# Topological superconductor candidates: materials and devices

Compare bulk crystals, superconducting surfaces, vortex spectra and semiconductor devices while preserving the boundary between candidate signals, topological interpretation and demonstrated operations.

## Scope

Topological superconductivity concerns the paired quasiparticle state, not merely normal-state band topology next to a superconducting transition. The target and dimensionality must be stated before interpreting a probe:

| Target | What the evidence addresses | What it does not establish by itself |
| --- | --- | --- |
| Bulk paired phase | A gapped BdG state with specified symmetries and an appropriate invariant | A superconducting transition plus normal-state band inversion does not determine the pairing topology |
| Superconducting surface | Surface spin texture and a superconducting surface gap, interpreted with a surface model | A surface result does not classify all three-dimensional bulk pairing or demonstrate a localized vortex mode |
| Localized vortex state | A defect solution or spatially resolved low-energy spectroscopy | A model zero mode or local spectral peak does not demonstrate non-Abelian operations |

This comparison separates mathematical classification, probe evidence and proposed operations. The classification reference treats non-interacting or mean-field Hamiltonians and discusses stability to interactions small compared with the gap; it is not a complete classification of strongly interacting or nodal superconductivity. [R1](#ref-source-37986d2488e4cace) [R2](#ref-source-4e92d11d59fb3441) [R3](#ref-source-7dc64372df6ba886) [R8](#ref-source-e2865b9f2461bc7f)

For a fully gapped Bogoliubov–de Gennes description, topology depends on dimensionality, symmetries and the occupied quasiparticle states. This restricted framework does not exhaust interacting or nodal superconductivity. A local zero-bias peak is neither a universal requirement for every topological superconductor nor sufficient proof of a Majorana mode. Demonstrating non-Abelian operations is a further question; it is not the definition of every bulk topological superconducting phase. [R1](#ref-source-37986d2488e4cace) [R4](#ref-source-45c980d0afc5ac14) [R5](#ref-source-87a91e2d8cd79e95) [R11](#ref-source-dca781066e96eecc)

## Mathematical assumptions

One clean single-subband wire model uses $H(k)=\xi_k\tau_z+\alpha k\tau_z\sigma_y+V_Z\sigma_x+\Delta\tau_x$, where $\xi_k=\hbar^2k^2/(2m)-\mu$, $\sigma$ acts on spin and $\tau$ on particle–hole space. Here $\Delta$ is real induced pairing, spin–orbit and Zeeman axes are perpendicular, and $V_Z=|g\mu_B B|/2$ is an energy rather than a magnetic field. With $a=\alpha k$, diagonalization yields $E_\pm^2=\xi_k^2+a^2+V_Z^2+\Delta^2\pm2\sqrt{V_Z^2(\xi_k^2+\Delta^2)+\xi_k^2a^2}$. [R4](#ref-source-45c980d0afc5ac14) [R5](#ref-source-87a91e2d8cd79e95)

The zero-momentum gap closes at $V_Z^2=\mu^2+\Delta^2$. In the stated gapped wire model, the side with $V_Z^2>\mu^2+\Delta^2$ supports the topological phase. The inequality alone is not a material diagnostic: setting spin–orbit coupling to zero can leave finite-momentum gapless excitations even when it holds. Independent checks compare 54 matrix spectra, three closure points and that counterexample. They do not simulate multiband disorder, orbital depairing, interfaces or a complete material invariant. [R4](#ref-source-45c980d0afc5ac14) [R5](#ref-source-87a91e2d8cd79e95)

The Fu–Berg criterion assumes a three-dimensional, fully gapped, time-reversal-invariant BdG state with inversion symmetry and odd-parity pairing. Its normal-state Fermi-surface shortcut also assumes weak pairing and bands separated from the Fermi energy at the time-reversal-invariant momenta (TRIM).

- **Use the BdG inversion operator:** odd-parity pairing requires $\widetilde P=P\tau_z$. The invariant multiplies its negative-energy eigenvalues at the eight TRIM, taking one member of each Kramers pair. Multiplying both members would square each sign and lose the parity information.
- **Read the result as a sufficient test:** the parity $\nu=n\bmod2$ is only part of the three-dimensional integer classification. An odd result establishes nonzero $n$ within these assumptions. An even result does not by itself establish $n=0$; an even, nonzero invariant is not excluded. Counting enclosed TRIM neither measures the pairing parity nor certifies a doped specimen.
- **Keep surface and vortex states separate:** Fu–Kane's uniform proximitized helical surface has a gapped spectrum and preserves time reversal, despite its projected analogy with spinless $p_x+ip_y$ pairing. A vortex introduces a winding pairing phase, $\Delta(r,\theta)=\Delta_0(r)e^{\pm i\theta}$, and admits a localized zero mode in that ideal model. The vortex calculation is not an observation of a zero mode on every uniform surface, nor a demonstration of braiding.

