Two-site Kitaev sweet spots evolving into topological islands

J. Carlos Egues; Poliana H. Penteado; Rodrigo A. Dourado

REVIEW 3 major objections 6 minor 2 cited by

Two-site Kitaev sweet spots evolving into topological islands

T0 review · 3 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read As an artificial Kitaev chain grows past about 20 sites, its fragile two-site Majorana sweet spot expands into a protected 'topological island' readable as quantized conductance.

desk verdict Useful scaling result for QD-based Majorana chains, but the 'strictly zero-energy' disorder-robust claim needs a numerical-threshold rewrite. read the letter →

arxiv 2501.19376 v1 pith:S4HDOMVN submitted 2025-01-31 cond-mat.mes-hall

Rodrigo A. Dourado , J. Carlos Egues , Poliana H. Penteado This is my paper · ORCID

classification cond-mat.mes-hall

keywords MajoranaboundstatesKitaevchainquantumdotarraysPoorman'stopologicalislandzero-energyplateauspolarizationzero-biasconductance

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

This paper argues that the main experimental obstacle to Poor man's Majoranas—the need to sit precisely on a two-site sweet spot—relaxes as the artificial Kitaev chain is made longer. Using the Bogoliubov–de Gennes spectrum, the authors show that the zero-energy and maximal-Majorana-polarization lines crossing at the two-site sweet spot fan out as the number of sites $N$ grows, turning a point into a region whose area increases with $N$. For $N \geq 20$ this region becomes a plateau where the lowest mode is exactly self-adjoint ($\gamma^2 = 1/2$) and remains so under Anderson disorder of strength comparable to the pairing gap; they call this region a topological island. The same physics appears in a realistic spinful quantum-dot model, and the island is detectable as an $e^2/2h$ zero-bias conductance plateau in a side-coupled dot. If the claims hold, they map the crossover from fragile PMMs to protected Majorana bound states and give a concrete target length for experiments.

What carries the argument

Two diagnostics carry the argument. First, $\gamma^2 = \sum_j u_j v_j$ computed for the lowest Bogoliubov mode: particle-hole symmetry makes $\gamma^2 = 1/2$ if and only if the mode is exactly at zero energy and its Bogoliubov operator is self-adjoint (a Majorana operator), whereas finite-energy fermionic modes have $\gamma^2 = 0$. Second, the Majorana polarization $M_j = 2u_j v_j/(u_j^2 + v_j^2)$, whose magnitude is $1$ when the local electron and hole weights are equal, signaling edge-localized, non-overlapping Majorana wave functions. The structural reason for the sweet spot's growth is the analytic family of zero-energy lines from the diagonalization of the Kitaev model, Eq. (6), all converging at the two-site sweet spot; adding sites adds lines, so the zero-energy and maximal-polarization conditions are satisfied on an ever larger region.

What would settle it

A direct numerical check: for $N=25$ at the plateau center ($\mu=0$, $t=\Delta$) and $W=\Delta$, compute the minimum value of $\gamma^2$ and the minimum excitation gap over the 100 disorder realizations rather than the average; if any realization deviates from $\gamma^2 = 1/2$ by more than numerical tolerance, the strict zero-energy plateau claim fails. Experimentally, the predicted $e^2/2h$ flat conductance plateau must persist when the side-coupled dot is detuned across the island and disappear outside its boundaries in the $(\mu,t)$ plane; a purely Lorentzian dot conductance inside the island would falsify the detection claim.

