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arxiv: 2604.22553 · v2 · submitted 2026-04-24 · ❄️ cond-mat.mes-hall · cond-mat.supr-con· quant-ph

Corner Majorana states in semi-Dirac materials

M. Garc\'ia Olmos , Y. Baba , R. A. Molina , M. Amado This is my paper

Pith reviewed 2026-05-08 10:11 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-conquant-ph
keywords semi-Dirac materialsMajorana bound statestopological superconductivityedge statesRashba spin-orbit couplingZeeman fieldKitaev chainscorner modes
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The pith

Semi-Dirac materials realize Majorana bound states at the corners of finite samples through their non-chiral edge states.

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

The paper shows that two-dimensional semi-Dirac systems possess non-chiral edge states that propagate only along specific directions, forming separate one-dimensional channels. When Rashba spin-orbit coupling and a Zeeman field are applied together with proximity to an s-wave superconductor, these channels acquire an effective p-wave pairing. In finite geometries such as strips, each edge undergoes a topological phase transition independently, resulting in zero-energy Majorana modes localized at the four corners. At low energies, the system maps onto coupled Kitaev chains, which explains the origin and robustness of these corner states. This approach avoids the need for artificial nanostructures or higher-order topological mechanisms.

Core claim

Finite geometries of semi-Dirac materials with Rashba spin-orbit coupling, Zeeman field, and s-wave superconducting proximity support four zero-energy modes localized at the corners. These arise because the non-chiral edge states form independent one-dimensional channels that each transition into a topological p-wave superconductor. The low-energy subspace is described by coupled Kitaev chains.

What carries the argument

Non-chiral edge states of the semi-Dirac system, which form independent one-dimensional channels that undergo a topological transition to p-wave superconductivity under Rashba and Zeeman fields with proximity pairing.

If this is right

  • Each of the four edges in a rectangular geometry independently hosts a one-dimensional topological superconductor.
  • Zero-energy Majorana bound states appear localized at the corners of the finite sample.
  • The modes are tunable by adjusting the strength of the Rashba and Zeeman fields.
  • The system admits an exact mapping to coupled Kitaev chains at low energies.
  • Robustness follows from the topological nature of the 1D channels.

Where Pith is reading between the lines

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

  • Similar corner Majorana states might appear in other anisotropic semimetals with comparable edge-state structures.
  • Experimental realization could simplify device fabrication by using natural material edges instead of engineered junctions.
  • Transport signatures such as zero-bias peaks at corners would confirm the presence of these modes.

Load-bearing premise

The non-chiral edge states behave as independent one-dimensional channels that acquire effective p-wave pairing from the proximity effect without requiring further microscopic details.

What would settle it

Spectroscopic measurements on a finite semi-Dirac strip showing or lacking zero-energy states localized specifically at the corners when Rashba, Zeeman, and superconducting proximity are applied.

Figures

Figures reproduced from arXiv: 2604.22553 by M. Amado, M. Garc\'ia Olmos, R. A. Molina, Y. Baba.

Figure 1
Figure 1. Figure 1: FIG. 1. Electronic band structure of a semi-infinite system in view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Band dispersion of the semi-infinite system in the view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Panels (a) and (b) display the bands of a system with translational invariance along view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spatially resolved Majorana polarization (MP). The arrows indicate the real and imaginary part of MP in each site. view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Gap size view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Integrated LDOS of the four central states within the fixed corner mask as a function of the disorder strength view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Bulk spectrum of a semi-Dirac system in terms of the sgn( view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Electronic band structure of a semi-infinite system in view at source ↗
read the original abstract

Proximity-induced superconductivity in low-dimensional systems offers a powerful pathway to engineer topological superconducting phases in, otherwise, non-superconducting systems. These exotic phases are of fundamental and technological interest due to the presence of robust zero-energy modes, the Majorana bound states. In this work, we propose a theoretical framework to realize Majorana bound states from the edge states of a two-dimensional semi-Dirac system. This anisotropic system, under specific conditions, can host non-chiral edge states that propagate only along particular edges, effectively forming separated one-dimensional channels. We show that the interplay between Rashba spin-orbit coupling and a Zeeman field on this setup provides the right conditions to get an effective p-wave pairing between the edge states by proximity with a s-wave superconductor. In finite geometries, each edge can independently undergo a topological phase transition into a one-dimensional topological superconductor and give rise to four zero-energy modes localized at the strip corners. At low energies, the edge states subspace admits a description in terms of coupled Kitaev chains, providing a clear picture of the origin, robustness, and tunability of the corner Majorana modes. Our results establish semi-Dirac materials as a natural platform for realizing Majorana modes in two dimensions without relying on engineered nanostructures, vortices, or crystalline higher-order topology.

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 proposes realizing corner Majorana bound states in finite 2D semi-Dirac systems. Non-chiral edge states, under Rashba SOC, Zeeman field, and proximity s-wave superconductivity, are claimed to form independent 1D channels that each undergo a topological transition to an effective p-wave superconductor. At low energies these are mapped to coupled Kitaev chains, producing four zero-energy corner modes without engineered nanostructures, vortices, or crystalline higher-order topology.

