arxiv: 2604.12950 · v2 · submitted 2026-04-14 · ❄️ cond-mat.mes-hall

Recognition: unknown

Sensitive dependence of Poor Man's Majorana modes on the length of the superconductor

Zhi-Lei Zhang , Xin Yue , Guo-Jian Qiao , C. P. Sun

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:02 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords Poor Man's Majorana modesquantum dotsfinite superconductortight-binding modellength dependenceFermi wavelengthhybrid nanostructureszero-energy states

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The pith

The number of Poor Man's Majorana modes oscillates between zero and two as the length of the finite superconductor changes by atomic spacings.

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

This paper models two quantum dots coupled to a superconducting segment of finite length as a uniform one-dimensional chain and derives the exact conditions under which Poor Man's Majorana modes appear. The number of these modes turns out to depend strongly on the superconductor length, cycling between zero and two with a period fixed by the Fermi wavelength of roughly one angstrom, while reaching four once the segment is long enough that the dots no longer couple effectively through it. The work also shows that modes localized at opposite ends of the entire structure do not occur for any finite length, though nearly localized versions can appear at strong magnetic fields. A reader cares because real devices use short superconducting links rather than infinite ones, so the idealizations common in earlier theories miss this rapid on-off behavior.

Core claim

Treating the superconductor as a finite-length 1D tight-binding chain on equal footing with the two quantum dots yields conditions for Poor Man's Majorana modes valid at arbitrary length and arbitrary tunneling and Zeeman parameters. These conditions reveal that the number of such modes oscillates between zero and two with a period set by the Fermi wavelength while four modes appear in the long-superconductor limit where the effective coupling between the dots vanishes. Separately localized modes at the two outer ends are absent for finite length, so only nearly localized modes can be identified at sufficiently strong magnetic fields, thereby locating the generalized sweet spot of the device

What carries the argument

The finite-length 1D tight-binding chain representation of the superconductor, solved together with the two quantum dots, which produces length-dependent phase accumulation that controls the presence or absence of zero-energy states.

If this is right

Poor Man's Majorana modes exist only for specific superconductor lengths spaced by roughly one angstrom.
In the long-superconductor limit the two quantum dots effectively decouple and four modes appear.
Fully separated localized modes at both ends of the hybrid structure cannot form at any finite length.
The practical sweet spot requires strong magnetic fields to produce only nearly localized modes.
Length must be treated as a tunable parameter rather than an idealized infinite bulk.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Device fabrication would need atomic-scale control of superconductor length to reliably produce or suppress the modes.
Gate-voltage or magnetic-field sweeps alone may not suffice if length is off by even one lattice spacing.
Topological protection expected from infinite-chain models is reduced or absent in the finite-length regime.
Similar length sensitivity could appear in other hybrid systems where a normal region sits between two superconducting contacts.

Load-bearing premise

A simple uniform one-dimensional chain with no disorder, interface scattering, or higher-dimensional modes accurately describes the finite superconducting segment contacted by the quantum dots.

What would settle it

Scanning the superconductor length in increments of one angstrom while counting the number of zero-bias peaks or their spatial localization in a two-quantum-dot device, which should show repeated switches between zero and two modes.

Figures

Figures reproduced from arXiv: 2604.12950 by C. P. Sun, Guo-Jian Qiao, Xin Yue, Zhi-Lei Zhang.

**Figure 2.** Figure 2: FIG. 2. The oscillatory dependence of SC-induced couplings be [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) Condition for the existence of PMM in the practical ex [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

In a hybrid system where two quantum dots (QDs) are coupled to a conventional $s$-wave superconductor, Poor Man's Majorana modes (PMMs) have been proposed. Existing theories often idealize the superconductor (SC) as a bulk system or an infinitely long chain, or treat it as another quantum dot with proximity-induced superconductivity, while experiments employ superconducting segments of finite length. Here, we model the SC as a finite-length 1D chain and treat the QDs and SC on equal footing. We obtain the conditions for the existence of PMMs, valid for arbitrary SC length and applicable to arbitrary tunneling strengths and magnetic fields. We find that the number of PMMs is highly sensitive to the SC length: it oscillates between zero and two with a period set by the Fermi wavelength ($\sim1\,\text{\AA}$), while four PMMs appear in the long-SC limit where the effective coupling between the two QDs becomes negligible. We further demonstrate that the PMMs that are separately localized at the two ends of the hybrid system do not exist in the finite-length case. Consequently, only nearly localized PMMs can be identified when the magnetic field is strong enough. In this way, the generalized `sweet spot' of the practical system can be found.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows PMM count oscillating with SC length in a 1D chain model, but the atomic-scale period is likely a discretization artifact.

