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Giant and robust Josephson diode effect in multiband topological nanowires

T0 review · 2 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper predicts that in multiband Majorana nanowires, the coexistence of Majorana and Andreev bound states yields a large, stable Josephson diode effect far into the topological phase.

desk verdict Real new mechanism in multiband nanowire JDE, but the 'robust plateau' is filling-dependent per their own S-4; worthy of peer review with qualifications. read the letter →

arxiv 2510.05772 v3 pith:4KRHGS2S submitted 2025-10-07 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.supr-con

Bao-Zong Wang , Zi-Kai Li , Zhong-Da Li , Xiong-Jun Liu This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.supr-con
keywords JosephsondiodeeffectMajoranaboundstatesmultibandnanowireAndreev4π-periodicspin-paritybandexchangetopologicalsuperconductorsupercurrent
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 a quasi-one-dimensional superconducting nanowire with multiple transverse subbands can act as a highly efficient supercurrent diode—letting current flow more easily in one direction than the other—even deep inside the topological superconducting phase. The key is that two kinds of bound states coexist: Majorana bound states, which carry a 4π-periodic supercurrent, and ordinary Andreev bound states, which carry the usual 2π-periodic current. Their interplay produces a large diode asymmetry across most of the topological regime. The paper further identifies a spin-parity band exchange mechanism: as the longitudinal magnetic field is raised, subbands with opposite spin parity swap energy order, and after the swap the Fermi-momentum shifts of the different subbands balance, creating a robust plateau of high diode efficiency as a function of field. If correct, this makes multiband engineering a practical route to optimize the Josephson diode effect and offers a new signature of the topological phase in realistic nanowires.

What carries the argument

The central object is the multiband tight-binding Hamiltonian of a Rashba nanowire with Ny transverse orbital channels, and specifically the spin-parity Ps of each subband, defined as the sign of the eigenvalue of hxσx + (hy + 2αx sin kx ax)σy at ky = 0. The spin-parity labels which way the Fermi points of a subband shift under the inversion-breaking field hy: Ps = +1 bands shift left, Ps = −1 bands shift right. The diode efficiency is computed from the supercurrent I(ϕ) = (2e/ℏ)∂E/∂ϕ, and the relevant quantity is the ratio ξ = 2|(ΔkF)+1| / [(ΔkF)−1 + (ΔkF)−2], which quantifies the relative total shift of left- and right-moving Fermi points. The paper shows that after spin-parity subband exc

What would settle it

A concrete way to test the claim is to measure the diode efficiency η as a function of the longitudinal magnetic field hx in a multiband nanowire Josephson junction with a known number of occupied subbands: if η does not develop a saturated high-value plateau after the predicted spin-parity exchange field (around |μ1−μ2|/2), the mechanism is wrong. Alternatively, a numerical calculation with a 3-orbit model and only 3 occupied subbands already shows the plateau declining and dropping, so a single-band-like efficiency peak without a plateau in any filling regime would falsify the claim of robus

Watch

Extended reading notes

Core claim

The paper's central claim is that in the multiband regime of a Majorana nanowire Josephson junction, the total supercurrent contains both a fractional 4π-periodic component from Majorana bound states (I4π) and a conventional 2π-periodic component from Andreev bound states (I2π), and that the competition between these two components—rather than the near-critical-field resonance needed in single-band wires—sustains a strong diode effect across the entire topological phase. The novel mechanism is the spin-parity band exchange: upon increasing the Zeeman field hx, subbands with spin parity Ps = +1 and Ps = −1 shift in opposite directions in energy and exchange their ordering. After the exchange,

Load-bearing premise

The load-bearing premise is that the saturation of the Fermi-momentum shift ratio ξ is what balances the 4π- and 2π-periodic supercurrents, an assumption not quantitatively derived, and the high-efficiency plateau only appears for a specific subband filling (2Ny−1 spinful subbands).

