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arxiv: 2407.06925 · v3 · submitted 2024-07-09 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Confinement-induced Majorana modes in a nodal topological superconductor

Simone Traverso , Niccol\`o Traverso Ziani , Maura Sassetti , Fernando Dominguez This is my paper

Pith reviewed 2026-05-23 22:44 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords Majorana zero modesnodal topological superconductorquantum confinementHaldane modelequal-spin pairingquasi-one-dimensionaltopological invariantquantized conductance
0
0 comments X

The pith

Quantum confinement gaps bulk bands faster than edge states in a nodal topological superconductor, producing Majorana zero modes via edge hybridization in quasi-one-dimensional structures.

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

The paper examines an extension of the Haldane model that incorporates equal-spin-pairing superconductivity. In two dimensions this model supports a nodal topological superconducting phase featuring chiral Majorana modes on the edges of structures with cylindrical boundary conditions. These modes become unstable in rectangular geometries, where corner states appear instead. When the system is confined by shrinking one dimension, the bulk bands gap out more rapidly than the edge states. This differential gapping allows the edge states to hybridize and form Majorana zero modes, which are protected and produce a quantized conductance of 2e²/h. The result points to confinement as a way to realize quasi-one-dimensional topological superconductivity from two-dimensional nodal materials.

Core claim

An extension of the Haldane model with equal spin pairing superconductivity realizes a two-dimensional nodal topological superconducting phase. Under cylindrical boundary conditions this phase hosts propagating chiral Majorana modes along the edges of nanoribbons. In rectangular flakes the phase is unstable and corner states near zero energy emerge. Reducing one spatial dimension causes quantum confinement to open gaps in the bulk spectrum faster than in the edge spectrum. Consequently the edge states hybridize to produce Majorana zero modes. The phase is diagnosed by a topological invariant whose value yields a differential conductance quantized at 2e²/h in a normal-superconducting junction

What carries the argument

Quantum confinement that gaps the bulk bands more rapidly than the edge states, enabling their hybridization into Majorana zero modes

If this is right

  • In two dimensions the model exhibits a nodal phase with chiral Majorana edge modes for cylindrical boundaries.
  • Rectangular two-dimensional flakes instead host corner states close to zero energy.
  • Confinement to quasi-one dimension stabilizes Majorana zero modes through edge-state hybridization.
  • The topological phase is accompanied by a quantized conductance of 2e²/h in normal-superconducting junctions.

Where Pith is reading between the lines

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

  • Similar confinement strategies might apply to other two-dimensional nodal superconductors to engineer Majorana modes.
  • Device fabrication focusing on narrow ribbons could provide a practical route to observe these modes experimentally.
  • The approach highlights how geometry can control topological phases without altering material parameters.

Load-bearing premise

The equal-spin-pairing superconductivity and chosen boundary conditions produce a nodal phase whose edge states remain intact and hybridize appropriately under quantum confinement.

What would settle it

Measuring the energy spectrum in narrow ribbons carved from the two-dimensional material and checking whether zero-energy modes appear only when the width is reduced sufficiently to induce bulk gapping but not edge gapping.

read the original abstract

We investigate the topological phase diagram of {an extension of the Haldane model with equal spin pairing superconductivity}. In two dimensions, we find a topological nodal superconducting phase, which exhibits a chiral Majorana mode propagating along the edges of nanoribbons with cylindrical boundary conditions. This phase is however unstable in a finite two-dimensional rectangular-shaped lattice, yielding corner states close to zero energy in a flake with alternating zigzag and armchair edges. When we reduce one of the dimensions, quantum confinement gaps out the bulk bands faster than the edge states. In this scenario, hybridization between the edge states can then result in Majorana zero modes. Our results hence suggest quantum confinement as a crucial ingredient in building quasi-one-dimensional topological superconducting phases out of two-dimensional nodal topological superconductors. Furthermore, we characterize the emergence of this novel topological phase by means of its topological invariant, coinciding with a quantized conductance of $2 e^2/h$ in a normal-superconducting junction.

