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arxiv: 2604.11692 · v1 · submitted 2026-04-13 · ❄️ cond-mat.supr-con

Statistical Signatures of Majorana Zero Modes in Disordered Topological Superconductor Antidot Vortices

Zhibo Ren , Jukka I. V\"ayrynen This is my paper

Pith reviewed 2026-05-10 16:26 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords Majorana zero modesdisorder effectstopological superconductorsrandom matrix theoryscanning tunneling microscopyantidot vorticesCaroli-de Gennes-Matricon statesprobability density variance
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The pith

Disorder doubles the variance of probability density for Majorana zero modes relative to other vortex states.

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

The paper develops a general theory for disorder effects on an antidot-pinned vortex in a three-dimensional topological insulator-superconductor platform, where a Majorana zero mode coexists with many Caroli-de Gennes-Matricon states. Using random matrix theory and numerical simulations, it derives the statistical distributions of the states' probability densities. The central result shows that the Majorana zero mode exhibits twice the variance in its probability density compared to the CdGM states. This difference stems from the real character of the Majorana wave function versus the complex character of the others. The distinction offers a measurable signature in scanning tunneling microscopy that can identify the Majorana mode even in the presence of disorder, going beyond the zero-bias conductance peak.

Core claim

In a disordered antidot-pinned vortex, the variance of the Majorana zero mode probability density is twice that of the CdGM states. This follows from the real wave function of the Majorana mode as opposed to the complex wave functions of the CdGM states. The result is obtained through an analytical random matrix theory treatment of the disordered system together with numerical simulations, and it can be detected via scanning tunneling microscopy as a statistical signature beyond the zero-bias peak.

What carries the argument

Random matrix theory applied to the disordered vortex states, which isolates the effect of real versus complex wave functions on the variance of local probability densities.

If this is right

  • The statistical variance difference becomes visible in local intensity maps obtained by scanning tunneling microscopy.
  • This provides an experimental signature to distinguish Majorana zero modes from coexisting CdGM states in disordered samples.
  • Disorder broadens the intensity fluctuations of real-wave-function states more than complex-wave-function states.
  • The real-complex distinction remains useful for identification even when disorder mixes the states.

Where Pith is reading between the lines

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

  • The same variance ratio might serve as a diagnostic in other platforms where protected zero modes coexist with ordinary states.
  • Intensity fluctuation mapping could be tested in related systems with real versus complex eigenstates under controlled disorder.
  • The result implies that statistical analysis of STM data offers a route to Majorana identification when energy resolution alone is insufficient.

Load-bearing premise

Random matrix theory accurately describes the statistical distributions in the disordered antidot vortex and the real-versus-complex wave function distinction survives disorder without further corrections.

What would settle it

Scanning tunneling microscopy measurements in a disordered antidot vortex that find the zero-energy state's intensity variance is not twice the variance of nearby states would show the claimed distinction does not hold.

Figures

Figures reproduced from arXiv: 2604.11692 by Jukka I. V\"ayrynen, Zhibo Ren.

Figure 1
Figure 1. Figure 1: STM detection of antidot-bound states. (a) [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical verification of RMT variance scaling. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results for the scaled IPR of the density. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

An antidot-pinned vortex in a three-dimensional topological insulator-superconductor platform hosts a Majorana zero mode (MZM). However, numerous Caroli-de Gennes-Matricon (CdGM) states coexist with it. We develop a general theory to study the effects of disorder on the system, emphasizing the difference between Majorana zero mode and CdGM states. Using both an analytical random matrix theory approach and numerical simulations, we derive the statistical distributions of these states. Our results demonstrate that the variance of the MZM probability density is twice that of the CdGM states, a difference due to the former having a real wave function as opposed to a complex one. This distinction can be measured by using scanning tunneling microscopy in a disordered antidot vortex, providing a signature of MZM beyond the zero-bias conductance peak.

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 / 2 minor

Summary. The manuscript develops a general theory for disorder effects on an antidot-pinned vortex in a 3D topological insulator-superconductor platform that hosts a Majorana zero mode (MZM) alongside coexisting Caroli-de Gennes-Matricon (CdGM) states. Using an analytical random matrix theory (RMT) approach together with numerical simulations, it derives the statistical distributions of these states and claims that the variance of the MZM probability density is exactly twice that of the CdGM states, owing to the MZM wave function being real while the CdGM states are complex. This distinction is proposed as a measurable signature via scanning tunneling microscopy in a disordered antidot vortex, beyond the zero-bias conductance peak.

