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REVIEW 3 major objections 5 minor 82 references

Annular Majorana mode in a superconducting topological insulator

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Pith's one-line read A superconducting vortex in a topological insulator with 3-fold or 6-fold rotational symmetry and surface states of winding number 3 binds a Majorana zero mode whose probability density is ring-shaped, vanishing at the core and peaking on…

desk verdict A clean symmetry argument predicts a ring-shaped Majorana mode in the single-cone case, but the three-cone material-realized case needs a finite-chemical-potential robustness check before the prediction is trusted. read the letter →

arxiv 2608.05632 v1 pith:YLSYYIYG submitted 2026-08-06 cond-mat.supr-con

Shengshan Qin , Chi Wu , Lun-hui Hu , Tiantian Zhang , Jiangping Hu This is my paper · ORCID
classification cond-mat.supr-con
keywords Majoranazeromodeannulartopologicalinsulatorsuperconductingvortexsurfacestateswindingnumber3chiralf-wavepairingZrTi2
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 predicts that a superconducting vortex on the surface of a topological insulator can bind a Majorana zero-energy mode shaped like a ring: the mode's probability density vanishes at the vortex center and reaches its maximum on a surrounding circle. The arrangement is protected by the crystal's 3-fold or 6-fold rotational symmetry together with surface states that carry a total winding number 3, realized either as a single nonlinear Dirac cone at the zone center or as three linear Dirac cones at the three $\bar{\mathrm{M}}$ points. In the single-cone case the proximity-induced pairing acts as an effective chiral $f$-wave, while in the three-cone case the interference of three effective chiral $p$-wave Majoranas cancels at the core and leaves one ring-shaped survivor. If correct, the mode should be visible in scanning tunneling microscopy as a zero-bias conductance dip at the core surrounded by a zero-bias peak, and it provides a spatially extended Majorana platform beyond the usual core-localized wave packet.

What carries the argument

The load-bearing object is the winding number 3 carried by the surface Fermi surface, expressed through the cubic off-diagonal term $\tilde{A}(k_-^3 s_+ - k_+^3 s_-)$ in the surface Hamiltonian. That term fixes the angular-momentum content of vortex bound states when the vortex order parameter is $\Delta(r)=\Delta_0 \tanh(r/\xi)e^{i\theta}$: the Majorana zero mode must lie in the $l_z=0$ (or $l_z \bmod 6=0$) sector, and its wavefunction is a superposition of Bessel functions $J_m$ with nonzero $m$, which all vanish at $r=0$. In the three-Dirac-cone case, the analogous object is the triplet of valley Majoranas $\gamma_1,\gamma_2,\gamma_3$ and their rotation-symmetric hybridization; the surviving combination cancels at the core by destructive interference, producing the same ring. The winding number is what converts an ordinary core-localized vortex Majorana into an annular one.

What would settle it

Image a single vortex in a candidate material such as ZrTi$_2$ with scanning tunneling microscopy: the prediction fails if the zero-bias conductance is peaked at the vortex center (the conventional Majorana signature) or if no ring-shaped conductance feature appears. Numerically, the same check is to solve the Bogoliubov--de Gennes equations with a distorted or off-center vortex profile and measure the zero-energy density at $r=0$; any substantial weight at the core falsifies the exact ring.

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Extended reading notes

Core claim

The paper's central claim is that a vortex in a superconducting topological insulator whose surface Fermi surface carries winding number 3 hosts a Majorana zero mode with a ring-shaped wavefunction. For the single nonlinear Dirac cone, the cubic winding term in the surface Hamiltonian forces the zero-energy mode into the channel with angular momentum $l_z=0$ (or $l_z \bmod 6 = 0$ when a $\tilde{B}$ term breaks continuum rotation down to $C_6$), and the radial wavefunction is built from Bessel functions of nonzero order, so it must vanish at $r=0$. For the three linear Dirac cones at the $\bar{\mathrm{M}}$ points, each cone contributes one Majorana operator $\gamma_i$, and the rotation-symmetric hybridization $h_{\rm hyb}=i\eta(\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1)$ leaves the symmetric combination $\gamma=\gamma_1+\gamma_2+\gamma_3$, whose real-space wavefunction cancels at the core by the interfering phases $e^{2\pi i/3}$ and $e^{4\pi i/3}$. The paper verifies both scenarios numerically and analytically, shows that a nematic perturbation deforms and eventually splits the ring into lobes, and predicts that the material ZrTi$_2$, with three $\bar{\mathrm{M}}$-point Dirac cones on its (001) surface, should realize the effect.

