# Dimensionality: BKT coherence, phase slips and finite-size pairing

Compare two-dimensional films, narrow wires and small grains using phase stiffness, screening, sample geometry and explicit experimental criteria.

## Scope

Dimensionality compares geometry with physical length scales. A layered crystal, a two-dimensional electronic band and a superconducting film with a nearly uniform order parameter through its thickness are different descriptions. Compare thickness with the perpendicular coherence length, wire transverse dimensions with coherence lengths, and lateral size with magnetic screening lengths. Interlayer coupling, substrate, disorder and carrier density can change when layers are removed. Reduced thickness alone does not establish an ideal two-dimensional phase transition. [R1](#ref-source-408113812912d816) [R6](#ref-source-c80fb39cdf748a83)

For a short-range two-dimensional phase model, vortex–antivortex unbinding separates quasi-long-range order from a disordered phase. The relevant stiffness is renormalized by fluctuations. In a charged film, magnetic screening changes the vortex interaction at long distance: Kosterlitz and Thouless explicitly identified this limitation in their original paper. BKT-like behavior requires an appropriate range of length, current and temperature scales; do not transfer the neutral infinite-system limit to every finite superconducting sheet. [R1](#ref-source-408113812912d816) [R2](#ref-source-8f81510a80c35e3a)

## Mathematical assumptions

Define the sheet phase stiffness $J$ in energy units by $F_\theta=(J/2)\int d^2r\,|\nabla\theta|^2$. A single vortex has logarithmic energy $\pi J\ln(L/a)$ and positional entropy $2k_B\ln(L/a)$ within this model, where $a$ is a core cutoff. The universal jump is $J_R(T_{\rm BKT}^-)=2k_BT_{\rm BKT}/\pi$, with the long-distance **renormalized** stiffness $J_R$. The elementary energy–entropy balance explains the scale; it is not a replacement for the renormalization calculation. For a local sheet inductance $L_k$ per square, the corresponding stiffness convention is $J=\hbar^2/(4e^2L_k)$. Frequency, vortex response and finite size matter when comparing a measured inductance with the static jump. [R1](#ref-source-408113812912d816) [R2](#ref-source-8f81510a80c35e3a)

In a uniform narrow wire near its mean-field transition, the zero-current Ginzburg–Landau saddle gives $\Delta F_0=(8\sqrt2/3)u_c A\xi$, where $u_c=B_c^2/(2\mu_0)$ is the SI condensation-energy density and $A$ is cross-sectional area. A local check integrates the normalized saddle $f(y)=\tanh(y/\sqrt2)$, $y=x/\xi$, in $\int[(1-f^2)^2+2(df/dy)^2]dy$. The profile satisfies $f''+f-f^3=0$, approaches opposite real order-parameter signs at either end, and passes through zero at its centre. Substituting $z=f(y)$ reduces the integral to $2\sqrt2\int_{-1}^{1}(1-z^2)dz=8\sqrt2/3$. This independently checks the stationary profile and barrier factor under those assumptions, not a finite-current barrier, quantum tunneling action or the full LAMH prefactor. The original Langer–Ambegaokar and McCumber–Halperin full derivations have not been retrieved for this review. [R3](#ref-source-aa12d674a498e02d) [R4](#ref-source-866d4492c91f5b39)

For rare independent thermal phase slips, define $r_0$ explicitly as the zero-bias rate **per direction**. With $\Phi_0=h/(2e)$, detailed balance gives $r_\pm=r_0\exp[\pm I\Phi_0/(2k_BT)]$ to leading small-bias barrier tilt, $V=\Phi_0(r_+-r_-)$ and $R_{I\to0}=\Phi_0^2r_0/(k_BT)$. The thermal activation factor belongs inside $r_0$; this definition avoids confusing a total event rate with a directional one. Low-temperature resistance saturation alone does not identify coherent quantum phase slips. [R3](#ref-source-aa12d674a498e02d) [R4](#ref-source-866d4492c91f5b39) [R8](#ref-source-42e0dec220b7b3ea)