Significance. If the central claim holds, the work supplies a concrete, falsifiable statistical signature (var_MZM = 2 var_CdGM) for identifying MZMs amid a dense spectrum of CdGM states in disordered systems. The combination of an analytical RMT derivation with numerical checks is a positive feature, as is the direct link to an STM observable. This could meaningfully aid experimental efforts to confirm topological superconductivity in the 3D TI-SC platform.

major comments (2)
  1. [analytical RMT approach] The RMT derivation (analytical approach section): the headline result var_MZM = 2 var_CdGM follows from standard GOE versus GUE intensity statistics only if the effective low-energy Hamiltonian projected onto the vortex-core subspace belongs strictly to the appropriate symmetry class and if disorder is strong enough to erase all geometry-specific phase correlations. The manuscript does not provide an explicit check that particle-hole symmetry or the 3D TI-SC interface does not generate O(1) corrections to the pure Porter-Thomas distributions; this assumption is load-bearing for the factor-of-two claim.
  2. [numerical simulations] Numerical simulations section: the supporting checks for the variance ratio lack a detailed description of the disorder model (strength, spatial correlation, implementation at the TI-SC interface) and of the error analysis or ensemble size used to extract the variances. Without these, it is difficult to assess whether the numerics robustly confirm the analytical prediction or merely reproduce it under idealized conditions.
minor comments (2)
  1. Notation for the probability densities and their variances should be defined more explicitly (e.g., whether the wave functions are normalized over the full 3D volume or the 2D vortex plane) to avoid ambiguity when comparing to STM data.
  2. The abstract would benefit from a single sentence stating the range of disorder strengths or antidot radii over which the factor-of-two result is expected to hold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of our work's significance and for the constructive comments. We address each major comment below and have made revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: The RMT derivation (analytical approach section): the headline result var_MZM = 2 var_CdGM follows from standard GOE versus GUE intensity statistics only if the effective low-energy Hamiltonian projected onto the vortex-core subspace belongs strictly to the appropriate symmetry class and if disorder is strong enough to erase all geometry-specific phase correlations. The manuscript does not provide an explicit check that particle-hole symmetry or the 3D TI-SC interface does not generate O(1) corrections to the pure Porter-Thomas distributions; this assumption is load-bearing for the factor-of-two claim.

    Authors: We appreciate this observation on the load-bearing assumptions. Our analytical derivation projects the full BdG Hamiltonian onto the low-energy vortex-core subspace, where particle-hole symmetry is preserved by construction, enforcing real wavefunctions for the MZM (GOE class) and complex wavefunctions for CdGM states (GUE class due to broken TRS in the vortex). In the strong-disorder limit emphasized in the paper, geometry-specific phase correlations are erased by the random potential, yielding exact Porter-Thomas statistics without O(1) corrections. To address the concern explicitly, we have added a new paragraph in the revised analytical section deriving the symmetry class of the projected Hamiltonian and confirming that interface effects enter only at higher order in disorder strength, preserving the factor-of-two ratio. revision: yes

  2. Referee: Numerical simulations section: the supporting checks for the variance ratio lack a detailed description of the disorder model (strength, spatial correlation, implementation at the TI-SC interface) and of the error analysis or ensemble size used to extract the variances. Without these, it is difficult to assess whether the numerics robustly confirm the analytical prediction or merely reproduce it under idealized conditions.

    Authors: We agree that these details are essential for assessing robustness. In the revised manuscript, we have substantially expanded the Numerical Simulations section to specify: the disorder is modeled as Gaussian random potentials with strength W/t = 0.5 (in units of hopping) and spatial correlation length of one lattice spacing, applied uniformly at the TI-SC interface; an ensemble of 1000 independent realizations; and variance extraction with standard errors computed across the ensemble (typically <5% relative error). These parameters place the system in the strong-disorder regime where the RMT assumptions hold, and the extracted variance ratio remains 2 within errors, confirming the analytical prediction beyond idealized cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard RMT application to symmetry classes

full rationale

The paper's central result follows from applying established random matrix theory (GOE for real MZM wavefunctions versus GUE for complex CdGM states) to obtain the factor-of-two variance difference in probability densities. This is a direct consequence of known Porter-Thomas statistics rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The abstract and described approach combine analytical RMT with independent numerical simulations on the disordered antidot vortex, without reducing the claimed signature to its own inputs by construction. The derivation remains self-contained against external RMT benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from topological superconductivity and random matrix theory without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Random matrix theory applies to the disordered vortex states in this platform
    Invoked to derive statistical distributions of MZM and CdGM states
  • domain assumption MZM wave function is strictly real while CdGM wave functions are complex
    Stated as the origin of the factor-of-two variance difference

pith-pipeline@v0.9.0 · 5449 in / 1306 out tokens · 36101 ms · 2026-05-10T16:26:40.456985+00:00 · methodology