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

0 major / 3 minor

Summary. The manuscript extends the Haldane model by adding equal-spin-pairing superconductivity and maps its 2D topological phase diagram. It reports a nodal topological superconducting phase that supports a chiral Majorana edge mode on nanoribbons with cylindrical boundary conditions; the same phase is unstable on rectangular lattices with alternating zigzag/armchair edges, producing near-zero corner states. Reducing one spatial dimension induces quantum confinement that gaps the bulk bands faster than the edge states; hybridization of the surviving edge states then yields Majorana zero modes. The resulting quasi-1D phase is diagnosed by a topological invariant whose value coincides with a quantized differential conductance of 2e²/h in a normal-superconductor junction.

Significance. If the numerics and invariant calculations hold, the work supplies a concrete, confinement-based route to Majorana zero modes starting from a 2D nodal topological superconductor. The explicit link between the topological invariant and the quantized conductance provides a falsifiable experimental signature. The absence of free parameters in the central derivation and the use of standard boundary-condition comparisons are strengths that would make the result reproducible and testable.

minor comments (3)
  1. [Abstract] The abstract states that the nodal phase is 'unstable' in rectangular geometry but does not specify the minimal system sizes or the precise edge terminations at which the corner states appear; adding a short paragraph or figure caption with these parameters would clarify the claim.
  2. [Introduction / Model section] The phrase 'equal spin pairing superconductivity' is introduced without an immediate equation; inserting the explicit pairing term (e.g., the form of Δ(k)) at first mention would improve readability for readers outside the immediate subfield.
  3. [Abstract] The manuscript refers to 'the topological invariant' without naming it (e.g., Chern number, Pfaffian, or winding number) in the abstract; a single sentence identifying the invariant would strengthen the summary.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript on confinement-induced Majorana modes. Their recommendation for minor revision is noted. No specific major comments appear in the report, so we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper describes a standard model extension of the Haldane Hamiltonian with equal-spin superconducting pairing, followed by numerical diagonalization to identify nodal phases, edge modes, and confinement effects. Topological invariants are computed via standard formulas and shown to coincide with quantized conductance in the usual manner for NS junctions. No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the central claim rests on explicit Hamiltonian numerics rather than renaming or ansatz smuggling. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the extended Haldane model with equal-spin pairing and on the assumption that the chosen boundary conditions capture the dominant physics; no free parameters are named in the abstract.

axioms (1)
  • domain assumption The Haldane model plus equal-spin-pairing superconductivity produces a stable nodal topological phase whose edge states respond to confinement as described.
    This modeling choice is the foundation of the entire phase diagram and confinement argument.

pith-pipeline@v0.9.0 · 5707 in / 1191 out tokens · 33256 ms · 2026-05-23T22:44:51.228264+00:00 · methodology

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Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    & Green, D

    Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum hall effect. Phys. Rev. B 61, 10267–10297 (2000)

    work page 2000

  2. [2]

    Ivanov, D. A. Non-abelian statistics of half-quantum vortices in p-wave super- conductors. Phys. Rev. Lett. 86, 268–271 (2001)

    work page 2001

  3. [3]

    Kitaev, A. Y. Unpaired majorana fermions in quantum wires. Physics-Uspekhi 44, 131 (2001)

    work page 2001

  4. [4]

    T., Lee, P

    Law, K. T., Lee, P. A. & Ng, T. K. Majorana fermion induced resonant andreev reflection. Phys. Rev. Lett. 103, 237001 (2009). 19

    work page 2009

  5. [5]

    R., Medvedyeva, M

    Wimmer, M., Akhmerov, A. R., Medvedyeva, M. V., Tworzyd lo, J. & Beenakker, C. W. J. Majorana bound states without vortices in topological superconductors with electrostatic defects. Phys. Rev. Lett. 105, 046803 (2010)

    work page 2010

  6. [6]

    R., Dahlhaus, J

    Akhmerov, A. R., Dahlhaus, J. P., Hassler, F., Wimmer, M. & Beenakker, C. W. J. Quantized conductance at the majorana phase transition in a disordered superconducting wire. Phys. Rev. Lett. 106, 057001 (2011)

    work page 2011

  7. [7]