Editorial extensions

If this is right

  • In multiband nanowires, the high diode efficiency is not confined to the phase-transition boundary but extends deep into the topological phase.
  • The spin-parity band exchange gives a robust plateau of high diode efficiency as a function of magnetic field, for appropriate subband filling (2Ny − 1 occupied spinful subbands).
  • Subband engineering—controlling the number and filling of transverse modes—is a practical tool to optimize the Josephson diode effect.
  • The diode efficiency plateau provides a new signature of the topological phase and Majorana bound states in realistic nanowires.
  • The effect is expected to survive in a realistic parameter window (e.g., for common semiconductor-superconductor hybrids) with magnetic fields below about 1 Tesla.

Reading between the lines

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

  • As an extension beyond the paper, the plateau's dependence on the specific filling 2Ny−1 implies that for a fixed device, increasing the magnetic field beyond the exchange point will eventually degrade the diode effect—the plateau is bounded in field, not infinite.
  • If the spin-parity exchange mechanism is correct, the diode efficiency could be used experimentally as a probe for subband crossings, not just for topological phase identification.
  • A natural testable extension is to measure the differential conductance across the junction as a function of field: the appearance of a dominant 4π-periodic supercurrent component after the exchange would confirm the mechanism.
  • The mechanism might carry over to other quasi-one-dimensional platforms with multiple subbands and proximity-induced pairing, where the same balance of fractional and conventional currents could be engineered.
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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 / 6 minor

Summary. This manuscript predicts a large Josephson diode effect (JDE) in multiband topological Majorana nanowires. The authors model a quasi-1D Rashba nanowire (Ny=1,2,3 transverse orbitals) with Zeeman fields hx (time-reversal breaking) and hy (inversion breaking), compute the current-phase relation via the recursive Green's function method, and evaluate the diode efficiency eta. Their central findings are: (i) with three occupied subbands, eta remains large deep in the topological phase, in contrast to the single-band case where eta peaks only near the topological transition; and (ii) an increase of hx drives a 'spin-parity band exchange' among subbands of opposite spin parity, after which eta forms a high plateau (demonstrated for Ny=2 with 3 subbands and Ny=3 with 5 subbands). The authors attribute the plateau to saturation of a Fermi-shift ratio xi derived from the normal-state dispersion, Eq. (4). A supplementary section (S-4) shows, however, that for Ny=3 with 3 occupied subbands the plateau declines and then drops, and the authors concede that the high-efficiency platform will not extend infinitely.

Significance. If the results hold, this is a useful and timely contribution: real Majorana nanowire experiments operate in a multiband regime, and the paper provides realistic parameter mapping (S-1), a falsifiable prediction (high eta in the deep topological phase for the 2Ny-1 filling), and a concrete design rule (subband engineering; choosing the Fermi level to populate 2Ny-1 subbands). Credit is due for the following: the central eta curves are computed directly from the microscopic tight-binding model with no parameter fitted to the target result; the parameters are anchored to InAs/InSb-Al/Nb experiments; robustness is tested against tx, NN, alpha_x, and Ly (Fig. 2, S-3); and the paper itself discloses the counterexample in S-4. The two substantive weaknesses are (1) the 'robust/generic' language is overbroad relative to S-4 and must be qualified, and (2) the spin-parity exchange 'mechanism' is only a qualitative correlation with the numerics. Neither undermines the numerical prediction itself, but both affect the paper's central claims as currently written.