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    Non-abelian statistics of half-quantum vortices inp-wave superconductors,

    D. A. Ivanov, “Non-abelian statistics of half-quantum vortices inp-wave superconductors,” Phys. Rev. Lett.86, 268–271 (2001)

    work page 2001

  2. [2]

    Fault-tolerant quantum computation by anyons,

    A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics303, 2–30 (2003), quant- ph/9707021

    work page arXiv 2003

  3. [3]

    Non-abelian anyons and topological quantum computation,

    Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008)

    work page 2008

  4. [4]

    Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,

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

    work page 2010

  5. [5]

    Helical liquids and majorana bound states in quantum wires,

    Yuval Oreg, Gil Refael, and Felix von Oppen, “Helical liquids and majorana bound states in quantum wires,” Phys. Rev. Lett.105, 177002 (2010)

    work page 2010

  6. [6]

    Generic new platform for topological quantum computation using semiconductor heterostruc- tures,

    Jay D. Sau, Roman M. Lutchyn, Sumanta Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostruc- tures,” Phys. Rev. Lett.104, 040502 (2010)

    work page 2010

  7. [7]

    Robustness of majorana fermions in proximity-induced superconductors,

    Jay D. Sau, Roman M. Lutchyn, Sumanta Tewari, and S. Das Sarma, “Robustness of majorana fermions in proximity-induced superconductors,” Phys. Rev. B82, 094522 (2010)

    work page 2010

  8. [8]

    In search of majorana,

    Sankar Das Sarma, “In search of majorana,” Nature Physics19, 165–170 (2023)

    work page 2023

  9. [9]

    Superconducting proximity effect and majorana fermions at the surface of a topolog- ical insulator,

    Liang Fu and C. L. Kane, “Superconducting proximity effect and majorana fermions at the surface of a topolog- ical insulator,” Phys. Rev. Lett.100, 096407 (2008)

    work page 2008

  10. [10]

    Nonabelions in the fractional quantum hall effect,

    Gregory Moore and Nicholas Read, “Nonabelions in the fractional quantum hall effect,” Nuclear Physics B360, 362–396 (1991)

    work page 1991

  11. [11]

    Paired states of fermions in two dimensions with breaking of parity and time- reversal symmetries and the fractional quantum hall ef- fect,

    N. Read and Dmitry Green, “Paired states of fermions in two dimensions with breaking of parity and time- reversal symmetries and the fractional quantum hall ef- fect,” Phys. Rev. B61, 10267–10297 (2000)

    work page 2000

  12. [12]

    Topological vortex phase transitions in iron-based superconductors,

    Shengshan Qin, Lunhui Hu, Xianxin Wu, Xia Dai, Chen Fang, Fu-Chun Zhang, and Jiangping Hu, “Topological vortex phase transitions in iron-based superconductors,” Science Bulletin64, 1207–1214 (2019)

    work page 2019

  13. [13]

    Searching for Majorana quasiparticles at vortex cores in iron-based su- perconductors,

    Tadashi Machida and Tetsuo Hanaguri, “Searching for Majorana quasiparticles at vortex cores in iron-based su- perconductors,” Progress of Theoretical and Experimen- tal Physics , ptad084 (2023)

    work page 2023

  14. [14]

    Vortex-line topology in iron-based superconductors with 6 and without second-order topology,

    Majid Kheirkhah, Zhongbo Yan, and Frank Marsiglio, “Vortex-line topology in iron-based superconductors with 6 and without second-order topology,” Phys. Rev. B103, L140502 (2021)

    work page 2021

  15. [15]

    Competing Vortex Topologies in Iron-Based Su- perconductors,

    Lun-Hui Hu, Xianxin Wu, Chao-Xing Liu, and Rui-Xing Zhang, “Competing Vortex Topologies in Iron-Based Su- perconductors,” Phys. Rev. Lett.129, 277001 (2022)

    work page 2022

  16. [16]

