Pith. sign in

REVIEW 2 major objections 4 minor 43 references

A solvable non-Hermitian Dirac vortex gives closed-form frequencies, thresholds, and polarizations for topological-cavity surface-emitting lasers.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-11 15:50 UTC pith:VNRVRX3X

load-bearing objection Closed-form non-Hermitian Dirac-vortex formulas that actually give usable TCSEL design rules and match the polarization and mode-crossover experiments. the 2 major comments →

arxiv 2607.04646 v1 pith:VNRVRX3X submitted 2026-07-06 physics.optics

Non-Hermitian Dirac Vortex: Minimal Theory for Topological-Cavity Surface-Emitting Laser

Zong-Liang Li , Guang-Rui Li , Le-Chen Yang , Zhong Wang , Ling Lu This is my paper
classification physics.optics
keywords non-Hermitian Dirac vortextopological-cavity surface-emitting laserTCSELJackiw-Rossi modelvector-beam polarizationcomplex-mass windingabsorbing boundaryphotonic crystal laser
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Topological-cavity surface-emitting lasers (TCSELs) are large photonic-crystal devices that lase from a Dirac-vortex zero mode. Full-wave simulation of cavities thousands of periods across is impractical, so designers have relied on numerical coupled-wave theory. This paper constructs a minimal continuum model that extends the Jackiw–Rossi vortex and the neutrino-billiard boundary into the non-Hermitian regime: a complex-mass winding encodes vertical radiation loss while an infinite imaginary potential defines the absorbing edge of the active region. The model admits closed-form modal frequencies, boundary losses that scale differently for the zero mode and competing modes, and far-field vector-beam polarizations that can be rotated by the initial mass phase. Experiments on optically pumped devices confirm both the predicted polarization tuning and the crossover from zero-mode to unbound-singlet lasing when the normalized mass is reduced. The result supplies an analytical design tool that explains single-mode stability margins without heavy computation.

Core claim

A non-Hermitian Dirac Hamiltonian with complex-mass winding number w and an infinite-imaginary-potential disk of radius R yields analytic eigenvalues and spinors. After nondimensionalization by R, the zero-mode boundary loss scales as 16(mR)^2 exp(−4mR), the next modes scale more slowly, and the radiation operator maps the spinor to two cylindrical vector beams of charges +1 and −2 whose relative weight and common polarization angle are fixed by mR and the initial mass phase θ0. These expressions reproduce the measured three-lobe far fields and the observed mode crossover.

What carries the argument

Non-Hermitian Dirac vortex: the 4×4 continuum Hamiltonian with real mass m e^{i(wθ+θ0)}, imaginary mass −μ e^{−2i(wθ+θ0)}, and the absorbing boundary condition −τ_z(σ·n̂)|ψ⟩ = |ψ⟩ at r = R. Separation of variables plus first-order perturbation in μ produces the closed-form spectrum and radiation patterns.

Load-bearing premise

The continuum Dirac description plus the infinite-absorption boundary remain accurate for a real photonic-crystal slab whose absorption is large but finite and whose lattice is discrete.

What would settle it

Fabricate a series of devices with fixed radius while continuously varying the mass amplitude so that mR sweeps through 0.85; if the lasing far-field does not switch from the three-lobe zero-mode pattern to the unbound-singlet pattern exactly as predicted, or if measured threshold margins deviate strongly from the analytic ΔαR curves, the continuum model fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Designers can choose mR ≈ 1.5 to maximize the analytic threshold margin while keeping free spectral range and boundary loss favorable.
  • Output polarization (radial, azimuthal, or spiral) is set by the single geometric parameter θ0 without redesigning the cavity shape.
  • The same closed-form loss hierarchy applies to any C3-symmetric lattice that realizes a Dirac vortex, independent of microscopic lattice details.
  • The massless non-Hermitian billiard spectrum supplies a universal lower bound αR ≥ 0.5 that any competing whispering-gallery mode must respect.

