# Topological superconductivity

Connect symmetry, bulk invariants and boundary evidence while separating Majorana model constructions, experimental readout and unresolved identification.

## Scope

Topological superconductivity describes a paired state's quasiparticle structure under specified dimensionality, symmetry and gap conditions. It does not name an independent pairing interaction. This entry concentrates on fully gapped Bogoliubov–de Gennes (BdG) descriptions; topology of nodal superconductors requires a different invariant and boundary analysis. The noninteracting/BdG classification is not, by itself, a complete classification of interacting phases. [R5](#ref-source-37986d2488e4cace)

Read and Green distinguish weak and strong pairing in a specified two-dimensional spinless p-wave model. This establishes a theoretical phase distinction, not a classification of every material with an unusual tunnelling spectrum. A material claim needs a superconducting state compatible with the proposed invariant and the relevant boundary geometry. [R2](#ref-source-a33c639b7db4f05a)

## Mathematical assumptions

For a spinless chain, take positive nearest-neighbour hopping $t$, real pairing $\Delta$, chemical potential $\mu$ and unit lattice spacing. One Nambu convention gives

$$H(k)=(-\mu-2t\cos k)\tau_z+2\Delta\sin k\,\tau_y,\qquad E_\pm(k)=\pm\sqrt{(\mu+2t\cos k)^2+4\Delta^2\sin^2 k}.$$

The bulk is gapped for nonzero $\Delta$ away from $|\mu|=2t$. The particle-hole-protected parity index is $Q=\operatorname{sgn}(\mu^2-4t^2)$: $Q=-1$ identifies the nontrivial region. The real-parameter chain also has an additional effective time-reversal symmetry; retaining that symmetry gives the finer BDI classification. Using its class-D parity does not claim that its additional symmetry is absent. These are ideal mean-field assumptions, not a determination of an experimental device's parameters. [R1](#ref-source-ade538b429fdd847) [R5](#ref-source-37986d2488e4cace)

An independent finite-matrix check constructs the hopping and antisymmetric pairing blocks directly. Five periodic eight-site spectra agree with the analytic energies above; Hermiticity, particle-hole symmetry and eigenvector residuals are checked separately. For open chains with $\mu=t=\Delta=1$ in common energy units, the smallest positive eigenvalues at lengths 8, 12 and 16 are approximately 0.00585989, 0.000366211 and 0.0000228882. The corresponding normalized eigenvectors place about 94% of their weight in the outermost two sites at each end. This demonstrates finite-size splitting and end localization for these parameters, not a universal protection scale.

A second independent check illustrates why zero energy is insufficient. At $\mu=0$ and $t=\Delta=1$, an eight-site open chain has two exact zero eigenvalues; the zero-subspace projector has one unit of Nambu weight at each end. A trivial chain at $\mu=3$ also acquires two zero eigenvalues if its first site is deliberately disconnected and tuned to zero on-site energy. Both projector weights then sit at that same site. The sum of two is a Nambu subspace dimension, not a measured charge. No leads or conductance are simulated here; this constructed counterexample tests the inference from energy alone.

For centrosymmetric, time-reversal-invariant, fully gapped odd-parity pairing, the Fu–Berg weak-pairing criterion supplies a useful sufficient test: the Fermi surface encloses an odd number of time-reversal-invariant momenta. In their three-dimensional setting the test detects the parity of an integer invariant. An even result does not establish that the integer itself vanishes. The normal-state Fermi-surface reduction assumes weak pairing and normal bands away from the chemical potential at those momenta. Do not apply it to an unspecified nodal, inversion-broken or time-reversal-broken state. [R4](#ref-source-4e92d11d59fb3441)

## Experimental criteria

Distinguish intrinsic bulk order from an engineered interface. In the Fu–Kane proposal, ordinary s-wave proximity pairing gaps a topological-insulator surface. Its relation to a spinless p+ip description does not make the uniform surface a time-reversal-breaking chiral bulk superconductor: the uniform paired surface preserves time reversal, whereas a vortex changes the local setting. Surface participation in pairing and the actual defect or junction geometry matter. [R3](#ref-source-7dc64372df6ba886)