Load-bearing premise

The ring shape depends on the vortex being a rotationally symmetric defect with order parameter $\Delta_0 \tanh(r/\xi)e^{i\theta}$: if the core is off-center, distorted, or the pairing profile differs, the angular-momentum selection rule and the exact vanishing of the mode at $r=0$ can break down.

Editorial extensions

If this is right

  • Scanning tunneling microscopy on a vortex in a suitable material should show a zero-bias conductance dip at the vortex center and a zero-bias peak on a surrounding ring, the reverse of the conventional core-peaked signature.
  • The annular mode survives a moderate $C_6\to C_2$ nematic perturbation as a deformed ring, but a strong enough perturbation tears it into discrete lobes, so the ring shape itself is a diagnostic of the unperturbed rotational symmetry.
  • In the three-$\bar{\mathrm{M}}$-cone realization, the two partner Majoranas are split from the annular mode only by the small hybridization $\eta$ arising from large-momentum inter-valley scattering, so the modes can be nearly degenerate and require ultra-low temperature, high-resolution measurement.
  • Two-dimensional spin-1/2 and spin-3/2 Rashba electron gases with three $\bar{\mathrm{M}}$-point Fermi surfaces (or one $\bar{\Gamma}$-point Fermi surface in the spin-3/2 case), under a Zeeman field and conventional pairing, should support the same annular mode.
  • The material ZrTi$_2$ is predicted to be a concrete platform, with three $\bar{\mathrm{M}}$-point Dirac cones about 50--80 meV above the Fermi level depending on surface termination.

Reading between the lines

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

  • A spatially extended ring is a natural handle for coupling Majorana modes: the tunneling between two vortices would depend on ring-to-ring overlap rather than core-to-core distance, which the paper does not discuss but which would matter for braiding proposals.
  • A quantitative test beyond the paper would be to compute how the ring radius and the depth of the core dip vary with chemical potential, vortex size $\xi$, and pairing strength $\Delta_0$; those curves could be compared directly with STM maps.
  • The same angular-momentum selection argument plausibly extends to surface states with other winding numbers (such as 5), where zero modes might form multiple concentric rings or higher angular-momentum patterns; the paper does not explore this.
  • Because the three-cone ring relies on exact destructive interference of equal-amplitude valley Majoranas, disorder that scatters between $\bar{\mathrm{M}}$ valleys will partially fill in the core; measuring how the dip fills with impurity density would test the interference mechanism.
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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

3 major / 5 minor

Summary. The paper studies superconducting vortices on the surfaces of topological insulators that respect 3-fold or 6-fold rotational symmetry and whose surface states carry winding number 3. Two complementary scenarios are considered: a single nonlinear (cubic) Dirac cone at the Brillouin-zone center, and three linear Dirac cones at the three M-bar points. The authors argue that in both cases the vortex-bound Majorana zero mode has a ring-shaped probability density, vanishing exactly at the vortex core, and they dub this an annular Majorana mode. For the nonlinear cone, the ring follows from angular-momentum classification and the Bessel-function expansion of the vortex bound states in the l_z=0 (or l_z mod 6=0) sector. For the three-linear-cone case, the ring is obtained by superposing three equal-amplitude cone Majoranas with relative phases 1, e^{2\pi i/3}, and e^{4\pi i/3}, which interfere destructively at the vortex core. The paper also proposes ZrTi2 as a candidate material whose (001) surface hosts three linear Dirac cones at M-bar, and it suggests STM signatures: suppressed zero-bias conductance at the vortex core with a surrounding zero-bias enhancement.

Significance. If correct, the annular Majorana mode is a new and qualitatively distinct vortex-bound state, with a concrete experimental fingerprint that differs from the point-like Majorana wave packets studied previously. The nonlinear-cone derivation is a clean symmetry argument: the Bessel orders in the l_z=0 sector are all nonzero, so the zero-mode wave function vanishes at the core, and this remains true for all n mod 6=0 sectors when the B-tilde term is present. The paper is also commendably free of data-fitting: the central derivation is a symmetry-based consequence of the assumed winding-3 surface Hamiltonian, and the model parameters are illustrative. However, the three-cone case, which is the case realized by the proposed candidate material, is derived only at zero chemical potential with perfectly symmetric cones, and the paper does not establish robustness at finite chemical potential or against valley-asymmetry perturbations. The practical significance of the material prediction is therefore conditional on additional analysis.