These are different theoretical constructions. Applying either to a material requires checking its own symmetry, gap, geometry and disorder conditions; neither theorem replaces those measurements. [R2](#ref-source-4e92d11d59fb3441) [R3](#ref-source-7dc64372df6ba886)

## Experimental criteria

Hor's Cu-intercalated Bi2Se3 study reports a transition near 3.8 K. In the checked author manuscript, Cu0.12Bi2Se3 has a magnetic response roughly one-fifth of ideal diamagnetism at the lowest accessible temperature, which the authors treat as a lower bound because the transition is incomplete. The resistance does not reach zero. This establishes why sample preparation and connected superconducting paths matter; neither a universal superconducting volume nor topological pairing can be inferred from those numbers alone. [R6](#ref-source-0857b567ea17c2f2)

Butch's YPtBi crystals show a resistive transition near 0.77 K with an AC diamagnetic response. The critical-field curve uses a 50% resistive criterion, with 10–90% bounds, and field perpendicular to [001]. The often-quoted 1.5 T is a zero-temperature extrapolation. Converting it with $B_{c2}=\Phi_0/(2\pi\xi^2)$ gives about 14.8 nm as a consistency estimate. Superconductivity in a candidate topological semimetal does not itself identify its pairing topology or establish a calibrated superconducting volume. [R7](#ref-source-4d53a5f7cdff4a54)

Two iron-chalcogenide papers require separate attribution. Zhang's ARPES study resolves a spin-helical surface band and a superconducting surface gap around 1.8 meV, interpreted as surface topological superconductivity. Its low-temperature measurement at 2.4 K is not the reported 14.5 K transition temperature. The study's surface interpretation must not be replaced by a statement that all bulk pairing is topological. In its angular-gap figure, hollow mirrored markers are not additional independent measurements. [R8](#ref-source-e2865b9f2461bc7f)

Wang's distinct vortex-STM paper studies FeTe0.55Se0.45. Figure 2 spectra at 0.55 K and 0.5 T parallel to c are normalized by integrated spectral area; they cannot be read as an absolute quantized conductance plateau. The approximately 0.3 meV peak width contains instrumental and thermal broadening. Barrier and temperature scans elsewhere use different fields. The zero-energy signal is interpreted as Majorana evidence, while the displayed braiding sequence is a proposal, not a performed operation. [R9](#ref-source-a972be8eb742181f)

Xu's Bi2Te3/NbSe2 heterostructure study compares vortex spectra across film thicknesses. The checked 2013 author version underlying the 2015 publication distinguishes where split peaks become experimentally resolvable from the extrapolated origin of their splitting. The former is around 20 nm even for thinner films; the latter changes near four quintuple layers and supports the authors' Majorana interpretation. Treating every unsplit interval as a directly measured Majorana radius would discard both the extrapolation and approximately 0.2 meV resolution. [R10](#ref-source-6861a8084ed315a1)

Semiconductor readout advances also need their own evidence category. Aghaee's 2025 InAs–Al experiment measures single-shot parity interferometrically and explicitly retains an Andreev-state alternative. Van Loo's 2026 work demonstrates parity readout in a minimal Kitaev-chain device, with a two-site rather than asymptotically long-chain model. These are useful experimental capabilities; neither result, as scoped here, establishes braiding or fault-tolerant topological computation. Readout performance, spatial nonlocality and a protected bulk phase remain distinct questions. [R14](#ref-source-e3824a8727c3ec54) [R15](#ref-source-a3c2dc153700478a)

A 10 July 2026 preprint by Zatelli and colleagues reports coherent parity oscillations in coupled minimal Kitaev chains. The authors distinguish partial protection at the sweet spot from the unprotected degeneracy-lifting operations used for rotation. No journal acceptance was verified here; this is not a demonstrated braid or a certified gate fidelity. [R16](#ref-source-818818441532a8ec)

## Limits and disputes

A local zero-bias peak can arise from ordinary Andreev states, confinement or disorder. Pan and Das Sarma's simulations also show that end-to-end conductance correlations are not conclusive on their own: accidental correlations can occur in selected disordered examples, including long wires. Length must be considered relative to the relevant coherence scale; a fixed micrometre threshold is not a universal diagnostic. These model counterexamples do not identify the mechanism in every later device. Temperature, barrier, calibration, disorder and nonlocal controls must be interpreted together.

**Retracted quantization claim:** *Quantized Majorana conductance* (2018) was retracted on 8 March 2021. The author-institution copy of the notice reports:

- Previously undisclosed charge-jump corrections in Figure 2a and Extended Data Figure 4b, and a mislabelled axis in Figure 4b.
- Recalibration shifting the conductance plateaus by 8%, above $2e^2/h$; this is a reported calibration change, not an uncertainty interval.
- Replotting the full parameter range placing points outside the stated two-sigma error bars, so the authors withdrew the quantized-conductance conclusion.

