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REVIEW 1 major objections 4 minor 56 references

Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A 2D second-order topological superconductor can host zero-energy vortex-localized modes together with Majorana corner modes, provided the normal-state bulk spectrum has Dirac cones.

desk verdict A useful analytic step on vortex zero modes in 2D HOTSC, but the justification that the (k_x^2 - k_y^2) kinetic term leaves them gapless is weaker than the text suggests. read the letter →

arxiv 2411.14831 v2 pith:WECT65NZ submitted 2024-11-22 cond-mat.supr-con

A. D. Fedoseev , A. O. Zlotnikov This is my paper · ORCID
classification cond-mat.supr-con
keywords second-ordertopologicalsuperconductorMajoranacornermodesvortexzeroDiracconesspin-orbitcouplinginsulator-superconductorinterfacehigher-ordertopologysuperconducting
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

Second-order topological superconductors are expected to have gapped edge spectra, so vortex-bound states would be expected to acquire finite energy. This paper shows that in a two-dimensional second-order topological superconductor built from a normal layer with spin-orbit coupling and a superconducting layer, zero-energy vortex-localized modes can coexist with the Majorana corner modes that define the higher-order phase. The requirement is that the non-superconducting bulk spectrum be gapless and contain Dirac cones; the number of pairs of vortex zero modes equals the number of Dirac cones, and an explicit wavefunction is derived for them. If the normal-state spectrum is gapped, the vortex modes are gapped while corner Majoranas survive. The result matters because it gives concrete lattice-parameter conditions under which a single material could show both corner Majoranas and zero-bias vortex peaks in scanning tunneling experiments.

What carries the argument

The machinery is a continuum expansion around the Dirac points of the normal-state spectrum, combined with the standard zero-mode construction for vortices in topological systems. The key object is the zero-energy vortex wavefunction of Eq. (12), built from Bessel functions and the radial envelope $F(r)$, which solves the vortex equations when the chemical potential is finite. The argument hinges on the separation of the kinetic terms in Eq. (13): the combination $(\partial_x^2 - \partial_y^2)$ has zero overlap with the zero mode because of its angular dependence, whereas $(\partial_x^2 + \partial_y^2)$ would produce a finite gap. Vanishing of both the constant term and the $(\partial_x^2 + \partial_y^2)$ term in the expansion yields the coexistence condition Eq. (14). A second central mechanism is the absence of a boundary-localized zero-mode counterpart: in a first-order topological superconductor the increasing solution of the vortex equations corresponds to an edge Majorana, but in a second-order superconductor the edge spectrum is gapped, so only the vortex-localized solution survives.

What would settle it

Compute the exact lattice spectrum of the model with a vortex at the parameters satisfying Eq. (14) for a sequence of increasing system sizes and extract the lowest vortex-bound eigenvalue; if that energy approaches a nonzero saturation value rather than decaying to zero, the continuum zero mode is not exact. A complementary check is to include the next-order momentum terms omitted from Eq. (13) and test whether the vortex-mode energy becomes nonzero.

Watch

Extended reading notes

Core claim

The central claim is that zero-energy vortex modes are not forbidden in a second-order topological superconductor; they appear precisely when the normal (non-superconducting) layer has Dirac cones in its bulk spectrum. Working in a two-orbital square-lattice model with spin-orbit coupling and spin-singlet pairing, the authors expand the Hamiltonian near the spin-orbit nodal points and solve the superconducting vortex equations. The resulting zero-energy solution, Eq. (12), is a combination of Bessel functions $J_0(r\mu/2\lambda)$ and $J_1(r\mu/2\lambda)$ times a radial envelope $F(r)=\exp\left(-\int_0^r \Delta(\rho)/2|\lambda|\,d\rho\right)$, with only one spin projection selected by the vortex winding. The gapless modes survive when the kinetic parameters satisfy $t_1 = c_x \,\delta t\, \operatorname{sign} t_y / 2$ and $\Delta\varepsilon = -2c_x \,\delta t\, \operatorname{sign} t_x$, where $c_x = \pm 1$ labels the Dirac point and $\delta t = (|t_x| - |t_y|)/2$; then the number of zero-mode pairs equals the number of Dirac cones. When this condition is violated and the normal spectrum is gapped, the vortex modes acquire a gap while the Majorana corner modes remain, so the coexistence is controlled by lattice parameters rather than by topology alone.

Load-bearing premise

The derivation assumes that lattice-scale corrections to the kinetic energy, beyond the terms kept in the continuum expansion, do not shift the zero-energy vortex mode; the paper verifies this for one combination of second derivatives, showing it has zero overlap with the wavefunction, but does not prove the full kinetic operator exactly kills the state, so neglected higher-order terms could open a small gap.