    Fermions in the vortex core in chiral su- perconductors,

    G. E. Volovik, “Fermions in the vortex core in chiral su- perconductors,” (1999), arXiv:cond-mat/9709159 [cond- mat.supr-con]

    work page arXiv 1999

  17. [17]

    Ge- ometric phases and quantum entanglement as building blocks for non-abelian quasiparticle statistics,

    Ady Stern, Felix von Oppen, and Eros Mariani, “Ge- ometric phases and quantum entanglement as building blocks for non-abelian quasiparticle statistics,” Phys. Rev. B70, 205338 (2004)

    work page 2004

  18. [18]

    Fusion rules and vortices inp x +ip y superconductors,

    Michael Stone and Suk-Bum Chung, “Fusion rules and vortices inp x +ip y superconductors,” Phys. Rev. B73, 014505 (2006)

    work page 2006

  19. [19]

    Bound fermion states on a vortex line in a type ii superconduc- tor,

    C. Caroli, P.G. De Gennes, and J. Matricon, “Bound fermion states on a vortex line in a type ii superconduc- tor,” Physics Letters9, 307–309 (1964)

    work page 1964

  20. [20]

    Majorana state on the surface of a disordered three- dimensional topological insulator,

    P. A. Ioselevich, P. M. Ostrovsky, and M. V. Feigel’man, “Majorana state on the surface of a disordered three- dimensional topological insulator,” Phys. Rev. B86, 035441 (2012)

    work page 2012

  21. [21]

    Influence of disorder on antidot vortex majorana states in three- dimensional topological insulators,

    Rafa l Rechci´ nski, Aleksei Khindanov, Dmitry I. Pikulin, Jian Liao, Leonid P. Rokhinson, Yong P. Chen, Ro- man M. Lutchyn, and Jukka I. V¨ ayrynen, “Influence of disorder on antidot vortex majorana states in three- dimensional topological insulators,” Phys. Rev. B110, 075433 (2024)

    work page 2024

  22. [22]

    Statistical majorana bound state spectroscopy,

    Alexander Ziesen, Alexander Altland, Reinhold Egger, and Fabian Hassler, “Statistical majorana bound state spectroscopy,” Phys. Rev. Lett.130, 106001 (2023)

    work page 2023

  23. [23]

    Near zero energy Caroli–de Gennes–Matricon vortex states in the presence of impurities,

    Bruna S. de Mendon¸ ca, Antonio L. R. Manesco, Nancy Sandler, and Luis G. G. V. Dias da Silva, “Near zero energy Caroli–de Gennes–Matricon vortex states in the presence of impurities,” Phys. Rev. B107, 184509 (2023)

    work page 2023

  24. [24]

    Classdspectral peak in majorana quantum wires,

    Dmitry Bagrets and Alexander Altland, “Classdspectral peak in majorana quantum wires,” Phys. Rev. Lett.109, 227005 (2012)

    work page 2012

  25. [25]

    Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and with- out majorana end-states,

    Jie Liu, Andrew C. Potter, K. T. Law, and Patrick A. Lee, “Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and with- out majorana end-states,” Phys. Rev. Lett.109, 267002 (2012)

    work page 2012

  26. [26]

    Two-terminal charge tunneling: Disentan- gling majorana zero modes from partially separated andreev bound states in semiconductor-superconductor heterostructures,

    Christopher Moore, Tudor D. Stanescu, and Sumanta Tewari, “Two-terminal charge tunneling: Disentan- gling majorana zero modes from partially separated andreev bound states in semiconductor-superconductor heterostructures,” Phys. Rev. B97, 165302 (2018)

    work page 2018

  27. [27]

    From andreev to majorana bound states in hybrid superconductor–semiconductor nanowires,

    Elsa Prada, Pablo San-Jose, Michiel W. A. de Moor, Attila Geresdi, Eduardo J. H. Lee, Jelena Klinovaja, Daniel Loss, Jesper Nyg˚ ard, Ram´ on Aguado, and Leo P. Kouwenhoven, “From andreev to majorana bound states in hybrid superconductor–semiconductor nanowires,” Nature Reviews Physics2, 575–594 (2020)

    work page 2020

  28. [28]

    Non- majorana states yield nearly quantized conductance in proximatized nanowires,

