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Measuring coherence factors of states in superconductors through local current

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that two local current measurements can estimate the local Majorana polarization of a subgap state in a superconductor, and it benchmarks the recipe on two- and three-site Kitaev chains.

desk verdict A useful two-current estimator for local Majorana polarization at sweet spots, but the 'local' claim breaks down under left-right asymmetry and the paper understates how badly. read the letter →

arxiv 2507.20696 v2 pith:PVNMXF4O submitted 2025-07-28 cond-mat.mes-hall cond-mat.supr-con

Rodrigo A. Dourado , Jeroen Danon , Martin Leijnse , Rubén Seoane Souto This is my paper
classification cond-mat.mes-hallcond-mat.supr-con
keywords coherencefactorsMajoranapolarizationpoorman'sboundstatesKitaevchainsquantumdotarrayslocalcurrentsequentialtunnelingsubgap
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a transport-based way to measure the local electron-hole composition of a subgap state in a grounded superconductor using only currents through normal leads. The central relation is that, in the sequential-tunneling regime, the ratio of the current through one lead when the opposite lead is decoupled to the current when both leads are equally coupled equals $\left(2|u||v|/(|u|^2+|v|^2)\right)^2$, where $u$ and $v$ are the local coherence factors. From two such current readings one can estimate the local Majorana polarization, a quantitative measure of how Majorana-like a state is. The paper verifies this estimate numerically on minimal Kitaev chains of quantum dots, including finite Zeeman fields and electron-electron interactions, and finds good agreement across a wide parameter range. If the method works as claimed, it gives an experimental route to quantify the overlap and quality of Majorana states in short Kitaev chains without nonlocal measurements.

What carries the argument

The load-bearing object is the rate-equation expression for the local current through a spinless subgap state, Eq. (3), derived in the regime $V \gg T \gg \Gamma_{L,R}$. The current is determined by the local coherence factors $u_\nu, v_\nu$ and the tunnel rates $\Gamma_L, \Gamma_R$; taking the ratio of the current with the opposite lead pinched off ($\Gamma_R=0$) to the current with equal couplings ($\Gamma_R=\Gamma_L$) isolates the combination $2|u||v|/(|u|^2+|v|^2)$, whose square equals the Majorana polarization in the symmetric case. The estimator $M_\nu$, defined in Eq. (15), converts two current measurements into an experimental handle on this quantity, and the derivative $dI_L/d\Gamma_R$ provides a closely related probe that requires only small changes in the coupling.

What would settle it

Compare the estimator $M_\nu$ obtained from two local current measurements against an independent extraction of the Majorana polarization from nonlocal conductance or a probe quantum dot on the same two-site chain, while sweeping the Zeeman field from $V_z \gg \Delta$ down to $V_z \approx 0.25\Delta$; the predicted deviations of about 3% to 14%, as captured by the spinful expression in Eq. (17), would confirm or falsify the sequential-tunneling assumption.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a single spinless subgap state coupled to two normal leads, in the limit $V \gg T \gg \Gamma_{L,R}$ with first-order sequential tunneling, the local current obeys $I_L(\Gamma_R=0)/I_L(\Gamma_R=\Gamma_L)=\left(2|u||v|/(|u|^2+|v|^2)\right)^2$ when the coherence factors are left-right symmetric. This leads to the estimator $M_\nu = \sqrt{I_\nu(\Gamma_{\bar\nu}=0)/I_\nu(\Gamma_{\bar\nu}=\Gamma_\nu)}$, which the authors identify with the local Majorana polarization for symmetric Kitaev chains. When the state has local electron-hole symmetry, $|u|=|v|$, the local current becomes independent of the coupling to the other lead, so flatness of the current as that coupling is varied marks sweet spots. The authors benchmark this relation against numerical transport calculations for two- and three-site Kitaev chains described by a microscopic model with spin and interactions, showing that the estimator tracks the true polarization, with deviations around 3% at $V_z=0.75\Delta$ and 14% at $V_z=0.25\Delta$.

Load-bearing premise

The load-bearing premise is that a single non-spin-degenerate subgap state, tunneled through in the regime $V \gg T \gg \Gamma$, dominates the transport at the measurement bias; extra states, cotunneling, spin channels, or strong left-right asymmetry break the simple current-ratio formula.

Editorial extensions

If this is right

  • In a symmetric spinless setup, two local current measurements - one with the opposite lead pinched off and one with equal couplings - give the local Majorana polarization through $M_\nu = \sqrt{I_\nu(\Gamma_{\bar\nu}=0)/I_\nu(\Gamma_{\bar\nu}=\Gamma_\nu)}$.
  • Flatness of the local current as the other lead's coupling is varied signals local electron-hole symmetry and can be used to find and verify sweet spots without moving the system away from them.
  • The estimate continues to peak at the true polarization maximum even at low Zeeman fields and for asymmetric couplings, so it can still be used to optimize device tuning when it no longer equals $|M|$ quantitatively.
  • For three-site Kitaev chains, the same two-current recipe reproduces the numerically computed polarization map, showing that the method extends beyond the minimal two-site case.
  • The derivative $dI_L/d\Gamma_R$ offers an equivalent probe that requires only small changes in $\Gamma_R$, though it needs knowledge of the absolute tunnel rates.