For a homogeneous London film in free space with thickness $d\ll\lambda$, define the Pearl length explicitly as $\Lambda=2\lambda^2/d$. Pearl's Fourier denominator is $1+q\Lambda$; outside the core, the circulating sheet current crosses from $1/r$ for $r\ll\Lambda$ to $1/r^2$ for $r\gg\Lambda$. The far-field interaction energy therefore decays as $1/r$, rather than retaining a logarithm indefinitely. In the same local thin-film convention, $L_k=\mu_0\lambda^2/d$, so $\Lambda=2L_k/\mu_0$. A useful logarithmic window must fit between the core and both screening and lateral boundaries; this length comparison alone does not diagnose BKT behavior. The inductance conversion and numerical asymptote checks here are local model calculations, not measured film parameters. [R10](#ref-source-260bbd6c79b3d696) [R1](#ref-source-408113812912d816)

## Experimental criteria

Xi's encapsulated NbSe2 devices distinguish several transition definitions. In the checked author manuscript, the monolayer Aslamazov–Larkin fit gives $T_{c0}\approx3.0$ K, close to the 50% normal-resistance criterion; the inferred BKT transition lies near the 1% criterion. Those are separate quantities. Normal resistance was taken at 8 K. The four-terminal field series used a 50% criterion, while a separate two-terminal monolayer device used a rapid-resistance-drop criterion. The 0.3 K measurement floor is not another transition. Large in-plane critical fields support the authors' Ising-pairing interpretation under their model, rather than proving triplet pairing by themselves. The journal issue is from 2016; online publication was 9 November 2015. [R6](#ref-source-c80fb39cdf748a83)

Bezryadin's nanotube-templated amorphous Mo79Ge21 wires were approximately 10 nm or less in diameter and 95–185 nm long. The resistance drop near 5.5 K belongs to the electrodes, after which the wire resistance could be isolated. In the studied series, samples on opposite sides of a normal-resistance scale near $R_Q=h/(4e^2)\approx6.45$ kilohms showed decreasing or increasing resistance on cooling. One superconducting sample was checked down to 50 mK. These are sample-specific transport observations and a quantum-fluctuation interpretation, not a universal wire transition temperature or universal critical resistance for all geometries and environments. [R7](#ref-source-9ebba9ba0616123e)

Astafiev's InOx loop experiment used microwave spectroscopy to investigate coherent coupling between flux states. The checked author version describes a 35 nm film, a narrow segment about 40 nm wide and 400 nm long, and measurements at 40 mK. A two-state fit gave $E_S/h=4.9$ GHz and persistent current 24 nA for the illustrated device; resonator features distorted portions of the spectrum. The film's separately reported 2.7 K transition is distinct from the loop spectroscopy temperature. Coherent coupling evidence is stronger and different in kind from fitting a dissipative resistance tail; the selected readings do not reproduce the complete supplementary analysis. [R8](#ref-source-42e0dec220b7b3ea)

For aluminum nanoparticles, Black, Ralph and Tinkham reported even/odd differences in tunneling excitation thresholds and field suppression of a spectroscopic gap in sufficiently large particles, with radius roughly 5 nm or greater in their abstract. Radius is not diameter. Discrete levels, charging and parity enter the interpretation. Local pairing signatures do not demonstrate zero resistance across an isolated grain or global phase coherence in an array. This entry currently relies on the original publisher abstract, without a complete original-text audit. [R9](#ref-source-9fd1a118a8a9914b)