Where Pith is reading between the lines

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

  • Because the analytic thresholds depend only on the dimensionless product mR, the same formulas can be reused for electrically pumped or mid-infrared TCSELs once the effective mass is extracted from band structure.
  • The anti-PT pairing that isolates the zero mode may be portable to other non-Hermitian topological lasers that combine a mass defect with an absorbing boundary.
  • If lattice-scale corrections mix the valleys enough to spoil the shared vertical loss α⊥, the single-mode advantage would shrink even while the continuum topology remains intact—an effect testable by comparing devices with different supercell sizes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript constructs a non-Hermitian Dirac-vortex model that combines a complex-mass winding (real mass m with winding w and imaginary mass μ with winding −2w) with an infinite-imaginary-potential boundary of radius R. This extends the Jackiw–Rossi vortex and neutrino-billiard models into the dissipative regime and is presented as a minimal continuum theory for topological-cavity surface-emitting lasers (TCSELs). From the 4 imes4 Hamiltonian (Eq. 1) and the absorbing boundary condition (Eq. 2), the authors obtain closed-form modal frequencies and boundary losses (via Whittaker radial functions after neglecting μ for the eigenvalue problem), show that vertical radiation loss α⊥ = 2μ is mode-independent to first order (Eq. 5), and derive the radiation operator that yields θ0-tunable cylindrical vector beams (Eq. 8). Asymptotic scalings of α∥R (zero-mode ∼16(mR)^{2}e^{-4mR}, bound doublet exponentially suppressed, unbound singlet power-law) and the single-mode stability window near mR ≈ 1.5 are given in Fig. 3. End Matter solves the massless non-Hermitian billiard and reports experimental far-field polarization control and the zero-mode/unbound-singlet crossover.

Significance. If the continuum-plus-perturbation hierarchy holds at the claimed level of accuracy, the work supplies a rare, analytically closed non-Hermitian topological design theory for a practical large-area laser. Closed-form loss scalings, the identification of an optimal mR window, and the explicit θ0 control of vector-beam polarization are directly usable for device engineering and go beyond purely numerical coupled-wave theory. The experimental matches for polarization patterns (Fig. 6) and the lasing-mode crossover (Fig. 7) provide independent, falsifiable checks. The construction also cleanly unifies Jackiw–Rossi, neutrino-billiard and non-Hermitian optics, which is of broader interest to topological photonics.

major comments (2)
  1. The central claim that the closed-form thresholds and mode ordering are quantitatively predictive for real TCSELs rests on three linked approximations that are only scale-justified: continuum Dirac description of a discrete C3 photonic-crystal slab, replacement of finite exterior absorption (~600 cm^{-1}) by an infinite imaginary potential (justified by 600 ≫ 1/R ~ 20 cm^{-1}), and first-order treatment of μ ≪ m so that α⊥ is identical for every mode (Eq. 5) and the radiation operator (Eq. 8) is accurate. No error bound or direct comparison of the analytic ΔαR peak near mR ≈ 1.5 (Fig. 3) against full CWT or finite-absorption numerics is provided. If lattice-scale or soft-boundary corrections reorder the lowest-loss modes by an amount comparable to that peak, the claimed single-mode stability window ceases to be quantitative. A short numerical validation (or an explicit statement of the e
  2. The radiation operator (Eq. 8) and the assertion that α⊥ = 2μ is strictly mode-independent both rely on the same first-order perturbation in μ after the Hermitian vortex has been solved. Residual μ-induced mixing between the zero mode and the nearby unbound singlet/bound doublet is not estimated. Because the threshold margin ΔαR is itself of order the boundary-loss differences plotted in Fig. 3, even a modest second-order correction could shift the optimal operating point. A brief estimate of the size of these corrections (or a statement that they remain negligible throughout the recommended mR window) would strengthen the load-bearing claim.
minor comments (4)
  1. The factor of 1/2 that converts the complex eigenvalue into the intensity decay rate α is introduced without derivation in Eq. (1); a one-sentence reminder of the e^{iωt} convention would help non-specialist readers.
  2. Fig. 3(a) black dashed curve (minor-to-major component ratio) is useful but its definition appears only in the caption; a short inline definition would improve readability.
  3. End Matter Table I compares Hermitian and non-Hermitian billiards; the boundary-condition operators are written with slightly different conventions from the main-text Eq. (2). Aligning the notation would avoid confusion.
  4. Several key experimental references (e.g., the original TCSEL papers) are self-citations; a brief pointer to independent experimental realizations of related Dirac-vortex cavities would help place the work in the broader literature.

