pith. sign in

arxiv: 2606.23183 · v1 · pith:U3SB2RDBnew · submitted 2026-06-22 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Rotating Zeeman field as a tool for Majorana zero mode detection in topological superconducting wire

Piotr Stefa\'nski This is my paper

Pith reviewed 2026-06-26 07:10 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords Majorana zero modestopological phase transitionZeeman field rotationquantum dot spin polarizationsuperconducting nanowiretopological superconductivityspin polarization detection
0
0 comments X

The pith

Rotating the Zeeman field in a superconducting wire changes the attached quantum dot's Fermi-level spin polarization only when Majorana zero modes are present.

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

This paper shows that the spin polarization of a quantum dot coupled to a topological superconducting wire can detect Majorana zero modes through rotation of the Zeeman field in the wire while the field direction in the dot stays fixed. In the presence of a Majorana mode the polarization at the Fermi energy changes substantially when the field direction shifts from parallel to perpendicular to the wire axis. In the trivial state the polarization remains nearly constant during the same rotation. The method also locates the topological phase transition by a nonlinear variation of the polarization with field strength at the critical point. These signatures hold for different coupling strengths between wire and dot and for different field angles.

Core claim

The paper establishes that when a Majorana zero mode exists in the wire the effective spin polarization of the attached quantum dot at the Fermi energy varies markedly upon rotating the Zeeman field from parallel to perpendicular orientation, whereas in the trivial phase this polarization remains essentially unchanged. A nonlinear dependence of the polarization on the magnitude of the magnetic field at its critical value further marks the topological phase transition, and this behavior holds irrespective of the wire-dot coupling strength or the angle of the Zeeman field.

What carries the argument

The differential response of the quantum dot spin polarization at the Fermi energy to the orientation of the Zeeman field in the wire, which is sensitive to the presence of a Majorana zero mode.

Load-bearing premise

The quantum dot spin polarization at the Fermi energy can be isolated from effects of disorder, finite temperature, or additional sub-gap states that might produce similar responses.

What would settle it

Measuring no significant change in quantum dot spin polarization upon rotating the Zeeman field in a wire confirmed to host Majorana zero modes would falsify the detection scheme.

Figures

Figures reproduced from arXiv: 2606.23183 by Piotr Stefa\'nski.

Figure 1
Figure 1. Figure 1: Spectral densities of the dot and its spin polarization at Fermi energy [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spectral densities of the dot at Fermi energy and spin polarization vs. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spectral densities of the dot and its spin polarization at Fermi energy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectral densities of the dot vs. energy for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spectral densities of the dot vs. energy and its spin polarization vs. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

We demonstrate that analysis of the spin polarization of a quantum dot (QD) attached to the topological wire can provide valuable insights into Majorana zero mode (MZM) formation and topological phase transition. Detection is realized by rotation of the Zeeman field in the wire, while retaining the Zeeman field direction in the dot intact. In the presence of Majorana mode, the effective QD spin polarization at Fermi energy changes significantly when the direction of the Zeeman field in the wire changes from parallel to perpendicular to the wire axis. It can be opposed to the wire in its trivial state, when spin polarization remains practically constant while the magnetic field is rotated. Similar unaltered spin polarization is observed when QD spin sub-level at Fermi energy mimics MZM. Moreover, the characteristic non-linear dependence of the spin polarization on the magnetic field magnitude at its critical value identifies a topological phase transition in the wire. This feature is observed independently on the coupling strength of the wire to the dot and the angle of the Zeeman field.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes using rotation of the Zeeman field direction in a topological superconducting wire (while keeping the QD Zeeman field fixed) to detect Majorana zero modes via changes in the attached quantum dot's spin polarization at the Fermi energy. It claims that the presence of an MZM produces a significant change in QD polarization when the wire field rotates from parallel to perpendicular to the wire axis, in contrast to the trivial regime (and to cases where a QD sub-level mimics an MZM) where polarization remains essentially constant. A non-linear dependence of polarization on field magnitude at the critical value is also reported to identify the topological phase transition, independent of wire-dot coupling strength and field angle.