The linked notice and [raw-data archive](https://doi.org/10.5281/zenodo.4545577) document the correction trail; this review has not independently recalibrated the apparatus or reproduced that archive's analysis. The retracted result remains historical context, never positive confirmation of later devices. Nor does retracting this result decide the status of every other platform. [R11](#ref-source-dca781066e96eecc) [R12](#ref-source-6a7f18eec743e89d) [R13](#ref-source-f127bb2d99183698)

Complete scientific acceptance remains open. This article checks selected primary passages, figures and restricted models. Earlier author versions are identified; complete final-version comparisons, supplements, raw-data analysis, calibration and exclusion of competing interpretations remain unfinished. These limits concern the present review and should not be inflated into a claim that every platform has the same evidential status. [R4](#ref-source-45c980d0afc5ac14) [R5](#ref-source-87a91e2d8cd79e95) [R6](#ref-source-0857b567ea17c2f2) [R7](#ref-source-4d53a5f7cdff4a54) [R8](#ref-source-e2865b9f2461bc7f) [R9](#ref-source-a972be8eb742181f) [R10](#ref-source-6861a8084ed315a1)

## Discovery implications

Specify the target first: bulk odd-parity superconductivity, a proximitized topological surface, a wire end state or a controllable parity device. Then record composition, geometry, temperature, field, superconducting criterion and competing-state tests for that target. Compare complementary probes on characterized samples and preserve failures and retractions. Strong spin–orbit coupling or an attractive band calculation can motivate an experiment, but cannot replace identification of the actual paired phase and its protected observables. [R2](#ref-source-4e92d11d59fb3441) [R3](#ref-source-7dc64372df6ba886) [R5](#ref-source-87a91e2d8cd79e95) [R11](#ref-source-dca781066e96eecc) [R13](#ref-source-f127bb2d99183698)

## Related topics

[Topological superconductivity](/scwiki/topological-superconductivity), [Majorana zero modes](/scwiki/majorana-zero-modes), [Heavy-fermion superconductors](/scwiki/heavy-fermion-superconductors), and [Spin–orbit coupling](/scwiki/spin-orbit-coupling).

For a dated comparison of spectroscopy, control and exchange criteria, see [the Majorana identification question](/scwiki/open-questions/n4).

## Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.

| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |
|---|---|---|---|---|---|---|
| FeTe0.55Se0.45 (aliases: FeTeSe, iron chalcogenide); Wang vortex spectra, Figure2,0.5T||c; area-normalized | 0.55 K; 0.55K vortex-STM measurement; not a superconducting transition temperature. | measurement; vortex-stm-measurement | Scanning tunneling spectroscopy | ambient; unknown GPa | unknown | experiment; [source](#ref-source-a972be8eb742181f) |
| Cu0.12Bi2Se3 (aliases: CuxBi2Se3, copper intercalated bismuth selenide); Hor Cu-intercalated crystal, author2009 Figures2/3 | 3.8 K; Reported magnetic/resistive transition near3.8K; resistance did not reach zero; partial diamagnetic response. | transition; onset | Resistivity and magnetization | ambient; unknown GPa | unknown | experiment; [source](#ref-source-0857b567ea17c2f2) |

## Sources

### Ref source-37986d2488e4cace

[R1] Classification of topological insulators and superconductors in three spatial dimensions (2008). [DOI 10.1103/PhysRevB.78.195125](https://doi.org/10.1103/PhysRevB.78.195125). Publication: journal-article. Check: selected-author-classification-scope. Checked: 2026-10-20. Authorv3 submitted15June2009; selectedpp2–3/TableI re-read,p2visuallyinspected. Classification concerns single-particle/noninteracting orBdG systems; small-interaction/gap qualification retained. PDFcompiledate is not publicationdate. Fullfinal/interacting classification not certified.
### Ref source-4e92d11d59fb3441

[R2] Odd-Parity Topological Superconductors: Theory and Application to CuxBi2Se3 (2010). [DOI 10.1103/PhysRevLett.105.097001](https://doi.org/10.1103/PhysRevLett.105.097001). Publication: journal-article. Check: selected-author-criterion-and-proof. Checked: 2026-10-20. Authorv1,17December2009;selectedpp1–2 and proofEqs2–4 re-read, bothpagesvisuallyinspected. BdG inversionP*tau_z, onefactorperKramerspair, weakpairing awayfromTRIM energies andnu=n mod2. Sufficient-only criterion; fullfinalcomparison remainsopen.
### Ref source-7dc64372df6ba886

