Non-Majorana-origin of the half-integer conductance quantization elucidated by multi-terminal superconductor-quantum anomalous Hall insulator heterostructure

Alexey A. Taskin; Anjana Uday; Bibek Bhujel; Gertjan Lippertz; Yoichi Ando

REVIEW 3 major objections 5 minor 49 references

Non-Majorana-origin of the half-integer conductance quantization elucidated by multi-terminal superconductor-quantum anomalous Hall insulator heterostructure

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The half-integer two-terminal conductance in a superconductor–quantum anomalous Hall insulator heterostructure is shown to arise from edge-state potential equilibration at the superconducting electrode, not from chiral Majorana edge modes.

desk verdict A careful experimental study showing the half-integer plateau is a trivial equilibration effect; the floating-SC boundary condition in the theory needs tightening, but the trench control carries the claim. read the letter →

arxiv 2411.14903 v2 pith:DJ2IMP66 submitted 2024-11-22 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-elcond-mat.supr-con

Anjana Uday , Gertjan Lippertz , Bibek Bhujel , Alexey A. Taskin , Yoichi Ando This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.str-elcond-mat.supr-con

keywords quantumanomalousHallinsulatorchiralMajoranaedgemodeshalf-integerconductancequantizationLandauer-Büttikerformalismsuperconductingproximityeffectedge-stateequilibrationcrossedAndreevreflectionV-doped(BiSb)2Te3

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

Superconductor–quantum anomalous Hall insulator (QAHI) heterostructures, in which a ferromagnetic topological insulator thin film carries a chiral one-dimensional edge state, have been proposed to host chiral Majorana edge modes, with a two-terminal conductance of $e^2/(2h)$ proposed as their smoking-gun signature. This paper argues that the plateau is not a Majorana signature: in a Landauer–Büttiker description of the multi-terminal device, the conductance is $\sigma_{2T}=(e^2/(2h))(1+k)$, and $k=0$—the value giving the plateau—occurs whenever the superconducting electrode equilibrates the potentials of the incoming edge states, exactly as a normal metal would. The authors confirm this in V-doped $(\mathrm{Bi,Sb})_2\mathrm{Te}_3$ films with micrometer-scale Nb strips by measuring the individual edge potentials, finding $k\approx 0$, and they show the plateau survives even when the film is trenched underneath the superconductor so no longitudinal Majorana transmission is possible. If the paper is right, the $e^2/(2h)$ plateau cannot serve as evidence for topological superconductivity in QAHI systems, and searches must shift to contacts comparable to the superconducting coherence length.

What carries the argument

The load-bearing object is the Landauer–Büttiker transport description of an eleven-terminal QAHI Hall bar with a superconducting strip (contact 11), expressed through electron and hole transmission coefficients $T^{ee}_L$, $T^{eh}_L$, $T^{ee}_T$, $T^{eh}_T$ and single-particle tunneling $T_D$. Its key identity is Eq. (5), $\sigma_{2T}=(e^2/(2h))(1+k)$ with $k\equiv(T^{ee}_L-T^{eh}_L)-(T^{ee}_T-T^{eh}_T)$, which collapses all possible microscopic processes into one number; for the floating superconducting electrode, Eq. (8), $V_{11}=(V_1+V_6)/2$, shows that the electrode equilibrates the incoming edge-state potentials regardless of the transmission coefficients. These relations let the multi-terminal measurements determine which processes actually carry the current, and they show that $k=0$—and hence the half-integer plateau—is produced by equilibration rather than by a single chiral Majorana mode.

What would settle it

Replace the superconducting strip by an identical normal-metal strip: the paper's mechanism predicts the same $e^2/(2h)$ plateau with $k\approx 0$ and the same edge potentials, whereas any Majorana-based explanation predicts the plateau disappears when superconductivity is absent. A second test is to measure the same multi-terminal resistances in a device known to be in the $N=1$ topological phase; if $R_{3-4}$ and $R_{9-8}$ remain exactly $h/e^2$, the plateau carries no information about the phase.

