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Gap reopening as a possible signature of coupling between Majorana zero modes in Sn-(Bi,Sb)2(Te,S)3-based Josephson trijunctions

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Two adjacent Josephson trijunctions on a topological insulator show a minigap that reopens where the Fu-Kane phase diagram predicts coexisting Majorana zero modes, a possible signature of coupling between them.

desk verdict Carefully hedged first step toward coupling MZMs in adjacent trijunctions; worth refereeing despite a narrow evidence base. read the letter →

arxiv 2507.06474 v2 pith:HP4W2WBL submitted 2025-07-09 cond-mat.supr-con cond-mat.mes-hall

Duolin Wang , Xiang Zhang , Yunxiao Zhang , Heng Zhang , Fucong Fei , Xiang Wang , Bing Li , Xiaozhou Yang
show 16 more authors
Yukun Shi Zhongmou Jia Enna Zhuo Yuyang Huang Anqi Wang Zenan Shi Zhaozheng Lyu Xiaohui Song Peiling Li Bingbing Tong Ziwei Dou Jie Shen Guangtong Liu Fanming Qu Fengqi Song Li Lu
This is my paper
classification cond-mat.supr-concond-mat.mes-hall
keywords MajoranazeromodesJosephsontrijunctionsFu-KanemodeltopologicalinsulatorminigapreopeningAndreevboundstatesquantumcomputationSn-(BiSb)2(TeS)3
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 reports a possible experimental signature of coupling between Majorana zero modes in a topological-insulator device. The authors built two adjacent Josephson trijunctions on a Sn-(Bi,Sb)2(Te,S)3 flake, with flux-loop areas arranged so that a single sweep of the global magnetic field carries both trijunctions through the regions of the Fu-Kane phase diagram where their Majorana zero modes should coexist. If the two modes were independent, the minigap at each trijunction center would stay closed in those regions; if they couple, it should reopen. The measured minigap line shapes in two of more than ten devices matched the coupled-mode prediction, including a three-lobe oscillation over two periods at the upper trijunction, while equal-area control devices showed uncoupled behavior. Because coupled Majorana zero modes are the ingredient needed for braiding and fusion in the Fu-Kane topological quantum computation scheme, this qualitative match matters even though alternative interpretations are not fully excluded.

What carries the argument

The device is the central object: four superconducting electrodes form five single Josephson junctions and two trijunction centers, with three flux loops whose areas satisfy $S_1=2S_2=2S_3$. The Fu-Kane rule for a single trijunction says a Majorana zero mode nucleates at the center when the minigaps of an odd number of its single junctions are negative; the minigap is half the spacing between the electron-like and hole-like Andreev bound states. Sweeping the global magnetic field moves each trijunction along a diagonal line in its two-dimensional phase diagram, and the area ratio makes the two trijunctions enter their Majorana-coexistence regions at the same field values. When the two zero modes couple, the expected closure of each center's minigap turns into a reopening. The measurement channel is the contact conductance of normal-metal electrodes placed only at the trijunction centers, a quantity previously established to track the local minigap.

What would settle it

Fabricate the same $S_1=2S_2=2S_3$ two-trijunction geometry with a controlled series of inter-trijunction separations: the reopened minigap should shrink with increasing separation in the way expected for overlapping Majorana wavefunctions. Reopening that is independent of separation, or that appears when local flux control keeps one trijunction outside its Majorana region while the other is inside, would refute the coupling explanation.

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Extended reading notes

Core claim

The central claim is that two adjacent Fu-Kane Josephson trijunctions on one topological-insulator surface can host Majorana zero modes that couple to each other, and that this coupling is observable as a reopening of the minigap at both trijunction centers exactly where the single-trijunction phase diagram predicts simultaneous Majorana presence. In the device with flux-loop areas $S_1=2S_2=2S_3$, a global magnetic field moves the upper and lower trijunction phase points along the same diagonal trajectory; in the overlapping purple regions, each trijunction would show a closed minigap if it were isolated. Instead, the upper trijunction showed three minigap lobes over two periods and the lower trijunction showed local maxima in the overlap regions, in qualitative agreement with the expected line shapes once partial wavefunction penetration between the centers is included. In control devices with equal loop areas $S_1=S_2=S_3$, where no simultaneous Majorana coexistence occurs, the minigaps evolved as in individual trijunctions and no reopening appeared. The authors conclude that the observations support the Fu-Kane picture and provide a possible signature of inter-trijunction Majorana coupling rather than a definitive demonstration.