Significance. If the central claims hold under realistic conditions, the method would supply a distinct experimental signature for MZM presence and topological phase transitions that complements conductance-based probes and is insensitive to coupling strength. The contrast between topological and trivial regimes under field rotation is a potentially falsifiable prediction. However, the idealized clean-model setting limits immediate experimental impact.

major comments (2)
  1. [Abstract and numerical results] The central claim (Abstract) requires that the angular dependence of QD spin polarization at E_F is markedly different in the topological versus trivial regimes. The analysis is performed in an idealized Kitaev-like wire + QD Hamiltonian at zero temperature with no disorder; the manuscript does not quantify how the reported contrast survives when disorder, finite-temperature broadening, or additional Andreev states are included, any of which can hybridize the zero-energy feature and flatten or induce spurious angular variation.
  2. [Abstract] The claim that the non-linear dependence on magnetic-field magnitude at the critical value 'identifies a topological phase transition' independently of coupling strength and angle (Abstract) is load-bearing for the detection proposal. Without explicit checks against disorder or sub-gap states that could shift or broaden the critical point, it is unclear whether this feature remains a reliable indicator.
minor comments (1)
  1. [Abstract] The abstract states expected signatures but supplies no model Hamiltonian, equations, or figure references; the full manuscript should include these in the main text or supplementary material for reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and numerical results] The central claim (Abstract) requires that the angular dependence of QD spin polarization at E_F is markedly different in the topological versus trivial regimes. The analysis is performed in an idealized Kitaev-like wire + QD Hamiltonian at zero temperature with no disorder; the manuscript does not quantify how the reported contrast survives when disorder, finite-temperature broadening, or additional Andreev states are included, any of which can hybridize the zero-energy feature and flatten or induce spurious angular variation.

    Authors: We agree that the calculations are performed in an idealized clean model at zero temperature. This establishes the proposed signature under controlled conditions, which is the standard starting point for such theoretical proposals. We will revise the abstract and add a dedicated paragraph in the discussion to explicitly state the idealized nature of the model and to note that quantitative assessment of robustness to disorder, finite temperature, and additional Andreev states lies beyond the present scope. revision: partial

  2. Referee: [Abstract] The claim that the non-linear dependence on magnetic-field magnitude at the critical value 'identifies a topological phase transition' independently of coupling strength and angle (Abstract) is load-bearing for the detection proposal. Without explicit checks against disorder or sub-gap states that could shift or broaden the critical point, it is unclear whether this feature remains a reliable indicator.

    Authors: Within the clean model the non-linear dependence on field magnitude at the critical point is observed and is independent of coupling strength and angle. We concur that its reliability as an indicator has not been tested against disorder or sub-gap states. We will revise the abstract to qualify the claim as applying to the idealized model and will add a corresponding statement in the conclusions. revision: yes

standing simulated objections not resolved
  • Quantitative checks of the angular contrast and non-linear field dependence under disorder, finite temperature, or additional Andreev states.

Circularity Check

0 steps flagged

No circularity; model-based distinction is independent of inputs

full rationale

The paper computes QD spin polarization at E_F from an explicit Kitaev-like wire + QD Hamiltonian under rotating Zeeman field. The reported contrast (large angular variation only in topological regime) follows directly from the model's zero-energy mode structure and is not obtained by fitting parameters to the target observable, redefining inputs, or invoking self-citations. No load-bearing step reduces to its own output by construction. The derivation remains self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or audited from the given text.

pith-pipeline@v0.9.1-grok · 5708 in / 1110 out tokens · 20626 ms · 2026-06-26T07:10:44.102256+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 22 canonical work pages · 4 internal anchors

  1. [1]

    Majorana, A symmetric theory of electrons and positrons, Nouvo Cim

    E. Majorana, A symmetric theory of electrons and positrons, Nouvo Cim. 14 (3) (1937) 171–184.doi: 10.1007/978-3-540-48095-2_10

    work page doi:10.1007/978-3-540-48095-2_10 1937

  2. [2]

    R. M. Lutchyn, E. P. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, Y . Oreg, Majorana zero modes in superconductor-semiconductor heterostructures, Nat. Rev. Mater. 3 (5) (2018) 52.arXiv:1707.04899, doi:10.1038/s41578-018-0003-1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/s41578-018-0003-1 2018

  3. [3]

    R. M. Lutchyn, J. D. Sau, S. Das Sarma, Ma- jorana fermions and a topological phase transition in semiconductor-superconductor heterostructures, Phys. Rev. Lett. 105 (7) (2010) 077001.arXiv:1002.4033, doi:10.1103/PhysRevLett.105.077001. 7