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Extended reading notes

Core claim

On its own terms, the paper's central result is a degeneracy: the two-terminal conductance $\sigma_{2T}$ of a superconducting-strip QAHI device is $\sigma_{2T}=(e^2/(2h))(1+k)$ with $k=(T^{ee}_L-T^{eh}_L)-(T^{ee}_T-T^{eh}_T)$, so the Majorana prediction of $k=0$ is only one of many transmission combinations that produce the half-integer plateau. In the experimentally realized floating-superconductor configuration, the formalism gives $V_{11}=(V_1+V_6)/2$ independent of all transmission coefficients, meaning the superconducting strip pins its potential to the average of the two incoming edge states; the measured resistances across the strip then give $k\approx 0$. The multi-terminal data also show $T^{ee}_T\approx T^{eh}_T$ and $T^{ee}_L\approx T^{eh}_L$, no negative nonlocal resistances at low bias, and an unchanged plateau in a trenched device where the QAHI film is interrupted beneath the superconductor. The paper concludes that the half-integer quantization is a trivial edge-state equilibration effect, not a signature of chiral Majorana edge modes, and that superconducting proximity signatures in these films require electrode dimensions of the order of the induced coherence length.

Load-bearing premise

The conclusion rests on treating the floating superconductor as an ideal voltage node that perfectly averages the incoming edge potentials; if its internal charging dynamics or non-equilibrium electronic state instead matter, the measured potentials would not directly give the transmission coefficients, and the plateau could have a different cause.

Editorial extensions

If this is right

The two-terminal $e^2/(2h)$ plateau is a degenerate signature: many transmission combinations, including a normal-metal-like superconducting contact with $T_D\approx 1$, produce it, so it cannot certify the $N=1$ topological superconducting state.
Micrometer-scale superconducting electrodes on V-doped (Bi,Sb)$_2$Te$_3$ equilibrate incoming chiral edge potentials and show no negative nonlocal resistances, implying crossed Andreev reflection is not active over length scales much larger than the coherence length.
The grounded-superconductor configurations $R_{3-4}$ and $R_{8-4}$ isolate the transverse and longitudinal transmission coefficients, giving future experiments a way to search for genuine Andreev processes rather than relying on the two-terminal plateau.
A device with the QAHI film trenched underneath the superconductor still shows the same edge potentials and $k\approx 0$, showing the plateau does not require transmission of a chiral Majorana edge mode beneath the strip.
Observing superconducting proximity signatures in QAHI films will require superconducting electrodes with dimensions comparable to the induced coherence length, as in the sub-micrometer devices that showed negative downstream potentials.

Reading between the lines

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

The same degeneracy likely applies to other proposed Majorana transport observables whenever a floating superconducting island can equilibrate the incoming edge potentials; a direct check would be to compare noise or thermal conductance on samples with normal-metal versus superconducting strips of identical geometry.
The voltage-node assumption behind Eq. (8) suggests the plateau may become sensitive to the microscopic state of the superconducting island (charging energy, phase fluctuations, quasiparticle population) at smaller scales; probing with a gate-tunable or Josephson-coupled island could reveal deviations from $k=0$.
If equilibration is caused by in-gap states at the superconductor–QAHI interface, then cleaner interfaces or different barrier materials should change the transmission coefficients while preserving the plateau; a systematic barrier-thickness study could separate interface effects from intrinsic topological superconductivity.

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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,

This paper is a solid experimental contribution to the ongoing demolition of the e2/2h plateau as a Majorana smoking gun. The genuinely new pieces are the multi-terminal Landauer–Büttiker formulas, the direct edge-potential measurements, and the trench-device control that rules out chiral Majorana modes by construction. The above-Hc2 control, where nothing changes when Nb becomes normal, is convincing and sticks with you. The authors deserve credit for being upfront that their main conclusion echoes Kayyalha et al. and for pushing further with a more complete edge-potential picture.