| Question and scale | Relevant evidence | What must remain separate |
| --- | --- | --- |
| Is the film in a two-dimensional phase-coherence regime? | Thickness, coherence lengths, lateral size and magnetic screening; renormalized stiffness in the ideal jump relation. [R1](#ref-source-408113812912d816) [R2](#ref-source-8f81510a80c35e3a) [R10](#ref-source-260bbd6c79b3d696) | A thin geometry or finite-frequency stiffness measurement alone is not an infinite-system BKT diagnosis. |
| Which NbSe2 transition is reported? | Xi author version: roughly 3 K fluctuation-conductivity fit, distinct BKT/resistance criteria and an 8 K normal reference. [R6](#ref-source-c80fb39cdf748a83) | Pairing-fit temperature, phase-coherence inference, resistance threshold and measurement floor are not interchangeable. |
| What is a nanowire resistance drop? | Bezryadin: electrode transition near 5.5 K; wire response subsequently isolated; one sample measured to 50 mK. [R7](#ref-source-9ebba9ba0616123e) | Electrode and wire temperatures, and sample-specific resistance scaling versus a universal threshold. |
| Are phase slips coherent? | Astafiev: flux-state microwave spectroscopy at 40 mK, with resonator features and a two-state fit. [R8](#ref-source-42e0dec220b7b3ea) | Coherent state coupling, dissipative resistance tails and the separate film transition at 2.7 K. |
| Is a nanoparticle locally paired? | Black: parity-dependent tunneling thresholds and field evolution of a spectroscopic gap. [R9](#ref-source-9fd1a118a8a9914b) | Local pairing signatures versus global phase coherence; radius versus diameter; abstract-only review scope. |

## Limits and disputes

Use several diagnostics for a BKT interpretation: nonlinear current–voltage behavior, stiffness and resistance scaling over justified windows, together with finite-size, heating and inhomogeneity checks. A convenient fit does not establish its asymptotic assumptions. Likewise, Giordano's early indium-wire work reported low-temperature deviations from thermal-activation predictions consistent with quantum tunneling; it should not be generalized into a rule that every residual resistance tail proves quantum phase slips. Weak links, contacts, normal regions and electromagnetic environment need independent controls. [R1](#ref-source-408113812912d816) [R2](#ref-source-8f81510a80c35e3a) [R5](#ref-source-93c2e7d6280c7bbc) [R7](#ref-source-9ebba9ba0616123e) [R8](#ref-source-42e0dec220b7b3ea)

Selected original KT passages, NK journal text and author-version experimental sections have been checked. The NK stiffness definition, recursion relations and jump equation have now also been visually checked on journal pages 1202–1203. Pearl's original pages 65–66 were read for the thin-sheet assumption and current asymptotes; the entire electromagnetic solution was not independently reproduced. The original LA, McCumber–Halperin, Giordano and nanoparticle full texts remain unreviewed; final-version comparisons for Xi, Bezryadin and Astafiev and complete supplements/raw-data analyses also remain open. Local algebra checks and successful software tests do not complete scientific acceptance. [R1](#ref-source-408113812912d816) [R2](#ref-source-8f81510a80c35e3a) [R3](#ref-source-aa12d674a498e02d) [R4](#ref-source-866d4492c91f5b39) [R5](#ref-source-93c2e7d6280c7bbc) [R6](#ref-source-c80fb39cdf748a83) [R7](#ref-source-9ebba9ba0616123e) [R8](#ref-source-42e0dec220b7b3ea) [R9](#ref-source-9fd1a118a8a9914b) [R10](#ref-source-260bbd6c79b3d696)

## Discovery implications

Build thickness and width series with documented oxidation protection, substrate, contacts and disorder. Store pairing-fit, phase-coherence, resistance and local-spectroscopy criteria separately, retaining the actual sample and measurement temperature. A route that strengthens pairing can simultaneously reduce phase stiffness; dimensional reduction is a tuning parameter, not a guarantee of a higher usable superconducting transition. [R6](#ref-source-c80fb39cdf748a83) [R7](#ref-source-9ebba9ba0616123e) [R8](#ref-source-42e0dec220b7b3ea) [R9](#ref-source-9fd1a118a8a9914b)

## Related topics

Compare [disorder and impurities](/scwiki/disorder-and-impurities), [magnetic fields](/scwiki/magnetic-field) and [critical current](/scwiki/critical-current).