[R3] Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator (2008). [DOI 10.1103/PhysRevLett.100.096407](https://doi.org/10.1103/PhysRevLett.100.096407). Publication: journal-article. Check: selected-published-version-author-manuscript-check. Checked: 2026-10-20. Authorv3,28April2008,listed as publishedversion;selectedpp1–2/Eqs2–3 re-read,p1visuallyinspected. Uniformgapped TRIsurface differsfrom spinlesschiral analogy andphase-windingvortex zero mode. No actual device, braiding orcompletejunction/rawdata validation.
### Ref source-45c980d0afc5ac14

[R4] Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures (2010). [DOI 10.1103/PhysRevLett.105.077001](https://doi.org/10.1103/PhysRevLett.105.077001). Publication: journal-article. Check: selected-author-Hamiltonian-and-phase-boundary. Checked: 2026-10-19. Authorv2,13August2010;selectedpp1–2read,p2visuallychecked. Nambu convention, single occupied channel, Zeeman g muB B/2, induced pairing and C0 boundary checked. PDF compiled26November2024 is not publication date. Full junction derivation/final comparison open.
### Ref source-87a91e2d8cd79e95

[R5] Helical Liquids and Majorana Bound States in Quantum Wires (2010). [DOI 10.1103/PhysRevLett.105.177002](https://doi.org/10.1103/PhysRevLett.105.177002). Publication: journal-article. Check: selected-author-spectrum-check. Checked: 2026-10-19. Authorv2,10June2010;selectedpp1–2read,p2visuallychecked. Eq1 spin/particle-hole convention, Eq2 spectrum and Eq3 zero-momentum closure; B denotes Zeeman energy here. No full network/braiding derivation or final comparison.
### Ref source-0857b567ea17c2f2

[R6] Superconductivity in CuxBi2Se3 and its Implications for Pairing in the Undoped Topological Insulator (2010). [DOI 10.1103/PhysRevLett.104.057001](https://doi.org/10.1103/PhysRevLett.104.057001). Publication: journal-article. Check: selected-author-magnetization-and-resistance. Checked: 2026-10-19. Authorv1,15September2009;selectedpp3–4read andvisuallychecked. Cu0.12Bi2Se3 about3.8K magnetic/resistive transition;10OeZFC response~20percent ideal at lowest accessible temperature is author lower bound,not calibrated universal volume. Resistance did not reach zero. Final APS download401;full processing/final/raw audit open.
### Ref source-4d53a5f7cdff4a54

[R7] Superconductivity in the topological semimetal YPtBi (2011). [DOI 10.1103/PhysRevB.84.220504](https://doi.org/10.1103/PhysRevB.84.220504). Publication: journal-article. Check: selected-final-methods-and-critical-field. Checked: 2026-10-19. Final selectedpp1–3read,p3Figure3visuallychecked. Bi-fluxcrystal,0.77Kresistive transition/ACdiamagnetism,transport/susceptibilitydown20mK. Hc2curve50percentresistance,bars10/90percent,fieldperpendicular[001];1.5Tzero-Textrapolation. No fullACvolume/raw audit.
### Ref source-e2865b9f2461bc7f

[R8] Observation of topological superconductivity on the surface of an iron-based superconductor (2018). [DOI 10.1126/science.aan4596](https://doi.org/10.1126/science.aan4596). Publication: journal-article. Check: selected-author-ARPES-and-gap. Checked: 2026-10-19. Authorv2,12March2018;selectedpp1–4read,p9Figure4visuallychecked. HRARPES1.4meV/SARPES5.5meVresolution;2.4Kmeasurementnot14.5KreportedTc;surfacegap~1.8meV. Hollowangularmarkersmirrorfilledpoints. Surface interpretation not whole3Dbulk;notvortexSTM. Fullfinal/supplement/rawfit open.
### Ref source-a972be8eb742181f

[R9] Evidence for Majorana bound states in an iron-based superconductor (2018). [DOI 10.1126/science.aao1797](https://doi.org/10.1126/science.aao1797). Publication: journal-article. Check: selected-final-vortex-spectra-and-conditions. Checked: 2026-10-19. Final selectedprinted333–335results/captionsread;Figure2p334visuallychecked. FeTe.55Se.45,0.55K,0.5T||c,Fig2area-normalized;~.3meVFWHM includesresolution/thermal~.28meVcombined. Fig3barrier2.5Tversustemperature4T;braidingpanelproposal. Online16August2018;PDFfooter/issue date discrepancy unresolved. Fullsupplement/trivial-state/raw audit open.
### Ref source-6861a8084ed315a1

