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arxiv: 2505.17980 · v4 · pith:3246XI2Rnew · submitted 2025-05-23 · ❄️ cond-mat.supr-con

Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators

Xun-Jiang Luo , Mingliang Tian This is my paper

Pith reviewed 2026-05-22 13:05 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords Majorana vortex end modestime-reversal invariant topological insulatorshigher-order topologysuperconducting proximitytopologically trivial insulatorsZ2 protectionvortex phases
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The pith

Majorana vortex end modes persist in time-reversal invariant systems with fully gapped surfaces even when topologically trivial.

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

The paper constructs time-reversal invariant higher-order topological insulators from two copies of a topological insulator, adding mass terms that preserve time-reversal symmetry while gapping surface states anisotropically. These mass terms act as weak perturbations that leave the vortex phase transitions of the underlying double TI essentially unchanged. As a result, a Z2-protected window opens between the two critical chemical potentials where Majorana vortex end modes appear. The same window survives in models where all surfaces receive the same-sign gap, making the bulk topologically trivial by both first-order and second-order invariants. This widens the range of solid-state platforms that can host Majorana vortex phases to include trivial insulators.

Core claim

Z2-protected Majorana vortex end modes emerge when the chemical potential lies between the critical chemical potentials of the two component TI vortex transitions. These modes remain present even when all surfaces are gapped with the same sign, rendering the system topologically trivial in both first- and second-order classifications.

What carries the argument

Double TI construction with time-reversal symmetry preserving mass terms that anisotropically gap the surface states.

Load-bearing premise

The mass terms that anisotropically gap the surface states of the double TI have a negligible impact on the vortex phase transitions and induce no additional topological phase transitions.

What would settle it

Varying the magnitude of the surface-gapping mass terms and observing whether the chemical-potential window supporting MVEMs stays fixed or acquires new phase boundaries.

Figures

Figures reproduced from arXiv: 2505.17980 by Mingliang Tian, Xun-Jiang Luo.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematical illustration for (a) double TIs residing in the subspace [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Schematical illustration of MVEMs for the case [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Schematical illustration of HHMs in THOTI Bi. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) and(e) Schematical illustration of trivial insulators for double TIs and TI, respectively, with uniform Dirac mass [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) and (b) Vortex energy spectra of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Majorana vortex phases have been extensively studied in topological materials with conventional superconducting pairing. Inspired by recent experimental progress in realizing time-reversal invariant higher-order topological insulators (THOTIs) and inducing superconducting proximity effects, we investigate Majorana vortex phases in these systems. We construct THOTIs as two copies of a topological insulator (TI) with time-reversal symmetry-preserving mass terms that anisotropically gap the surface states. We find that these mass terms have a negligible impact on the vortex phase transitions of double TIs when treated as perturbations, and no additional topological phase transitions are induced. Consequently, $\mathbb{Z}_2$-protected Majorana vortex end modes (MVEMs) emerge when the chemical potential lies between the critical chemical potentials $\mu_c^{(1)}$ and $\mu_c^{(2)}$ of the two TI vortex phase transitions. We demonstrate this behavior across multiple THOTI models, including rotational symmetry-protected THOTI, inversion symmetry-protected THOTI, rotational and inversion symmetries-protected THOTI bismuth, and extrinsic THOTI. Remarkably, MVEMs persist even when all surfaces are gapped with the same sign, rendering the system topologically trivial in both first- and second-order classifications. Our findings establish that MVEMs can be realized in time-reversal invariant systems with fully gapped surfaces, encompassing both topologically nontrivial and trivial insulators, thus significantly broadening the solid state material platforms for hosting Majorana vortex phases.

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

0 major / 3 minor

Summary. The paper claims that Z2-protected Majorana vortex end modes (MVEMs) persist in time-reversal invariant higher-order topological insulators (THOTIs) constructed as double TIs plus TRS-preserving mass terms, as well as in topologically trivial insulators. Explicit lattice models are built for rotational, inversion, bismuth-type, and extrinsic cases; the mass terms are treated as perturbations that anisotropically gap surfaces but leave the vortex critical chemical potentials μ_c^(1) and μ_c^(2) essentially unchanged, with no additional transitions. Consequently MVEMs appear in the window between those critical points even when all surfaces are gapped with uniform sign, rendering the system trivial in both first- and second-order classifications.

Significance. If the results hold, the work substantially broadens candidate platforms for Majorana vortex phases to include fully gapped, time-reversal invariant systems that are trivial in both first- and second-order topology. Credit is due for the concrete lattice Hamiltonians, phase diagrams, and mode spectra supplied for multiple symmetry classes; these constitute explicit, reproducible constructions that directly illustrate the claimed persistence of MVEMs.

minor comments (3)
  1. [§3] §3 (model Hamiltonians): the statement that the mass terms induce 'no additional topological phase transitions' would benefit from an explicit statement of the criterion used to detect transitions (e.g., closing of the bulk gap or change in the Z2 invariant of the vortex sector).
  2. [Figure 4] Figure 4 (phase diagrams): axis labels and color scales should be repeated on all panels for immediate readability when comparing the double-TI and mass-gapped cases.
  3. [Conclusion] The abstract and introduction cite the persistence of MVEMs in the trivial regime; a brief sentence in the conclusion summarizing the range of chemical potentials over which this holds for each model would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. The referee's summary accurately captures the central claims and constructions presented in the work.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit model constructions

full rationale

The paper builds concrete lattice Hamiltonians for rotational, inversion, and bismuth-type THOTIs plus the trivial insulator case. It treats the anisotropic surface-gapping mass terms as perturbations and reports (via phase diagrams and mode spectra) that vortex critical points μ_c^(1) and μ_c^(2) remain essentially unchanged with no extra transitions. The persistence of Z2-protected MVEMs in the window between those points, even when all surfaces are gapped with uniform sign, follows directly from these model calculations rather than from any quantity defined in terms of itself, any fitted parameter renamed as a prediction, or any load-bearing self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard time-reversal symmetry, the existence of a superconducting proximity effect, and the assumption that the added mass terms remain perturbative; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Time-reversal symmetry is preserved by the added surface mass terms.
    Stated in the model-construction sentence of the abstract.
  • domain assumption The superconducting pairing is conventional and proximity-induced.
    Implicit in the study of Majorana vortex phases.

pith-pipeline@v0.9.0 · 5797 in / 1247 out tokens · 32436 ms · 2026-05-22T13:05:26.559952+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We construct THOTIs as two copies of a topological insulator (TI) with time-reversal symmetry-preserving mass terms that anisotropically gap the surface states... MVEMs persist even when all surfaces are gapped with the same sign, rendering the system topologically trivial in both first- and second-order classifications.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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