Majorana Hinge Modes in Topological Superconductors

• Majorana hinge modes are one-dimensional Bogoliubov excitations localized at codimension-2 interfaces in 3D superconductors, arising from surface mass sign changes.
• They manifest as chiral or helical channels depending on symmetry breaking, with realizations in iron-based superconductors and engineered topological insulator heterostructures.
• Experimental probes like STM/STS and thermal transport, alongside diverse fabrication approaches, address their unique higher-order topological properties.

Majorana hinge modes are one-dimensional Bogoliubov boundary excitations localized on codimension-2 boundaries of three-dimensional superconducting systems. In the canonical second-order topological-superconductor setting, the bulk is gapped, the relevant two-dimensional boundary surfaces are gapped by symmetry-breaking or pairing-induced masses, and a gapless Majorana channel remains where adjacent surfaces meet. Across the literature, the term encompasses several distinct boundary phenomena: chiral class-D hinge channels, helical class-DIII hinge channels, flat zero-energy hinge bands, and symmetry-protected nodal hinge states that are not fully gapped one-dimensional Majorana channels in the usual sense (Zhang et al., 2018, Fu et al., 2020, Wu et al., 2019, Na et al., 2023).

1. Definition, scope, and taxonomy

Majorana hinge modes are best understood as “boundary-of-boundary” states. In a three-dimensional higher-order topological superconductor, a topological bulk does not necessarily leave gapless two-dimensional surface states; instead, lower-dimensional gapless excitations survive only on selected hinges. The most standard cases are chiral Majorana hinge modes, which propagate unidirectionally because time-reversal symmetry is broken, and helical Majorana hinge modes, which form Kramers-related counterpropagating pairs in time-reversal-invariant settings (Wu et al., 2019, Zhang et al., 2018).

The literature also uses nearby but nonidentical notions. In doped superconducting topological insulators, one finds mirror-protected nodal hinge states with symmetry-protected zero-energy crossings at finite hinge momentum; these are Majorana in the BdG sense, but they are not the same as a fully gapped second-order phase supporting isolated chiral or helical hinge channels (Ghorashi et al., 2022). In type-II Dirac semimetals and second-order Dirac superconductors, the hinge spectrum can appear as a flat zero-energy band or “hinge arc” spanning a finite or even the entire hinge Brillouin zone, coexisting with gapless surface structure rather than replacing it (Xie et al., 2023, Ghorashi et al., 2019).

A recurring misconception is that “Majorana hinge mode” always implies a fully surface-gapped second-order topological superconductor. Several constructions instead realize hybrid phases in which hinge modes coexist with surface Bogoliubov-Dirac cones, helical Majorana surface cones, or nodal superconductivity (Xie et al., 2023, Ghorashi et al., 2019, Kheirkhah et al., 2021). Another source of ambiguity is spatial terminology: in some higher-order topological insulators proximitized by magnetism and superconductivity, the low-energy Majorana channels can split away from a geometric hinge and circulate around proximitized or magnetized surface regions near that hinge, rather than remaining pinned to the hinge line itself (Queiroz et al., 2018).

2. Boundary-mass domain walls as the central mechanism

The dominant mechanism is a Dirac-mass sign reversal on neighboring surfaces. A particularly transparent prototype is the surface Majorana theory of a class-DIII topological superconductor,

$$H=-iv(\partial_x\sigma_y-\partial_y\sigma_x)+V(y)\sigma_z,\qquad V(y)=\operatorname{sgn}(y)V_z,$$

for which the sign change at $y=0$ binds a chiral Majorana channel by the usual Jackiw-Rebbi logic (Na et al., 2023). The same logic underlies many later constructions: start from a gapless surface Dirac or Majorana cone, gap adjacent surfaces with masses of opposite sign, and the hinge becomes the one-dimensional domain wall.

