Quasi-two-dimensional Majorana zero modes from finite-size-coupled chiral hinge states

Abstract: Majorana zero modes (MZMs) in topological superconductors have attracted broad research interest for their potential applications in topological quantum computation. In this work, we propose a quasi-two-dimensional route to realize spatially separated MZMs in a chiral higher-order topological insulator (HOTI) proximitized by a conventional $s$-wave superconductor through a theoretical model study. In three dimensions, the chiral HOTI hosts gapless hinge states along the $z$ direction, arising from a mass term that anisotropically gaps the surface Dirac cones of a topological insulator. By confining the sample along the $x$ direction while keeping it extended along $y$ and finite along $z$, opposite $z$-directed chiral hinge states hybridize and effectively form one-dimensional helical channels. Incorporating the superconducting proximity effect into this quasi-two-dimensional system induces effective $p$-wave pairing in these helical channels, thereby opening a topological gap. A fully open-boundary sample then hosts four localized MZMs, one at each endpoint of the helical channels, realizing a second-order topological superconductor characterized by Majorana corner modes. In addition to MZMs, we also find that superconducting pairing in this model produces extended Majorana hinge modes in three dimensions. Furthermore, representative disorder calculations indicate that these Majorana corner modes are robust against weak-to-moderate disorder, provided the excitation gap remains open. These results establish finite-size-coupled chiral hinge states as a promising platform for engineering multiple MZMs via conventional superconducting proximity effect.

• The paper presents a theoretical model showing that finite-size hybridization of chiral hinge states in a HOTI yields spatially separated Majorana zero modes.
• The method employs numerical diagonalization under s-wave superconductivity to reveal a robust second-order topological superconducting phase with four localized zero modes.
• The study demonstrates the spectral isolation and moderate disorder resilience of the MZMs, underscoring their potential for scalable topological quantum computing.

Quasi-Two-Dimensional Majorana Zero Modes from Finite-Size-Coupled Chiral Hinge States

Overview

This paper presents a theoretical construction for realizing spatially separated Majorana zero modes (MZMs) in a quasi-2D geometry by leveraging finite-size-coupled chiral hinge states of a higher-order topological insulator (HOTI) proximitized by a conventional $s$-wave superconductor. The realized system exhibits a rich interplay between finite-size hybridization and superconducting pairing, enabling robust second-order topological superconductivity and corner-localized MZMs. The model elucidates the spectral, spatial, and disorder-properties of these emergent MZMs and establishes parameter regimes supporting their existence.

Model and Mechanism

The primary structure comprises a HOTI slab, thin along $x$, extended along $y$, and finite along $z$, in proximity to an $s$-wave superconductor.

Figure 1: Schematic of the quasi-two-dimensional HOTI-superconductor geometry, showing the formation of hinge states and their hybridization across the thin $x$ direction.

The normal-state HOTI is captured by a four-band Dirac model with an additional anisotropic mass term that gaps the conventional surface Dirac cones while enforcing mass domain walls at the hinges, producing chiral $z$-propagating hinge states. The key technical element is the symmetry-protected mass configuration, governed by combined $\mathcal{C}_{4z} \mathcal{T}$ symmetry, generating a mass contrast between adjacent facets.

Numerical diagonalization in a $z$-periodic geometry reveals that the hinge states are spectrally isolated from the bulk and strictly localized to system corners in the $x$-$x$0 plane.

Figure 2: Spectrum and probability densities show diagonal hinge localization for states marked A and B, emphasizing the HOTI boundary structure.

Upon application of $x$1-wave superconductivity, the induced pairing gaps the $x$2 surface, retaining the chiral Majorana modes on side hinges, which are robust due to their topological origin and remain delocalized (propagating), as dictated by the unpaired chiral structure:

Figure 3: Spectrum of the BdG Hamiltonian illustrating chiral Majorana hinge states in the proximitized geometry.