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Extended reading notes

Core claim

The central claim is that the isolated sweet spot of a two-site Kitaev chain is not a special accident but the $N = 2$ member of a family of zero-energy lines, $\mu^{(n)} = 2\sqrt{t^2 - \Delta^2}\,\cos(n\pi/(N+1))$, that all converge at the same point. As $N$ increases, convergence of these lines produces an extended region—the topological island—where the lowest Bogoliubov excitation has zero energy and the Majorana polarization is maximal. In long chains ($N \geq 20$) the island becomes a strictly zero-energy plateau: $\gamma^2 = \sum_j u_j v_j$ equals $1/2$, the value forced by particle-hole symmetry for an exact zero mode, and the plateau survives disorder averaging over 50–100 realizations for disorder strengths up to the gap. The paper further claims that a spin-polarized quantum dot side-coupled to the chain shows a zero-bias conductance of $e^2/2h$ exactly when $\gamma^2 = 1/2$ and a flat conductance plateau as a function of the dot level, so the topological island can be observed in transport.

Load-bearing premise

The load-bearing premise is that a disorder-averaged numerical quantity, $\gamma^2 = 1/2$, computed for the lowest mode of a finite chain, genuinely establishes an exact zero-energy Majorana mode that is protected in every disorder realization, not just on average.

Editorial extensions

If this is right

Adding sites from $N=2$ to $N=9$ already converts the sweet spot into a finite 'sweet region' where energy stays near zero and $|M|$ stays near 1 under simultaneous variation of all chemical potentials and hoppings, so 4–5 site chains should be significantly easier to tune than 2-site ones.
For $N \geq 20$, the zero-energy plateau survives disorder strengths up to the pairing gap, implying that longer quantum-dot arrays enter a genuinely protected regime rather than simply extending the fragile two-site device.
A zero-bias conductance of $e^2/2h$ through the side-coupled dot occurs exactly when $\gamma^2 = 1/2$, so transport can certify the topologically protected regime; outside it, conductance is set by the dot alone.
Detuning the probe-dot level $\varepsilon_d$ sharpens the conductance map of the island at finite temperature, giving an experimental dial that trades signal magnitude for resolution.
Within the topological island the derivative of the energy splitting with respect to parameter fluctuations vanishes, which should lengthen dephasing times in Majorana-based qubits.

Reading between the lines

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

Beyond the paper, the same $\gamma^2 = 1/2$ plateau diagnostic could be applied to perturbations the authors did not test, such as correlated disorder or pairing-phase fluctuations, to see whether the topological island survives them.
Beyond the paper, because the plateau width grows with $N$ and is bounded by the $|\mu| < 2t$ topological phase boundary, the island area should saturate; sweeping plateau width versus $N$ between 5 and 20 would yield a direct experimental curve for the PMM-to-MBS crossover.
Beyond the paper, the flatness of the conductance plateau as the dot level $\varepsilon_d$ is varied could serve as a calibration of residual Majorana hybridization, since the detuning at which flatness is lost sets an upper bound on the hybridization energy $E_M$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper delivers a genuinely useful numerical message: the two-site Kitaev sweet spot widens into a finite parameter-space region as N grows, and the effect is already visible at N=5-9. The systematic N-dependent maps of excitation energy and Majorana polarization, the extension to a spinful QD model, and the side-coupled QD conductance scheme with an analytic formula tying G to γ² are all solid. The conductance derivation in App. A checks out, and the numerics match the analytic sweet-spot results. Credit where due: this is a careful study of standard models, and the scaling idea is directly relevant to the experimental roadmap for few-dot Majorana devices.

The soft spot is real but not fatal. The phrase 'strictly zero-energy plateaus' is an overclaim. γ²=1/2 only occurs for an exactly degenerate zero eigenmode; in a finite disordered chain there is no symmetry forcing exact zero, so all you can claim is that the splitting is below the numerical threshold (or machine precision). The paper should state that threshold and use 'exponentially small' or 'numerically zero' rather than 'strictly zero.' Relatedly, the disorder robustness is shown as an averaged γ² over 50-100 realizations, without variance or histograms. Since γ² is an all-or-nothing quantity, the average is hard to interpret unless essentially all realizations give 1/2. If that is the case, one histogram would settle it. The central message—sweet spots expand and long chains become more robust—does not depend on exact zero, so the fix is mostly wording plus one additional figure.