[R10] Experimental Detection of a Majorana Mode in the Core of a Magnetic Vortex inside a Topological Insulator-Superconductor Bi2Te3/NbSe2 Heterostructure (2015). [DOI 10.1103/PhysRevLett.114.017001](https://doi.org/10.1103/PhysRevLett.114.017001). Publication: journal-article. Check: selected-earlier-author-spatial-profile. Checked: 2026-10-19. Authorv1,26December2013,earlier title;selectedpp5–8andfigurecaptionsread,p14Figure2visuallychecked. MBE1–6QLfilms;resolved splitting near20nm acrossfilms differs from extrapolated splitting-origin offset,4QLcrossover. Resolution~.2meV;fullsupplement/2015final comparison open. Do not confuse with1312.3713/PRL112217001.
### Ref source-dca781066e96eecc

[R11] Physical mechanisms for zero-bias conductance peaks in Majorana nanowires (2020). [DOI 10.1103/PhysRevResearch.2.013377](https://doi.org/10.1103/PhysRevResearch.2.013377). Publication: journal-article. Check: selected-author-correlation-counterexamples. Checked: 2026-10-20. Authorv2submitted2April2020; selectedpp13–14 correlation/disorder discussion re-read,p14visuallyinspected. Accidental correlations possible even in selected long-wire disordered examples; coherence-length condition retained. No transport solver/raw fit or universal current-device diagnosis.
### Ref source-6a7f18eec743e89d

[R12] RETRACTED ARTICLE: Quantized Majorana conductance (2018). [DOI 10.1038/nature26142](https://doi.org/10.1038/nature26142). Publication: journal-article. Retracted: 2021-03-08. [Retraction notice](https://crossmark.crossref.org/dialog/?doi=10.1038%2Fnature26142). Check: official-retraction-identity-check. Checked: 2026-10-20. Crossmark rechecked originalDOI and8March2021retraction; original2018paper retained only as retractedhistory, not positive evidence.
### Ref source-f127bb2d99183698

[R13] Retraction Note: Quantized Majorana conductance (2021). [DOI 10.1038/s41586-021-03373-x](https://doi.org/10.1038/s41586-021-03373-x). Publication: journal-article. Check: author-institution-retraction-text. Checked: 2026-10-20. Full author-institution notice text re-read; Nature confirms8March2021publication/DOI but notice body subscription-gated. Figure2a/ED4bcharge-jump corrections,Fig4baxis,8percent recalibration andfull-range points outside2sigma stated. Zenodo4545577archive linked bypublisher; no independent archiveanalysis or instrumentcalibration.
### Ref source-e3824a8727c3ec54

[R14] Interferometric single-shot parity measurement in InAs–Al hybrid devices (2025). [DOI 10.1038/s41586-024-08445-2](https://doi.org/10.1038/s41586-024-08445-2). Publication: journal-article. Check: selected-author-hosted-published-text. Checked: 2026-10-18. Published19February2025. Selected first page651 and p654: readout result, optimized assignment error and explicit alternative Andreev-state caveat; both-end and coupling constraints retained. Full datasets/supplement not audited.
### Ref source-a3c2dc153700478a

[R15] Single-shot parity readout of a minimal Kitaev chain (2026). [DOI 10.1038/s41586-025-09927-7](https://doi.org/10.1038/s41586-025-09927-7). Publication: journal-article. Check: selected-accepted-author-model-and-readout. Checked: 2026-10-18. Published online11February2026 per publisher-deposited Crossref metadata, print12February2026. Author v2 uploaded6October2026; selected abstract/pp2–4 and ED6caption; p4 visually checked after PDF extraction warnings. Two-site equation check reproduced; full final-version comparison and datasets open.
### Ref source-818818441532a8ec

[R16] Majorana parity qubit in coupled minimal Kitaev chains (2026). [DOI 10.48550/arXiv.2607.09511](https://doi.org/10.48550/arXiv.2607.09511). Publication: preprint. Check: selected-preprint-results-and-conclusion. Checked: 2026-10-19. v1 submitted 10 July 2026. Selected pp. 1–6 and p. 9; p. 5 Figure 2 and fitted trace visually checked. No journal acceptance verified; full methods, raw data and independent device replication remain open.

## Claim evidence and conditions

### scwiki-d12-claim-scope-1

Topological superconductivity concerns the paired quasiparticle state, not merely normal-state band topology next to a superconducting transition. The target and dimensionality must be stated before interpreting a probe:

| Target | What the evidence addresses | What it does not establish by itself |
| --- | --- | --- |
| Bulk paired phase | A gapped BdG state with specified symmetries and an appropriate invariant | A superconducting transition plus normal-state band inversion does not determine the pairing topology |
| Superconducting surface | Surface spin texture and a superconducting surface gap, interpreted with a surface model | A surface result does not classify all three-dimensional bulk pairing or demonstrate a localized vortex mode |
| Localized vortex state | A defect solution or spatially resolved low-energy spectroscopy | A model zero mode or local spectral peak does not demonstrate non-Abelian operations |

This comparison separates mathematical classification, probe evidence and proposed operations. The classification reference treats non-interacting or mean-field Hamiltonians and discusses stability to interactions small compared with the gap; it is not a complete classification of strongly interacting or nodal superconductivity.