Load-bearing premise

The central interpretation stands or falls on the assumption that the minigap reopening seen in two devices is caused by coupling between Majorana zero modes, rather than by another gap effect, and that the failure to see it in most devices is a material-quality issue rather than evidence against the mechanism.

Editorial extensions

If this is right

  • If the reopening is genuine Majorana coupling, the Fu-Kane trijunction platform supports interacting Majorana zero modes rather than only isolated ones, a necessary step toward braiding and fusion operations.
  • The equal-area control devices confirm that without simultaneous Majorana coexistence the two trijunctions behave independently even when wavefunctions penetrate between their centers.
  • The observed line shapes, especially the three-lobe pattern over two magnetic-field periods, give a concrete experimental fingerprint for diagnosing coupled Majorana modes in future multi-trijunction devices.
  • Because the effect appeared in only two of more than ten devices, the results imply that Majorana coherence length and topological-insulator material quality are the limiting factors in realizing the Fu-Kane scheme in practice.

Reading between the lines

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

  • We infer that the low success rate points to Majorana coherence length, not geometry, as the bottleneck; devices made from cleaner topological-insulator films should show the reopening in a larger fraction of devices.
  • We infer that the reopened-gap amplitude should fall off with inter-trijunction separation, so comparing 1 μm and 2 μm devices with more statistics would turn the qualitative signature into a quantitative test of the expected distance dependence.
  • We infer that adding parity readout to this two-trijunction geometry could distinguish coupling from fusion and test the non-Abelian statistics, since the current experiment cannot tell whether two coupled Majorana modes form a fermion or the vacuum state.
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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

3 major / 4 minor

Summary. This manuscript reports measurements on two adjacent Josephson trijunctions fabricated on a Sn-(Bi,Sb)2(Te,S)3 flake, with normal-metal electrodes probing the centers of the two trijunctions. In devices with loop areas satisfying S1 = 2S2 = 2S3, the authors observe a three-lobe pattern in the magnetic-field dependence of the minigap at the upper trijunction and a reopening pattern at the lower trijunction, which they interpret as a possible signature of coupling between Majorana zero modes localized at the two trijunction centers. The observations are compared with hand-drawn expected line shapes that incorporate a phenomenological 1/4 change in apparent minigap amplitude due to state penetration between the two centers. Control devices with equal loop areas (S1 = S2 = S3) show no such reopening, and a second device type with local flux control provides evidence for wavefunction overlap between the two centers. The authors explicitly acknowledge that the reopening was observed in only two of more than ten devices, that the lower-trijunction minigap is inferred indirectly from contact resistance, and that alternative interpretations cannot be fully ruled out.

Significance. If the interpretation holds, this would be a meaningful experimental step toward the Fu-Kane topological quantum computation scheme by demonstrating inter-trijunction coupling of Majorana zero modes. The paper's strengths are its modest and honest framing, the inclusion of control devices, the presentation of failure cases in the Supplemental Material, an independent local-flux-control check of state penetration, and public data availability. The reopening-versus-non-reopening distinction is not achieved by fitting free parameters, which gives the central claim a degree of falsifiability. However, the significance is limited by the low device yield, the indirect measurement of the lower-trijunction minigap, and the absence of a microscopic calculation that would make the expected line shapes specific rather than schematic.