Different papers implement the required mass pattern in different ways. In iron-based superconductors, projecting extended $s_\pm$ pairing onto topological surface Dirac states yields an orientation-dependent superconducting mass $\Delta_{\rm eff}(\theta)$; a hinge hosts helical Majorana modes whenever

$$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$

so that adjacent facets inherit opposite pairing signs (Zhang et al., 2018). In topological-insulator heterostructures with conventional $s$-wave pairing, an in-plane Zeeman field modifies the effective surface masses differently on different faces; for a 3D TI, $h_x>\Delta_0$ drives a surface-mass inversion and produces chiral hinge modes propagating along $z$ (Wu et al., 2019). In superconducting Dirac materials with intrinsic $s+id$ pairing, the $d_{xy}$ component gaps side-surface Majorana cones with alternating sign, generating chiral hinge modes without external fields (Fu et al., 2020).

The same principle extends beyond flat crystalline facets. On a spherical surface of a class-DIII topological superconductor, a uniform Zeeman field along the polar axis produces a local normal component $y=0$0, positive on one hemisphere and negative on the other, so the equator becomes a closed hinge supporting a single chiral Majorana mode (Na et al., 2023). In antiferromagnetic topological insulators proximitized by $y=0$1-wave superconductors, type-F surfaces are magnetically gapped while type-A surfaces are superconductively gapped; their common hinge is then a magnetic–superconducting mass interface carrying a chiral Majorana channel (Peng et al., 2018).

A lower-dimensional prototype appears in S/TI/S $y=0$2-junctions. There, an in-plane Zeeman field gaps helical Majorana edge modes with edge-dependent masses; corners where the mass flips sign bind Majorana zero modes. This two-dimensional corner mechanism is the codimension-reduced analogue of hinge Majoranas in three dimensions (Chen et al., 2022).

3. Symmetry classes and topological characterization

The symmetry class depends on whether time reversal survives. Chiral hinge modes typically occur in class D after explicit or spontaneous time-reversal breaking, as in Zeeman-gapped TI heterostructures, $y=0$3 Dirac superconductors, AFMTI/superconductor structures, and the curved-surface sphere construction (Wu et al., 2019, Fu et al., 2020, Peng et al., 2018, Na et al., 2023). Helical hinge modes occur in class DIII, as in iron-based superconductors with $y=0$4 pairing and superconducting Dirac semimetals with time-reversal-invariant $y=0$5 order (Zhang et al., 2018, Kheirkhah et al., 2021).

No single bulk invariant covers all known mechanisms. Instead, different models admit different higher-order diagnostics. In superconducting Dirac materials with intrinsic $y=0$6 pairing, the higher-order phase is characterized by a $y=0$7-dependent quadrupole moment

$y=0$8

and its winding

$y=0$9

with $s_\pm$0 in the nontrivial phase (Fu et al., 2020). In second-order Dirac superconductors, a quadrupole-like invariant on fixed $s_\pm$1 slices,

$s_\pm$2

diagnoses flat-band hinge Majoranas and their coexistence with surface states (Ghorashi et al., 2019).

Momentum-slice topology is especially important in semimetal-derived superconductors. In type-II Dirac semimetals, the $s_\pm$3 planes carry a class-DIII $s_\pm$4 invariant, while generic gapped $s_\pm$5 slices are characterized by a nested Wilson-loop corner charge $s_\pm$6; stacking those slices yields hinge Majorana flat bands (Xie et al., 2023). In superconducting Dirac semimetals, the phase boundary can be read geometrically from the relative placement of the band-inversion surface, pairing-node surface, and Fermi surface, while the hinge theory itself reduces to a helical Majorana Hamiltonian $s_\pm$7 (Kheirkhah et al., 2021).

Nodal hinge states require separate treatment. In doped superconducting topological insulators, the finite-momentum hinge crossings are protected only when mirror symmetry, particle-hole symmetry, and translation along the hinge are all preserved; the resulting hinge-node classification is $s_\pm$8, not an integer multiplicity of propagating hinge channels (Ghorashi et al., 2022). This distinction is essential when comparing “hinge states” across different superconducting phases.

