---
title: Majorana Hinge Modes in Topological Superconductors
url: https://www.emergentmind.com/topics/majorana-hinge-modes
type: topic
---

# Majorana Hinge Modes in Topological Superconductors

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 [1812.10493, 2010.15633, 1905.08896, 2310.09258].

## 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 [1905.08896, 1812.10493].

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 [2211.00682]. 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 [2303.11729, 1901.07579].

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 [2303.11729, 1901.07579, 2107.02811]. 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 [1807.04141].

## 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 [2310.09258]. 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 [1812.10493]. 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 \(z\) [1905.08896]. 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 [2010.15633].

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 \(V_\perp=V\cos\theta\), positive on one hemisphere and negative on the other, so the equator becomes a closed hinge supporting a single chiral Majorana mode [2310.09258]. In antiferromagnetic topological insulators proximitized by \(s\)-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 [1809.09112].

A lower-dimensional prototype appears in S/TI/S \(\pi\)-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 [2209.14885].

## 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, \(s+id\) Dirac superconductors, AFMTI/superconductor structures, and the curved-surface sphere construction [1905.08896, 2010.15633, 1809.09112, 2310.09258]. **Helical hinge modes** occur in class DIII, as in iron-based superconductors with \(s_\pm\) pairing and superconducting Dirac semimetals with time-reversal-invariant \(s_\pm\) order [1812.10493, 2107.02811].

No single bulk invariant covers all known mechanisms. Instead, different models admit different higher-order diagnostics. In superconducting Dirac materials with intrinsic \(s+id\) pairing, the higher-order phase is characterized by a \(k_z\)-dependent quadrupole moment
$$
q_{xy}(k_z)= \frac{1}{2\pi} \mathrm{Im}\log\!\left[ \mathrm{Det}\!\left(U_{k_z}^\dagger Q U_{k_z}\right) \sqrt{\mathrm{Det}Q^\dagger} \right],
$$
and its winding
$$
\Delta q_{xy} = \int_{-\pi}^{\pi} dk_z\, \partial_{k_z} q_{xy}(k_z),
$$
with \(\Delta q_{xy}=1\) in the nontrivial phase [2010.15633]. In second-order Dirac superconductors, a quadrupole-like invariant on fixed \(k_z\) slices,
$$
q_{xy}(k_z)\equiv \bigl(P_x(k_z)+P_y(k_z)-Q_c(k_z)\bigr)\bmod 1,
$$
diagnoses flat-band hinge Majoranas and their coexistence with surface states [1901.07579].

Momentum-slice topology is especially important in semimetal-derived superconductors. In type-II Dirac semimetals, the \(k_z=0,\pi\) planes carry a class-DIII \(\mathbb Z_2\) invariant, while generic gapped \(k_z\) slices are characterized by a nested Wilson-loop corner charge \(Q_c=0.5\); stacking those slices yields hinge Majorana flat bands [2303.11729]. 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 \(H_h(k_y)=\lambda k_y s_z\) [2107.02811].

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 \(\mathbb Z_2\), not an integer multiplicity of propagating hinge channels [2211.00682]. 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\(_{1-x}\)Se\(_x\) are modeled as topological-band-inverted materials with sign-changing \(s_\pm\) pairing, yielding helical hinge Majoranas without vortices [1812.10493]. Orthorhombic \(D_{2h}\) Dirac materials can realize a second-order superconducting \(s+id\) state with chiral hinge modes and gapless top and bottom surfaces [2010.15633]. Type-II Dirac semimetals with dominant inter-orbital \(B_{1u}/B_{2u}\) 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\(_3\), KMgBi, HfInPd\(_2\), and YPd\(_2\)Sn [2303.11729]. Superconducting Dirac semimetals with \(C_{4z}\) symmetry and \(s_\pm\)-wave pairing realize helical hinge modes when the normal side surfaces host both Fermi arcs and surface Dirac cones [2107.02811].

A second line uses **engineered heterostructures**. In proximitized 3D topological insulators, conventional \(s\)-wave pairing plus an in-plane Zeeman field generates chiral hinge modes once the Zeeman energy exceeds the induced pairing gap on selected surfaces [1905.08896]. 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 [1907.02070]. Antiferromagnetic topological insulators such as the MnBi\(_2\)Te\(_4\) 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 \(\sigma_{12}=e^2/2h\) [1809.09112].

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 [2310.09258]. In chiral higher-order topological insulators proximitized by ordinary \(s\)-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 [2607.04358]. Two-dimensional S/TI/S \(\pi\)-junctions and Majorana-corner platforms remain important because they provide a directly solvable lower-dimensional analogue of the same edge-mass mechanism [2209.14885].

## 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 [2310.09258]. In FeTe\(_{1-x}\)Se\(_x\)-type models, rod geometries show linearly dispersing hinge bands localized at the intersections of top/bottom and side surfaces [1812.10493]. In type-II Dirac semimetals, the hinge spectrum can be a flat zero-energy band across the full \(z\)-directed hinge Brillouin zone, connected to the projections of surface helical Majorana cones at \(k_z=0,\pi\) [2303.11729]. 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 [1901.07579].

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 \(H_{\rm hinge}(k_z)=k_z^2\sigma^z\) at criticality, with zero modes only at \(k_z^\ast=\pm\sqrt{|\mu-1|}\) below the transition [2211.00682]. 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 [1812.10493, 2005.03603]. 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 \(\sigma_{12}=e^2/2h\) in a clean idealized setup [1809.09112]. The curved-surface proposal points toward equatorial thermal transport and higher-order Fabry-Pérot or Mach-Zehnder-type interferometers as natural follow-ups [2310.09258].

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 [1907.02070]. 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 [2607.04358]. 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 [2101.05978].

## 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 [2105.11489]. 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 [2211.00682].

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 [2310.09258]. The iron-based-superconductor proposal uses a minimal model and leaves more realistic band structures, disorder, and spectroscopy largely open [1812.10493]. AFMTI/superconductor hinge proposals model proximity pairing phenomenologically and do not resolve interface-specific microscopic details [1809.09112]. 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 \(\mathsf C_{2n}\mathcal T\) 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 [1905.11421]. More recently, coupled-nanowire constructions have generalized the noninteracting helical Majorana hinge phase to interacting \(\mathbb Z_{2q}\) parafermionic hinge states; the Majorana case is recovered in the noninteracting limit \(q=1\) [2604.07313].

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.

Source: https://www.emergentmind.com/topics/majorana-hinge-modes