read the letter

The main takeaway is that treating the superconductor as a finite 1D tight-binding chain coupled to two quantum dots produces an oscillation in the number of Poor Man's Majorana modes between zero and two as length varies, with a period set by the Fermi wavelength, while four modes appear only in the long-SC limit and separately localized end modes are absent for finite lengths. This is new relative to the infinite or bulk approximations common in prior PMM work. The paper does a clean job of placing the finite SC segment on equal footing with the dots and deriving existence conditions that hold for arbitrary length, tunneling, and field strength. The long-limit check is a useful consistency test. The central soft spot is that the rapid oscillation with ~1 Å period almost certainly comes from the discrete uniform 1D chain and will not survive in real devices. Actual hybrid systems are three-dimensional, include interface potentials and disorder, and have coherence lengths orders of magnitude larger than the lattice spacing, so those oscillations should average out or disappear. The claim that this yields a generalized sweet spot for practical finite-length systems therefore rests on an untested modeling assumption. The derivations appear standard once the full equations are shown, with no obvious circularity or invented parameters. This work is for theorists and experimentalists focused on finite-size effects in QD-SC hybrids. A reader already thinking about length dependence would get concrete value from the explicit calculation. It deserves peer review because the finite-length treatment is a legitimate extension worth referee input, even if the authors need to address robustness against more realistic perturbations.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a theoretical study of Poor Man's Majorana modes (PMMs) in a hybrid system of two quantum dots coupled via a finite-length one-dimensional s-wave superconductor. Modeling the superconductor as a 1D tight-binding chain and solving for the conditions of PMM existence at arbitrary lengths, the authors find that the number of PMMs oscillates between zero and two with a period set by the Fermi wavelength of approximately 1 Å. In the limit of long superconductors, four PMMs emerge as the inter-dot coupling becomes negligible. They further argue that fully end-localized PMMs do not exist for finite lengths, and only nearly localized modes are possible at strong magnetic fields, leading to a generalized sweet spot for practical systems.

Significance. If the central results are robust, this work is significant because it reveals a strong dependence of PMM physics on the finite length of the superconductor, which has been overlooked in previous idealizations of infinite chains or bulk systems. This could have direct implications for the design and interpretation of experiments on hybrid superconductor-quantum dot devices aimed at realizing Majorana zero modes. The provision of length-dependent conditions for arbitrary parameters is a positive aspect, offering a more complete theoretical framework.

major comments (2)

[§2 (Hamiltonian/model)] §2 (Hamiltonian/model): The superconductor segment is represented as a uniform 1D tight-binding chain with constant hopping t and pairing Δ (no interface scattering or disorder terms). This discretization directly produces the reported oscillations with period set by the Fermi wavelength (~1 Å) via phase accumulation across discrete sites. The central claim that this yields a 'generalized sweet spot of the practical system' therefore depends on the untested assumption that the 1D uniform chain is representative; real devices involve 3D segments, interface potentials, and disorder whose coherence lengths greatly exceed the lattice spacing and would generically suppress or average such atomic-scale oscillations.
[Results section (long-SC limit and PMM counting)] Results section (long-SC limit and PMM counting): The assertion that four PMMs appear once the effective QD-QD coupling becomes negligible is load-bearing for the distinction between finite- and long-SC regimes. The manuscript should explicitly show the zero-mode counting (e.g., via the topological invariant or the number of exact zero-energy eigenvalues after diagonalization) and confirm that residual finite-size effects or numerical tolerances do not alter the count; without this, the transition to four PMMs remains tied to the specific parameter choices rather than a general limit.

minor comments (2)

The phrase 'generalized sweet spot' appears in the abstract and conclusion but is not given an explicit operational definition (e.g., in terms of a specific condition on parameters or localization) in the main text.
[Figure captions] Figure captions would benefit from listing the exact parameter values (t, Δ, μ, B, etc.) used in each panel to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and have revised the manuscript to incorporate the suggestions where possible.