4. Material and platform realizations

One line of work seeks intrinsic higher-order superconductors. Iron-based superconductors such as FeTe$s_\pm$9Se$\Delta_{\rm eff}(\theta)$0 are modeled as topological-band-inverted materials with sign-changing $\Delta_{\rm eff}(\theta)$1 pairing, yielding helical hinge Majoranas without vortices (Zhang et al., 2018). Orthorhombic $\Delta_{\rm eff}(\theta)$2 Dirac materials can realize a second-order superconducting $\Delta_{\rm eff}(\theta)$3 state with chiral hinge modes and gapless top and bottom surfaces (Fu et al., 2020). Type-II Dirac semimetals with dominant inter-orbital $\Delta_{\rm eff}(\theta)$4 pairing support a hybrid first-/second-order phase with surface helical Majorana cones and hinge Majorana flat bands, and candidate materials discussed in that context include VAl$\Delta_{\rm eff}(\theta)$5, KMgBi, HfInPd$\Delta_{\rm eff}(\theta)$6, and YPd$\Delta_{\rm eff}(\theta)$7Sn (Xie et al., 2023). Superconducting Dirac semimetals with $\Delta_{\rm eff}(\theta)$8 symmetry and $\Delta_{\rm eff}(\theta)$9-wave pairing realize helical hinge modes when the normal side surfaces host both Fermi arcs and surface Dirac cones (Kheirkhah et al., 2021).

A second line uses engineered heterostructures. In proximitized 3D topological insulators, conventional $$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$0-wave pairing plus an in-plane Zeeman field generates chiral hinge modes once the Zeeman energy exceeds the induced pairing gap on selected surfaces (Wu et al., 2019). Replacing the parent TI by a second-order topological insulator yields SOTI/superconductor heterostructures in which the preexisting higher-order boundary structure is converted directly into chiral Majorana hinge channels (Yan, 2019). Antiferromagnetic topological insulators such as the MnBi$$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$1Te$$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$2 family provide a materials-based magnetic platform in which superconducting type-A surfaces meet magnetically gapped type-F surfaces, generating chiral hinge Majoranas and a predicted two-terminal conductance $$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$3 (Peng et al., 2018).

A third line exploits boundary engineering and geometry. On a sphere, magnetic impurities and their RKKY-driven radial polarization create the sign-changing Majorana mass texture needed for an equatorial chiral hinge mode while mitigating orbital magnetic effects (Na et al., 2023). In chiral higher-order topological insulators proximitized by ordinary $$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$4-wave superconductors, three-dimensional superconducting geometries support extended Majorana hinge modes, whereas thin samples can hybridize opposite hinge states into effective helical channels that terminate in four Majorana zero modes in the quasi-two-dimensional limit (Liang et al., 5 Jul 2026). Two-dimensional S/TI/S $$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$5-junctions and Majorana-corner platforms remain important because they provide a directly solvable lower-dimensional analogue of the same edge-mass mechanism (Chen et al., 2022).

5. Spectral phenomenology, probes, and device concepts

Spectrally, Majorana hinge modes appear as hinge-localized in-gap branches. On the curved sphere, a single chiral branch traverses the surface gap, the direct spectral manifestation of one equatorial Majorana channel (Na et al., 2023). In FeTe$$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$6Se$$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$7-type models, rod geometries show linearly dispersing hinge bands localized at the intersections of top/bottom and side surfaces (Zhang et al., 2018). In type-II Dirac semimetals, the hinge spectrum can be a flat zero-energy band across the full $$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$8-directed hinge Brillouin zone, connected to the projections of surface helical Majorana cones at $$\Delta_{\rm eff}(\theta_1)\Delta_{\rm eff}(\theta_2)<0,$$9 (Xie et al., 2023). In second-order Dirac superconductors, some pairing channels yield flat zero-energy hinge bands, while others generate Zeeman-induced dispersive helical hinge states that occupy only two of the four hinges and switch to the opposite hinge pair when the sign of the Zeeman term is reversed (Ghorashi et al., 2019).

Not every hinge-localized superconducting spectrum is a canonical hinge Majorana channel. In doped superconducting topological insulators, the hinge spectrum may instead consist of finite-momentum nodal crossings described by $s$0 at criticality, with zero modes only at $s$1 below the transition (Ghorashi et al., 2022). This nodal phenomenology has different symmetry requirements and different experimental implications from a single chiral or helical hinge branch crossing an otherwise open gap.