The soft spots are real but not disqualifying. The treatment of the floating SC contact as an ideal voltage node with a free supercurrent in Eq. (16) is delicate. For a truly floating island the physical constraint is zero total current; the paper does not fully justify why I_SC_11 can be an unconstrained parameter, and this affects the derivation of Eq. (8) and the k≈0 extraction. However, the trench device—where no CMEM can exist and the same edge potentials still appear—carries the main argument without leaning on that boundary condition. The grounded configuration and the field-independent resistances above Hc2 also stand on their own. So the central claim survives.

Minor issues: the appendix omits several nonzero a_ij coefficients, which makes the algebra needlessly hard to audit, and the quantitative dismissal of Huang et al. rests on a fitted T_S parameter that is somewhat ad hoc. The contact-resistance interpretation for the grounded configuration is plausible but inferred rather than directly measured.

Bottom line: this paper deserves serious peer review. The theory section should be tightened—the floating-SC condition needs a careful justification or a fix—but the experimental evidence for the non-Majorana origin of the half-integer plateau is strong and independently supported by the trench control. I would accept it after minor-to-moderate revision.

Referee Report

3 major / 5 minor

Summary. The paper addresses the debated origin of the half-integer two-terminal conductance plateau observed in superconductor/quantum-anomalous-Hall-insulator (SC/QAHI) heterostructures. Using a Landauer-Büttiker description of a multi-terminal device with a micrometre-scale Nb strip, the authors show that the two-terminal conductance depends only on a combination k of the electron and hole transmission coefficients across and along the SC electrode. They report measurements on V-doped (Bi,Sb)2Te3 Hall bars with floating and grounded Nb electrodes, and infer k ≈ 0 from the observed resistances. Together with a trench-device control in which no chiral Majorana mode can propagate, they conclude that the half-integer plateau in their devices arises from equilibration of the incoming edge-state potentials at the SC electrode, not from chiral Majorana edge modes. The paper also argues that two-terminal measurements alone cannot distinguish Majorana physics from this trivial mechanism, and it critiques a recent claim of Majorana signatures in similar devices.

Significance. If the central claim holds, the paper provides a clear resolution to a long-standing controversy: the half-integer conductance plateau in SC/QAHI devices is not a smoking gun for chiral Majorana modes. The theoretical observation that the two-terminal conductance is determined by a single combination k of four transmission coefficients is simple and important, and the experimental control (the trench device, where no longitudinal Majorana transmission is possible) is a strong and appropriate test. The manuscript also ships its data on Zenodo, which is a concrete reproducibility strength. The main result is a useful contribution to the Majorana search literature, and it aligns with earlier work by Kayyalha et al. while extending the analysis to a multi-terminal framework.

major comments (3)

[Appendix B, Eq. (16) and Section III-B, Eq. (8)] The derivation of Eq. (8) is presented as following from Eq. (16), but as written Eq. (16) imposes no constraint on the voltages because I_SC_11 is an unconstrained variable that can absorb any value of I11. This makes the treatment of the floating SC electrode look like an ideal voltage-node assumption. In fact, Eq. (8) follows from the zero-current conditions at contacts 3 and 8 [Eq. (15)] together with the current boundary conditions at contacts 1 and 6, and it does not require Eq. (16) at all for k ≠ 1. The authors should rewrite the derivation to show this explicitly, and either remove Eq. (16) or state clearly that it merely defines the balancing supercurrent after the voltage solution is obtained. This is load-bearing because the experimental inference V3 ≈ V8 ≈ V1/2 and hence k ≈ 0 depends directly on Eq. (8).
[Section III-B, paragraph after Eq. (8)] The statement that Eq. (8) is independent of the choice of T_D, T_ee_L, T_eh_L, T_ee_T, and T_eh_T is not strictly correct. For k = 1 (for example, T_ee_L = 1 with all other transmission coefficients zero), the linear system for the voltages becomes singular and Eq. (8) is not enforced. The statement should be restricted to k ≠ 1, which is the experimentally relevant regime (k ≈ 0). The same caveat applies to the broad claim in the Conclusion that 'any experimental results on the two-terminal configuration can be explained by the SC electrode equilibrating all the chiral edge state potentials'; this should be qualified to avoid overstatement.
[Section VII-D, Eqs. (22)-(25)] The critique of Ref. [24] relies on an ad hoc parameter T_S representing an electric short across the SC strip, and the conclusion that the kinks observed by Huang et al. are 'most likely not related to the N=1 topological SC state' is speculative. The argument is not needed for the central claim of this manuscript, and the strength of the wording should be tempered, or the analysis should be supported by a more direct model or by data from the present devices. As written, this section may be read as overreach and could distract from the otherwise well-supported main result.