    H., Stern, A., Freedman, M

    Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)

    work page 2008

  8. [8]

    New directions in the pursuit of majorana fermions in solid state systems

    Alicea, J. New directions in the pursuit of majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012)

    work page 2012

  9. [9]

    Search for majorana fermions in superconductors

    Beenakker, C. Search for majorana fermions in superconductors. Annual Review of Condensed Matter Physics 4, 113–136 (2013)

    work page 2013

  10. [10]

    Majorana quasiparticles in condensed matter

    Aguado, R. Majorana quasiparticles in condensed matter. Riv. Nuovo Cimento 40, 523 (2017)

    work page 2017

  11. [11]

    & Recher, P

    Schuray, A., Frombach, D., Park, S. & Recher, P. Transport signatures of majo- rana bound states in superconducting hybrid structures. The European Physical Journal Special Topics 229, 593 (2020)

    work page 2020

  12. [12]

    Prada, E. et al. From andreev to majorana bound states in hybrid superconduc- tor–semiconductor nanowires. Nat. Rev. Phys. 2, 575 (2020)

    work page 2020

  13. [13]

    & Stern, A

    Flensberg, K., von Oppen, F. & Stern, A. Engineered platforms for topological superconductivity and majorana zero modes. Nat. Rev. Mat. 6, 944 (2021)

    work page 2021

  14. [14]

    & Kane, C

    Fu, L. & Kane, C. L. Josephson current and noise at a superconductor quantum spin hall insulator fsuperconductor junction. Phys. Rev. B 79, 161408(R) (2009)

    work page 2009

  15. [15]

    M., Sau, J

    Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010)

    work page 2010

  16. [16]

    & von Oppen, F

    Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010)

    work page 2010

  17. [17]

    Mourik, V. et al. Signatures of majorana fermions in hybrid superconductor- semiconductor nanowire devices. Science 336, 1003 (2012)

    work page 2012

  18. [18]

    Deng, M. T. et al. Anomalous zero-bias conductance peak in a nb–insb nanowire–nb hybrid device. Nano Letters 12, 6414–6419 (2012). 20

    work page 2012

  19. [19]

    Das, A. et al. Zero-bias peaks and splitting in an al-inas nanowire topological superconductor as a signature of majorana fermions. Nat. Phys. 8, 887 (2012)

    work page 2012

  20. [20]

    Nichele, F. et al. Scaling of majorana zero-bias conductance peaks. Phys. Rev. Lett. 119, 136803 (2017)

    work page 2017

  21. [21]

    P., Liu, X

    Rokhinson, L. P., Liu, X. & Furdyna, J. K. The fractional a.c. josephson effect in a semiconductor–superconductor nanowire as a signature of majorana particles. Nat. Phys. 1038, 2429 (2012)

    work page 2012

  22. [22]

    Wiedenmann, J. et al. 4π-periodic josephson supercurrent in hgte-based topolog- ical josephson junctions. Nature Communications 7, 10303 (2016)

    work page 2016

  23. [23]

    Li, C. et al. 4π-periodic andreev bound states in a dirac semimetal. Nat. Materials 17, 880 (2018)

    work page 2018

  24. [24]

    Bocquillon, E. et al. Gapless andreev bound states in the quantum spin hall insulator hgte. Nat. Nano. 12, 137 (2016)

    work page 2016

  25. [25]

    Deacon, R. S. et al. Josephson radiation from gapless andreev bound states in hgte-based topological junctions. Phys. Rev. X 7, 021011 (2017)

    work page 2017

  26. [26]

    Laroche, D. et al. Observation of the 4 π-periodic josephson effect in indium arsenide nanowires. Nat. Comm. 10, 245 (2019)

    work page 2019

  27. [27]

    D., Stanescu, T

    Liu, C.-X., Sau, J. D., Stanescu, T. D. & Das Sarma, S. Andreev bound states versus majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks. Phys. Rev. B 96, 075161 (2017)

    work page 2017

  28. [28]