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.105.077001 2010

  4. [4]

    Y . Oreg, G. Refael, F. von Oppen, Helical liquids and Ma- jorana bound states in quantum wires, Phys. Rev. Lett. 105 (17) (2010) 177002.arXiv:1003.1145,doi:10. 1103/PhysRevLett.105.177002

    Pith/arXiv arXiv 2010

  5. [5]

    S. D. Sarma, M. Freedman, C. Nayak, Majorana zero modes and topological quantum computation, npj Quan- tum Inf. 1 (1) (2015) 15001.doi:10.1038/npjqi. 2015.1

    work page doi:10.1038/npjqi 2015

  6. [6]

    Marra, Majorana nanowires for topological quantum computation, J

    P. Marra, Majorana nanowires for topological quantum computation, J. Appl. Phys. 132 (2022) 231101.arXiv: 2206.14828,doi:10.1063/5.0102999

    work page doi:10.1063/5.0102999 2022

  7. [7]

    Aghaee, A

    Microsoft Azure Quantum, M. Aghaee, A. Al- caraz Ramirez, Z. Alam, R. Ali, M. Andrzejczuk, A. Antipov, M. Astafev, A. Barzegar, B. Bauer, et al., Interferometric single-shot parity measurement in InAs– Al hybrid devices, Nature 638 (8051) (2025) 651–655. doi:10.1038/s41586-024-08445-2

    work page doi:10.1038/s41586-024-08445-2 2025

  8. [9]

    Aghaee, et al., InAs-Al Hybrid Devices Passing the Topological Gap Protocol (2022).arXiv:2103.12217

    M. Aghaee, et al., InAs-Al Hybrid Devices Passing the Topological Gap Protocol (2022).arXiv:2103.12217

    arXiv 2022

  9. [10]

    20 Second Parity Lifetime in an InAs--Pb Tetron Device

    M. Aghaee, Z. Alam, M. Andrzejczuk, A. Antipov, et al., 20 Second Parity Lifetime in an InAs–Pb Tetron Device, submitted 2 June 2026; revised 3 June 2026 (2026).arXiv:2606.03884,doi:10.48550/arXiv. 2606.03884

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2026

  10. [11]

    J. Chen, B. D. Woods, P. Yu, M. Hocevar, D. Car, S. R. Plissard, E. P. A. M. Bakkers, T. D. Stanescu, S. M. Frolov, Ubiquitous non-majorana zero-bias conductance peaks in nanowire devices, Phys. Rev. Lett. 123 (2019) 107703.doi:10.1103/PhysRevLett.123.107703

    work page doi:10.1103/physrevlett.123.107703 2019

  11. [12]

    Valentini, F

    M. Valentini, F. Peñaranda, A. Hofmann, M. Brauns, R. Hauschild, P. Krogstrup, P. San-Jose, E. Prada, R. Aguado, G. Katsaros, Non-topological zero bias peaks in full-shell nanowires induced by flux tunable Andreev states, Science 373 (2021) 82.arXiv:2008.02348,doi: 10.1126/science.abf1513

    work page doi:10.1126/science.abf1513 2021

  12. [13]

    Das Sarma, H

    S. Das Sarma, H. Pan, Disorder-induced zero-bias peaks in majorana nanowires, Phys. Rev. B 103 (2021) 195158. doi:10.1103/PhysRevB.103.195158

    work page doi:10.1103/physrevb.103.195158 2021

  13. [14]

    P. Yu, J. Chen, M. Gomanko, G. Badawy, E. P. A. M. Bakkers, K. Zuo, V . Mourik, S. M. Frolov, Non-Majorana states yield nearly quantized conduc- tance in proximatized nanowires, Nature Physics 17 (4) (2021) 482–488.arXiv:2004.08583,doi:10.1038/ s41567-020-01107-w

    arXiv 2021

  14. [15]