major comments (2)
  1. [Abstract; 'Spin-parity band exchange mechanism'; S-4] The central 'robust/generic' plateau claim is at odds with the authors' own S-4. The main text states: 'The predicted high diode efficiency plateau is not limited to Ny=2, but a generic result applicable to the multiband regime' (paragraph after Fig. 3), and the abstract advertises a 'robust high efficiency plateau.' However, S-4, Fig. S3(b) shows that for Ny=3 with three occupied subbands—a filling the authors call 'more realistic and reasonable'—eta declines after the first spin-parity exchange and drops after the second, ending in a 'relatively low platform.' S-4 concludes that this 'provides a counter-argument to the conjecture we made in the text' and that 'the high-efficiency platform will not extend infinitely.' Since three occupied subbands in a 3-orbit model is squarely within the stated scope (Ny<=3), the result is filling-selective (2Ny-1 subbands) and field-window-limited, no
  2. [Eqs. (2)-(4); Fig. 4] The causal role of the spin-parity band exchange is asserted, not established. The text claims that saturation of the normal-state Fermi-shift ratio xi 'signals the balance between the two types of Josephson currents and the high diode efficiency plateau,' and that the blue bands 'mainly contribute I2pi' while the red band 'predominantly contributes I4pi.' But no relation is derived between xi (or Eq. (4)'s Delta kF) and the current amplitudes I2pi, I4pi, or the efficiency eta computed from Eqs. (2)-(3). The mechanism is thus a correlation between two features in parameter space (xi saturation and eta plateau) rather than a demonstrated causal chain. The authors should either provide a quantitative link (e.g., a short-junction analysis expressing the ABS/MBS current amplitudes in terms of Fermi data) or present the spin-parity exchange as a phenomenological marker, not as an established
minor comments (6)
  1. [Fig. 1(c) caption] Typo: 'Majonara' should be 'Majorana'.
  2. [Eq. (1); S-2] The alpha_y term couples effective orbital chains (the i_y index), which is not a real-space y hopping; this notation should be clarified. Also, the spin-parity Ps is defined from h_x sigma_x + (h_y + 2 alpha_x sin k_x a_x) sigma_y, neglecting the k-independent interband SOI alpha_y; the statement that 'the interband SOI vanishes' at k_y=0 needs justification, or Ps should be labeled approximate.
  3. [S-4] S-4 refers to 'the conjecture we made in the text' about populating 2Ny-1 spinful subbands, but this rule is never stated explicitly in the main text; it should be stated and then qualified by the S-4 counterexample in the main text.
  4. [Eq. (2) vs Eq. (3)] Eq. (2) is presented as the defining current formula, but the actual computation uses the Green's function expression, Eq. (3); the relation between the two (bound-state sum vs full spectrum) should be clarified.
  5. [Fig. 2(a)] The specific alpha_x values used in the comparison are not given in the caption; please list them.
  6. [Conclusion] The claim that the multiband JDE 'offer[s] a new probe for identifying topological phase' is not supported by discussion of how eta would distinguish topological from trivial multiband regimes; Fig. 2(d) region II shows a finite, though weak, eta in a trivial multiband region, so the criterion needs to be made more specific.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: central η curves are direct numerical outputs; the only self-citations are minor references to a standard symmetry condition, and S-4 is a robustness caveat, not a circular step.

full rationale

The central numerical prediction is self-contained: the diode efficiency η is obtained by evaluating the explicit tight-binding Hamiltonian (1) with the recursive Green's function current formula (3), with no parameter fitted to the target η curves. The same calculation yields all η(hx) curves, including the single-band comparison and the multiband plateau. The spin-parity band-exchange mechanism is presented as an interpretive explanation: Eq. (4) derives Fermi-point shifts from the normal-state dispersion, and the saturation of the ratio ξ is correlated with, not used as input to, the computed η. There is no equation in which a predicted quantity is defined through the target, and no fitted parameter is renamed as a prediction. The only self-citations are refs [49,50] for the standard condition that simultaneous breaking of inversion and time-reversal symmetry makes I_c^+ ≠ I_c^-; this is a well-established symmetry criterion and the paper's own Hamiltonian explicitly includes hy, so the central claim does not rest on unverified self-cited work. One internal limitation should be noted for correctness rather than circularity: Supplementary S-4 shows that for the Ny=3 model with 3 occupied subbands the plateau tilts downward and then drops, concluding that 'the high-efficiency platform will not extend infinitely'; this qualifies the robustness claim but is an honest caveat, not a circular step. Therefore no circularity is found.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on (i) the multiband tight-binding model with parameters derived from InAs/InSb experiments, (ii) the short-junction recursive Green's function computation, and (iii) the interpretive assumption that normal-state Fermi-point shifts control the 4π/2π current balance. No new physical entities are introduced.