    & Trauzettel, B

    Fleckenstein, C., Dom´ ınguez, F., Traverso Ziani, N. & Trauzettel, B. Decaying spectral oscillations in a majorana wire with finite coherence length. Phys. Rev. B 97, 155425 (2018)

    work page 2018

  29. [29]

    Moore, C., Zeng, C., Stanescu, T. D. & Tewari, S. Quantized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological majorana zero modes. Phys. Rev. B 98, 155314 (2018)

    work page 2018

  30. [30]

    & Nitta, M

    Marra, P. & Nitta, M. Topologically nontrivial andreev bound states. Phys. Rev. B 100, 220502 (2019)

    work page 2019

  31. [31]

    & Klinovaja, J

    Dmytruk, O., Loss, D. & Klinovaja, J. Pinning of andreev bound states to zero energy in two-dimensional superconductor- semiconductor rashba heterostruc- tures. Phys. Rev. B 102, 245431 (2020)

    work page 2020

  32. [32]

    M., Aguado, R

    Cayao, J., San-Jose, P., Black-Schaffer, A. M., Aguado, R. & Prada, E. Majorana splitting from critical currents in josephson junctions. Phys. Rev. B 96, 205425 (2017). 21

    work page 2017

  33. [33]

    A., Cayao, J

    Awoga, O. A., Cayao, J. & Black-Schaffer, A. M. Supercurrent detection of topologically trivial zero-energy states in nanowire junctions. Phys. Rev. Lett. 123, 117001 (2019)

    work page 2019

  34. [34]

    & Oreg, Y

    Haim, A., Berg, E., von Oppen, F. & Oreg, Y. Signatures of majorana zero modes in spin-resolved current correlations. Phys. Rev. Lett. 114, 166406 (2015)

    work page 2015

  35. [35]

    T., Calzona, A., Sassetti, M

    Fleckenstein, C., Ziani, N. T., Calzona, A., Sassetti, M. & Trauzettel, B. Forma- tion and detection of majorana modes in quantum spin hall trenches. Phys. Rev. B 103, 125303 (2021)

    work page 2021

  36. [36]

    Pakizer, J. D. & Matos-Abiague, A. Signatures of topological transitions in the spin susceptibility of josephson junctions. Phys. Rev. B 104, L100506 (2021)

    work page 2021

  37. [37]

    Dominguez, F., Novik, E. G. & Recher, P. Fraunhofer pattern in the presence of majorana zero modes. Phys. Rev. Res. 6, 023304 (2024)

    work page 2024

  38. [38]

    & Recher, P

    Bittermann, L., Dominguez, F. & Recher, P. Photonic cross-noise spectroscopy of majorana bound states. Phys. Rev. B 110, 045429 (2024)

    work page 2024

  39. [39]

    Nadj-Perge, S. et al. Observation of majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014)

    work page 2014

  40. [40]

    Jeon, S. et al. Distinguishing a majorana zero mode using spin-resolved measurements. Science 358, 772–776 (2017)

    work page 2017

  41. [41]

    Suominen, H. J. et al. Zero-energy modes from coalescing andreev states in a two-dimensional semiconductor-superconductor hybrid platform. Phys. Rev. Lett. 119, 176805 (2017)

    work page 2017

  42. [42]

    L., Aguado, R., Guinea, F

    San-Jose, P., Lado, J. L., Aguado, R., Guinea, F. & Fern´ andez-Rossier, J. Majorana zero modes in graphene. Phys. Rev. X 5, 041042 (2015)

    work page 2015

  43. [43]

    & San-Jose, P

    Finocchiaro, F., Guinea, F. & San-Jose, P. Topological π junctions from crossed andreev reflection in the quantum hall regime. Phys. Rev. Lett. 120, 116801 (2018)

    work page 2018

  44. [44]

    & San-Jose, P

    Pe˜ naranda, F., Aguado, R., Prada, E. & San-Jose, P. Majorana bound states in encapsulated bilayer graphene. SciPost Phys. 14, 075 (2023)

    work page 2023

  45. [45]