Minor points: the 'topological island' terminology is a bit loose for a finite-size region, but they do define it; and the claim of consistency with experiments is credible but not essential. The citations to the zero-energy line formula [30] are appropriate; no circularity issue.

Who is this for? People building few-dot Majorana devices, and theorists wanting a concrete scaling argument. It deserves a serious referee, but I would send it back for a revision that tightens the zero-energy language and shows the disorder statistics properly.

Referee Report

3 major / 6 minor

Summary. The paper investigates how the two-site Kitaev-chain sweet spot, at which Poor man's Majorana bound states appear, evolves as the number of sites N increases. Using exact diagonalization of the BdG Hamiltonian, the authors study the lowest-energy excitation EM and the Majorana polarization MP for both the spinless Kitaev model and a more realistic spinful quantum-dot array. They define a numerical diagnostic γ² = Σ_j u_j v_j for the lowest Bogoliubov mode, argue that γ² = 1/2 identifies a self-adjoint (Majorana) operator, and use this to map out regions of parameter space where the energy is claimed to be strictly zero. For short chains (N < 10) they find that the sweet spot expands into a region of low EM and high MP; for N ≥ 20 they report 'strictly zero-energy plateaus' robust against Anderson disorder, which they call topological islands. They also derive an analytic conductance formula for a quantum dot side-coupled to the chain, showing G = e²/2h when γ² = 1/2 and G = 0 otherwise, and verify this numerically.

Significance. The paper addresses a timely experimental question: whether extending two-site Kitaev chains to longer arrays relaxes the fine-tuning required for Majorana-like states. The main conceptual message—that sweet spots grow into protected regions as N increases—is physically plausible and is supported by the convergence of known zero-energy lines (Ref. [30]) and by the numerical trends in Figs. 1 and 3. A clear strength is the analytic conductance derivation in Appendix A: the scattering-matrix formalism yields closed-form expressions for the e²/2h plateau, and the comparison with numerics in Fig. 4 is convincing, with no fitting parameters. The paper also makes a falsifiable prediction: the zero-bias conductance through a side-coupled dot can track the topological island (Eq. (26)). These positive features make the manuscript a potentially useful contribution, but the central claim of 'strictly zero-energy plateaus' requires more careful statistical and numerical analysis.

major comments (3)

[Sec. VI, Fig. 3] The evidence for 'strictly zero-energy plateaus robust against disorder' is not established by the presented data. The quantity γ² plotted in Fig. 3(a) is binary per disorder realization: for an exact eigenvector at E ≠ 0, γ² = 0 (as the paper itself notes in Sec. III.A), while γ² = 1/2 only for an exactly degenerate zero mode. Averaging this binary quantity over 50 realizations does not reveal the fraction of realizations with γ² = 1/2; a region with average γ² ≈ 1/2 could mean either that all realizations have γ² = 1/2 or that a fraction do. The authors should report the percentage of realizations yielding γ² = 1/2, or show the distribution of the lowest positive eigenvalue EM, to support the claim that the zero-energy region survives in every realization.
[Sec. III.A, Eq. (10) and Sec. VI] The identification 'γ² = 1/2 implies an exact zero-energy mode' is only exact for an eigenvector at E = 0. In a finite chain with nonzero EM, the exact BdG eigenvector satisfies γ² = 0, so the numerical observation of γ² = 1/2 indicates that EM is below the solver's numerical threshold, not that EM is exactly zero. The paper does not specify the diagonalization tolerance or the numerical precision, and the 'topological island' boundaries in Fig. 3(a) are therefore defined by that threshold rather than by a physical condition. The authors should either provide direct evidence of exact zero energy (e.g., a symmetry argument), or revise the wording to 'numerically zero' or 'exponentially small' and show, for instance, log EM versus N to demonstrate the exponential suppression that underlies the plateau.
[Sec. VI, Figs. 3(c) and 3(d)] The disorder robustness claim is made solely on the basis of disorder-averaged γ² curves. For N = 20, the plateau is 'significantly depleted' at W = 0.5Δ and absent at W = Δ, but the average could hide realizations where γ² = 1/2 still occurs with lower probability. For N = 40, the plateau appears unchanged, but again the average does not show whether every realization retains γ² = 1/2. The authors should provide per-realization statistics, such as the fraction of realizations with γ² = 1/2 or the median and variance of EM, to substantiate the claim that the zero-energy mode is robust in each realization, not merely on average.