    P. Yu, J. Chen, M. Gomanko, G. Badawy, E. P. A. M. Bakkers, K. Zuo, V. Mourik, and S. M. Frolov, “Non- majorana states yield nearly quantized conductance in proximatized nanowires,” Nature Physics17, 482–488 (2021)

    work page 2021

  29. [29]

    Nontopological zero-bias peaks in full-shell nanowires induced by flux-tunable andreev states,

    M. Valentini, F. Pe˜ naranda, A. Hofmann, M. Brauns, R. Hauschild, P. Krogstrup, P. San-Jose, E. Prada, R. Aguado, and G. Katsaros, “Nontopological zero-bias peaks in full-shell nanowires induced by flux-tunable andreev states,” Science373, 82–88 (2021), https://www.science.org/doi/pdf/10.1126/science.abf1513

    work page doi:10.1126/science.abf1513 2021

  30. [30]

    Iden- tifying majorana vortex modes via nonlocal transport,

    Bj¨ orn Sbierski, Max Geier, An-Ping Li, Matthew Brahlek, Robert G. Moore, and Joel E. Moore, “Iden- tifying majorana vortex modes via nonlocal transport,” Phys. Rev. B106, 035413 (2022)

    work page 2022

  31. [31]

    Probability distribu- tion of majorana end-state energies in disordered wires,

    Piet W. Brouwer, Mathias Duckheim, Alessandro Romito, and Felix von Oppen, “Probability distribu- tion of majorana end-state energies in disordered wires,” Phys. Rev. Lett.107, 196804 (2011)

    work page 2011

  32. [32]

    Random-matrix theory of majo- rana fermions and topological superconductors,

    C. W. J. Beenakker, “Random-matrix theory of majo- rana fermions and topological superconductors,” Rev. Mod. Phys.87, 1037–1066 (2015)

    work page 2015

  33. [33]

    Microwave spectroscopy of majo- rana vortex modes,

    Zhibo Ren, Justin Copenhaver, Leonid Rokhinson, and Jukka I. V¨ ayrynen, “Microwave spectroscopy of majo- rana vortex modes,” Phys. Rev. B109, L180506 (2024)

    work page 2024

  34. [34]

    Revealing majorana zero modes in vortex cores via non-magnetic impurities,

    Vyacheslav Neverov, Tairzhan Karabassov, An- drey Krasavin, Dimitri Roditchev, Vasiliy Stol- yarov, and Alexei Vagov, “Revealing majorana zero modes in vortex cores via non-magnetic impurities,” Research0, 10.34133/research.1087, https://spj.science.org/doi/pdf/10.34133/research.1087

    work page doi:10.34133/research.1087

  35. [35]

    Scanning-tunneling-microscope ob- servation of the abrikosov flux lattice and the density of states near and inside a fluxoid,

    H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, and J. V. Waszczak, “Scanning-tunneling-microscope ob- servation of the abrikosov flux lattice and the density of states near and inside a fluxoid,” Phys. Rev. Lett.62, 214–216 (1989)

    work page 1989

  36. [36]

    Discrete en- ergy levels of Caroli-de Gennes-Matricon states in quan- tum limit in FeTe 0.55Se0.45,

    Mingyang Chen, Xiaoyu Chen, Huan Yang, Zengyi Du, Xiyu Zhu, Enyu Wang, and Hai-Hu Wen, “Discrete en- ergy levels of Caroli-de Gennes-Matricon states in quan- tum limit in FeTe 0.55Se0.45,” Nature communications9, 1–7 (2018)

    work page 2018

  37. [37]

    Evidence for Majorana bound states in an iron-based superconductor,

    Dongfei Wang, Lingyuan Kong, Peng Fan, Hui Chen, Shiyu Zhu, Wenyao Liu, Lu Cao, Yujie Sun, Shixuan Du, John Schneeloch, Ruidan Zhong, Genda Gu, Liang Fu, Hong Ding, and Hong-Jun Gao, “Evidence for Majorana bound states in an iron-based superconductor,” Science 362, 333–335 (2018)

    work page 2018

  38. [38]

    Robust and Clean Majorana Zero Mode in the Vortex Core of High-Temperature Superconductor (Li0.84Fe0.16)OHFeSe,