A zero-bias conductance peak alone does not uniquely identify a Majorana mode. Pan and Das Sarma construct alternatives involving quantum dots, spatially varying potentials and disorder. Their analysis also limits end-to-end correlation as a diagnostic: trivial correlations can occur, particularly in short wires, and accidental correlations require scrutiny even in longer examples. Length must be assessed relative to the relevant localization scales. Their model comparisons do not diagnose every experimental device. [R6](#ref-source-dca781066e96eecc)

As an editorial review protocol, for an experimental claim, retain the full gate and field ranges, calibration, temperature, tunnel coupling and device-selection procedure. Compare local spectra with independent bulk-gap and nonlocal constraints. A measured gap reopening, a near-quantized peak or an isolated boundary feature must be assessed together with competing models; a checklist of individually ambiguous signatures cannot substitute for a state-specific argument. The model comparison and retraction supply context for this protocol, rather than independently validating it. [R6](#ref-source-dca781066e96eecc) [R7](#ref-source-f127bb2d99183698)

A recent readout result illustrates a narrower, measurable advance. Van Loo and colleagues, published on 11 February 2026, report quantum-capacitance parity readout in a minimal two-site Kitaev chain, with switching lifetimes exceeding a millisecond and charge-sensing checks. Their abstract explicitly limits the protection of these minimal-chain modes relative to longer chains. Parity readout is not a demonstration of fault-tolerant braiding or a bulk material's superconducting transition. Here the published abstract and bibliographic record were checked; the full methods and raw data remain to be reviewed. [R8](#ref-source-a3c2dc153700478a)

## Limits and disputes

The 2018 claim titled “Quantized Majorana conductance” was retracted in 2021. The authors' retraction describes undisclosed charge-jump corrections, a mislabeled axis and recalibration that moved plateau values above the originally claimed quantized value. They withdrew the quantization claim. The original positive claim must not be reused as confirmation; the retraction does not by itself disprove every Majorana platform. [R7](#ref-source-f127bb2d99183698)

Protection is conditional on the relevant gap, symmetries and isolation. Finite-size overlap, uncontrolled perturbations and changes of state need explicit treatment. The finite checks above establish neither a material invariant nor transport statistics, parity lifetime, fusion rules or a fault-tolerant operation. This entry does not promote a candidate compound to confirmed topological superconductivity. Full final-version comparisons and complete scientific acceptance remain open.

## Discovery implications

As an editorial research workflow, start from a specified superconducting order and its dimensionality. Compute the appropriate invariant with convergence checks, predict boundary or defect responses in the actual geometry, and test alternatives against the same data range. Keep an engineered interface result separate from a host bulk-phase claim. Record negative results, corrections and the distinction between theoretical construction, compatible measurement and unique experimental identification.

## Related topics

Connect the paired-state description to [BCS theory](/scwiki/bcs-weak-coupling), [Andreev reflection](/scwiki/andreev-reflection), [ferromagnetic fluctuations](/scwiki/ferromagnetic-fluctuations) and [multiband pairing](/scwiki/multiband-pairing).

## Material cases

No sample-specific material records are included in this article. Model results are not experimental transition temperatures.