minor comments (5)

[Appendix A] The list of nonzero proportionality coefficients a_ij is incomplete; for example, coefficients connecting contacts along the chiral edge (such as a_{i,i-1}) are not listed. The authors should provide the full matrix or explicitly state the convention for how the chiral edge connects the contacts.
[Appendix B, Eq. (16)] The phrase 'supercurrent flowing into the device from contact 11' is ambiguous about sign conventions. The authors should define the direction of I_SC_11 clearly, for instance by stating that a positive value corresponds to a current entering the SC electrode from the external circuit.
[Section III-B, Fig. 2] The vertical axis label 'R (h/e2)' in Fig. 2(c-f) is unconventional; the authors should define it in the caption or use 'Resistance (h/e²)' to avoid confusion.
[Abstract and Conclusion] The word 'unambiguously' is used in the abstract and conclusion. Given the caveat about the k = 1 exceptional case and the model assumptions, a more cautious formulation such as 'consistently shows' would be more appropriate.
[Section III-B, Eq. (9)] The statement that R3-4 and R9-8 'cannot become negative, since -1 ≤ k ≤ 1' is correct within the model, but it would be helpful to note that this follows from the unitarity constraints on the transmission coefficients, which are not explicitly stated in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the half-integer conductance conclusion follows from a stated Landauer–Büttiker model, with k inferred from measured resistances rather than imposed.

full rationale

The central derivation is self-contained. Equation (5) follows algebraically from the stated transmission-coefficient model, and Eq. (9) is used to infer k from measured four-terminal resistances (R3-4 ≈ h/e2 implies k ≈ 0); the inferred k is then interpreted as edge-state equilibration, not inserted as a fit. The floating-superconductor treatment in Appendix B, Eq. (16), introduces the supercurrent I_SC_11 as an explicit modeling boundary condition; while physically debatable, this is an assumption about the electrode, not a reduction of the conclusion to its inputs, and changing that assumption would affect the model's validity rather than make the derivation circular. The T_S parameter in Section VII-D is used only in the secondary reanalysis of Ref. [24] as a consistency check and does not support the main non-Majorana conclusion. Self-citation of Ref. [18] supports the fabrication/proximitization assumption but is not load-bearing for the logical claim that a half-integer plateau is not a unique Majorana signature. No equation reduces by construction to the claimed result, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim requires no invented particles or forces. It relies on the standard Landauer-Buttiker framework, an ideal floating voltage-node treatment of the SC electrode, and the assumption that the devices are proximitized. The main fitted parameters are secondary: T_S is used only in the reinterpretation of Huang et al. data, and R_c,11 is used to model the grounded configuration. The parameter T_D is left undetermined by experiment, which is a genuine ambiguity in the microscopic origin of k=0, though not in the non-Majorana conclusion.

free parameters (3)