major comments (3)
  1. [Expected line shapes (Figs. 2(d)-2(g))] The predicted line shapes are constructed by an ad hoc procedure: the text states that state penetration causes a '1/4 change in the apparent amplitude' and that MZM coupling 'will turn the local minima ... into local maxima.' No microscopic model of two coupled trijunctions is given, so the specificity of the predicted line shape is unknown. Any mechanism that suppresses the minimum near the purple regions could produce the same three-lobe pattern. The authors should either provide a concrete model (e.g., a Bogoliubov-de Gennes or scattering-matrix calculation with an explicit inter-trijunction coupling term) that generates the expected line shapes, or clearly label the curves in Figs. 2(f,g) as purely schematic and soften the claim that the data are 'in agreement' with a quantitative prediction.
  2. [Lower trijunction data (Fig. 2(m))] The minigap at the lower trijunction is not extracted from dI/dV peak positions; instead, the normalized contact resistance at a single bias current (I_bias = -15 nA) is folded over six periods and used as a surrogate. This is load-bearing because the lower trijunction is one of the two observation sites. The contact-conductance-to-minigap correspondence is drawn from prior work, but for this device it is not shown that the correspondence holds at the chosen bias current. The authors should provide dI/dV conversion data for the lower trijunction on at least one successful device, or explicitly discuss how the finite-bias surrogate could distort the line shape.
  3. [Limitations paragraph and Supplemental Material §4] The central claim rests on only two devices out of more than ten. The authors attribute the absence of reopening in the majority of devices to short Majorana coherence length and material quality, but this is a post hoc explanation; the same material and architecture could produce a rare non-Majorana three-lobe effect. To make the statistics more informative, the authors should examine whether the two successful devices correlate with any measurable parameter (e.g., junction transparency, disorder, inter-trijunction distance, or normal-metal contact properties). Without such a correlation, the low success rate is consistent with the effect being an outlier artifact and does not itself support the MZM-coupling interpretation.
minor comments (4)
  1. [Introduction and text near Fig. 2(d)] There are several typos and grammatical errors: 'just what happened in previously studies' should be 'just as happened in previous studies,' and 'after considered state penetration' should be 'after considering state penetration.'
  2. [Fig. 2(k) and caption] The text states that the data demonstrate three lobes in two periods, but the expected line shape in Fig. 2(f) is described as turning local minima into maxima; please clarify whether the expected pattern has three maxima in two periods and ensure the horizontal axes in Figs. 2(k) and 2(f) are normalized consistently.
  3. [Fig. 2(m)] The choice of the horizontal linecut at I_bias = -15 nA is not physically motivated; please state why this bias current was selected and whether the resulting line shape is robust across nearby bias currents.
  4. [References and Data Availability] Reference [28] is a self-referential note rather than a standard citation; consider moving it to a footnote. The Zenodo DOI appears both in the Data Availability section and as reference [33]; please consolidate.

Circularity Check

0 steps flagged · score 2.0 of 10

Low circularity: the reopening-vs-closure distinction is a falsifiable prediction not forced by the ad hoc 1/4 state-penetration parameter, and the same-group citations for the conductance-minigap correspondence are supporting rather than load-bearing.

full rationale

The paper's central derivation is not circular. The expected minigap line shapes in the absence of coupling come from the external Fu-Kane single-trijunction model (Ref. [21]), and the claimed signature of MZM coupling is the reopening of a minigap where individual-trijunction theory predicts closure. This is a falsifiable, binary distinction rather than a quantity fitted to the data. The '1/4 change in apparent amplitude' due to state penetration is admittedly ad hoc and phenomenological (the paper itself states that the analysis 'needs more rigorous theoretical studies'), but it does not by itself produce the three-lobe structure; the coupling prediction is that local minima become local maxima, and the comparison to data is qualitative. The paper also supplies a control configuration (S1 = S2 = S3) where no MZM coexistence is expected and no reopening is observed, providing an independent contrast that would fail under a purely circular construction. The self-citations to Refs. [23,26,27,29] are used for the empirical contact-conductance/minigap correspondence and for prior verification of the single-trijunction phase diagram; these are prior experimental results, not parameters fitted in the present work, and the paper directly demonstrates the correspondence for the upper trijunction in Fig. 2(i). No equation or constructed quantity reduces by definition to the data it is said to predict. The explicit limitations (reopening seen in only two of more than ten devices, qualitative evidence only) further indicate that the authors are not presenting a forced or definitional match. Overall, the derivation chain is self-contained against the external Fu-Kane framework, with only minor reliance on same-group prior work, so the circularity score is low.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Fu-Kane prediction, the conductance-minigap correspondence, and the state-penetration assumption. The only hand-chosen quantity is the 1/4 penetration factor. No new entities are introduced.

free parameters (1)
  • State penetration amplitude factor = 1/4 (assumed, not fitted)
    Used to modify the apparent minigap amplitude in the expected line shapes of Figs. 2(d,e) and 3(d,e); the value is chosen ad hoc to improve qualitative agreement with data.
assumptions (4)
  • domain assumption Fu-Kane theory predicts that a trijunction hosts a Majorana zero mode when an odd number of single-junction minigaps are negative, and that coupling between MZMs reopens a closed minigap.
    The expected line shapes and the interpretation of reopening rely on this theory, cited as Ref. [21] and previously supported by Ref. [23].
  • domain assumption The contact conductance at a trijunction center reflects the local density of states and hence the local minigap.
    Used to convert measured contact resistance into minigap behavior; established in prior work [23,26,27,29].
  • domain assumption Electronic states from one trijunction center penetrate to the other, altering the apparent local minigap.
    The 1/4 amplitude modification and the cross-response in Fig. 4 are interpreted as evidence for such penetration.
  • standard math The single junctions are in the small-junction limit with complete flux penetration, so phase differences are determined by loop areas and applied magnetic field.
    Used to map magnetic field to phase-space trajectories; stated as experimentally confirmed by standard Fraunhofer patterns.