## 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 |
|---|---|---|---|---|---|---|
| NbSe2 (aliases: niobium diselenide, Ising NbSe2); Xi encapsulated monolayer | 3 K; Approximate fit temperature from author Figure2; not the distinct BKT or zero-resistance criterion. | transition; aslamazov-larkin-fit | Four-terminal transport fluctuation-conductivity fit | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-c80fb39cdf748a83) |
| NbSe2 (aliases: niobium diselenide, Ising NbSe2); Xi high-field measurement floor | 0.3 K; Measurement floor only; separate device criteria retained inthe article. | measurement; measurement-floor | Magnetotransport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-c80fb39cdf748a83) |
| NbSe2 (aliases: niobium diselenide, Ising NbSe2); Xi normal-resistance reference | 8 K; Reference temperature for Rn; not Tc. | normal-state; normal-resistance-reference | Transport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-c80fb39cdf748a83) |
| Mo79Ge21 (aliases: MoGe nanowire, amorphous MoGe); Bezryadin superconducting wire measurement floor | 0.05 K; 50mK checked in one superconducting sample. The5.5K electrode transition is not assigned to the wire. | measurement; low-temperature-resistance | Low-bias transport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-9ebba9ba0616123e) |
| InOx (aliases: indium oxide, InOx film); Astafiev superconducting film | 2.7 K; Film transition reported separately from loop spectroscopy; exact operational threshold not established. | transition; transition-unspecified | DC transport of film | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-42e0dec220b7b3ea) |
| InOx (aliases: indium oxide, InOx loop); Astafiev loop with narrow wire | 0.04 K; 40mK measurement temperature, not a loop transition. | measurement; coherent-phase-slip-spectroscopy | Microwave spectroscopy | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-42e0dec220b7b3ea) |

## Sources

### Ref source-408113812912d816

[R1] Ordering, metastability and phase transitions in two-dimensional systems (1973). [DOI 10.1088/0022-3719/6/7/010](https://doi.org/10.1088/0022-3719/6/7/010). Publication: journal-article. Check: selected-original-superfluid-and-screening-sections. Checked: 2026-10-20. Earlier selected original section5 pp1197–1198 text review retained; charged-film screening paragraph reread this follow-up. It explicitly bounds the neutral vortex argument. No new complete KT derivation or visual scan review claimed.
### Ref source-8f81510a80c35e3a

[R2] Universal Jump in the Superfluid Density of Two-Dimensional Superfluids (1977). [DOI 10.1103/PhysRevLett.39.1201](https://doi.org/10.1103/PhysRevLett.39.1201). Publication: journal-article. Check: selected-original-publisher-scan-and-equation-review. Checked: 2026-10-20. Publisher PDF acquired. Original pp1202–1203 visually checked: bare K versus renormalized KR, recursion equations8 and universal limit10. Scan also includes neighboring articles; only the NK article is used. Independent RG proof and finite charged-film diagnosis remain open; historical mass-density coefficient is not silently replaced with modern-constant arithmetic.
### Ref source-aa12d674a498e02d

[R3] Intrinsic Resistive Transition in Narrow Superconducting Channels (1967). [DOI 10.1103/PhysRev.164.498](https://doi.org/10.1103/PhysRev.164.498). Publication: journal-article. Check: publisher-abstract-only. Checked: 2026-10-18. Original abstract read; full text access unsuccessful. Displayed zero-current GL barrier independently integrated locally; original full derivation remains open.
### Ref source-866d4492c91f5b39

[R4] Time Scale of Intrinsic Resistive Fluctuations in Thin Superconducting Wires (1970). [DOI 10.1103/PhysRevB.1.1054](https://doi.org/10.1103/PhysRevB.1.1054). Publication: journal-article. Check: publisher-abstract-only. Checked: 2026-10-18. Original abstract identifies revised time-dependent GL prefactor. Full original derivation not retrieved; no numerical LAMH attempt-frequency claim made. Directional-rate detailed balance is a separate local calculation.
### Ref source-93c2e7d6280c7bbc

[R5] Evidence for Macroscopic Quantum Tunneling in One-Dimensional Superconductors (1988). [DOI 10.1103/PhysRevLett.61.2137](https://doi.org/10.1103/PhysRevLett.61.2137). Publication: journal-article. Check: publisher-abstract-only. Checked: 2026-10-18. Indium-wire low-temperature deviation and quantum interpretation checked in original abstract only; no full data audit or quantitative transition record.
### Ref source-c80fb39cdf748a83