Reading between the lines

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

  • A natural extension the authors do not develop: attaching leads at intermediate sites of a longer Kitaev chain would turn the same ratio into a spatially resolved map of coherence factors rather than an end-point probe.
  • Because the estimator peaks whenever the true polarization peaks even when its quantitative accuracy degrades, it could serve as a cheap in-situ cost function for autonomous tuning; the paper mentions machine-learning tuning but does not test this role.
  • Comparing the left- and right-lead estimates in an asymmetric device could bound the asymmetry parameter and provide an error bar on the inferred polarization; the paper only analyzes fixed asymmetric points.
  • The spinful version of the ratio, Eq. (17), suggests a separate diagnostic: sweeping the Zeeman field while monitoring the current ratio could isolate the weight of the unwanted spin channel directly.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes an experimental protocol for inferring local coherence factors and the local Majorana polarization (MP) of a subgap state in a grounded superconductor. The method uses two local current measurements: the current through one lead with the opposite lead pinched off and with the two leads equally coupled, whose ratio is claimed to give the squared local MP. An analytic expression is derived from first-order sequential-tunneling rate equations for a single spinless subgap state, with an extension to the spinful case, and the result is benchmarked against scattering-matrix and rate-equation numerics for 2-site and 3-site minimal Kitaev chains, including electron-electron interactions and finite Zeeman fields.

Significance. If the central claim holds, the protocol offers a simple, spatially resolved complement to nonlocal conductance measurements for quantifying Majorana overlap in poor-man's Majorana chains. The analytic derivation in App. C is internally consistent, and the numerical benchmarks use an independently defined MP (Eq. 13) with no fitted parameters, which is a notable strength. The paper also explicitly acknowledges some limitations at low Zeeman fields and in asymmetric setups. However, the central 'local probe' claim is currently established only under left-right symmetry of the coherence factors, and the manuscript does not supply a quantitative validity region for asymmetric chains. The claimed quantitative accuracy for asymmetric parameter space is therefore not yet supported.

major comments (2)
  1. [§II, Eq. (6), and §III, Eq. (15)] The estimator Mν in Eq. (15) is derived from Eq. (6) under the explicit assumption |u_L|=|u_R| and |v_L|=|v_R|. For a single spinless ABS with unequal left and right coherence factors, the ratio IL(ΓR=0)/IL(ΓR=ΓL) from Eq. (C12) is a joint function of both ends and is not equal to M_L^2. For example, with equal local weights, δ_R=0, and |M_L|=0.6, the ratio is 0.529, so Eq. (15) returns 0.73 rather than 0.60. Section III B states that the conclusions 'hold for asymmetric setups, see App. D,' but App. D only scans the one-parameter family µ1=aµ2 and concedes overestimation for a<0.5; no general validity bound is given. The central claim of a local probe therefore needs either a quantitative asymmetry bound or a revised, more qualified statement.
  2. [§III D and Eq. (17)] The extension to finite Zeeman fields is handled by Eq. (17), which shows that for two spin channels the current ratio is not expressible in terms of the MP of Eq. (14). The paper reports deviations of about 3% at Vz=0.75∆ and about 14% at Vz=0.25∆ for particular parameter points, and notes that interactions reduce the error, but no general error estimate or clear threshold Vz≳∆ is established. Because the microscopic model in Sec. III C is the main numerical benchmark, the paper should either bound the error as a function of Vz/∆ and the spin-up weight or restrict the quantitative MP claim to the spin-polarized regime.
minor comments (4)
  1. [§III D title] The section title 'Dependence on the pin-splitting' appears to be a typo for 'spin-splitting'.
  2. [§I] The phrases 'Sec. (III)', 'Sec. (III B)', and similar contain redundant parentheses; consider writing 'Sec. III' and 'Sec. III B'.
  3. [Fig. 1 caption] In the caption, 'u = 2 v, u = v, and v = 4u' is ambiguous; please clarify the intended ordering and notation (e.g., u=2v, u=v, v=4u).
  4. [§III D] The sentence 'we have simulated the effective 2-site Kitaev chain using the microscopic model' is ambiguous because the microscopic model is not the effective Kitaev chain; please rephrase to indicate that the microscopic model is used to realize the effective chain.