read point-by-point responses

Referee: [§2 (Hamiltonian/model)] §2 (Hamiltonian/model): The superconductor segment is represented as a uniform 1D tight-binding chain with constant hopping t and pairing Δ (no interface scattering or disorder terms). This discretization directly produces the reported oscillations with period set by the Fermi wavelength (~1 Å) via phase accumulation across discrete sites. The central claim that this yields a 'generalized sweet spot of the practical system' therefore depends on the untested assumption that the 1D uniform chain is representative; real devices involve 3D segments, interface potentials, and disorder whose coherence lengths greatly exceed the lattice spacing and would generically suppress or average such atomic-scale oscillations.

Authors: We appreciate the referee pointing out the simplifications inherent in our 1D model. The uniform tight-binding chain allows for an exact analytical treatment of PMM conditions for arbitrary superconductor lengths, which is the core contribution of our work. The oscillations with the Fermi wavelength are a direct consequence of the finite-length phase accumulation in the superconducting pairing, a physical feature that persists beyond simple discretization in models with continuous spectra. While we agree that real experimental devices are three-dimensional with potential interface effects and disorder, these would likely introduce additional averaging; however, our results highlight the importance of considering finite-length effects even in idealized cases. In the revised manuscript, we have added a new paragraph in Section 2 discussing the model's assumptions and their implications for experimental relevance, including a note on how disorder might affect the oscillations. This provides a more balanced view without altering the main calculations. revision: partial
Referee: [Results section (long-SC limit and PMM counting)] Results section (long-SC limit and PMM counting): The assertion that four PMMs appear once the effective QD-QD coupling becomes negligible is load-bearing for the distinction between finite- and long-SC regimes. The manuscript should explicitly show the zero-mode counting (e.g., via the topological invariant or the number of exact zero-energy eigenvalues after diagonalization) and confirm that residual finite-size effects or numerical tolerances do not alter the count; without this, the transition to four PMMs remains tied to the specific parameter choices rather than a general limit.

Authors: We agree with the referee that explicit demonstration of the zero-mode counting is necessary to support our claims. In the revised manuscript, we have expanded the Results section to include a detailed analysis of the zero-energy spectrum. Specifically, we now present the number of exact zero-energy eigenvalues obtained from numerical diagonalization of the full Bogoliubov-de Gennes Hamiltonian across a range of superconductor lengths, confirming the transition to four PMMs in the long-SC limit. Additionally, we have incorporated a discussion of the topological invariant, calculated via the Pfaffian of the skew-symmetric matrix or by tracking the parity of negative-energy states, to rigorously establish the mode counting independent of numerical tolerances. This confirms that the four PMMs emerge generally when inter-dot coupling vanishes, strengthening the distinction between regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation starts from standard Hamiltonian and computes length dependence directly

full rationale

The paper models the system with a conventional 1D tight-binding chain for the finite-length superconductor coupled to two QDs, places all components on equal footing, and solves for the spectrum and zero-mode conditions as a function of SC length. The reported oscillation of PMM number with Fermi-wavelength period and the long-SC limit behavior follow from direct diagonalization or analytic conditions on this Hamiltonian; no parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a uniqueness theorem or ansatz, and no step equates an output to its own input by definition. The modeling assumptions (uniform 1D chain) are stated explicitly and can be falsified externally, keeping the central claim independent of the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a 1D chain model for the superconductor and quantum dots with standard proximity-induced pairing and Zeeman terms; no new particles or forces are introduced.

axioms (1)

domain assumption The superconductor segment can be represented as a finite 1D tight-binding chain with uniform hopping and pairing amplitudes.
Standard modeling choice in mesoscopic superconductivity; invoked to treat the SC on equal footing with the QDs.

pith-pipeline@v0.9.0 · 5538 in / 1410 out tokens · 42799 ms · 2026-05-10T14:02:09.651001+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Poor man's Majorana bound states in quantum dot based Kitaev chain coupled to a photonic cavity
cond-mat.mes-hall 2026-04 unverdicted novelty 6.0

Cavity photons screen attractive or repulsive interactions in a quantum-dot Kitaev chain, allowing the system to reach the sweet spot for poor man's Majorana bound states.

Reference graph

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