Editorial extensions

If this is right

  • In a sample in the second-order topological superconducting phase with $t_1 = c_x \,\delta t\, \operatorname{sign} t_y / 2$ and $\Delta\varepsilon = -2c_x \,\delta t\, \operatorname{sign} t_x$, scanning tunneling spectroscopy should simultaneously show zero-bias conductance peaks at vortex cores and at corners.
  • The number of zero-energy vortex pairs is fixed by the number of Dirac cones in the normal-state spectrum: one pair for a single cone and two pairs for two cones, so the vortex-mode degeneracy directly probes the normal-state band structure.
  • When the normal spectrum is gapped, the same model produces only gapped vortex modes while Majorana corner modes persist, so the absence of a zero-bias vortex peak does not exclude a second-order topological superconducting phase.
  • The vortex zero modes are not protected by a topological invariant, so approaching a boundary or corner hybridizes them with edge and corner states; near a corner one pair becomes corner-localized and may remain gapless or acquire a small gap depending on $\Delta_0$ and other parameters.
  • The coexistence condition is a precise relation among hopping parameters and on-site energies, so it can be tuned by lattice deformations or gate potentials to switch between gapped and gapless vortex modes without destroying the corner Majoranas.

Reading between the lines

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

  • Because the zero mode is not topologically protected, real materials will generically have small corrections that split it from zero; the practical prediction is a narrow but not exactly pinned zero-bias peak whose residual energy is set by the deviation from Eq. (14).
  • The Dirac-cone-to-vortex-mode counting may extend beyond this two-orbital lattice: any normal state with several symmetry-related Dirac points should contribute one vortex-mode pair per cone, provided the same kind of kinetic cancellation can be arranged.
  • The condition Eq. (14) can be read as a design rule: strain that changes $|t_x| - |t_y|$ or gates that shift $\Delta\varepsilon$ provide a knob to turn vortex zero modes on and off while leaving corner Majoranas intact, which could be used to manipulate the low-energy state content of a device.
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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

1 major / 4 minor

Summary. The paper studies a two-orbital lattice model of a two-dimensional second-order topological superconductor with Rashba spin-orbit coupling and singlet pairing. It derives an approximate continuum solution for zero-energy vortex-localized modes when the normal-state bulk spectrum has Dirac cones, giving the explicit wavefunction in Eq. (12) and the parameter condition (14) for gapless vortex modes. It reports numerical spectra showing that these zero modes can coexist with Majorana corner modes when the superconducting bulk and Wannier gaps are open, and it analyzes how the modes hybridize when the vortex approaches an edge or a corner.

Significance. If the claims hold, the paper provides a useful and nontrivial extension of vortex zero-mode physics to higher-order topological superconductors: unlike first-order systems, the edge spectrum is gapped, yet zero-energy vortex modes can still appear and coexist with corner Majoranas, with the number of pairs set by the number of Dirac cones. The analytic construction is self-contained, the parameter condition (14) is explicit, and the finite-size numerics in Figs. 2 and 3 support the coexistence scenario. The main caveat is the treatment of the quadratic kinetic term, which is the subject of the major comments.

major comments (1)
  1. [Sec. IV, Eq. (13) and Appendix B] The statement that the quadratic term t cx sign tx in Eq. (13) 'does not influence' the zero-energy solution (12) is not established. Appendix B proves only that the diagonal matrix element Psi^dag (d_x^2 - d_y^2) Psi vanishes for state (12); it does not show that the operator annihilates Psi or that all off-diagonal matrix elements to other states vanish. Acting on (12) with the operator in Eq. (B1) generates components with cos 2phi and sin 2phi angular factors that are not proportional to the original state, so the state is not an eigenstate when this term is retained. Generically this produces a second-order energy shift, so the vortex mode is zero only to leading order in the continuum expansion. Consequently, condition (14), which cancels only the other terms in Eq. (13), may not be sufficient to guarantee truly gapless vortex modes in the full lattice model. The numerical demonstrations at specific parameter points are consistent with the claim, but they do not establish Eq. (14) as an exact condition. I request a proof of the vanishing of the off-diagonal couplings, an estimate of the residual gap, or a concrete finite-size scaling study showing that the lowest positive eigenvalue tends to zero at the parameters of Fig. 2.
minor comments (4)
  1. [Eq. (4)] The notation for the vortex phase uses arg[z((R_f+R_m)/2) - z(R_v)] for bond-centered pairings; please clarify the convention for the bond midpoint and the branch of arg.
  2. [Abstract and throughout] The term 'gapless vortex modes' is potentially confusing because the modes are localized and do not form a band; 'zero-energy vortex modes' would be more precise throughout.
  3. [Figs. 2 and 3] The system size N is never stated; please give the lattice size and the numerical tolerance used to identify zero-energy states.
  4. [Appendix A, Eq. (A1)] The phase theta introduced in Eq. (A1) is not defined before it is used; please specify its relation to the sign choices in beta.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the vortex-mode derivation is self-contained and the only self-citations are non-load-bearing.