Circularity Check

1 steps flagged

No load-bearing circularity in the closed-form spectra or polarizations; self-citations supply only the TCSEL device platform and motivation, not the analytic results.

specific steps
  1. self citation load bearing [Introduction (TCSEL paragraph) and End Matter experimental section]
    "Our non-Hermitian Dirac-vortex model provides a minimal theory for the emerging topological-cavity surface-emitting laser (TCSEL) [10–15] … To experimentally validate our theoretical predictions, we implement the non-Hermitian Dirac vortex in optically pumped topological-cavity surface-emitting lasers, following our previous work [11]."

    The claim that the model is the minimal theory for TCSEL, and the experimental platform itself, rest on citations whose author lists overlap with the present paper. This supplies device context and motivation but is not load-bearing for the closed-form eigenvalues, loss scalings or radiation operator, which are derived independently from Eqs. (1)–(8).

full rationale

The derivation chain is self-contained: the 4 imes4 non-Hermitian Dirac Hamiltonian (Eq. 1) with complex-mass winding is obtained from CWT couplings, the infinite-imaginary-potential boundary (Eq. 2) is imposed by scale separation, angular-momentum separation yields Whittaker radial solutions whose eigenvalues and asymptotic boundary losses (e.g. zero-mode α∥R∼16(mR)2e−4mR) follow by direct solution, and the radiation operator (Eq. 8) maps the spinor to far-field vector beams whose θ0 dependence is fixed by the C3 phases. These steps do not reduce to their inputs by construction, nor are parameters fitted to data and then re-presented as predictions. Experimental far-field patterns and the unbound-singlet crossover (End Matter Figs. 6–7) serve as independent qualitative checks. Self-citations [10–15,11] identify the TCSEL platform and prior observations but are not used to justify uniqueness theorems, force the spectral formulas, or smuggle ansatze; the mathematics stands alone. Hence only minor non-load-bearing self-citation, consistent with score 2.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 2 invented entities

The central claim rests on the continuum non-Hermitian Dirac Hamiltonian extracted from CWT, the infinite-imaginary-potential boundary, the scale hierarchy μ ≪ m and absorption ≫ 1/R, and C3-enforced winding relations. Free parameters are the design knobs mR and θ0 (and overall μ scale for radiation strength); they are not fitted to force the spectral formulas. No new particles or forces are postulated—the “non-Hermitian Dirac vortex” and “unbound singlet” are named solutions of the stated PDE. Background axioms are standard Dirac/Jackiw–Rossi mathematics plus domain assumptions about photonic-crystal radiation and quantum-well absorption.

free parameters (3)
  • normalized real mass mR
    Dimensionless design parameter that sets the entire complex spectrum; chosen by lattice modulation strength and cavity radius. Not fitted to force agreement—experiments scan effective mR via pump size.
  • initial mass phase θ0
    Geometric angle defining where Arg(m)=0 on the vortex; freely chosen in lattice design and used to predict polarization rotation, not fitted post hoc.
  • imaginary-mass magnitude μ (and overall vertical loss α⊥=2μ)
    Sets common vertical radiation loss; treated perturbatively. Absolute scale is material/geometry dependent and not needed for relative thresholds or polarization patterns.
axioms (5)
  • domain assumption Bulk TCSEL physics reduces to the 4×4 non-Hermitian Dirac Hamiltonian Eq. (1) with real mass m from first-order K–K′ coupling and imaginary mass μ from second-order coupling via Γ radiation.
    Stated as derived from coupled-wave theory in SM Sec. I; continuum long-wavelength limit of the photonic crystal.
  • domain assumption Unpumped exterior may be replaced by an infinite imaginary potential, giving the compact boundary condition −τz(σ·n̂)|ψ⟩=|ψ⟩ at r=R.
    Justified by absorption ∼600 cm^{−1} ≫ 1/R ∼20 cm^{−1}; non-Hermitian analogue of MIT-bag / neutrino-billiard walls.
  • domain assumption μ ≪ m and windings differ (μ winds as −2w), so eigenfunctions may be solved with μ=0 and radiation treated by first-order perturbation, yielding identical α⊥=2μ for all modes.
    Used after Eq. (4) and in Eq. (5); enables analytical separation of variables.
  • standard math Odd winding implies anti-PT (particle-hole) symmetry pairing ±ω+iα/2, leaving the zero mode unpaired.
    Eq. (3) and surrounding text; standard non-Hermitian symmetry classification.
  • domain assumption C3 lattice symmetry forces mass windings w and −2w and restricts far-field topological charges to C3-compatible values (+1, −2, …).
    Fig. 1 and radiation operator discussion; links microscopic lattice to continuum phases.
invented entities (2)
  • Non-Hermitian Dirac vortex (complex-mass winding + infinite imaginary potential) independent evidence
    purpose: Minimal analytically solvable model that unifies Jackiw–Rossi vortex confinement with dissipative boundary and radiation loss for TCSEL.
    Named construction of the paper; not a new particle but a new solvable PDE setup. Independent handle is experimental far-field and mode-crossover tests.
  • Unbound singlet mode independent evidence
    purpose: Labels the low-loss singlet that is not a Hermitian bound state, competes with the zero mode at small mR, and is observed when the pump is shrunk.
    Emerges from the massless non-Hermitian billiard spectrum (End Matter); experimental far field matches prediction.