Significance. If the effective low-energy description is valid, the work identifies semi-Dirac anisotropy as a route to 2D Majorana modes that exploits intrinsic material properties rather than external engineering. It extends standard BdG/Kitaev constructions to a new class of anisotropic 2D systems and could motivate experimental searches in candidate materials such as certain transition-metal dichalcogenides or engineered heterostructures.

major comments (2)
  1. [low-energy effective model / edge-state subspace description] The central claim that proximity-induced pairing yields an effective momentum-independent p-wave gap on the edge states (leading to independent Kitaev chains) is load-bearing yet lacks an explicit projection onto the edge subspace. The semi-Dirac anisotropy implies direction-dependent penetration depths and density of states; without a calculation of the induced pairing matrix elements (e.g., via integration over bulk states or numerical diagonalization of the full BdG Hamiltonian), residual bulk hybridization or momentum dependence cannot be ruled out. This directly affects the asserted topological phase transition and corner-mode robustness.
  2. [finite-geometry results / corner-mode localization] The manuscript asserts that the four corner modes remain protected and decoupled across the parameter regime where each edge is topological. However, no explicit check is provided for inter-edge coupling or finite-size hybridization when the strip width is varied, nor for the effect of the quadratic dispersion direction on the localization length of the Majorana wavefunctions. Such a check is required to confirm that the modes are truly corner-localized zero-energy states rather than gapped or delocalized.
minor comments (2)
  1. [Abstract] The abstract states that the edge states 'propagate only along particular edges'; a brief clarification of which crystallographic directions support these non-chiral modes would aid readability.
  2. [Model Hamiltonian] Notation for the semi-Dirac Hamiltonian parameters (e.g., the coefficients of linear vs. quadratic terms) should be defined once at first use and used consistently in the effective-model section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below, providing clarifications and indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [low-energy effective model / edge-state subspace description] The central claim that proximity-induced pairing yields an effective momentum-independent p-wave gap on the edge states (leading to independent Kitaev chains) is load-bearing yet lacks an explicit projection onto the edge subspace. The semi-Dirac anisotropy implies direction-dependent penetration depths and density of states; without a calculation of the induced pairing matrix elements (e.g., via integration over bulk states or numerical diagonalization of the full BdG Hamiltonian), residual bulk hybridization or momentum dependence cannot be ruled out. This directly affects the asserted topological phase transition and corner-mode robustness.

    Authors: We agree that an explicit projection onto the edge subspace would strengthen the presentation of the effective model. In the original manuscript, we derived the low-energy effective Hamiltonian by focusing on the non-chiral edge states and incorporating the effects of Rashba SOC, Zeeman field, and proximity-induced s-wave pairing, leading to the mapping to coupled Kitaev chains. However, to address this concern rigorously, we will add in the revised version an explicit calculation of the induced pairing matrix elements. This will involve projecting the full Bogoliubov-de Gennes Hamiltonian onto the edge-state subspace, confirming the momentum-independent p-wave gap and demonstrating that bulk hybridization effects are negligible in the relevant parameter regime. We believe this addition will solidify the validity of the topological phase transition and the robustness of the corner modes. revision: yes

  2. Referee: [finite-geometry results / corner-mode localization] The manuscript asserts that the four corner modes remain protected and decoupled across the parameter regime where each edge is topological. However, no explicit check is provided for inter-edge coupling or finite-size hybridization when the strip width is varied, nor for the effect of the quadratic dispersion direction on the localization length of the Majorana wavefunctions. Such a check is required to confirm that the modes are truly corner-localized zero-energy states rather than gapped or delocalized.

    Authors: We acknowledge the importance of verifying the localization and decoupling of the corner modes in finite geometries. The manuscript presents results for finite systems showing four zero-energy modes at the corners, but we agree that additional checks would be beneficial. In the revised manuscript, we will include systematic studies varying the strip width to examine inter-edge coupling and finite-size effects on the energy splitting. Additionally, we will analyze the localization length of the Majorana wavefunctions, particularly considering the anisotropic dispersion, by computing the decay profiles along the edges. These additions will confirm that the modes remain localized at the corners and protected in the topological regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper applies the standard Bogoliubov-de Gennes formalism and effective Kitaev-chain mapping to the non-chiral edge states of a semi-Dirac Hamiltonian under Rashba+Zeeman fields plus s-wave proximity. The low-energy reduction to coupled 1D topological superconductors follows directly from projecting the microscopic Hamiltonian onto the edge subspace; no equation equates a derived quantity to a fitted parameter or prior self-citation by construction. The topological phase transition and corner-mode counting are standard consequences of the Kitaev model once the effective p-wave pairing is assumed, and the paper supplies no load-bearing uniqueness theorem or ansatz imported from the authors' prior work. The derivation is therefore self-contained against external benchmarks (known 1D topological superconductivity) and receives the default non-circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal relies on standard condensed-matter assumptions about edge states in semi-Dirac systems and the validity of effective 1D topological models; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption Existence of non-chiral, direction-selective edge states in 2D semi-Dirac systems under appropriate conditions
    Invoked to form separated 1D channels that can host independent topological transitions.
  • standard math Standard Bogoliubov-de Gennes treatment of proximity-induced superconductivity
    Used to obtain effective p-wave pairing from s-wave superconductor plus Rashba and Zeeman terms.

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