Evidence: editorial-analysis. Sources: source-37986d2488e4cace, source-4e92d11d59fb3441, source-7dc64372df6ba886, source-e2865b9f2461bc7f.

Source relationship: context-for-editorial-analysis. Editorial comparison of classification, Fu–Berg bulk criterion, Fu–Kane surface/vortex construction and Zhang surface spectroscopy. The table organizes distinct targets; it does not assign a measured invariant to a specimen.

Conditions: Dimension, gap and symmetry class belong to the paired Hamiltonian. Distinguish bulk phase, superconducting surface and localized vortex/operation evidence.

Verification scope: Selected Schnyder authorv3pp2–3/TableI; Fu–Bergv1/Fu–Kanev3pp1–2; Zhang authorv2abstract/selected surface text/Fig4. No complete interacting classification or material-invariant reproduction.

### scwiki-d12-claim-scope-2

For a fully gapped Bogoliubov–de Gennes description, topology depends on dimensionality, symmetries and the occupied quasiparticle states. This restricted framework does not exhaust interacting or nodal superconductivity. A local zero-bias peak is neither a universal requirement for every topological superconductor nor sufficient proof of a Majorana mode. Demonstrating non-Abelian operations is a further question; it is not the definition of every bulk topological superconducting phase.

Evidence: hypothesis. Sources: source-37986d2488e4cace, source-45c980d0afc5ac14, source-87a91e2d8cd79e95, source-dca781066e96eecc.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-mathematical-assumptions-1

One clean single-subband wire model uses $H(k)=\xi_k\tau_z+\alpha k\tau_z\sigma_y+V_Z\sigma_x+\Delta\tau_x$, where $\xi_k=\hbar^2k^2/(2m)-\mu$, $\sigma$ acts on spin and $\tau$ on particle–hole space. Here $\Delta$ is real induced pairing, spin–orbit and Zeeman axes are perpendicular, and $V_Z=|g\mu_B B|/2$ is an energy rather than a magnetic field. With $a=\alpha k$, diagonalization yields $E_\pm^2=\xi_k^2+a^2+V_Z^2+\Delta^2\pm2\sqrt{V_Z^2(\xi_k^2+\Delta^2)+\xi_k^2a^2}$.

Evidence: theory. Sources: source-45c980d0afc5ac14, source-87a91e2d8cd79e95.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-mathematical-assumptions-2

The zero-momentum gap closes at $V_Z^2=\mu^2+\Delta^2$. In the stated gapped wire model, the side with $V_Z^2>\mu^2+\Delta^2$ supports the topological phase. The inequality alone is not a material diagnostic: setting spin–orbit coupling to zero can leave finite-momentum gapless excitations even when it holds. Independent checks compare 54 matrix spectra, three closure points and that counterexample. They do not simulate multiband disorder, orbital depairing, interfaces or a complete material invariant.

Evidence: theory. Sources: source-45c980d0afc5ac14, source-87a91e2d8cd79e95.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-mathematical-assumptions-3

The Fu–Berg criterion assumes a three-dimensional, fully gapped, time-reversal-invariant BdG state with inversion symmetry and odd-parity pairing. Its normal-state Fermi-surface shortcut also assumes weak pairing and bands separated from the Fermi energy at the time-reversal-invariant momenta (TRIM).

- **Use the BdG inversion operator:** odd-parity pairing requires $\widetilde P=P\tau_z$. The invariant multiplies its negative-energy eigenvalues at the eight TRIM, taking one member of each Kramers pair. Multiplying both members would square each sign and lose the parity information.
- **Read the result as a sufficient test:** the parity $\nu=n\bmod2$ is only part of the three-dimensional integer classification. An odd result establishes nonzero $n$ within these assumptions. An even result does not by itself establish $n=0$; an even, nonzero invariant is not excluded. Counting enclosed TRIM neither measures the pairing parity nor certifies a doped specimen.
- **Keep surface and vortex states separate:** Fu–Kane's uniform proximitized helical surface has a gapped spectrum and preserves time reversal, despite its projected analogy with spinless $p_x+ip_y$ pairing. A vortex introduces a winding pairing phase, $\Delta(r,\theta)=\Delta_0(r)e^{\pm i\theta}$, and admits a localized zero mode in that ideal model. The vortex calculation is not an observation of a zero mode on every uniform surface, nor a demonstration of braiding.

These are different theoretical constructions. Applying either to a material requires checking its own symmetry, gap, geometry and disorder conditions; neither theorem replaces those measurements.

Evidence: theory. Sources: source-4e92d11d59fb3441, source-7dc64372df6ba886.

Source relationship: context-for-editorial-analysis. The two primary theoretical constructions supply the criterion and surface/vortex solutions. The even-parity-result limitation, Kramers double-counting warning and material-evidence comparison are explicit logical consequences and editorial explanation, not a new invariant calculation or experimental validation.