## Sources

### Ref source-ade538b429fdd847

[R1] Unpaired Majorana fermions in quantum wires (2001). [DOI 10.1070/1063-7869/44/10S/S29](https://doi.org/10.1070/1063-7869/44/10S/S29). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-19. Selected pp.3–7; p.4 Eq.(4) and p.7 Eq.(13) visually rechecked. Spinless toy model, bulk spectrum and boundary assumptions; local finite calculations are separately attributed.
### Ref source-a33c639b7db4f05a

[R2] Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect (2000). [DOI 10.1103/PhysRevB.61.10267](https://doi.org/10.1103/PhysRevB.61.10267). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Abstract-scoped weak/strong pairing distinction; no full-paper claim.
### 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: 2026-10-18. Selected pp1–2 uniform time-reversal preservation, surface proximity and vortex distinction.
### Ref source-4e92d11d59fb3441

[R4] 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-earlier-author-manuscript-check. Checked: 2026-10-18. Selected pp1–2, visually checked p1 sufficient-not-necessary criterion and parity of integer invariant; weak-pairing proof conditions retained.
### Ref source-37986d2488e4cace

[R5] 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-manuscript-check. Checked: 2026-10-18. Selected classification discussion pp3–5: noninteracting/BdG symmetry scope; not a full interacting classification.
### Ref source-dca781066e96eecc

[R6] 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-manuscript-check. Checked: 2026-10-19. Abstract and selected pp13–14 correlation/disorder caveats; no transport reproduction or universal experimental diagnosis.
### Ref source-f127bb2d99183698

[R7] 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: institution-hosted-retraction-text-check. Checked: 2026-10-19. Full retraction text reproduced by author institution read; publisher endpoint unavailable. Retracted original is not positive evidence.
### Ref source-a3c2dc153700478a

[R8] 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: published-abstract-and-date-check. Checked: 2026-10-18. Published abstract read; publication date11February2026 corroborated by PubMed41673287. Institution PDF403; methods/raw data not checked.

## Claim evidence and conditions

### scwiki-b11-claim-scope-1

Topological superconductivity describes a paired state's quasiparticle structure under specified dimensionality, symmetry and gap conditions. It does not name an independent pairing interaction. This entry concentrates on fully gapped Bogoliubov–de Gennes (BdG) descriptions; topology of nodal superconductors requires a different invariant and boundary analysis. The noninteracting/BdG classification is not, by itself, a complete classification of interacting phases.

Evidence: theory. Sources: source-37986d2488e4cace.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-scope-2

Read and Green distinguish weak and strong pairing in a specified two-dimensional spinless p-wave model. This establishes a theoretical phase distinction, not a classification of every material with an unusual tunnelling spectrum. A material claim needs a superconducting state compatible with the proposed invariant and the relevant boundary geometry.

Evidence: theory. Sources: source-a33c639b7db4f05a.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-mathematical-assumptions-1

For a spinless chain, take positive nearest-neighbour hopping $t$, real pairing $\Delta$, chemical potential $\mu$ and unit lattice spacing. One Nambu convention gives

Evidence: theory. Sources: source-ade538b429fdd847.

Source relationship: mixed-source-and-local-derivation. Kitaev author manuscript v2, p.4 Eq.(4) and p.7 Eq.(13), supplies the spinless chain and bulk spectrum. The displayed Nambu matrix uses the local real-pairing convention; finite checks do not establish an experimental device phase.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-mathematical-assumptions-2

$$H(k)=(-\mu-2t\cos k)\tau_z+2\Delta\sin k\,\tau_y,\qquad E_\pm(k)=\pm\sqrt{(\mu+2t\cos k)^2+4\Delta^2\sin^2 k}.$$

Evidence: theory. Sources: source-ade538b429fdd847.

Source relationship: mixed-source-and-local-derivation. Kitaev author manuscript v2, p.4 Eq.(4) and p.7 Eq.(13), supplies the spinless chain and bulk spectrum. The displayed Nambu matrix uses the local real-pairing convention; finite checks do not establish an experimental device phase.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-mathematical-assumptions-3

The bulk is gapped for nonzero $\Delta$ away from $|\mu|=2t$. The particle-hole-protected parity index is $Q=\operatorname{sgn}(\mu^2-4t^2)$: $Q=-1$ identifies the nontrivial region. The real-parameter chain also has an additional effective time-reversal symmetry; retaining that symmetry gives the finer BDI classification. Using its class-D parity does not claim that its additional symmetry is absent. These are ideal mean-field assumptions, not a determination of an experimental device's parameters.