free parameters (4)
  • μ0 (background chemical potential) = set to give 3 (or 5) occupied spinful subbands
    Chosen to place the Fermi level halfway between transverse modes; the efficiency plateau depends on this filling.
  • h_y (in-plane magnetic field) = 0.8Δ
    Breaks inversion symmetry and controls Fermi-point shifts; the diode asymmetry depends on it.
  • α_y (effective transverse SOI) = 1.3Δ
    Estimated from the infinite-well model in S-1; affects the band structure and exchange transitions.
  • V = E(n_y=1) (subband spacing) = 2.5Δ
    Sets the exchange threshold h_x ≈ |μ1−μ2|/2; derived from L_y = 130 nm and m* = 0.04 m_e.
assumptions (6)
  • domain assumption N_z=1: only the lowest transverse mode in z-direction is occupied; L_z ≪ L_y ≪ L_x.
    Standard approximation for nanowires, from refs [11,12]; invoked in 'The multiband model'.
  • domain assumption Transverse modes are described by an infinite square well with energies E(i_y)=i_y^2 π²ℏ²/(2m* L_y²).
    Used to set μ_i and α_y; derived in Supplementary S-1.
  • standard math Josephson current is I = (2e/ℏ) ∂E/∂φ with E = −1/2 Σ_{E_i≥0} E_i.
    Beenakker formula [92]; used to define the diode efficiency η.
  • ad hoc to paper The normal-state Fermi-point shifts (Eq. (4)) determine the balance between I4π and I2π.
    The paper asserts ξ saturation corresponds to η plateau but does not derive this from the superconducting current.
  • domain assumption For the 2-orbit model, after h_x exceeds h_c, no further topological phase transition occurs (no gap closure).
    Stated in S-6; used to claim the deep topological regime is maintained.
  • domain assumption Short-junction limit N_N=2 captures the physics; results checked for N_N up to ~10.
    Used in the numerical setup; S-3 shows some robustness.

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Cite this review

Pith. "Pith review of Giant and robust Josephson diode effect in multiband topological nanowires." pith.science (2026). https://pith.science/paper/4KRHGS2S

@misc{pith2026251005772,
  author       = {Pith},
  title        = {Pith review of: Giant and robust Josephson diode effect in multiband topological nanowires},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KRHGS2S}},
  note         = {Machine review of arXiv:2510.05772}
}
abstract

We theoretically predict the giant and robust Josephson diode effect in quasi-one-dimensional topological Majorana nanowires in the regime with multiple subbands, which is expected to be relevant for the real experiment. In the multiband regime, the Majorana bound states and conventional Andreev bound states can naturally coexist, and respectively contribute to the fractional and conventional parts in the Josephson effect, with the former/latter having 4$\pi$/2$\pi$-periodicity. We show that the interplay between the two types of bound modes can produce a robust and giant diode effect in the deep topological phase regime. Notably, we unveil a novel spin parity exchange mechanism, occurring only in the multiband regime, which leads to a robust high efficiency plateau of the giant diode effect. This effect is a nontrivial consequence of the balanced Fermi moment shifts of the multiple subbands in tuning the external magnetic field. Our finding highlights the subband engineering as a powerful tool to optimize the Josephson diode effect realistically and provides a new feasible signature to identify topological phase regime in superconducting nanowires.