minor comments (6)

[Sec. VI, Fig. 3 caption] The caption says 'random fluctuations in the hopping and chemical potential of 5% of Δ', but the text in Sec. VI says 'random fluctuations of 5% of the gap'. Please use a consistent definition of the disorder strength and specify whether W = 0.05Δ for the data in Fig. 3(a).
[Sec. V] In the spinful model, the parameters are set as Vz = 2.5δ, t = 0.5δ, and tso = 0.2t, but it is not explicitly stated that t in 'tso = 0.2t' is the spin-conserving hopping of Eq. (2). Clarify to avoid confusion with the effective Kitaev hopping.
[Sec. VII, Eq. (31)] The formula Γ_eff = Γ t0² / (εd² + 2t0²) is presented without parentheses in the text; please format it as Γ_eff = Γ t0² / (εd² + 2t0²) to avoid ambiguity.
[Sec. III.A] The statement that γ²(E ≠ 0) = 0 for 'fermionic operators' relies on the choice of a real gauge for the BdG eigenvectors. The paper does not state whether the numerical diagonalization returns real eigenvectors or whether a gauge transformation is applied. This should be clarified because the diagnostic is central to the rest of the work.
[Sec. IV, Fig. 1] The figure caption refers to 'EM → 0' and '|M| → 1' in the white regions, but the color scale is not shown. Adding a color bar or explicitly stating the color-to-value mapping would improve readability.
[Sec. VII, Eq. (26)] The claim G = 0 for γ² = 0 is derived in the limit T → 0 and for the specific voltage-drop parameter α = 1/2 in Eq. (25). The paper later discusses general α, but Eq. (26) should explicitly state that it applies at T = 0 and for the symmetric voltage configuration, as is done in the text.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central numerical maps and the conductance derivation are self-contained, and the only self-citations are contextual or rederived.

full rationale

The derivation chain is self-contained. The zero-energy line formula, Eq. (6), is imported from an external analytical diagonalization (Ref. [30]) and then used as input to the numerical maps; it is not fitted to the paper's own outputs. The short-chain maps in Figs. 1-2 and the long-chain plateau maps in Fig. 3 are direct BdG diagonalizations, with disorder included as stated, not as parameters fitted to produce the claimed plateaus. The conductance result G = e^2/2h for EM = 0 is derived analytically in Appendix A (Eqs. 21-25) and only afterward compared with the earlier result of Ref. [32], so the self-citation to Ref. [32] is confirmatory rather than load-bearing. The identification of gamma^2 = 1/2 with an exact zero-energy mode (Eq. (10) and Sec. III.A) is a mathematical consequence of particle-hole symmetry for exact BdG eigenstates, not a definition that smuggles in the conclusion; whether finite disordered chains actually produce exact zero eigenvalues to numerical precision is a correctness and validation concern, not a circularity. No fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. The paper contains self-citations (Refs. [12] and [32]), but they are contextual or rederived, so they do not raise the circularity score above 1.

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

The central results are computed from standard Hamiltonians taken from prior literature; the paper's contribution is the numerical characterization and the conductance scheme, not a new derivation. The ledger therefore contains no fitted free parameters and no invented physical entities. The axioms listed are the background modeling choices the claims inherit from the cited literature.