Evidence: theory. Sources: source-ade538b429fdd847, source-37986d2488e4cace.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-mathematical-assumptions-4

An independent finite-matrix check constructs the hopping and antisymmetric pairing blocks directly. Five periodic eight-site spectra agree with the analytic energies above; Hermiticity, particle-hole symmetry and eigenvector residuals are checked separately. For open chains with $\mu=t=\Delta=1$ in common energy units, the smallest positive eigenvalues at lengths 8, 12 and 16 are approximately 0.00585989, 0.000366211 and 0.0000228882. The corresponding normalized eigenvectors place about 94% of their weight in the outermost two sites at each end. This demonstrates finite-size splitting and end localization for these parameters, not a universal protection scale.

Evidence: theory. Sources: source-ade538b429fdd847.

Source relationship: context-for-local-derivation. The finite lengths, numerical eigenvalues and deliberately decoupled trivial dot are local explanatory calculations in tools/check_topological_models.py. Kitaev supplies model context, not these particular numerical examples; no transport or material assignment is calculated.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-mathematical-assumptions-5

A second independent check illustrates why zero energy is insufficient. At $\mu=0$ and $t=\Delta=1$, an eight-site open chain has two exact zero eigenvalues; the zero-subspace projector has one unit of Nambu weight at each end. A trivial chain at $\mu=3$ also acquires two zero eigenvalues if its first site is deliberately disconnected and tuned to zero on-site energy. Both projector weights then sit at that same site. The sum of two is a Nambu subspace dimension, not a measured charge. No leads or conductance are simulated here; this constructed counterexample tests the inference from energy alone.

Evidence: theory. Sources: source-ade538b429fdd847.

Source relationship: context-for-local-derivation. The finite lengths, numerical eigenvalues and deliberately decoupled trivial dot are local explanatory calculations in tools/check_topological_models.py. Kitaev supplies model context, not these particular numerical examples; no transport or material assignment is calculated.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-mathematical-assumptions-6

For centrosymmetric, time-reversal-invariant, fully gapped odd-parity pairing, the Fu–Berg weak-pairing criterion supplies a useful sufficient test: the Fermi surface encloses an odd number of time-reversal-invariant momenta. In their three-dimensional setting the test detects the parity of an integer invariant. An even result does not establish that the integer itself vanishes. The normal-state Fermi-surface reduction assumes weak pairing and normal bands away from the chemical potential at those momenta. Do not apply it to an unspecified nodal, inversion-broken or time-reversal-broken state.

Evidence: theory. Sources: source-4e92d11d59fb3441.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-experimental-criteria-1

Distinguish intrinsic bulk order from an engineered interface. In the Fu–Kane proposal, ordinary s-wave proximity pairing gaps a topological-insulator surface. Its relation to a spinless p+ip description does not make the uniform surface a time-reversal-breaking chiral bulk superconductor: the uniform paired surface preserves time reversal, whereas a vortex changes the local setting. Surface participation in pairing and the actual defect or junction geometry matter.

Evidence: hypothesis. Sources: source-7dc64372df6ba886.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-experimental-criteria-2

A zero-bias conductance peak alone does not uniquely identify a Majorana mode. Pan and Das Sarma construct alternatives involving quantum dots, spatially varying potentials and disorder. Their analysis also limits end-to-end correlation as a diagnostic: trivial correlations can occur, particularly in short wires, and accidental correlations require scrutiny even in longer examples. Length must be assessed relative to the relevant localization scales. Their model comparisons do not diagnose every experimental device.

Evidence: hypothesis. Sources: source-dca781066e96eecc.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-experimental-criteria-3

As an editorial review protocol, for an experimental claim, retain the full gate and field ranges, calibration, temperature, tunnel coupling and device-selection procedure. Compare local spectra with independent bulk-gap and nonlocal constraints. A measured gap reopening, a near-quantized peak or an isolated boundary feature must be assessed together with competing models; a checklist of individually ambiguous signatures cannot substitute for a state-specific argument.