pith-pipeline@v1.1.0-grok45 · 17524 in / 3984 out tokens · 49243 ms · 2026-07-11T15:50:35.805948+00:00 · methodology

0 comments
read the original abstract

We construct a non-Hermitian Dirac-vortex model that combines a complex-mass winding with an infinite-imaginary-potential boundary, extending the Jackiw-Rossi and neutrino-billiard models to the dissipative regime. Moreover, this model serves as a minimal theory for the recently proposed topological-cavity surfaceemitting laser (TCSEL): the imaginary mass encodes vertical radiation loss and the absorbing boundary defines the active region. We derive closed-form expressions for the modal frequencies, thresholds, and tunable vectorbeam polarizations, which are validated experimentally. Our work provides a rare example in which an analytical non-Hermitian topological theory captures the essential physics for engineering practical optoelectronic devices.

Figures

Figures reproduced from arXiv: 2607.04646 by Guang-Rui Li, Le-Chen Yang, Ling Lu, Zhong Wang, Zong-Liang Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Non-Hermitian Dirac vortex. (a) Momentum-space cou [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Eigen-modes to the Hermitian and non-Hermitian vortices. (a) Real spectrum of the Hermitian vortex, showing an infinite series of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Scalings of boundary losses and mode separation. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Polarization control of the vector-beam output by the initial [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Eigenvalue spectrum of the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Lattice design and experimental validation. (a) Generalized Kekul [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Observation of the unbound singlet. The lasing mode [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

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

43 extracted references · 2 canonical work pages

  1. [1]

    Chodos, R

    A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V . F. Weis- skopf, New extended model of hadrons, Physical Review D9, 3471 (1974)

    1974

  2. [2]

    M. V . Berry and R. J. Mondragon, Neutrino billiards: time- reversal symmetry-breaking without magnetic fields, Proceed- ings of the Royal Society of London. A. Mathematical and Physical Sciences412, 53 (1987)

    1987

  3. [3]

    A. R. Akhmerov and C. W. J. Beenakker, Boundary conditions for dirac fermions on a terminated honeycomb lattice, Physical Review B77, 085423 (2008)

    2008

  4. [4]

    Christensen, W

    T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, Classical and quantum plasmonics in graphene nan- odisks: Role of edge states, Physical Review B90, 241414 (2014)

    2014

  5. [5]

    Jackiw and C

    R. Jackiw and C. Rebbi, Solitons with fermion number ½, Phys- ical Review D13, 3398 (1976)

    1976

  6. [6]

    Jackiw and P

    R. Jackiw and P. Rossi, Zero modes of the vortex-fermion sys- tem, Nuclear Physics B190, 681 (1981)

    1981

  7. [7]

    Cheng, J

    H. Cheng, J. Yang, Z. Wang, and L. Lu, Observation of monopole topological mode, Nature Communications15, 7327 (2024)

    2024

  8. [8]

    Fu and C

    L. Fu and C. L. Kane, Superconducting proximity effect and majorana fermions at the surface of a topological insulator, Physical Review Letters100, 096407 (2008)

    2008

  9. [9]

    J. C. Y . Teo and C. L. Kane, Majorana fermions and non- abelian statistics in three dimensions, Physical Review Letters 104, 046401 (2010)

    2010

  10. [10]