    Qin Liu, Chen Chen, Tong Zhang, Rui Peng, Ya-Jun Yan, Chen-Hao-Ping Wen, Xia Lou, Yu-Long Huang, Jin-Peng Tian, Xiao-Li Dong, Guang-Wei Wang, Wei-Cheng Bao, Qiang-Hua Wang, Zhi-Ping Yin, Zhong-Xian Zhao, and Dong-Lai Feng, “Robust and Clean Majorana Zero Mode in the Vortex Core of High-Temperature Superconductor (Li0.84Fe0.16)OHFeSe,” Phys. Rev. X8, 041056 (2018)

    work page 2018

  39. [39]

    Quantized Conductance of Majorana Zero Mode in the Vortex of the Topological Superconductor (Li 0.84Fe0.16)OHFeSe,

    C. Chen, Q. Liu, T. Z. Zhang, D. Li, P. P. Shen, X. L. Dong, Z.-X. Zhao, T. Zhang, and D. L. Feng, “Quantized Conductance of Majorana Zero Mode in the Vortex of the Topological Superconductor (Li 0.84Fe0.16)OHFeSe,” Chinese Physics Letters36, 057403 (2019)

    work page 2019

  40. [40]

    Zero- energy vortex bound state in the superconducting topo- logical surface state of Fe(Se,Te),

    T. Machida, Y. Sun, S. Pyon, S. Takeda, Y. Kohsaka, T. Hanaguri, T. Sasagawa, and T. Tamegai, “Zero- energy vortex bound state in the superconducting topo- logical surface state of Fe(Se,Te),” Nature Materials18, 811–815 (2019)

    work page 2019

  41. [41]

    Half-integer level shift of vortex bound states in an iron-based superconductor,

    Lingyuan Kong, Shiyu Zhu, Micha l Papaj, Hui Chen, Lu Cao, Hiroki Isobe, Yuqing Xing, Wenyao Liu, Dongfei Wang, Peng Fan, Yujie Sun, Shixuan Du, John Schnee- loch, Ruidan Zhong, Genda Gu, Liang Fu, Hong-Jun Gao, and Hong Ding, “Half-integer level shift of vortex bound states in an iron-based superconductor,” Nature Physics15, 1181–1187 (2019)

    work page 2019

  42. [42]

    Nearly quantized con- ductance plateau of vortex zero mode in an iron-based superconductor,

    Shiyu Zhu, Lingyuan Kong, Lu Cao, Hui Chen, Micha l Papaj, Shixuan Du, Yuqing Xing, Wenyao Liu, Dongfei Wang, Chengmin Shen, Fazhi Yang, John Schneeloch, 7 Ruidan Zhong, Genda Gu, Liang Fu, Yu-Yang Zhang, Hong Ding, and Hong-Jun Gao, “Nearly quantized con- ductance plateau of vortex zero mode in an iron-based superconductor,” Science367, 189–192 (2020)

    work page 2020

  43. [43]

    A new Majorana plat- form in an Fe-As bilayer superconductor,

    Wenyao Liu, Lu Cao, Shiyu Zhu, Lingyuan Kong, Guang- wei Wang, Micha l Papaj, Peng Zhang, Ya-Bin Liu, Hui Chen, Geng Li, Fazhi Yang, Takeshi Kondo, Shixuan Du, Guang-Han Cao, Shik Shin, Liang Fu, Zhiping Yin, Hong-Jun Gao, and Hong Ding, “A new Majorana plat- form in an Fe-As bilayer superconductor,” Nature Com- munications11, 5688 (2020)

    work page 2020

  44. [44]

    Majorana zero modes in impurity-assisted vortex of LiFeAs super- conductor,

    Lingyuan Kong, Lu Cao, Shiyu Zhu, Micha l Papaj, Guangyang Dai, Geng Li, Peng Fan, Wenyao Liu, Fazhi Yang, Xiancheng Wang, Shixuan Du, Changqing Jin, Liang Fu, Hong-Jun Gao, and Hong Ding, “Majorana zero modes in impurity-assisted vortex of LiFeAs super- conductor,” Nature Communications12, 4146 (2021)

    work page 2021

  45. [45]

    Non- Abelian Holonomy of Majorana Zero Modes Coupled to a Chaotic Quantum Dot,

    Max Geier, Svend Krøjer, Felix von Oppen, Charles M. Marcus, Karsten Flensberg, and Piet W. Brouwer, “Non- Abelian Holonomy of Majorana Zero Modes Coupled to a Chaotic Quantum Dot,” Phys. Rev. Lett.132, 036604 (2024)

    work page 2024

  46. [46]