T_S: transverse electric short fraction across the SC strip = approximately 0.26 for the N=2 interpretation and 0.42 for the N=1 interpretation of the Huang et al. values
Introduced in Section VII-D to reconcile Huang et al.'s observed sigma2T values (0.74 and 0.57-0.59 e2/h) with Chung-type N=1/N=2 topological SC states. T_S is chosen to match those values, so the critique depends on an assumed short rather than a measured one.
T_D: subgap single-particle transmission into the SC electrode
Appears in Eq. (2) and is central to the argument that a normal-metal-like SC contact with T_D close to 1 gives e2/2h. The experiment does not extract T_D; the observation k=0 is compatible both with T_D close to 1 and with equal electron and hole transmission amplitudes.
R_c,11: contact resistance of the grounded SC electrode = lower bound approximately 2 kOhm
Used in the non-ideal grounded-contact model in Appendix C to explain nonzero R3-4 and R8-4 in the grounded configuration. It is not independently measured; it is only bounded from the same data that it is used to explain.

assumptions (3)

standard math Validity of linear-response Landauer-Buttiker formalism with electron and hole transmission coefficients, Eq. (2).
The entire derivation of Eqs. (5), (8)-(14) rests on this standard transport framework applied to normal and superconducting contacts.
domain assumption The floating SC electrode can be described as an ideal voltage node with a freely adjusting supercurrent I_SC_11 in the current-conservation condition Eq. (16).
This assumption underlies Eq. (8), V11=(V1+V6)/2, which is used to infer the edge potentials and the value of k. If the supercurrent or charging dynamics are not captured by this voltage-node model, the inference could fail.
domain assumption The V-doped (BiSb)2Te3 film underneath the micron-scale Nb electrodes is well proximitized because the same fabrication recipe showed proximity effects in Ref. [18].
Stated in Section III A. The negative result is stronger if the film is actually proximitized; if it is not, the absence of Majorana signatures in these large electrodes would be expected for a trivial reason.

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Pith. "Pith review of Non-Majorana-origin of the half-integer conductance quantization elucidated by multi-terminal superconductor-quantum anomalous Hall insulator heterostructure." pith.science (2026). https://pith.science/paper/DJ2IMP66

@misc{pith2026241114903,
  author       = {Pith},
  title        = {Pith review of: Non-Majorana-origin of the half-integer conductance quantization elucidated by multi-terminal superconductor-quantum anomalous Hall insulator heterostructure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJ2IMP66}},
  note         = {Machine review of arXiv:2411.14903}
}

abstract

Chiral one-dimensional transport can be realized in thin films of a surface-insulating ferromagnetic topological insulator called quantum anomalous Hall insulator (QAHI). When superconducting (SC) pairing correlations are induced in the surface of such a material by putting an $s$-wave superconductor on the top, the resulting topological superconductivity gives rise to chiral Majorana edge-modes. A quantized two-terminal conductance of $\frac{1}{2}(e^2/h)$ was proposed as a smoking-gun evidence for the topological SC phase associated with a single chiral Majorana edge-mode. There have been experiments to address this proposal, but the conclusion remains unclear. Here, we formulate the edge transport in a multi-terminal superconductor-QAHI heterostructure using the Landauer-B\"uttiker formalism. Compared to the original proposal for the $\frac{1}{2}(e^2/h)$-quantization based on a simple two-terminal model, our formalism allows for deeper understanding of the origin of the quantization. The analysis of our experiments on multi-terminal devices unambiguously shows that the half-integer conductance quantization arises from the equilibration of the potentials of the incoming edge states at the SC electrode, and hence it is not of Majorana origin.

Figures

Figures reproduced from arXiv: 2411.14903 by the authors.

**Figure 2.** FIG. 2. (a) Schematic of the floating and grounded configurations, with the allowed edge potentials indicated near the arrows to [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4. Differential resistances at 17 mK measured as a func [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: FIG. 3. (a-b) The magnetic-field dependencies of the resis [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 5.** Figure 5: FIG. 5. The apparent transverse resistance [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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