assumptions (6)

standard math The zero-energy line formula Eq. (6), μ(n) = 2√(t² − Δ²) cos(nπ/(N+1)), taken from Refs. [30,40], is exact for the homogeneous N-site Kitaev chain.
Used in Secs. III.A and IV to interpret the growth of the sweet spot; the present paper does not derive it.
domain assumption The effective Kitaev Hamiltonian Eq. (1) captures the physics of the quantum-dot array after projection of the superconducting dots.
Invoked in Secs. I and V via Refs. [20,21,22,23]; the paper does not re-derive the projection and relies on it for all Kitaev-model results.
domain assumption The spinful quantum-dot model Eq. (2) with neglected Coulomb interactions adequately describes the experimental quantum-dot arrays at the parameters used, V_z = 2.5δ.
Stated in Sec. V: 'we have neglected Coulomb interaction terms, which become more relevant in the regime of weak magnetic fields (V_z < δ)'; the validity at V_z = 2.5δ is assumed, not verified.
domain assumption Anderson-type disorder in all parameters, with uniform distribution in [-W,W], is the relevant model for parameter fluctuations in the experimental system.
Introduced in Sec. II.A and used in Sec. VI; it is a modeling choice against which the claimed robustness is measured.
ad hoc to paper The numerical diagnostic γ2 = 1/2 faithfully identifies an exactly zero-energy Majorana mode, and is used to define the topological island and its disorder robustness.
Introduced in Sec. III.A and applied in Sec. VI; the 'strictly zero-energy plateaus' are inferred from this precision-based proxy rather than from a direct proof of exact degeneracy.
domain assumption The wide-band limit, symmetric lead coupling Γ1 = Γ2 = Γ, and the low-energy effective model Eq. (4) are valid for the conductance calculation.
Used in Sec. VII and App. A; these are standard approximations, but they restrict the conductance results to the T → 0, linear-response regime.

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Cite this review

Pith. "Pith review of Two-site Kitaev sweet spots evolving into topological islands." pith.science (2026). https://pith.science/paper/S4HDOMVN

@misc{pith2026250119376,
  author       = {Pith},
  title        = {Pith review of: Two-site Kitaev sweet spots evolving into topological islands},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4HDOMVN}},
  note         = {Machine review of arXiv:2501.19376}
}

abstract

Artificial Kitaev chains based on arrays of quantum dots are promising platforms for realizing Majorana Bound States (MBSs). In a two-site Kitaev chain, it is possible to find these non-Abelian zero-energy excitations at certain points in parameter space (sweet spots). These states, commonly referred to as Poor man's Majorana bound states (PMMs), are challenging to find and stabilize experimentally. In this work, we investigate the evolution of the sweet spots as we increase the number of sites of the Kitaev chain. To this end, we use the Bogoliubov-de Gennes representation to study the excitations of the system, and the scattering matrix and Green functions formalisms to calculate the zero-bias conductance. Our results show that the sweet spots evolve into a region that grows bigger and becomes gradually more protected as the number of sites $N$ increases. Due to the protection of the MBSs, we refer to this region as a topological island. We obtain similar results by considering a realistic spinful model with finite magnetic fields in a chain of normal-superconducting quantum dots. For long chains, $N \geq 20$, we show the emergence of strictly zero-energy plateaus robust against disorder. Finally, we demonstrate that the topological islands can be observed by performing conductance measurements via a quantum dot side-coupled to the Kitaev chain. Our work shows that the fine-tuning required to create and detect PMMs in a 2-site Kitaev chain is significantly relaxed as the length of the chain increases and details how PMMs evolve into MBSs. Our results are consistent with experimental reports for 2 and 3-site chains.

Figures

Figures reproduced from arXiv: 2501.19376 by the authors.

**Figure 1.** FIG. 1. Lowest energy excitation of the Bogoliubov-de [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Energy of the first excited state [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Figures from the paper (3 more)

**Figure 3.** Figure 3: FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]

**Figure 4.** Figure 4: FIG. 4. (a) Schematic of a Kitaev chain with an arbitrary [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) Conductance as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Measuring coherence factors of states in superconductors through local current
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A two-point local current ratio, sqrt(I_L(Γ_R=0)/I_L(Γ_R=Γ_L)), estimates the local Majorana polarization in short quantum-dot Kitaev chains.
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