Figures

Figures reproduced from arXiv: 2510.05772 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic plot of the quasi-1D Josephson junc [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. High diode efficiency plateau by spin-parity sub [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Underlying mechanism of high diode efficiency [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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Reference graph

Works this paper leans on

90 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Alicea, Reports on progress in physics75, 076501 (2012)

    J. Alicea, Reports on progress in physics75, 076501 (2012)

    2012

  2. [2]

    Kitaev, Annals of Physics303, 2 (2003)

    A. Kitaev, Annals of Physics303, 2 (2003)

    2003

  3. [3]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys.80, 1083 (2008)

    2008

  4. [4]

    D. A. Ivanov, Phys. Rev. Lett.86, 268 (2001)

    2001

  5. [5]

    He, J.-S

    Y.-P. He, J.-S. Hong, and X.-J. Liu, Acta Physica Sinica 69(2020)

    2020

  6. [6]

    C. Chan, L. Zhang, T. F. J. Poon, Y.-P. He, Y.-Q. Wang, and X.-J. Liu, Phys. Rev. Lett.119, 047001 (2017)

    2017

  7. [7]

    R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett.105, 077001 (2010)

    2010

  8. [8]

    Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010)

    2010

Show all 90 references
  1. [9]

    J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, and S. Das Sarma, Phys. Rev. B82, 214509 (2010)

    2010

  2. [10]

    Laubscher and J

    K. Laubscher and J. Klinovaja, Journal of Applied Physics130, 081101 (2021)

    2021

  3. [11]

    T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. B84, 144522 (2011)

    2011

  4. [12]

    R. M. Lutchyn, T. D. Stanescu, and S. Das Sarma, Phys. Rev. Lett.106, 127001 (2011)

    2011

  5. [13]

    G.-J. Qiao, X. Yue, and C. P. Sun, Phys. Rev. Lett.133, 266605 (2024)

    2024

  6. [14]

    Prada, P

    E. Prada, P. San-Jose, M. W. de Moor, A. Geresdi, E. J. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L. P. Kouwenhoven, Nature Reviews Physics2, 575 (2020)

    2020

  7. [15]

    Sasaki, M

    S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, and Y. Ando, Phys. Rev. Lett.107, 217001 (2011)

    2011

  8. [16]

    M. Deng, C. Yu, G. Huang, M. Larsson, P. Caroff, and H. Xu, Nano letters12, 6414 (2012)

    2012

  9. [17]

    A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Nature Physics8, 887 (2012)

    2012

  10. [18]

    Mourik, K

    V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science336, 1003 (2012)

    2012

  11. [19]

    H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Phys. Rev. B87, 241401 (2013)

    2013

  12. [20]

    H.-J. Kwon, V. M. Yakovenko, and K. Sengupta, Low Temperature Physics30, 613 (2004)

    2004

  13. [21]

    Fu and C

    L. Fu and C. L. Kane, Phys. Rev. B79, 161408 (2009)

    2009

  14. [22]

    H.-J. Kwon, K. Sengupta, and V. M. Yakovenko, The European Physical Journal B-Condensed Matter and Complex Systems37, 349 (2004)

    2004

  15. [23]

    Laroche, D

    D. Laroche, D. Bouman, D. J. van Woerkom, A. Prout- ski, C. Murthy, D. I. Pikulin, C. Nayak, R. J. van Gulik, J. Nygård, P. Krogstrup,et al., Nature communications 10, 245 (2019)

    2019

  16. [24]

    Wiedenmann, E

    J. Wiedenmann, E. Bocquillon, R. S. Deacon, S. Hartinger, O. Herrmann, T. M. Klapwijk, L. Maier, C. Ames, C. Brüne, C. Gould,et al., Nature communi- cations7, 10303 (2016)

    2016

  17. [25]

    D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch, R. Zhong, G. Gu, L. Fu, H. Ding, and H.-J. Gao, Science362, 333 (2018)

    2018

  18. [26]

    S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, and C. Marcus, Nature531, 206 (2016)

    2016

  19. [27]

    M. Deng, S. Vaitiek˙ enas, E. B. Hansen, J. Danon, M. Lei- jnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, Science354, 1557 (2016)

    2016

  20. [28]

    M. A. Quantum, M. Aghaee, A. Alcaraz Ramirez, Z. Alam, R. Ali, M. Andrzejczuk, A. Antipov, M. Astafev, A. Barzegar, B. Bauer,et al., Nature638, 651 (2025)