[R6] Ising pairing in superconducting NbSe2 atomic layers (2016). [DOI 10.1038/nphys3538](https://doi.org/10.1038/nphys3538). Publication: journal-article. Check: selected-author-results-methods-and-figure. Checked: 2026-10-18. Author v1 selected pp1–3, p8 methods and p10 Figure2 caption read; p10 visually checked. AL fit near3.0K, distinct resistance/BKT criteria, 8K normal resistance and0.3K floor retained. Publisher online9November2015; issue2016. Final PDF not obtained; final comparison and supplement open.
### Ref source-9ebba9ba0616123e

[R7] Quantum suppression of superconductivity in ultrathin nanowires (2000). [DOI 10.1038/35010060](https://doi.org/10.1038/35010060). Publication: journal-article. Check: selected-author-transport-and-methods. Checked: 2026-10-18. Selected author pp1–4 read; p4 visually checked. Mo79Ge21 wire geometry, electrode5.5K transition and50mK measurement floor distinguished. Final Nature404,971–974,27April2000; final comparison and raw-data review open.
### Ref source-42e0dec220b7b3ea

[R8] Coherent quantum phase slip (2012). [DOI 10.1038/nature10930](https://doi.org/10.1038/nature10930). Publication: journal-article. Check: selected-author-two-state-model-and-spectroscopy. Checked: 2026-10-18. Selected author pp2–7 read: flux-state model, geometry,40mK spectroscopy, film2.7K transition,4.9GHz fit and resonator distortion. Nature484,355–358(2012). Full supplement, final comparison and raw-data analysis remain open.
### Ref source-9fd1a118a8a9914b

[R9] Spectroscopy of the Superconducting Gap in Individual Nanometer-Scale Aluminum Particles (1996). [DOI 10.1103/PhysRevLett.76.688](https://doi.org/10.1103/PhysRevLett.76.688). Publication: journal-article. Check: publisher-abstract-only. Checked: 2026-10-18. Original publisher abstract: radius approximately5nm or greater, parity-dependent tunneling thresholds and field suppression. Full text not retrieved. Distinct from the1995 discrete-level paper; no Tc assigned.
### Ref source-260bbd6c79b3d696

[R10] Current distribution in superconducting films carrying quantized fluxoids (1964). [DOI 10.1063/1.1754056](https://doi.org/10.1063/1.1754056). Publication: journal-article. Check: original-scan-thin-sheet-assumptions-and-current-asymptotes. Checked: 2026-10-20. Original journal pp.65–66 read and visually inspected: d<<lambda sheet approximation, denominator1+2q lambda²/d and Eq12 radial-current asymptotes. DOI/title/pages checked against the Technion research record. Local normalized-kernel and SI-inductance checks do not independently reproduce all boundary conditions or the full historical electromagnetic prefactors.

## Claim evidence and conditions

### scwiki-e8-claim-scope-1

Dimensionality compares geometry with physical length scales. A layered crystal, a two-dimensional electronic band and a superconducting film with a nearly uniform order parameter through its thickness are different descriptions. Compare thickness with the perpendicular coherence length, wire transverse dimensions with coherence lengths, and lateral size with magnetic screening lengths. Interlayer coupling, substrate, disorder and carrier density can change when layers are removed. Reduced thickness alone does not establish an ideal two-dimensional phase transition.

Evidence: theory. Sources: source-408113812912d816, source-c80fb39cdf748a83.

Source relationship: context-for-editorial-analysis. Editorial comparison of physical scales, observables and source-review boundaries; the synthesis is not attributed as a single result to every cited source.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-scope-2

For a short-range two-dimensional phase model, vortex–antivortex unbinding separates quasi-long-range order from a disordered phase. The relevant stiffness is renormalized by fluctuations. In a charged film, magnetic screening changes the vortex interaction at long distance: Kosterlitz and Thouless explicitly identified this limitation in their original paper. BKT-like behavior requires an appropriate range of length, current and temperature scales; do not transfer the neutral infinite-system limit to every finite superconducting sheet.

Evidence: theory. Sources: source-408113812912d816, source-8f81510a80c35e3a.