    X. Gao, L. Yang, H. Lin, L. Zhang, J. Li, F. Bo, Z. Wang, and L. Lu, Dirac-vortex topological cavities, Nature Nanotechnol- ogy15, 1012 (2020)

    2020

  11. [11]

    L. Yang, G. Li, X. Gao, and L. Lu, Topological-cavity surface- emitting laser, Nature Photonics16, 279 (2022)

    2022

  12. [12]

    J. Ma, T. Zhou, M. Tang, H. Li, Z. Zhang, X. Xi, M. Mar- tin, T. Baron, H. Liu, Z. Zhang, S. Chen, and X. Sun, Room- temperature continuous-wave topological dirac-vortex micro- cavity lasers on silicon, Light: Science & Applications12, 255 (2023)

    2023

  13. [13]

    S. Han, Y . Chua, Y . Zeng, B. Zhu, C. Wang, B. Qiang, Y . Jin, Q. Wang, L. Li, A. G. Davies, E. H. Linfield, Y . Chong, B. Zhang, and Q. J. Wang, Photonic majorana quantum cascade 6 laser with polarization-winding emission, Nature Communica- tions14, 707 (2023)

    2023

  14. [14]

    J. Liu, Y . Xu, R. Li, Y . Sun, K. Xin, J. Zhang, Q. Lu, N. Zhuo, J. Liu, L. Wang, F. Cheng, S. Liu, F. Liu, and S. Zhai, High- power electrically pumped terahertz topological laser based on a surface metallic dirac-vortex cavity, Nature Communications 15, 4431 (2024)

    2024

  15. [15]

    Zhong, X

    M. Zhong, X. Bi, M. Song, N. Mou, D. Zhang, X. Zhuo, J. Hu, B. Xie, X. Ao, and J. Guan, Plasmonic dirac-vortex lasers via three-dimensional photonic mass vortices engineering, Nature Communications 10.1038/s41467-026-70833-1 (2026)

    work page doi:10.1038/s41467-026-70833-1 2026

  16. [16]

    Morthier and P

    G. Morthier and P. Vankwikelberge,Handbook of distributed feedback laser diodes(Artech House, 2013)

    2013

  17. [17]

    B. D. Padullaparthi, J. Tatum, and K. Iga,VCSEL industry: communication and sensing(John Wiley & Sons, 2021)

    2021

  18. [18]

    Yoshida, S

    M. Yoshida, S. Katsuno, T. Inoue, J. Gelleta, K. Izumi, M. D. Zoysa, K. Ishizaki, and S. Noda, High-brightness scalable continuous-wave single-mode photonic-crystal laser, Nature 618, 727 (2023)

    2023

  19. [19]

    S. Noda, M. Yoshida, T. Inoue, M. D. Zoysa, K. Ishizaki, and R. Sakata, Photonic-crystal surface-emitting lasers, Nature Re- views Electrical Engineering1, 802 (2024)

    2024

  20. [20]

    X. Gao, S. Chang, Z. Li, and T. Zhai, Far fields of two- dimensional tm modes in dirac-vortex topological cavity, Op- tics Letters50, 245 (2025)

    2025

  21. [21]

    Kogelnik and C

    H. Kogelnik and C. V . Shank, Coupled-wave theory of dis- tributed feedback lasers, Journal of Applied Physics43, 2327 (1972)

    1972

  22. [22]

    Kazarinov and C

    R. Kazarinov and C. Henry, Second-order distributed feedback lasers with mode selection provided by first-order radiation losses, IEEE Journal of Quantum Electronics21, 144 (1985)

    1985

  23. [23]

    Liang, C

    Y . Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Three- dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polariza- tion: finite-size effects, Optics Express20, 15945 (2012)

    2012

  24. [24]

    Liang, C

    Y . Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y . Tanaka, K. Kitamura, and S. Noda, Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization, Optics Express21, 565 (2013)

    2013

  25. [25]

    Inoue, M

    T. Inoue, M. Yoshida, J. Gelleta, K. Izumi, K. Yoshida, K. Ishizaki, M. D. Zoysa, and S. Noda, General recipe to re- alize photonic-crystal surface-emitting lasers with 100-w-to-1- kw single-mode operation, Nature Communications13, 3262 (2022)

    2022

  26. [26]

    C.-Y . Hou, C. Chamon, and C. Mudry, Electron fractionaliza- tion in two-dimensional graphenelike structures, Physical Re- view Letters98, 186809 (2007)