    2025

  21. [29]

    H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp,et al., Nature569, 93 (2019)

    2019

  22. [30]

    F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Nature 584, 373 (2020)

    2020

  23. [31]

    Miyasaka, R

    Y. Miyasaka, R. Kawarazaki, H. Narita, F. Ando, Y. Ikeda, R. Hisatomi, A. Daido, Y. Shiota, T. Moriyama, Y. Yanase,et al., Applied Physics Express 14, 073003 (2021)

    2021

  24. [32]

    N. F. Q. Yuan and L. Fu, Proceedings of the National Academy of Sciences118, e2019063118 (2021)

    2021

  25. [33]

    J. J. He, Y. Tanaka, and N. Nagaosa, New Journal of Physics24, 053014 (2022)

    2022

  26. [34]

    Narita, J

    H. Narita, J. Ishizuka, R. Kawarazaki, D. Kan, Y. Sh- iota, T. Moriyama, Y. Shimakawa, A. V. Ognev, A. S. Samardak, Y. Yanase,et al., Nature Nanotechnology17, 823 (2022)

    2022

  27. [36]

    Daido, Y

    A. Daido, Y. Ikeda, and Y. Yanase, Phys. Rev. Lett. 128, 037001 (2022)

    2022

  28. [37]

    Zinkl, K

    B. Zinkl, K. Hamamoto, and M. Sigrist, Phys. Rev. Res. 4, 033167 (2022)

    2022

  29. [38]

    Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ilić, O. Glezakou- Elbert, A. Varambally, F. S. Bergeret, A. Kamra, L. Fu, P. A. Lee, and J. S. Moodera, Phys. Rev. Lett.131, 027001 (2023). 6

    2023

  30. [39]

    L. D. Anh, K. Ishihara, T. Hotta, K. Inagaki, H. Maki, T. Saeki, M. Kobayashi, and M. Tanaka, Nature Com- munications15, 8014 (2024)

    2024

  31. [41]

    J. Hu, C. Wu, and X. Dai, Phys. Rev. Lett.99, 067004 (2007)

    2007

  32. [43]

    Symmetry con- straints on direct-current josephson diodes,

    D. Wang, Q.-H. Wang, and C. Wu, “Symmetry con- straints on direct-current josephson diodes,” (2022), arXiv:2209.12646 [cond-mat.supr-con]

    2022 arXiv

  33. [44]

    M.Davydova, S.Prembabu, andL.Fu,ScienceAdvances 8, eabo0309 (2022)

    2022

  34. [48]

    Zhang, Y

    Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, Phys. Rev. X12, 041013 (2022)

    2022

  35. [49]

    Liu and A

    X.-J. Liu and A. M. Lobos, Phys. Rev. B87, 060504 (2013)

    2013

  36. [50]

    Liu, Phys

    X.-J. Liu, Phys. Rev. Lett.109, 106404 (2012)

    2012

  37. [51]

    Andreev bound states and their signatures,

    J. Sauls, “Andreev bound states and their signatures,” (2018)

    2018

  38. [52]

    Mizushima and K

    T. Mizushima and K. Machida, Philosophical Transac- tions of the Royal Society A: Mathematical, Physical and Engineering Sciences376, 20150355 (2018)

    2018

  39. [53]

    M. Hays, G. de Lange, K. Serniak, D. J. van Woerkom, D. Bouman, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, Phys. Rev. Lett.121, 047001 (2018)

    2018

  40. [54]

    L. Tosi, C. Metzger, M. F. Goffman, C. Urbina, H. Poth- ier, S. Park, A. L. Yeyati, J. Nygård, and P. Krogstrup, Phys. Rev. X9, 011010 (2019)

    2019

  41. [55]

    Nichele, E

    F. Nichele, E. Portolés, A. Fornieri, A. M. Whiticar, A. C. C. Drachmann, S. Gronin, T. Wang, G. C. Gard- ner, C. Thomas, A. T. Hatke, M. J. Manfra, and C. M. Marcus, Phys. Rev. Lett.124, 226801 (2020)