Source relationship: context-for-editorial-analysis. Editorial comparison of physical scales, observables and source-review boundaries; the synthesis is not attributed as a single result to every cited source.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-mathematical-assumptions-1

Define the sheet phase stiffness $J$ in energy units by $F_\theta=(J/2)\int d^2r\,|\nabla\theta|^2$. A single vortex has logarithmic energy $\pi J\ln(L/a)$ and positional entropy $2k_B\ln(L/a)$ within this model, where $a$ is a core cutoff. The universal jump is $J_R(T_{\rm BKT}^-)=2k_BT_{\rm BKT}/\pi$, with the long-distance **renormalized** stiffness $J_R$. The elementary energy–entropy balance explains the scale; it is not a replacement for the renormalization calculation. For a local sheet inductance $L_k$ per square, the corresponding stiffness convention is $J=\hbar^2/(4e^2L_k)$. Frequency, vortex response and finite size matter when comparing a measured inductance with the static jump.

Evidence: theory. Sources: source-408113812912d816, source-8f81510a80c35e3a.

Source relationship: mixed-source-and-local-derivation. Original source supplies the stated model context; local notation, SI conversion and bounded calculations are identified separately. No complete RG solution, LAMH prefactor, finite-current dynamics or quantum action is reproduced.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-mathematical-assumptions-2

In a uniform narrow wire near its mean-field transition, the zero-current Ginzburg–Landau saddle gives $\Delta F_0=(8\sqrt2/3)u_c A\xi$, where $u_c=B_c^2/(2\mu_0)$ is the SI condensation-energy density and $A$ is cross-sectional area. A local check integrates the normalized saddle $f(y)=\tanh(y/\sqrt2)$, $y=x/\xi$, in $\int[(1-f^2)^2+2(df/dy)^2]dy$. The profile satisfies $f''+f-f^3=0$, approaches opposite real order-parameter signs at either end, and passes through zero at its centre. Substituting $z=f(y)$ reduces the integral to $2\sqrt2\int_{-1}^{1}(1-z^2)dz=8\sqrt2/3$. This independently checks the stationary profile and barrier factor under those assumptions, not a finite-current barrier, quantum tunneling action or the full LAMH prefactor. The original Langer–Ambegaokar and McCumber–Halperin full derivations have not been retrieved for this review.

Evidence: theory. Sources: source-aa12d674a498e02d, source-866d4492c91f5b39.

Source relationship: context-for-local-derivation. Original source supplies the stated model context; local notation, SI conversion and bounded calculations are identified separately. No complete RG solution, LAMH prefactor, finite-current dynamics or quantum action is reproduced.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-mathematical-assumptions-3

For rare independent thermal phase slips, define $r_0$ explicitly as the zero-bias rate **per direction**. With $\Phi_0=h/(2e)$, detailed balance gives $r_\pm=r_0\exp[\pm I\Phi_0/(2k_BT)]$ to leading small-bias barrier tilt, $V=\Phi_0(r_+-r_-)$ and $R_{I\to0}=\Phi_0^2r_0/(k_BT)$. The thermal activation factor belongs inside $r_0$; this definition avoids confusing a total event rate with a directional one. Low-temperature resistance saturation alone does not identify coherent quantum phase slips.

Evidence: theory. Sources: source-aa12d674a498e02d, source-866d4492c91f5b39, source-42e0dec220b7b3ea.

Source relationship: context-for-local-derivation. Original source supplies the stated model context; local notation, SI conversion and bounded calculations are identified separately. No complete RG solution, LAMH prefactor, finite-current dynamics or quantum action is reproduced.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-mathematical-assumptions-4

For a homogeneous London film in free space with thickness $d\ll\lambda$, define the Pearl length explicitly as $\Lambda=2\lambda^2/d$. Pearl's Fourier denominator is $1+q\Lambda$; outside the core, the circulating sheet current crosses from $1/r$ for $r\ll\Lambda$ to $1/r^2$ for $r\gg\Lambda$. The far-field interaction energy therefore decays as $1/r$, rather than retaining a logarithm indefinitely. In the same local thin-film convention, $L_k=\mu_0\lambda^2/d$, so $\Lambda=2L_k/\mu_0$. A useful logarithmic window must fit between the core and both screening and lateral boundaries; this length comparison alone does not diagnose BKT behavior. The inductance conversion and numerical asymptote checks here are local model calculations, not measured film parameters.