Circularity Check

0 steps flagged · score 0.0 of 10

The current-ratio Majorana-polarization estimator is derived from explicit rate equations and checked against independently computed wavefunction quantities; no fitted input or self-referential definition is load-bearing, so no significant circularity is found.

full rationale

The derivation chain is self-contained. The paper starts from the tunneling Hamiltonian (2) and obtains the local current (3)/(C12) from first-order rate equations in the V >> T >> Gamma regime, with the local coherence factors u and v entering as model parameters. In the symmetric spinless case u_L = u_R and v_L = v_R, Eq. (C15) gives the ratio IL(Gamma_R=0)/IL(Gamma_R=Gamma_L) = (2|u||v|/(|u|^2+|v|^2))^2. The Majorana polarization is defined independently in Eq. (13) and, in the noninteracting limit, in Eq. (14) as M = 2uv/(u^2+v^2). The identification of the square root of the current ratio with |M| is therefore a derived identity, not an input assumption or a fit. The numerical benchmarks compute the current from the scattering matrix (Appendix A) and the MP from ground-state wavefunctions, so the two quantities are evaluated separately; no parameter is tuned to force agreement. The paper explicitly reports discrepancies in regimes outside the derivation (low Zeeman field in Sec. III D and strong asymmetry in Appendix D), which confirms that the comparison is an honest test rather than a construction. Self-citations such as Refs. [36,37] supply the effective Kitaev-chain model and sweet-spot conditions, but the central current-to-MP relation does not depend on those citations for its validity; the rate-equation derivation and the numerical S-matrix benchmarks stand independently. The asymmetry caveat is a limitation on applicability, not a circular step, because Eq. (15) remains a well-defined measured ratio whose approximate relation to the independently defined MP is the paper's claim rather than its definition.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation has no fitted free parameters; coherence factors, tunnel rates, and the Majorana polarization are defined independently. The assumptions are the usual weak-coupling, single-level, large-bias approximations of sequential-tunneling theory, plus the left-right symmetry needed for the clean estimator. No new particles, forces, or conserved quantities are postulated.

assumptions (6)
  • domain assumption Transport is dominated by a single spinless subgap state well separated from other states.
    Used in Sec. II and App. C to write Hs = ξ α†α and derive Eq. (3); contributions from excited states are neglected in the analytic formula.
  • domain assumption Sequential first-order tunneling with V ≫ T ≫ Γ_L,R captures the local current.
    Stated before Eq. (3) and used throughout App. C to compute occupation probabilities and rates; if cotunneling or higher-order processes dominate, the simple ratio fails.
  • domain assumption Left-right symmetric coherence factors (|u_L|=|u_R|, |v_L|=|v_R|) for Eq. (6) and the MP estimator.
    Used in Sec. II to obtain Eq. (6); App. D shows asymmetric deviations are modest but non-negligible.
  • domain assumption An effective Kitaev chain describes the QD-superconductor array after integrating out superconducting segments.
    Assumed in Sec. III A following Refs. [36,37]; the microscopic model (Eqs. 7-11) is used in numerics to test this reduction.
  • domain assumption Majorana polarization Eq. (13)/(14) correctly quantifies local Majorana quality.
    Defined following Refs. [37,60-64]; the paper does not derive this measure but uses it as ground truth for benchmarking.
  • domain assumption Rate-equation occupation probabilities for the spinful case (App. C2) remain valid with the large-bias assumption V larger than the charging energy.
    Used to derive Eq. (C31) and Eq. (17); the steady-state equations assume incoherent sequential tunneling.

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Pith. "Pith review of Measuring coherence factors of states in superconductors through local current." pith.science (2026). https://pith.science/paper/PVNMXF4O

@misc{pith2026250720696,
  author       = {Pith},
  title        = {Pith review of: Measuring coherence factors of states in superconductors through local current},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVNMXF4O}},
  note         = {Machine review of arXiv:2507.20696}
}
abstract

The coherence factors of quasiparticles in a superconductor determine their properties, including transport and susceptibility to electric fields. In this work, we propose a way to infer the local coherence factors using local transport to normal leads. Our method is based on measuring the local current through a lead as the coupling to a second one is varied: the shape of the current is determined by the ratio between the local coherence factors, becoming independent of the coupling to the second lead in the presence of local electron-hole symmetry, {\it i.e.} coherence factors $|u|=|v|$. We apply our method to minimal Kitaev chains: arrays of quantum dots coupled via narrow superconducting segments. These chains feature Majorana-like quasiparticles (zero-energy states with $|u|=|v|$) at discrete points in parameter space. We demonstrate that the local current allows us to estimate the local Majorana polarization (MP) -- a measurement of the local Majorana properties of the state. We derive an analytical expression for the MP in terms of local currents and benchmark it against numerical calculations for 2- and 3-sites chains that include a finite Zeeman field and electron-electron interactions. These results provide a way to quantitatively assess the quality of Majorana states in short Kitaev chains.

Figures

Figures reproduced from arXiv: 2507.20696 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Sketch of the system considered, where a grounded [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Results for the microscopic model of an effective 2- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (a) Overlapping Majorana wavefunctions at the left [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Results for the microscopic model for an effective [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Results for finite [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Nonlocal conductance [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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