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Cite this review

Pith. "Pith review of Gap reopening as a possible signature of coupling between Majorana zero modes in Sn-(Bi,Sb)2(Te,S)3-based Josephson trijunctions." pith.science (2026). https://pith.science/paper/HP4W2WBL

@misc{pith2026250706474,
  author       = {Pith},
  title        = {Pith review of: Gap reopening as a possible signature of coupling between Majorana zero modes in Sn-(Bi,Sb)2(Te,S)3-based Josephson trijunctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HP4W2WBL}},
  note         = {Machine review of arXiv:2507.06474}
}
read the original abstract

In the past two decades, enormous efforts have been made to search for possible platforms and schemes to implement topological quantum computation (TQC). In exploring the Fu-Kane scheme of TQC based on Josephson trijunctions constructed on topological insulators, the predicted Majorana phase diagram of an individual trijunction has already been verified experimentally. If Majorana zero modes indeed exist in this kind of trijunction, coupling between them in multiple trijunction devices should be further expected. In this study, we fabricated Josephson devices containing two adjacent Josephson trijunctions on the surface of Sn-(Bi, Sb)2(Te, S)3 and observed a possible signature of the coupling effect manifesting as the reopening of a minigap in both trijunctions where a closure would otherwise be expected if the trijunctions existed individually. While alternative interpretations cannot be fully ruled out, our findings provide experimental support for the validity of the Fu-Kane theory and provide further motivation for advancing the TQC scheme proposed by Fu and Kane.

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    The length scale that governs the spatial distribution of supercurrents and magnetic flux in this system is the Josephson penetration depth J. J is estimated to be much larger than the size of our devices, leading to the experimentally confirmed standard Fraunhofer pattern o...

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    Gap reopening as a possible signature of coupling between Majorana zero modes in Sn-(Bi,Sb)2(Te,S)3-based Josephson trijunctions

    The data that support the findings of this study are openly available in Zenodo at 10.5281/zenodo.20427617. Figure Captions Fig. 1. False-color SEM images of a device containing two Josephson trijunctions separated by 1 m, constructed on the surface of a Sn -(Bi,Sb)2(Te,S)3 f...

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    Another device in the 𝑆1 = 2𝑆2 = 2𝑆3 configuration that exhibited the MZM coupling effect

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    Conversion of d𝑉/d𝐼bias data to d𝐼/d𝑉bias data

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    A direct comparison between the phase diagram and the experimental data showing the results of minigap reopening

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    An example of failure to observe the coupling effect in the 𝑆1 = 2𝑆2 = 2𝑆3 configuration

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    Another device in the 𝑺𝟏 = 𝟐𝑺𝟐 = 𝟐𝑺𝟑 configuration that exhibited the MZM coupling effect Besides the data shown in Fig. 2 of the main manuscript supporting the existence of inter -trijunction MZM coupling, here we present additional supporting evidence obtained on another dev...

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    For this purpose, we can convert the vertical linecuts, namely the d𝑉/d𝐼bias − 𝐼bias curves, in Fig

    Conversion of 𝐝𝑽/𝐝𝑰bias data to 𝐝𝑰/𝐝𝑽bias data While discussing Andreev Bound State -related physics, it would be more appropriate to use the d𝐼/d𝑉bias data instead of the d𝑉/d𝐼bias data. For this purpose, we can convert the vertical linecuts, namely the d𝑉/d𝐼bias − 𝐼bias curv...

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    S3 are the phase diagram for the 𝑆1 = 2𝑆2 = 2𝑆3 configuration, aligned together with the data shown in Figs

    A direct comparison between the phase diagram and the experimental data showing the results of minigap reopening Shown in Fig. S3 are the phase diagram for the 𝑆1 = 2𝑆2 = 2𝑆3 configuration, aligned together with the data shown in Figs. (k, m) of the main manuscript. It allows ...

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    More often, no obvious coupling effect could be recognized

    An example of failure to observe the coupling effect in the 𝑺𝟏 = 𝟐𝑺𝟐 = 𝟐𝑺𝟑 configuration As we have mentioned in the main manuscript, the coupling effect was observed on only two out of more than ten devices in the 𝑆1 = 2𝑆2 = 2𝑆3 configuration. More often, no obvious coupling ...

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Reviewed August 6, 2026 · model on record in the stance chip above.