Evidence: theory. Sources: source-260bbd6c79b3d696, source-408113812912d816.

Source relationship: mixed-source-and-local-derivation. Original source supplies the stated model context; local notation, SI conversion and bounded calculations are identified separately. No complete RG solution, LAMH prefactor, finite-current dynamics or quantum action is reproduced.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-experimental-criteria-1

Xi's encapsulated NbSe2 devices distinguish several transition definitions. In the checked author manuscript, the monolayer Aslamazov–Larkin fit gives $T_{c0}\approx3.0$ K, close to the 50% normal-resistance criterion; the inferred BKT transition lies near the 1% criterion. Those are separate quantities. Normal resistance was taken at 8 K. The four-terminal field series used a 50% criterion, while a separate two-terminal monolayer device used a rapid-resistance-drop criterion. The 0.3 K measurement floor is not another transition. Large in-plane critical fields support the authors' Ising-pairing interpretation under their model, rather than proving triplet pairing by themselves. The journal issue is from 2016; online publication was 9 November 2015.

Evidence: experiment. Sources: source-c80fb39cdf748a83.

Source relationship: direct-support. Source-specific observed quantities and the authors' interpretation remain separated; author-version or abstract-only limitations are retained, without a new whole-paper audit.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-experimental-criteria-2

Bezryadin's nanotube-templated amorphous Mo79Ge21 wires were approximately 10 nm or less in diameter and 95–185 nm long. The resistance drop near 5.5 K belongs to the electrodes, after which the wire resistance could be isolated. In the studied series, samples on opposite sides of a normal-resistance scale near $R_Q=h/(4e^2)\approx6.45$ kilohms showed decreasing or increasing resistance on cooling. One superconducting sample was checked down to 50 mK. These are sample-specific transport observations and a quantum-fluctuation interpretation, not a universal wire transition temperature or universal critical resistance for all geometries and environments.

Evidence: experiment. Sources: source-9ebba9ba0616123e.

Source relationship: direct-support. Source-specific observed quantities and the authors' interpretation remain separated; author-version or abstract-only limitations are retained, without a new whole-paper audit.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-experimental-criteria-3

Astafiev's InOx loop experiment used microwave spectroscopy to investigate coherent coupling between flux states. The checked author version describes a 35 nm film, a narrow segment about 40 nm wide and 400 nm long, and measurements at 40 mK. A two-state fit gave $E_S/h=4.9$ GHz and persistent current 24 nA for the illustrated device; resonator features distorted portions of the spectrum. The film's separately reported 2.7 K transition is distinct from the loop spectroscopy temperature. Coherent coupling evidence is stronger and different in kind from fitting a dissipative resistance tail; the selected readings do not reproduce the complete supplementary analysis.

Evidence: experiment. Sources: source-42e0dec220b7b3ea.

Source relationship: direct-support. Source-specific observed quantities and the authors' interpretation remain separated; author-version or abstract-only limitations are retained, without a new whole-paper audit.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-experimental-criteria-4

For aluminum nanoparticles, Black, Ralph and Tinkham reported even/odd differences in tunneling excitation thresholds and field suppression of a spectroscopic gap in sufficiently large particles, with radius roughly 5 nm or greater in their abstract. Radius is not diameter. Discrete levels, charging and parity enter the interpretation. Local pairing signatures do not demonstrate zero resistance across an isolated grain or global phase coherence in an array. This entry currently relies on the original publisher abstract, without a complete original-text audit.

Evidence: experiment. Sources: source-9fd1a118a8a9914b.