Evidence: hypothesis. Sources: source-dca781066e96eecc, source-f127bb2d99183698.

Source relationship: context-for-editorial-analysis. Pan and Das Sarma v2, pp.13–14, provide model-based nonlocal-correlation and selection caveats; the 2021 retraction supplies calibration and omitted-range context. The combined review protocol is editorial guidance, not a protocol directly reported or validated by either source.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-experimental-criteria-4

A recent readout result illustrates a narrower, measurable advance. Van Loo and colleagues, published on 11 February 2026, report quantum-capacitance parity readout in a minimal two-site Kitaev chain, with switching lifetimes exceeding a millisecond and charge-sensing checks. Their abstract explicitly limits the protection of these minimal-chain modes relative to longer chains. Parity readout is not a demonstration of fault-tolerant braiding or a bulk material's superconducting transition. Here the published abstract and bibliographic record were checked; the full methods and raw data remain to be reviewed.

Evidence: experiment. Sources: source-a3c2dc153700478a.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-limits-and-disputes-1

The 2018 claim titled “Quantized Majorana conductance” was retracted in 2021. The authors' retraction describes undisclosed charge-jump corrections, a mislabeled axis and recalibration that moved plateau values above the originally claimed quantized value. They withdrew the quantization claim. The original positive claim must not be reused as confirmation; the retraction does not by itself disprove every Majorana platform.

Evidence: experiment. Sources: source-f127bb2d99183698.

Conditions: Author retraction concerns the 2018 conductance report and its calibration, figure corrections and disclosed data ranges; it is not a universal statement about other devices.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-limits-and-disputes-2

Protection is conditional on the relevant gap, symmetries and isolation. Finite-size overlap, uncontrolled perturbations and changes of state need explicit treatment. The finite checks above establish neither a material invariant nor transport statistics, parity lifetime, fusion rules or a fault-tolerant operation. This entry does not promote a candidate compound to confirmed topological superconductivity. Full final-version comparisons and complete scientific acceptance remain open.

Evidence: hypothesis. Sources: source-ade538b429fdd847.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

### scwiki-b11-claim-discovery-implications-1

Start from a specified superconducting order and its dimensionality. Compute the appropriate invariant with convergence checks, predict boundary or defect responses in the actual geometry, and test alternatives against the same data range. Keep an engineered interface result separate from a host bulk-phase claim. Record negative results, corrections and the distinction between theoretical construction, compatible measurement and unique experimental identification.

Evidence: hypothesis. Sources: source-ade538b429fdd847.

Source relationship: context-for-editorial-analysis. Editorial research workflow informed by the model and evidence limits in this entry; the linked Kitaev paper is theoretical context, not a validated experimental acceptance checklist.

Conditions: State-specific BdG assumptions; independent finite spectra/projectors do not reproduce transport or establish a material phase.

Verification scope: Selected sources and independent chain checks; full scientific acceptance open.

## Revision and review

Article ID: scwiki-b11. Revision: 196ae2aa3286e45e1202. Updated: 2026-10-19T00:43:12.842953+00:00.

Review scope: Selected primary manuscripts, published abstracts, institution-hosted retraction text and independent finite-chain calculations. Follow-up distinguishes local chain calculations and editorial experimental-review advice from direct source results.

- Complete final-version, methods and raw-data review remains open.
- Finite-chain spectra and projectors do not establish experimental conductance or a material invariant.
- 2026 readout source reviewed at abstract scope only; it does not demonstrate fault-tolerant braiding.
- No candidate compound is certified as topological.

Source: [B_mechanisms/B11_topological_pairing.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/B_mechanisms/B11_topological_pairing.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 49d2d5071645f0596f86e3470a9fe227d9b5a3b888a5902c3a7fccb6b3a8253a.