    2007

  27. [27]

    M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Chiral tun- nelling and the klein paradox in graphene, Nature Physics2, 620 (2006)

    2006

  28. [28]

    Antoine, E

    X. Antoine, E. Lorin, and Q. Tang, A friendly review of ab- sorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations, Molecular Physics115, 1861 (2017)

    2017

  29. [29]

    Tumulka, Detection time distribution for dirac particles, arXiv.1601.04571 (2016)

    R. Tumulka, Detection time distribution for dirac particles, arXiv.1601.04571 (2016)

    Pith/arXiv arXiv 2016

  30. [30]

    Engquist and A

    B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computa- tion31, 629 (1977)

    1977

  31. [31]

    L. A. Coldren, S. W. Corzine, and M. L. Ma ˇsanovi´c,Diode Lasers and Photonic Integrated Circuits(Wiley, 2012)

    2012

  32. [32]

    Kawabata, S

    K. Kawabata, S. Higashikawa, Z. Gong, Y . Ashida, and M. Ueda, Topological unification of time-reversal and particle- hole symmetries in non-hermitian physics, Nature Communica- tions10, 297 (2019)

    2019

  33. [33]

    Ge and H

    L. Ge and H. E. T ¨ureci, Antisymmetric pt -photonic structures with balanced positive- and negative-index materials, Physical Review A88, 053810 (2013)

    2013

  34. [34]

    Seradjeh, Midgap spectrum of the fermion–vortex system, Nuclear Physics B805, 182 (2008)

    B. Seradjeh, Midgap spectrum of the fermion–vortex system, Nuclear Physics B805, 182 (2008)

    2008

  35. [35]

    A. L. Rakhmanov, A. V . Rozhkov, and F. Nori, Majorana fermions in pinned vortices, Physical Review B84, 075141 (2011)

    2011

  36. [36]

    R. S. Akzyanov, A. V . Rozhkov, A. L. Rakhmanov, and F. Nori, Tunneling spectrum of a pinned vortex with a robust majorana state, Physical Review B89, 085409 (2014)

    2014

  37. [37]

    Liu,Photonic Crystal Cavity Systems Described by Dirac Equation(The University of Wisconsin-Madison, 2023)

    B. Liu,Photonic Crystal Cavity Systems Described by Dirac Equation(The University of Wisconsin-Madison, 2023)

    2023

  38. [38]

    D. S., M. Abramowitz, and I. A. Stegun, Handbook of mathe- matical functions with formulas, graphs, and mathematical ta- bles., Mathematics of Computation20, 167 (1966)

    1966

  39. [39]

    Inoue, K

    T. Inoue, K. Maeda, M. Yoshida, J. Gelleta, S. Katsuno, K. Ishizaki, M. D. Zoysa, and S. Noda, Influence of band-edge frequency non-uniformity in ultra-large-area photonic-crystal surface-emitting lasers, IEEE Journal of Selected Topics in Quantum Electronics31, 1 (2025)

    2025

  40. [40]

    Zhan, Cylindrical vector beams: from mathematical con- cepts to applications, Advances in Optics and Photonics1, 1 (2009)

    Q. Zhan, Cylindrical vector beams: from mathematical con- cepts to applications, Advances in Optics and Photonics1, 1 (2009)

    2009

  41. [41]

    B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇci´c, Topo- logical nature of optical bound states in the continuum, Physical Review Letters113, 257401 (2014)

    2014

  42. [42]

    H. Li, J. Yu, J. Fan, S. Kang, B. Hou, Z. Lin, H. Chen, J. Jiang, and X. Wu, Unveiling spin-orbital angular momentum locking in photonic dirac vortex cavities, Laser & Photonics Reviews 10.1002/lpor.202502844 (2026)

    work page doi:10.1002/lpor.202502844 2026

  43. [43]

    Neutrino Billiard

    Supplemental Material in the full manuscript (2026), See the full manuscript version for Supplemental Material. 7 END MATTER Non-Hermitian “Neutrino Billiard” We solve the non-Hermitian Dirac vortex in the massless limit (𝑚=𝜇=0), namely a Dirac fermion confined by the infinite imaginary potential of radius𝑅. This differs from the “neutrino billiard” model...

    2026