Source relationship: direct-support. Source-specific observed quantities and the authors' interpretation remain separated; author-version or abstract-only limitations are retained, without a new whole-paper audit.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-limits-and-disputes-1

Use several diagnostics for a BKT interpretation: nonlinear current–voltage behavior, stiffness and resistance scaling over justified windows, together with finite-size, heating and inhomogeneity checks. A convenient fit does not establish its asymptotic assumptions. Likewise, Giordano's early indium-wire work reported low-temperature deviations from thermal-activation predictions consistent with quantum tunneling; it should not be generalized into a rule that every residual resistance tail proves quantum phase slips. Weak links, contacts, normal regions and electromagnetic environment need independent controls.

Evidence: experiment. Sources: source-408113812912d816, source-8f81510a80c35e3a, source-93c2e7d6280c7bbc, source-9ebba9ba0616123e, source-42e0dec220b7b3ea.

Source relationship: context-for-editorial-analysis. Editorial comparison of physical scales, observables and source-review boundaries; the synthesis is not attributed as a single result to every cited source.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-limits-and-disputes-2

Selected original KT passages, NK journal text and author-version experimental sections have been checked. The NK stiffness definition, recursion relations and jump equation have now also been visually checked on journal pages 1202–1203. Pearl's original pages 65–66 were read for the thin-sheet assumption and current asymptotes; the entire electromagnetic solution was not independently reproduced. The original LA, McCumber–Halperin, Giordano and nanoparticle full texts remain unreviewed; final-version comparisons for Xi, Bezryadin and Astafiev and complete supplements/raw-data analyses also remain open. Local algebra checks and successful software tests do not complete scientific acceptance.

Evidence: experiment. Sources: source-408113812912d816, source-8f81510a80c35e3a, source-aa12d674a498e02d, source-866d4492c91f5b39, source-93c2e7d6280c7bbc, source-c80fb39cdf748a83, source-9ebba9ba0616123e, source-42e0dec220b7b3ea, source-9fd1a118a8a9914b, source-260bbd6c79b3d696.

Source relationship: context-for-editorial-analysis. Editorial comparison of physical scales, observables and source-review boundaries; the synthesis is not attributed as a single result to every cited source.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

### scwiki-e8-claim-discovery-implications-1

Build thickness and width series with documented oxidation protection, substrate, contacts and disorder. Store pairing-fit, phase-coherence, resistance and local-spectroscopy criteria separately, retaining the actual sample and measurement temperature. A route that strengthens pairing can simultaneously reduce phase stiffness; dimensional reduction is a tuning parameter, not a guarantee of a higher usable superconducting transition.

Evidence: theory. Sources: source-c80fb39cdf748a83, source-9ebba9ba0616123e, source-42e0dec220b7b3ea, source-9fd1a118a8a9914b.

Source relationship: context-for-editorial-analysis. Editorial comparison of physical scales, observables and source-review boundaries; the synthesis is not attributed as a single result to every cited source.

Conditions: Distinguish geometry, coherence and magnetic-screening lengths; measured frequency and current windows; pairing, transport and local spectra; wire/electrode/sample identity and source access scope.

Verification scope: Selected original NK/Pearl equations now visually checked; bounded local London-kernel and zero-current GL checks. Earlier experimental reading retained; complete independent scientific acceptance remains open.

## Revision and review

Article ID: scwiki-e8. Revision: acfcd0ce34404975275c. Updated: 2026-10-20.

Review scope: Selected original NK journal equations and Pearl thin-sheet/asymptote scan, charged-film KT paragraph and new bounded London/GL calculations; earlier Xi/Bezryadin/Astafiev author-version checks retained, not newly repeated.

- Complete scientific acceptance, full supplements and raw-data reanalysis remain open.
- LA, McCumber–Halperin, Giordano and Black original full texts not retrieved; publisher abstract scope only.
- Xi, Bezryadin and Astafiev full final-version comparisons remain open. NK pp1202–1203 equations now visually checked; Pearl selected equations do not amount to a complete electromagnetic reconstruction.
- Finite-size/screening and measurement frequency limit direct comparison with the ideal renormalized BKT jump. No universal nanowire critical resistance asserted.

Source: [E_tuning_parameters/E8_dimensionality.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/E_tuning_parameters/E8_dimensionality.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 8a1bc46aef3230356d75fc3e1f6ecbc6afae2da1d3521aa6b1f7bbf9490f4b8c.