Conditions: Fu–Berg:3D fully gapped TRI centrosymmetric odd-parity BdG state; weak-pairing shortcut with normal bands away from chemical potential atTRIM. One factor perKramers pair, BdG inversionP*tau_z. Fu–Kane:uniform helical TI surface versus phase-winding vortex in ideal proximity model.

Verification scope: Fu–Berg authorv1 selectedpp1–2/proofEqs2–4; Fu–Kane published-version authorv3 selectedpp1–2/Eqs2–3. Three paper pages visually inspected. No complete final comparison, global invariant computation or experimental reproduction.

### scwiki-d12-claim-experimental-criteria-1

Hor's Cu-intercalated Bi2Se3 study reports a transition near 3.8 K. In the checked author manuscript, Cu0.12Bi2Se3 has a magnetic response roughly one-fifth of ideal diamagnetism at the lowest accessible temperature, which the authors treat as a lower bound because the transition is incomplete. The resistance does not reach zero. This establishes why sample preparation and connected superconducting paths matter; neither a universal superconducting volume nor topological pairing can be inferred from those numbers alone.

Evidence: experiment. Sources: source-0857b567ea17c2f2.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-experimental-criteria-2

Butch's YPtBi crystals show a resistive transition near 0.77 K with an AC diamagnetic response. The critical-field curve uses a 50% resistive criterion, with 10–90% bounds, and field perpendicular to [001]. The often-quoted 1.5 T is a zero-temperature extrapolation. Converting it with $B_{c2}=\Phi_0/(2\pi\xi^2)$ gives about 14.8 nm as a consistency estimate. Superconductivity in a candidate topological semimetal does not itself identify its pairing topology or establish a calibrated superconducting volume.

Evidence: experiment. Sources: source-4d53a5f7cdff4a54.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-experimental-criteria-3

Two iron-chalcogenide papers require separate attribution. Zhang's ARPES study resolves a spin-helical surface band and a superconducting surface gap around 1.8 meV, interpreted as surface topological superconductivity. Its low-temperature measurement at 2.4 K is not the reported 14.5 K transition temperature. The study's surface interpretation must not be replaced by a statement that all bulk pairing is topological. In its angular-gap figure, hollow mirrored markers are not additional independent measurements.

Evidence: experiment. Sources: source-e2865b9f2461bc7f.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-experimental-criteria-4

Wang's distinct vortex-STM paper studies FeTe0.55Se0.45. Figure 2 spectra at 0.55 K and 0.5 T parallel to c are normalized by integrated spectral area; they cannot be read as an absolute quantized conductance plateau. The approximately 0.3 meV peak width contains instrumental and thermal broadening. Barrier and temperature scans elsewhere use different fields. The zero-energy signal is interpreted as Majorana evidence, while the displayed braiding sequence is a proposal, not a performed operation.

Evidence: experiment. Sources: source-a972be8eb742181f.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-experimental-criteria-5

Xu's Bi2Te3/NbSe2 heterostructure study compares vortex spectra across film thicknesses. The checked 2013 author version underlying the 2015 publication distinguishes where split peaks become experimentally resolvable from the extrapolated origin of their splitting. The former is around 20 nm even for thinner films; the latter changes near four quintuple layers and supports the authors' Majorana interpretation. Treating every unsplit interval as a directly measured Majorana radius would discard both the extrapolation and approximately 0.2 meV resolution.

Evidence: experiment. Sources: source-6861a8084ed315a1.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-experimental-criteria-6

Semiconductor readout advances also need their own evidence category. Aghaee's 2025 InAs–Al experiment measures single-shot parity interferometrically and explicitly retains an Andreev-state alternative. Van Loo's 2026 work demonstrates parity readout in a minimal Kitaev-chain device, with a two-site rather than asymptotically long-chain model. These are useful experimental capabilities; neither result, as scoped here, establishes braiding or fault-tolerant topological computation. Readout performance, spatial nonlocality and a protected bulk phase remain distinct questions.

Evidence: experiment. Sources: source-e3824a8727c3ec54, source-a3c2dc153700478a.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-limits-and-disputes-1

A local zero-bias peak can arise from ordinary Andreev states, confinement or disorder. Pan and Das Sarma's simulations also show that end-to-end conductance correlations are not conclusive on their own: accidental correlations can occur in selected disordered examples, including long wires. Length must be considered relative to the relevant coherence scale; a fixed micrometre threshold is not a universal diagnostic. These model counterexamples do not identify the mechanism in every later device. Temperature, barrier, calibration, disorder and nonlocal controls must be interpreted together.