    2020

  42. [56]

    Ciaccia, R

    C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Linde- mann, M. J. Manfra, C. Schrade, and C. Schönenberger, Phys. Rev. Res.5, 033131 (2023)

    2023

  43. [57]

    Banerjee, M

    A. Banerjee, M. Geier, M. A. Rahman, C. Thomas, T. Wang, M. J. Manfra, K. Flensberg, and C. M. Mar- cus, Physical Review Letters131, 196301 (2023)

    2023

  44. [58]

    Matsuo, T

    S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Linde- mann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Nature Physics19, 1636 (2023)

    2023

  45. [59]

    Trahms, L

    M. Trahms, L. Melischek, J. F. Steiner, B. Mahendru, I. Tamir, N. Bogdanoff, O. Peters, G. Reecht, C. B. Winkelmann, F. von Oppen,et al., Nature615, 628 (2023)

    2023

  46. [60]

    B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju,et al., Nature physics18, 1228 (2022)

    2022

  47. [61]

    Baumgartner, L

    C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan,et al., Nature nanotech- nology17, 39 (2022)

    2022

  48. [62]

    Díez-Mérida, A

    J. Díez-Mérida, A. Díez-Carlón, S. Yang, Y.-M. Xie, X.- J. Gao, J. Senior, K. Watanabe, T. Taniguchi, X. Lu, A. P. Higginbotham,et al., Nature Communications14, 2396 (2023)

    2023

  49. [63]

    H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, Nature604, 653 (2022)

    2022

  50. [64]

    Cheng and Q.-F

    Q. Cheng and Q.-F. Sun, Phys. Rev. B107, 184511 (2023)

    2023

  51. [65]

    Davydova, S

    M. Davydova, S. Prembabu, and L. Fu, Science Advances8, eabo0309 (2022), https://www.science.org/doi/pdf/10.1126/sciadv.abo0309

    2022 doi

  52. [66]

    J. Hu, C. Wu, and X. Dai, Physical review letters99, 067004 (2007)

    2007

  53. [67]

    Misaki and N

    K. Misaki and N. Nagaosa, Phys. Rev. B103, 245302 (2021)

    2021

  54. [68]

    B. Lu, S. Ikegaya, P. Burset, Y. Tanaka, and N. Nagaosa, Phys. Rev. Lett.131, 096001 (2023)

    2023

  55. [69]

    R. S. Souto, M. Leijnse, and C. Schrade, Phys. Rev. Lett.129, 267702 (2022)

    2022

  56. [70]

    Halterman, M

    K. Halterman, M. Alidoust, R. Smith, and S. Starr, Phys. Rev. B105, 104508 (2022)

    2022

  57. [71]

    T. H. Kokkeler, A. A. Golubov, and F. S. Bergeret, Phys. Rev. B106, 214504 (2022)

    2022

  58. [72]

    Y. V. Fominov and D. S. Mikhailov, Phys. Rev. B106, 134514 (2022)

    2022

  59. [73]

    A. A. Kopasov, A. G. Kutlin, and A. S. Mel’nikov, Phys. Rev. B103, 144520 (2021)

    2021

  60. [74]

    Gupta, G

    M. Gupta, G. V. Graziano, M. Pendharkar, J. T. Dong, C. P. Dempsey, C. Palmstrøm, and V. S. Pribiag, Nature communications14, 3078 (2023)

    2023

  61. [75]

    Y.-F. Sun, Y. Mao, and Q.-F. Sun, Phys. Rev. B108, 214519 (2023)

    2023

  62. [76]

    Jiang, M

    J. Jiang, M. Milošević, Y.-L. Wang, Z.-L. Xiao, F. Peeters, and Q.-H. Chen, Phys. Rev. Appl.18, 034064 (2022)

    2022

  63. [77]