major comments (3)
  1. [Eqs. (7)-(9) and Fig. 3] The ring prediction in the three-cone case rests on the exact cancellation |gamma(0)|=0, which requires equal envelope amplitudes and the specific relative phases 1, e^{2\pi i/3}, e^{4\pi i/3} shown in Eq. (7). Eq. (7) is explicitly derived only for mu=0 and B-tilde=0, with identical Dirac velocities and identical pairing at the three M-bar points. The proposed candidate ZrTi2 has surface Dirac cones about 50 meV above the Fermi level on the Ti-terminated surface, so the chemical potential is not at the Dirac point; no finite-mu calculation is presented. Moreover, any valley-asymmetry perturbation (unequal velocities, strain, or disorder) makes the three envelope amplitudes unequal and would give a nonzero |gamma(0)|, so the annular shape is not robust in the regime that is actually demonstrated. Since ZrTi2 realizes the three-cone case, this is a load-bearing gap for the material prediction. Please supply the finite-mu zero-mode wavefunctions or direct numerical vortex simulations for the three-cone model, and quantify the tolerance to valley-asymmetry perturbations; the single-cone nematic test in Fig. 4 does not address this case.
  2. [Annular Majorana mode from linear surface Dirac cone, Eq. (7) and Fig. 3] The derivation of the three-cone result is not self-contained. Eq. (7) is introduced as following from a 'more detailed analysis in the SM', and Fig. 3 is plotted from Eq. (9) using Delta(r)=Delta_0 rather than the vortex profile Delta_0 tanh(r/xi)e^{i\theta} used elsewhere in the paper. It is therefore not clear how the vortex phase winding enters the envelope functions gamma_i(r), or why the constant phases in Eq. (7) remain unchanged at finite mu. Please include the derivation in an appendix and state explicitly how the nonlocal operators gamma_i in Eq. (7) are converted into the local wavefunction weights gamma_i(r) in Eq. (9), including how the azimuthal part of the vortex is accounted for in the plotted quantity |gamma(r)|^2.
  3. [Candidate materials and Supplemental Material] The main text relies on the Supplemental Material for several load-bearing checks: the analytical derivation leading to Eq. (7), the first-principles identification of ZrTi2 as a Z2=1 topological insulator with three M-bar Dirac cones, and the numerical verification of the Rashba-electron-gas extensions. The SM is not provided with the submitted manuscript, so these claims cannot be verified from the submission as it stands. Please supply the SM or move the essential steps into an appendix, at minimum the derivation of Eq. (7) and the ZrTi2 bulk invariants with the computed surface band structure.
minor comments (5)
  1. [Abstract and Introduction] The phrase 'decaying exponentially offthe vortex core' should read 'off the vortex core'.
  2. [Fig. 4 caption and text] The caption labels the panels as (a)-(c), but the main text refers to 'Figs. 4(b)-(d)'; the panel labels should be made consistent.
  3. [Introduction] The sentence 'Majorana mode whose antimode is the mode itself' is unclear; a Majorana fermion is normally described as a particle that is its own antiparticle.
  4. [Annular Majorana mode from nonlinear surface Dirac cone, Eq. (5)] The statement that MZMs 'must exist in the subspace with l_z mod 6=0 or 3' is followed by a numerical zero mode found only in the l_z mod 6=0 sector; the reason why the l_z mod 6=3 sector does not also contribute a zero mode should be clarified.
  5. [Throughout] The notation eC6z and eT for the symmetry operators is nonstandard; using \hat{C}_{6z} and \hat{T} would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the annular Majorana mode is derived from the assumed surface Hamiltonian and vortex ansatz, not from fitted parameters or self-citation chains.

full rationale

The central derivation is self-contained within the stated model. Given h_surf in Eq.(1) with the k^3 winding term, the BdG Hamiltonian Eq.(3), and the vortex order parameter Delta(r)=Delta0 tanh(r/xi)e^{i theta}, the MZM wave function is expanded in Bessel functions (Eq.(4)); the l_z=0 channel has only J_n with n not equal to 0, so |phi_0(0)|=0. This is a mathematical consequence of the model, not a restatement of an input. The three-cone result follows from the single-cone MZM solutions (Eq.(7)) and C3-symmetric hybridization (Eq.(8)); the equal amplitudes and phases 1, e^{2 pi i/3}, e^{4 pi i/3} are symmetry-dictated at mu=0 and are not fitted to produce the ring. The numerical simulations in Figs. 2 and 4 use illustrative parameters only. Candidate material ZrTi2 is characterized by first-principles DFT, an external benchmark independent of the fitted model parameters. Self-citations to the Supplemental Material and to previous papers are used for derivation details and context, not as unverified inputs that force the conclusion. The paper explicitly notes limitations, including the mu=0 derivation of Eq.(7), the small hybridization energy in the three-cone case, and the sensitivity of the ring to nematic perturbations; these are correctness or robustness concerns rather than circularity. No identified step reduces to its own input by construction, so the circularity score is 0.