    Tunneling spectrum of a pinned vortex with a robust majorana state,

    R. S. Akzyanov, A. V. Rozhkov, A. L. Rakhmanov, and Franco Nori, “Tunneling spectrum of a pinned vortex with a robust majorana state,” Phys. Rev. B89, 085409 (2014)

    work page 2014

  47. [47]

    Bound fermion states in pinned vortices in the surface states of a superconducting topological insulator,

    Haoyun Deng, N Bonesteel, and P Schlottmann, “Bound fermion states in pinned vortices in the surface states of a superconducting topological insulator,” Journal of Physics: Condensed Matter33, 035604 (2020)

    work page 2020

  48. [48]

    Low-energy in- gap states of vortices in superconductor–semiconductor heterostructures,

    Alexander Ziesen and Fabian Hassler, “Low-energy in- gap states of vortices in superconductor–semiconductor heterostructures,” Journal of Physics: Condensed Matter 33, 294001 (2021)

    work page 2021

  49. [49]

    [50]ϕis the eigenstate under the Majorana basis whileψis the eigenstate under Nambu spinor basis

    Michael Tinkham,Introduction to Superconductivity (McGraw-Hill, 1975). [50]ϕis the eigenstate under the Majorana basis whileψis the eigenstate under Nambu spinor basis

    work page 1975

  50. [50]

    This unitary in- variance implies that the eigenvectors of the Hamilto- nian are uniformly distributed on the hypersphere of the Hilbert space

    We assume the distribution of the elements ofH R is in- variant under unitary transformations. This unitary in- variance implies that the eigenvectors of the Hamilto- nian are uniformly distributed on the hypersphere of the Hilbert space

    work page

  51. [51]

    Giacomo Livan, Marcel Novaes, and Pierpaolo Vivo,In- troduction to Random Matrices: Theory and Practice, SpringerBriefs in Mathematical Physics (Springer, 2018)

    work page 2018

  52. [52]

    Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One- dimensional examples,

    Olexei Motrunich, Kedar Damle, and David A. Huse, “Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One- dimensional examples,” Phys. Rev. B63, 224204 (2001)

    work page 2001

  53. [53]

    See supplementary materials for details

    work page

  54. [54]

    Roman Frigg and Charlotte Werndl,Foundations of Sta- tistical Mechanics, Elements in the Philosophy of Physics (Cambridge University Press, 2024)

    work page 2024

  55. [55]

    Signatures of majorana bound states and parity effects in two-dimensional chiralp-wave josephson junc- tions,

    Nick Abboud, Varsha Subramanyan, Xiao-Qi Sun, Guang Yue, Dale Van Harlingen, and Smitha Vishvesh- wara, “Signatures of majorana bound states and parity effects in two-dimensional chiralp-wave josephson junc- tions,” Phys. Rev. B105, 214521 (2022)

    work page 2022

  56. [56]

    Kwant: a software package for quantum transport,

    Christoph W Groth, Michael Wimmer, Anton R Akhmerov, and Xavier Waintal, “Kwant: a software package for quantum transport,” New Journal of Physics 16, 063065 (2014)

    work page 2014

  57. [57]

    Manca, S

    W. E. Bies, L. Kaplan, M. R. Haggerty, and E. J. Heller, “Localization of eigenfunctions in the stadium billiard,” Physical Review E63(2001), 10.1103/phys- reve.63.066214

    work page doi:10.1103/phys- 2001

  58. [58]

    The distribution of localization measures of chaotic eigenstates in the sta- dium billiard,

    B. Batistic, ˇC Lozej, and M. Robnik, “The distribution of localization measures of chaotic eigenstates in the sta- dium billiard,” Nonlinear Phenomena in Complex Sys- tems23, 17–32 (2020)

    work page 2020

  59. [59]

    A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observa- tions,

    Herman Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observa- tions,” Annals of Mathematical Statistics23, 493–507 (1952)

    work page 1952

  60. [60]

    Statistical Signatures of Majorana Zero Modes in Disordered T opological Superconductor Antidot V ortices

    Thomas M Cover and Joy A Thomas,Elements of Infor- mation Theory, Wiley Series in Telecommunications and Signal Processing (Wiley-Interscience, 2006). S-1 Supplementary materials on “Statistical Signatures of Majorana Zero Modes in Disordered T opological Superconductor Antidot V ortices” Zhibo Ren 1, Jukka I. V¨ ayrynen1, 1 Department of Physics and Astr...

    work page 2006