    Ribeiro, R. C. B., Correa, J. H., Ricco, L. S., Seridonio, A. C. & Figueira, M. S. Spin-polarized majorana zero modes in double zigzag honeycomb nanoribbons. Phys. Rev. B 105, 205115 (2022)

    work page 2022

  46. [46]

    Bento Ribeiro, R. C. et al. Spin-polarized majorana zero modes in proximitized superconducting penta-silicene nanoribbons. Scientific Reports 13, 17965 (2023). 22

    work page 2023

  47. [47]

    Haldane, F. D. M. Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988)

    work page 2015

  48. [48]

    Chen, Z., Li, X. & Ng, T. K. Exactly solvable bcs-hubbard model in arbitrary dimensions. Phys. Rev. Lett. 120, 046401 (2018)

    work page 2018

  49. [49]

    Exact solutions for two-dimensional topological superconductors: Hub- bard interaction induced spontaneous symmetry breaking

    Ezawa, M. Exact solutions for two-dimensional topological superconductors: Hub- bard interaction induced spontaneous symmetry breaking. Phys. Rev. B 97, 241113 (2018)

    work page 2018

  50. [50]

    & Zhang, F.-C

    Miao, J.-J., Xu, D.-H., Zhang, L. & Zhang, F.-C. Exact solution to the haldane- bcs-hubbard model along the symmetric lines: Interaction-induced topological phase transition. Phys. Rev. B 99, 245154 (2019)

    work page 2019

  51. [51]

    Schnyder, A. P. & Brydon, P. M. R. Topological surface states in nodal superconductors. Journal of Physics: Condensed Matter 27, 243201 (2015)

    work page 2015

  52. [52]

    S., Wu, X

    Zhang, R.-X., Cole, W. S., Wu, X. & Das Sarma, S. Higher-order topology and nodal topological superconductivity in fe(se,te) heterostructures. Phys. Rev. Lett. 123, 167001 (2019)

    work page 2019

  53. [53]

    & Wang, Y

    Wu, Z. & Wang, Y. Nodal higher-order topological superconductivity from a ⌋4-symmetric dirac semimetal. Phys. Rev. B 106, 214510 (2022)

    work page 2022

  54. [54]

    & Traverso Ziani, N

    Traverso, S., Sassetti, M. & Traverso Ziani, N. Emerging topological bound states in haldane model zigzag nanoribbons. npj Quantum Materials 9, 9 (2024)

    work page 2024

  55. [55]

    R., Nilsson, J

    Akhmerov, A. R., Nilsson, J. & Beenakker, C. W. J. Electrically detected inter- ferometry of majorana fermions in a topological insulator. Phys. Rev. Lett. 102, 216404 (2009)

    work page 2009

  56. [56]

    & Saint-James, D

    Caroli, C., Combescot, R., Nozieres, P. & Saint-James, D. Direct calculation of the tunneling current. Journal of Physics C: Solid State Physics 4, 916 (1971)

    work page 1971

  57. [57]

    Reis, F. et al. Bismuthene on a sic substrate: A candidate for a high-temperature quantum spin hall material. Science 357, 287–290 (2017)

    work page 2017

  58. [58]

    Bampoulis, P. et al. Quantum spin hall states and topological phase transition in germanene. Phys. Rev. Lett. 130, 196401 (2023)

    work page 2023

  59. [59]

    Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling

    St¨ uhler, R.et al. Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling. Nat. Comm. 13, 3480 (2022)

    work page 2022

  60. [60]

    Klaassen, D. J. et al. Realization of a one-dimensional topological insulator in ultrathin germanene nanoribbons (2024). 2411.18156. 23

    work page arXiv 2024

  61. [61]

    C., Mart´ ın-Rodero, A

    Cuevas, J. C., Mart´ ın-Rodero, A. & Yeyati, A. L. Hamiltonian approach to the transport properties of superconducting quantum point contacts. Phys. Rev. B 54, 7366–7379 (1996)

    work page 1996

  62. [62]

    Sancho, M. P. L., Sancho, J. M. L., Sancho, J. M. L. & Rubio, J. Highly convergent schemes for the calculation of bulk and surface green functions.Journal of Physics F: Metal Physics 15, 851 (1985). 24

    work page 1985