**Retracted quantization claim:** *Quantized Majorana conductance* (2018) was retracted on 8 March 2021. The author-institution copy of the notice reports:

- Previously undisclosed charge-jump corrections in Figure 2a and Extended Data Figure 4b, and a mislabelled axis in Figure 4b.
- Recalibration shifting the conductance plateaus by 8%, above $2e^2/h$; this is a reported calibration change, not an uncertainty interval.
- Replotting the full parameter range placing points outside the stated two-sigma error bars, so the authors withdrew the quantized-conductance conclusion.

The linked notice and [raw-data archive](https://doi.org/10.5281/zenodo.4545577) document the correction trail; this review has not independently recalibrated the apparatus or reproduced that archive's analysis. The retracted result remains historical context, never positive confirmation of later devices. Nor does retracting this result decide the status of every other platform.

Evidence: editorial-analysis. Sources: source-dca781066e96eecc, source-6a7f18eec743e89d, source-f127bb2d99183698.

Source relationship: context-for-editorial-analysis. The original theoretical paper supplies counterexamples and the author retraction supplies its own calibration and figure corrections. The separation of those evidence types and non-transfer of conclusions to other devices is editorial analysis, not a new disorder simulation or calibration.

Conditions: Pan authorv2 selected short/long-wire disorder comparisons, not a current census of experimental devices; retraction applies to DOI10.1038/nature26142. Reported8percent calibration shift is not an uncertainty.

Verification scope: Pan authorv2selectedpp13–14 withp14pixels; full author-institution notice text, Nature date/DOI and Crossmark retraction checked. Raw archive not independently reanalysed.

### scwiki-d12-claim-limits-and-disputes-2

Complete scientific acceptance remains open. This article checks selected primary passages, figures and restricted models. Earlier author versions are identified; complete final-version comparisons, supplements, raw-data analysis, calibration and exclusion of competing interpretations remain unfinished. These limits concern the present review and should not be inflated into a claim that every platform has the same evidential status.

Evidence: hypothesis. Sources: source-45c980d0afc5ac14, source-87a91e2d8cd79e95, source-0857b567ea17c2f2, source-4d53a5f7cdff4a54, source-e2865b9f2461bc7f, source-a972be8eb742181f, source-6861a8084ed315a1.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-discovery-implications-1

Specify the target first: bulk odd-parity superconductivity, a proximitized topological surface, a wire end state or a controllable parity device. Then record composition, geometry, temperature, field, superconducting criterion and competing-state tests for that target. Compare complementary probes on characterized samples and preserve failures and retractions. Strong spin–orbit coupling or an attractive band calculation can motivate an experiment, but cannot replace identification of the actual paired phase and its protected observables.

Evidence: hypothesis. Sources: source-4e92d11d59fb3441, source-7dc64372df6ba886, source-87a91e2d8cd79e95, source-dca781066e96eecc, source-f127bb2d99183698.

Conditions: Separate bulk, surface, wire and device claims; preserve composition, field, temperature, resolution, transport criterion and publication/retraction status.

Verification scope: Selected primary text/figures and restricted model checks; full scientific acceptance remains open.

### scwiki-d12-claim-july2026-parity-control

A 10 July 2026 preprint by Zatelli and colleagues reports coherent parity oscillations in coupled minimal Kitaev chains. The authors distinguish partial protection at the sweet spot from the unprotected degeneracy-lifting operations used for rotation. No journal acceptance was verified here; this is not a demonstrated braid or a certified gate fidelity.

Evidence: experiment. Sources: source-818818441532a8ec.

Conditions: July 2026 preprint; partial sweet-spot protection and coherent rotation do not establish protected gates or braiding.

Verification scope: v1 submitted 10 July 2026. Selected pp. 1–6 and p. 9; p. 5 Figure 2 and fitted trace visually checked. No journal acceptance verified; full methods, raw data and independent device replication remain open.

## Revision and review

Article ID: scwiki-d12. Revision: 6463be9c84e5e700c29a. Updated: 2026-10-20.

Review scope: Selected Hor,Butch,Zhang,Wang,Xu,Lutchyn and Oreg primary text/figures; inherited explicitly scoped B11/F5 theory,readout and retraction checks; independent wire spectra and field-length conversion. Selected July 2026 coherent-parity preprint results and explicit protection limits. Follow-up Fu–Berg criterion/proof and Fu–Kane uniform-surface/vortex scope. Follow-up classification/target distinction and original retraction trail, with selected disorder-correlation counterexamples.

- Complete scientific acceptance remains open.
- Full final-version/supplement/raw-data and alternative-state/calibration reviews remain open.
- Selected author versions are not silently treated as complete final publications.
- Model checks do not establish material topology,braiding or a full experimental reproduction.

Source: [D_material_families/D12_topological_SC.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/D_material_families/D12_topological_SC.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 02e36865935c6d210334589f83b91605230cd0d63e2f26a255ac148a93f104cb.