The proposed probes reflect that distinction. For fully or partially surface-gapped hinge phases, hinge-selective STM/STS is the most direct local diagnostic: surfaces away from the hinge should be gapped, while the hinge should remain gapless (Zhang et al., 2018, Wu et al., 2020). Thermal transport and interferometry recur in the literature because a one-dimensional chiral Majorana channel is expected to carry neutral heat current along the hinge; the AFMTI proposal gives an explicit conductance interferometer and a half-quantized two-terminal value $s$2 in a clean idealized setup (Peng et al., 2018). The curved-surface proposal points toward equatorial thermal transport and higher-order Fabry-Pérot or Mach-Zehnder-type interferometers as natural follow-ups (Na et al., 2023).

Several device concepts exploit the relation between hinge channels and lower-dimensional Majoranas. In SOTI/superconductor heterostructures, increasing the pairing strength can convert two chiral Majorana modes per hinge into a single robust chiral hinge mode per hinge through a boundary transition (Yan, 2019). In the finite-size-coupled HOTI proposal, opposite chiral hinge states hybridize into two helical channels; once proximitized, each channel becomes a topological wire and contributes a Majorana zero mode at each endpoint, yielding four corner Majoranas in a fully open sample (Liang et al., 5 Jul 2026). A 2021 study on orthorhombic HYLION-12 interpreted coherent quantum phase slip, a constant conductance plateau, and zero-bias conductance peaks as signatures of Majorana hinge and corner modes without magnetic field and at room temperature, but this remains a claim specific to that work rather than an established consensus platform (Choi et al., 2021).

6. Conceptual boundaries, open issues, and interacting extensions

The term “Majorana hinge mode” is often used broadly, but the literature draws several careful boundaries. First, normal-state hinge states or superconducting bulk Majorana nodes are not automatically hinge Majorana channels. In SnTe nanowires, ordinary hinge states appear in the normal state, and superconductivity produces inversion-protected gapless bulk Majorana modes that gap into end-localized Majorana zero modes after inversion breaking; that work is adjacent to hinge-Majorana physics but is not a canonical realization of propagating superconducting hinge Majorana channels (Nguyen et al., 2021). Second, nodal hinge states at finite momentum are distinct from fully gapped higher-order phases, even though both can be hinge-localized and Majorana in BdG language (Ghorashi et al., 2022).

Recurring open problems are also clear. Materials-specific proposals often rely on simplified low-energy Dirac theories, phenomenological pairing terms, idealized surface terminations, or semiclassical magnetic textures. The spherical magnetic-impurity construction, for example, treats impurity moments semiclassically and does not solve the orbital screening problem self-consistently (Na et al., 2023). The iron-based-superconductor proposal uses a minimal model and leaves more realistic band structures, disorder, and spectroscopy largely open (Zhang et al., 2018). AFMTI/superconductor hinge proposals model proximity pairing phenomenologically and do not resolve interface-specific microscopic details (Peng et al., 2018). This suggests that the field’s central conceptual mechanism is now well established, whereas quantitative materials validation remains highly platform dependent.

Interactions extend the subject beyond free Majorana channels. A striking result is that chiral Majorana hinge modes protected by $s$3 can be completely gapped, while preserving the protecting symmetry, by placing alternating time-reversal-conjugate non-Abelian surface topological orders on the side surfaces. In that setting, the hinge anomaly is absorbed by the anomalous surface topological order rather than by a free gapless Majorana channel (Tiwari et al., 2019). More recently, coupled-nanowire constructions have generalized the noninteracting helical Majorana hinge phase to interacting $s$4 parafermionic hinge states; the Majorana case is recovered in the noninteracting limit $s$5 (Girdhar et al., 8 Apr 2026).

Taken together, these developments place Majorana hinge modes at the intersection of higher-order bulk-boundary correspondence, surface mass engineering, superconducting symmetry class, and crystalline geometry. What remains constant across otherwise diverse realizations is the codimension-2 principle: a three-dimensional topological superconductor can convert a pattern of gapped surfaces into a one-dimensional neutral channel on the line where incompatible surface terminations meet.