    R. Hess, H. F. Legg, D. Loss, J. Klinovaja, Trivial Andreev band mimicking topological bulk gap reopening in the non-local conductance of long Rashba nanowires, Phys. Rev. Lett. 130 (2023) 207001.arXiv:2210.03507,doi: 10.1103/PhysRevLett.130.207001

    work page doi:10.1103/physrevlett.130.207001 2023

  15. [16]

    interferometric single-shot par- ity measurement in inas-al hybrid devices

    H. F. Legg, Comment on “interferometric single-shot par- ity measurement in inas-al hybrid devices”, microsoft quantum, nature 638, 651–655 (2025) (2025).arXiv: 2503.08944,doi:10.48550/arXiv.2503.08944

    work page doi:10.48550/arxiv.2503.08944 2025

  16. [17]

    inas-al hybrid devices passing the topological gap protocol

    H. F. Legg, Comment on "inas-al hybrid devices passing the topological gap protocol", microsoft quantum, phys. rev. b 107, 245423 (2023) (2025).arXiv:2502.19560, doi:10.48550/arXiv.2502.19560

    work page doi:10.48550/arxiv.2502.19560 2023

  17. [18]

    Castelvecchi, Microsoft upgrades controver- sial quantum chip – researchers are still scep- tical, NatureNature News (Jun

    D. Castelvecchi, Microsoft upgrades controver- sial quantum chip – researchers are still scep- tical, NatureNature News (Jun. 2026).doi: 10.1038/d41586-026-01788-y

    work page doi:10.1038/d41586-026-01788-y 2026

  18. [19]

    Zhang, C

    F. Zhang, C. L. Kane, E. J. Mele, Time-reversal-invariant topological superconductivity and Majorana Kramers pairs, Phys. Rev. Lett. 111 (2013) 056402.arXiv:1212. 4232,doi:10.1103/PhysRevLett.111.056402

    work page doi:10.1103/physrevlett.111.056402 2013

  19. [20]

    Alicea, Majorana fermions in a tunable semiconductor device, Physical Review B 81 (2010) 125318.doi:10

    J. Alicea, Majorana fermions in a tunable semiconductor device, Physical Review B 81 (2010) 125318.doi:10. 1103/PhysRevB.81.125318

    2010

  20. [21]

    P. Stefa ´nski, Quantum dot detects Majorana modes of both chiralities, Journal of Magnetism and Magnetic Ma- terials 584 (2023) 171067.arXiv:2307.13366,doi: 10.1016/j.jmmm.2023.171067

    work page doi:10.1016/j.jmmm.2023.171067 2023

  21. [22]

    Yoneda, T

    J. Yoneda, T. Otsuka, T. Takakura, M. Pioro-Ladrière, R. Brunner, H. Lu, T. Nakajima, T. Obata, A. Noiri, C. J. Palmstrøm, A. C. Gossard, S. Tarucha, Robust micro- magnet design for fast electrical manipulations of single spins in quantum dots, Applied Physics Express 8 (2015) 084401.doi:10.7567/APEX.8.084401

    work page doi:10.7567/apex.8.084401 2015

  22. [23]

    N. I. D. Stuyck, F. A. Mohiyaddin, R. Li, M. Heyns, B. Govoreanu, I. P. Radu, Low dephasing and robust mi- cromagnet designs for silicon spin qubits, Applied Physics Letters 119 (2021) 094001.doi:10.1063/5.0059939

    work page doi:10.1063/5.0059939 2021

  23. [24]

    S. G. Philips, M. T. M˛ adzik, S. V . Amitonov, S. L. de Snoo, M. Russ, N. Kalhor, C. V olk, W. I. Lawrie, D. Brousse, L. Tryputen, B. P. Wuetz, A. Sammak, M. Veldhorst, G. Scappucci, L. M. Vandersypen, Uni- versal control of a six-qubit quantum processor in sil- icon, Nature 609 (2022) 919–924.doi:10.1038/ s41586-022-05117-x

    2022

  24. [25]

    Jiang, M

    Y . Jiang, M. Gupta, C. Riggert, M. Pendharkar, C. Dempsey, J. S. Lee, S. D. Harrington, C. J. Palm- strøm, V . S. Pribiag, S. M. Frolov, Zero-bias conductance peaks at zero applied magnetic field due to stray fields 8 from integrated micromagnets in hybrid nanowire quan- tum dots, SciPost Phys. 19 (2025) 030.doi:10.21468/ SciPostPhys.19.2.030

    2025

  25. [26]

    Interacting Dark Energy after DESI Baryon Acoustic Oscillation Measure- ments

    M. Bagchi, T. Y . Tounsi, A. Safeer, C. V . Efferen, A. Rosch, T. Michely, W. Jolie, T. A. Costi, J. Fischer, Spin-polarized scanning tunneling microscopy measure- ments of an anderson impurity, Physical Review Letters 133 (2024) 246701.doi:10.1103/PhysRevLett.133. 246701

    work page doi:10.1103/physrevlett.133 2024

  26. [27]