Finite-Size Hybridization and Majorana Localization

The pivotal step is reduction of the slab thickness along $x$3. When $x$4 is decreased to a few lattice constants, the $x$5-oriented hinge modes, originally spatially separated, hybridize via finite-size tunneling. This hybridization reconstructs the low-energy boundary spectrum, yielding two spatially separated helical channels—each localized near one of the $x$6-edges.

Figure 4: Evolution from decoupled chiral hinge states in the thick limit to helical channel formation via finite-size-coupling, underpinning the emergence of spatially separated MZMs.

Superconducting pairing projected onto these quasi-1D helical channels induces an effective $x$7-wave gap, analogous to the Kitaev chain. For open $x$8 boundaries, four MZMs emerge, each localized at a channel endpoint (corners of the $x$9-$y$0 plane), thus realizing second-order topological superconductivity.

Spectral Evolution, Phase Diagram, and Corner Localization

A systematic scan of chemical potential $y$1 and pairing amplitude $y$2 elucidates the topological phase boundaries:

• The topological superconducting (TSC) regime exhibits a gap closure and reopening as $y$3 is tuned across the reconstructed hinge bands.
• In the TSC, the open geometry hosts four near-zero-energy states, with densities localized at $y$4-$y$5 corners.

  Figure 5: (a)-(c) BdG spectral evolution at fixed $y$6 with increasing $y$7, showing topological transition; (d)-(g) Open-boundary spectra and projected densities confirming localization of MZMs at system corners.

The phase diagram in the $y$8 plane indicates a broad region supporting four robust MZMs.

Figure 6: TSC phase occupies a finite area in parameter space; gap-closing lines indicate transition boundaries to the nontopological SC regime.

Unlike conventional 1D or simple QAHC-based platforms, this HOTI construction inherently realizes four spatially non-overlapping MZMs, relevant for braiding and topological quantum computation.

Robustness to Disorder

Disorder resilience is crucial for practical applications. The study incorporates nonmagnetic onsite potential and orbital-mass disorder (random local chemical potential and band inversion strength) and analyzes their impact on the open-boundary spectra:

Figure 7: Representative low-energy spectra under increasing disorder strength; the MZMs persist and remain spectrally isolated for moderate disorder, with eventual collapse only at high disorder.

The results demonstrate that, as long as the excitation gap does not close, the MZMs are reasonably immune to both classes of disorder, a feature stemming from their topological and spatial separation.

Implications and Outlook

This construction leverages crystalline HOTI materials and conventional superconducting proximity—both accessible with current experimental resources (e.g., magnetically-doped topological insulators, axion insulators, and $y$9-wave superconductors). The modular architecture, yielding four robust and physically separated MZMs in a quasi-2D platform, presents several practical and theoretical implications:

• Spectroscopic signatures: Localized zero-bias peaks at the corners should be directly accessible via STM.
• Scalable architectures: The design admits generalizations to arrays and networked qubit geometries.
• Robustness to imperfections: Disorder tolerance and geometric flexibility facilitate device-level implementation.
• Analogy and beyond QAHC approaches: While reminiscent of narrow QAHC strips, the higher-order boundary structure delivers enhanced phase stability and multichannel MZMs, opening broader parameter windows than QAHC/standard TIs.
• Future directions: Further exploration is warranted on braiding protocols for the four-cornered MZMs, multi-terminal transport, and generalizations to interacting or multi-orbital HOTI platforms.

Conclusion

The paper establishes a concrete model for engineering quasi-2D, spatially isolated Majorana zero modes by exploiting finite-size-coupled chiral hinge states in a higher-order topological insulator with superconducting proximity. The approach synthesizes crystalline-symmetry-protected topology, finite-size spectral engineering, and conventional superconductivity to generate robust, disorder-resistant MZMs with direct implications for topological quantum computation architectures.

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Reference: "Quasi-two-dimensional Majorana zero modes from finite-size-coupled chiral hinge states" (2607.04358)