points 1-4 in Fig. 1(c)
Hand-picked to place domains in specified topological phases; central to creating the heterogeneous structure.
Superconducting gap amplitude Delta = 1.2 t_e in Fig. 4; self-consistent otherwise
Larger value chosen to shorten coherence length and make time-dependent simulation feasible.
Vortex transit time t_V = 20 tau_e
Chosen for adiabatic vortex motion; affects the non-equilibrium transfer dynamics shown in Fig. 4.
Vortex radius R_V = Not specified numerically
Defines the spatial profile of the suppressed superconducting order parameter in Eq. 20; chosen for numerical convenience.

assumptions (6)

standard math Bulk-boundary correspondence: a change in Chern number across a domain wall guarantees a Majorana edge mode.
Invoked in Results to infer a MEM at topological/trivial domain walls.
domain assumption Mean-field BdG treatment with s-wave pairing captures the relevant superconducting physics.
Used throughout for self-consistent order parameter and LDOS calculations.
standard math Peierls substitution with symmetric gauge correctly implements magnetic field and vortices.
Appendix B; standard lattice gauge coupling.
ad hoc to paper Rigid vortex profile for the time-dependent order parameter (Eqs. 20-22) approximates the self-consistent moving vortex.
Acknowledged in Appendix C as computationally necessary; central to the Fig. 4 transfer result.
domain assumption FeSe1-xTex possesses spatially separated topological and trivial domains due to variations in mu or J.
The proposal itself; motivated by ferromagnetism but not directly evidenced in the paper.
domain assumption Non-equilibrium density of states from the retarded Green's function is proportional to time-dependent dI/dV.
Appendix D, based on the authors' prior work [22].

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Cite this review

Pith. "Pith review of Majorana Zero Modes in a Heterogenous Structure of Topological and Trivial Domains in FeSe$_{1-x}$Te$_x$." pith.science (2026). https://pith.science/paper/5G7TXPXL

@misc{pith2026250515745,
  author       = {Pith},
  title        = {Pith review of: Majorana Zero Modes in a Heterogenous Structure of Topological and Trivial Domains in FeSe$_1-x$Te$_x$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5G7TXPXL}},
  note         = {Machine review of arXiv:2505.15745}
}

abstract

We propose that the existence of vortices in FeSe$_{1-x}$Te$_x$ with and without Majoarana zero modes (MZMs) can be explained by a heterogeneous mixture of strong topological and trivial superconducting domains, with only vortices in the former exhibiting MZMs. We identify the spectroscopic signatures of topological and trivial vortices and show that they are necessarily separated by a domain wall harboring Majorana edge modes. We demonstrate that when a vortex is moved from a trivial to a topological domain in real time, a domain wall Majorana edge mode is transferred to the vortex as an MZM.

Figures

Figures reproduced from arXiv: 2505.15745 by the authors.

**Figure 2.** FIG. 2. (a) Linecut of the LDOS between vortices in topolog [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Linecut of the LDOS across (a) three topological and [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. (a)-(f) [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

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X. Wu, S. B. Chung, C. Liu, and E.-A. Kim, Topological orders competing for the Dirac surface state in FeSeTe surfaces, Phys. Rev. Research3, 013066 (2021)

work page 2021

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work page 2020

[3] [1]

Zhang, K

P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D. Gu, H. Ding, and S. Shin, Observation of topological superconductivity on the surface of an iron-based superconductor, Science360, 182 (2018)

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J. D. Rameau, N. Zaki, G. D. Gu, P. D. Johnson, and M. Weinert, Interplay of paramagnetism and topology in the Fe-chalcogenide high-Tc superconductors, Phys. Rev. B99, 205117 (2019)

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