full rationale

I found no circular step in the derivation. The zero-energy vortex solution (12) is obtained by solving the continuum Bogoliubov–de Gennes equations with the standard Jackiw–Rossi procedure (Refs. [17,19,20,38–41]); it is not defined in terms of the coexistence it is used to demonstrate. The coexistence condition (14) is derived analytically by requiring the constant term ε~ and the radial quadratic term in the expanded tk (Eq. (13)) to vanish; it is a constraint on the lattice parameters, not a fitted output, and the numerical spectra in Figs. 2 and 3 are independent checks. The HOTSC phase is supported by the in-paper Wannier-band polarization and quadrupole calculation and by the external model of Ref. [34]; the self-citations [35,36] are used only as supplementary support and are not load-bearing for the vortex-mode derivation. Appendix B's assertion that (∂x^2−∂y^2) has no effect because Ψ†(∂x^2−∂y^2)Ψ=0 is a possible weakness in the analytic argument, but it is a correctness and rigor concern rather than circularity: the paper does not define the vortex-mode energy as that matrix element. Thus no circular step is present.

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

No parameters are fitted to data; all model parameters are physical inputs, and the coexistence condition (14) is analytically derived. The main axioms are the continuum linearization near Dirac points, the angular-momentum suppression of the quadratic kinetic term, the bulk topological invariants establishing the HOTSC phase, and the Jackiw-Rossi vortex-Dirac correspondence.

assumptions (4)
  • domain assumption The linearized continuum approximation around the spin-orbit nodal points captures the zero-energy vortex physics (Sec. III, Eqs. (5)-(8)).
    The derivation neglects quadratic and higher-order terms in the lattice dispersion near the Dirac points; this is standard but unquantified.
  • domain assumption The quadratic kinetic term proportional to (k_x^2 - k_y^2) does not couple to the zero-energy vortex solution (12); its matrix element vanishes by angular momentum conservation (Appendix B).
    Appendix B shows the expectation value vanishes for the specific angular structure, but does not show the operator annihilates the state; higher-order corrections are neglected.
  • standard math The model supports Majorana corner modes protected by time-reversal symmetry when sign(ΔxΔy) = - sign(tx ty) even at t1 = Δε = 0, established via Wannier band polarization and quadrupole moment (Sec. II).
    Uses bulk topological invariants from Refs. [8,37]; the authors calculate these for one parameter point.
  • standard math Jackiw-Rossi correspondence: a vortex in a Dirac fermion system binds zero modes, one per Dirac cone, under suitable winding (Sec. III, Ref. [41]).
    The analytical solution (12) relies on this established result extended to a two-band model.

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Cite this review

Pith. "Pith review of Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor." pith.science (2026). https://pith.science/paper/WECT65NZ

@misc{pith2026241114831,
  author       = {Pith},
  title        = {Pith review of: Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WECT65NZ}},
  note         = {Machine review of arXiv:2411.14831}
}
read the original abstract

Although the appearance of vortex-localized states with zero energy in first-order topological superconductors is well known, their possibility to form in the higher-order topological phase of 2D systems has not been completely uncovered yet. Here we demonstrate the coexistence of zero-energy vortex modes and Majorana corner modes in the model of a 2D second-order topological superconductor. The model describes an interface between a normal layer supporting the topological insulating phase and a superconducting layer, for which different symmetries of the spin-singlet superconducting order parameter are considered. We show that the gapless vortex modes can appear under certain conditions in the superconducting state with a vortex if the bulk energy spectrum of the normal (non-superconducting) state is gapless and has Dirac cones. The number of pairs of such vortex modes corresponds to the number of Dirac cones. It is essential that if the normal bulk spectrum becomes gapped and the system is in the state of a topological insulator, then the zero-energy vortex modes can not be realized, while Majorana corner modes hold in the superconducting state. The interaction of the vortex modes with the edge and topological corner modes is studied when the vortex appears near the boundaries.

Figures

Figures reproduced from arXiv: 2411.14831 by the authors.

Figure 1
Figure 1. FIG. 1. The topological phase diagram in variables chemical [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. a) The energy spectrum of the square-shaped system at [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Dependence of excitation energy of the square-shape [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

Works this paper leans on

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