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

The central claim rests on model Hamiltonians with winding-3 surface states, a symmetric vortex ansatz, and symmetry-based projection of s-wave pairing. The numerical inputs are illustrative model parameters, not fits to experimental data. The most fragile axioms are the existence of the nonlinear surface Dirac cone and the exact valley-symmetric interference in the three-cone case.

free parameters (4)
  • Surface Hamiltonian coefficients (A-tilde, B-tilde, M-tilde) in Eq. (1) = A-tilde=50, B-tilde=45, M-tilde=0 in Fig. 2; general values in text
    Chosen to illustrate the nonlinear Dirac cone; the ring result is argued to be independent of these values.
  • Chemical potential mu = mu=2 in Fig. 2; mu=0 in the analytic linear-cone derivation
    Set by hand; the linear-cone analytic wave functions are derived at mu=0.
  • Superconducting gap Delta0 and vortex core size xi = Delta0=1.5, xi=10 in Fig. 2; Delta0/A-tilde=0.8 in Fig. 3
    Model parameters for the vortex profile; not fitted to experimental data.
  • Hybridization strength eta in Eq. (8) = not given
    Introduced as the only C3-symmetric coupling among the three valley Majoranas; its value affects the near-degeneracy of the other two modes but not the existence of the zero mode.
assumptions (6)
  • standard math Bessel function expansion of BdG eigenfunctions in angular momentum channels (Eq. 4).
    Provides the basis for the single-cone zero-mode wave function and the vanishing at the core.
  • domain assumption A C6z-symmetric topological insulator can host a single nonlinear surface Dirac cone with Jz=+-3/2 and cubic dispersion, protected by time-reversal and C6z (Eq. 1).
    The nonlinear-cone leg of the main prediction rests on this model Hamiltonian; no concrete material is identified for this leg in the main text.
  • domain assumption Conventional s-wave pairing projected onto the surface band basis becomes effective chiral f-wave pairing (single cone) or chiral p-wave pairing (three cones).
    Links the normal-state winding number to the vortex Majorana physics.
  • domain assumption The vortex order parameter is Delta(r)=Delta0 tanh(r/xi)e^{i theta} and the system has rotational symmetry about the vortex core.
    Required for the integer angular momentum classification of bound states and for the exact ring shape.
  • domain assumption For the three-M-cone case, the system can be adiabatically tuned to B-tilde=0 in the superconducting state without closing the gap.
    Used to justify the analytic Majorana wave functions in Eq. (7); the adiabatic path is not demonstrated in the main text.
  • domain assumption ZrTi2 is a Z2=1 topological insulator with three linear surface Dirac cones at the M points, based on DFT parity calculations.
    The material candidate stands on first-principles calculations whose details appear in the Supplemental Material.

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Pith. "Pith review of Annular Majorana mode in a superconducting topological insulator." pith.science (2026). https://pith.science/paper/YLSYYIYG

@misc{pith2026260805632,
  author       = {Pith},
  title        = {Pith review of: Annular Majorana mode in a superconducting topological insulator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YLSYYIYG}},
  note         = {Machine review of arXiv:2608.05632}
}
abstract

When the surface states of a topological insulator becomes superconducting, topological superconductivity can be obtained, and each vortex on the surface can host one single Majorana zero-energy mode which is usually a wave packet decaying exponentially off the vortex core. Here, we predict stable Majorana zero-energy mode whose wave function is ring-shape, dubbed as annular Majorana mode, in the superconducting vortex in topological insulators respecting $3$-fold or $6$-fold rotational symmetry. Such topological insulators are featured with a single nonlinear Dirac cone located at $\bar{\Gamma}$ or three linear Dirac cones at $\bar{\text{M}}$ in the surface Brillouin zone. The annular Majorana mode originates from the effective chiral $f$-wave superconductivity on the nonlinear Dirac cone in the former case and the interference of the effective chiral $p$-wave superconductivity on the three linear Dirac cones in the latter. In both cases, the annular Majorana mode is stabilized by the rotational symmetry and the winding number $3$ carried by the surface states. Candidate materials supporting the annular Majorana mode are predicted. Our work provides new insights into the topological superconductivity in superconducting topological insulators.

Figures

Figures reproduced from arXiv: 2608.05632 by the authors.

Figure 1
Figure 1. FIG. 1. (color online) Annular Majorana mode can be realized in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (color online) (a) shows the low-energy superconducting [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (color online) The wave function profiles for the MZM [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (color online) (a) depicts the crystal structure of ZrTi [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. (color online) (a) [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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