    Tanaka, B

    Y. Tanaka, B. Lu, and N. Nagaosa, Phys. Rev. B106, 214524 (2022)

    2022

  64. [78]

    Maiani, K

    A. Maiani, K. Flensberg, M. Leijnse, C. Schrade, S. Vaitiek˙ enas, and R. Seoane Souto, Phys. Rev. B107, 245415 (2023)

    2023

  65. [79]

    Hu, Z.-T

    J.-X. Hu, Z.-T. Sun, Y.-M. Xie, and K. T. Law, Phys. Rev. Lett.130, 266003 (2023)

    2023

  66. [80]

    J. F. Steiner, L. Melischek, M. Trahms, K. J. Franke, and F. von Oppen, Phys. Rev. Lett.130, 177002 (2023)

    2023

  67. [81]

    Pillet, S

    J.-D. Pillet, S. Annabi, A. Peugeot, H. Riechert, E. Ar- righi, J. Griesmar, and L. Bretheau, Phys. Rev. Res.5, 033199 (2023)

    2023

  68. [82]

    R. Hess, H. F. Legg, D. Loss, and J. Klinovaja, Phys. Rev. B108, 174516 (2023)

    2023

  69. [83]

    Costa, O

    A. Costa, O. Kanehira, H. Matsueda, and J. Fabian, Phys. Rev. B111, L140506 (2025)

    2025

  70. [84]

    Design of supercurrent diode by vortex phase texture,

    Y. Fukaya, M. T. Mercaldo, D. Margineda, A. Crippa, E. Strambini, F. Giazotto, C. Ortix, and M. Cuoco, “Design of supercurrent diode by vortex phase texture,” (2024), arXiv:2403.04421

    2024

  71. [85]

    Debnath and P

    D. Debnath and P. Dutta, Journal of Physics: Condensed Matter37, 175301 (2025)

    2025

  72. [86]

    Z. Liu, L. Huang, and J. Wang, Phys. Rev. B110, 014519 (2024)

    2024

  73. [87]

    H. F. Legg, K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. B108, 214520 (2023)

    2023

  74. [88]

    Mondal, P.-H

    S. Mondal, P.-H. Fu, and J. Cayao, arXiv preprint arXiv:2503.08318 (2025). 7

    2025

  75. [89]

    J. Liu, A. C. Potter, K. Law, and P. A. Lee, Physical review letters109, 267002 (2012)

    2012

  76. [90]

    Further details are provided in the Supplementary Mate- rial,

  77. [91]

    N. F. Q. Yuan and L. Fu, Proceedings of the National Academy of Sciences119, e2119548119 (2022)

    2022

  78. [92]

    C. W. J. Beenakker, Phys. Rev. Lett.67, 3836 (1991)

    1991

  79. [93]

    J. Song, H. Liu, J. Liu, Y.-X. Li, R. Joynt, Q.-f. Sun, and X. C. Xie, Phys. Rev. B93, 195302 (2016)

    2016

  80. [94]

    Y. Mao, Q. Yan, Y.-C. Zhuang, and Q.-F. Sun, Phys. Rev. Lett.132, 216001 (2024)

    2024

  81. [95]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, Journal of the Physical Society of Japan74, 1674 (2005), https://doi.org/10.1143/JPSJ.74.1674. 8 Supplementary Material: Giant and robust Josephson diode effect in multiband topological nanowires S-1. FROM QUASI-1D NANOWIRE TO MUL TIBAND H...

    2005 doi

  82. [96]

    =i 2 yV. Meanwhile, the SOI in they-direction can be incorporated by the second quantization: 2αxax ⟨ψiy |∂y|ψi′y ⟩= 2α xax Z Ly 0 2 Ly sin iyπy Ly ∂y sin i′ yπy Ly dy= 2αxax Ly Aiyi′y =α y Aiyi′y A21 .(S4) One can calculateαy = 0.2meV, andA iyiy = 0,A 21 = 8/3,A 31 = 0,A 32 =...

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