    H. A. Nilsson, P. Caroff, C. Thelander, M. Larsson, J. B. Wagner, L.-E. Wernersson, L. Samuelson, H. Q. Xu, Gi- ant, level-dependentgfactors in InAs nanowire quan- tum dots, Nano Letters 9 (9) (2009) 3151–3156.doi: 10.1021/nl901234w

    work page doi:10.1021/nl901234w 2009

  27. [28]

    J. Shen, S. Heedt, F. Borsoi, B. van Heck, S. Gaz- ibegovic, R. L. M. Op het Veld, D. Car, J. A. Logan, M. Pendharkar, S. J. J. Ramakers, G. Wang, D. Xu, D. Bouman, B. Shojaei, D. E. Lee, L. P. Kouwenhoven, E. P. A. M. Bakkers, Parity transitions in the supercon- ducting ground state of hybrid InSb–Al Coulomb islands, Nature Communications 9 (2018) 4801.d...

    2018

  28. [29]

    Heedt, M

    S. Heedt, M. Quintero-Pérez, F. Borsoi, A. Fursina, N. van Loo, G. P. Mazur, M. P. Nowak, M. Ammerlaan, K. Li, S. Korneychuk, J. Shen, M. A. Y . van de Poll, G. Badawy, S. Gazibegovic, N. de Jong, P. Aseev, K. van Hoogdalem, E. P. A. M. Bakkers, L. P. Kouwenhoven, Shadow-wall lithography of ballistic superconductor– semiconductor quantum devices, Nature C...

    arXiv 2021

  29. [30]

    T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. D. ten Haaf, D. van Driel, F. Zatelli, X. Li, F. K. Malinowski, S. Gazibegovic, G. Badawy, E. P. A. M. Bakkers, M. Wimmer, L. P. Kouwenhoven, Realization of a minimal Kitaev chain in coupled quantum dots, Nature 614 (7948) (2023) 445–450.arXiv:2206.08045,doi: 10.1038/s41586-022-05585-1

    work page doi:10.1038/s41586-022-05585-1 2023

  30. [31]

    de Sousa, S

    R. de Sousa, S. Das Sarma, Gate control of spin dynam- ics in III-V semiconductor quantum dots, Phys. Rev. B 68 (2003) 1553301.arXiv:0306417,doi:10.1103/ PhysRevB.68.155330

    2003

  31. [32]

    E. M. Stoudenmire, J. Alicea, O. A. Starykh, M. P. Fisher, Interaction effects in topological superconducting wires supporting Majorana fermions, Phys. Rev. B 84 (1) (2011) 014503.arXiv:1104.5493,doi:10.1103/PhysRevB. 84.014503

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb 2011

  32. [33]

    G. Y . Huang, M. Leijnse, K. Flensberg, H. Q. Xu, Tunnel spectroscopy of Majorana bound states in topo- logical superconductor/quantum dot Josephson junctions, Phys. Rev. B 90 (21) (2014) 214507.doi:10.1103/ PhysRevB.90.214507

    2014

  33. [34]

    Rainis, L

    D. Rainis, L. Trifunovic, J. Klinovaja, D. Loss, To- wards a realistic transport modeling in a superconduct- ing nanowire with Majorana fermions, Phys. Rev. B 87 (2) (2013) 024515.arXiv:1207.5907,doi:10. 1103/PhysRevB.87.024515

    Pith/arXiv arXiv 2013

  34. [35]

    Stefa ´nski, Sub-gap Fano resonances in a topological su- perconducting wire with on-site Coulomb interactions, J

    P. Stefa ´nski, Sub-gap Fano resonances in a topological su- perconducting wire with on-site Coulomb interactions, J. Phys. Condens. Matter 33 (2021) 465602.doi:10.1088/ 1361-648x/ac1d6d

    2021

  35. [36]

    D. E. Liu, H. U. Baranger, Detecting a Majorana-fermion zero mode using a quantum dot, Physical Review B 84 (2011) 201308.arXiv:1107.4338,doi:10.1103/ PhysRevB.84.201308. 9

    Pith/arXiv arXiv 2011