Chiral Majorana Modes in Topological Superconductors

• Chiral Majorana modes are one-dimensional, charge-neutral quasiparticles that are their own antiparticles and propagate unidirectionally along edges or domain walls.
• They emerge in various platforms such as QAHI–superconductor heterostructures, higher-order systems, and pairing-density-wave phases, highlighting diverse topological mechanisms.
• Their coherent propagation enables non-Abelian operations and quantum interference, making them promising for topological quantum computing and advanced transport studies.

Chiral Majorana modes (CMMs) are one-dimensional, propagating Bogoliubov quasiparticles of topological superconducting systems that are their own antiparticles and move unidirectionally along an edge, hinge, domain wall, or vortex line. In the literature represented here, they appear in 2D chiral topological superconductors, at magnetic boundaries of superconducting topological-insulator surfaces, on hinges of second-order topological superconductors, on domain walls between opposite chiralities, and along vortex defects in pairing-density-wave phases. Their appeal is twofold: they encode nontrivial bulk topology through boundary dynamics, and their coherent propagation can implement non-Abelian transformations closely related to braiding protocols for Majorana zero modes (Hirayama et al., 2023, Lian et al., 2017, Fu et al., 2020, Chan et al., 2016).

1. Quasiparticle definition and elementary structure

The defining property of a Majorana quasiparticle is particle-hole self-conjugacy. In the BdG description used for CMMs, this is expressed as

$$\gamma(\epsilon)=\gamma^\dagger(-\epsilon), \qquad \gamma(0)=\gamma^\dagger(0),$$

so the zero-energy excitation is its own antiparticle (Hirayama et al., 2023). For a propagating chiral edge channel, the effective edge Hamiltonian can be written as

$H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$

with $\gamma(x)=\gamma^\dagger(x)$ and $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$ (Lian et al., 2017). This formulation makes explicit that a CMM is massless and unidirectional.

A recurring structural point is that a charged chiral fermion can be decomposed into two Majorana components. In QAHI-based constructions one writes, for example,

$\psi_A=\gamma_1+i\gamma_2,$

and the device physics follows from how those Majorana components split, propagate, and recombine (Lian et al., 2017). Scattering theory sharpens this further: in a normal lead written in a Majorana basis, each electron-hole channel becomes a pair of artificial Majorana modes, but only one Majorana mode from each pair couples to the intrinsic CMM, while its partner is fully reflected. Charge current is therefore not transported by a single neutral Majorana alone; it emerges from interference between Majorana pairs, through terms such as $\Re(r_1r_2^\ast)$ in the current formula (Li et al., 2012).

This immediately distinguishes CMMs from both ordinary chiral Dirac modes and localized Majorana zero modes. A chiral Dirac mode is a charged electronic channel, while a CMM is charge neutral in the BdG sense. A Majorana zero mode is a zero-dimensional bound state, typically at a vortex core or wire end, whereas a CMM is a one-dimensional propagating boundary excitation with linear dispersion (Chiu et al., 2016, Cao et al., 18 Sep 2025).

2. Topological mechanisms and invariants

In the 2D class-D setting, the standard bulk invariant is an integer Chern number, denoted ${\cal C}$, $\mathcal{N}$, or $N$ in the papers summarized here. The number of chiral Majorana edge channels equals the magnitude of that invariant, and phase boundaries occur when the bulk gap closes (Osca et al., 2018, He et al., 2019, Chiu et al., 2016). This is the organizing principle behind QAHI–superconductor heterostructures, superconducting topological-insulator surfaces with ferromagnetic overlayers, and higher-$N$ chiral topological superconductors with multiple co-propagating Majorana channels (Wang et al., 2018).

A distinct route uses crystalline topology rather than a strong $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$0 topological-insulator index. In hcp Tl, the $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$1 plane is a mirror plane, and spin-orbit coupling gaps a nodal-line structure to produce a topological crystalline insulating phase with nontrivial mirror Chern number

$H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$2

while the ordinary $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$3 invariant on that plane is trivial, $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$4. In that case the surface Dirac cones are enforced by mirror symmetry, and superconductivity plus a Zeeman field convert the surface into a topological superconducting phase that supports a hinge-localized CMM (Hirayama et al., 2023).

Higher-order realizations replace gapless 2D surfaces by gapped surfaces whose mass terms change sign. In superconducting Dirac materials with $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$5 symmetry, a mixed $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$6 ($H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$7) state is a second-order topological superconductor: the odd-parity $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$8-wave component yields Majorana surface states, the $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$9 component gaps side surfaces as a mass term, and sign changes of that mass bind chiral Majorana hinge modes. The bulk characterization is given by the winding of a quadrupole moment $\gamma(x)=\gamma^\dagger(x)$0, with $\gamma(x)=\gamma^\dagger(x)$1 in the topological phase (Fu et al., 2020). On a curved DIII superconducting surface, the same mass-domain-wall logic appears through an inversion-antisymmetric Zeeman mass $\gamma(x)=\gamma^\dagger(x)$2, which vanishes on the equator and leaves a 1D chiral Majorana channel on the nodal line (Na et al., 2023).

An even more unconventional mechanism appears in the pairing-density-wave phase of a time-reversal-breaking Weyl semimetal. There the fully gapped 3D bulk is topologically trivial in ordinary 3D class-D classification, but a vortex line in the PDW phase binds a CMM protected by an emergent second Chern number in synthetic $\gamma(x)=\gamma^\dagger(x)$3 parameter space: $\gamma(x)=\gamma^\dagger(x)$4 with the low-energy result $\gamma(x)=\gamma^\dagger(x)$5 (Chan et al., 2016).

3. Material and heterostructure platforms

The known proposals span proximity-based, intrinsic, higher-order, and defect-bound realizations.

Platform	Key ingredients	CMM location
QAHI + $\gamma(x)=\gamma^\dagger(x)$6-wave SC	QAH edge mode, BdG Chern phase	2D edge
STI surface + FI island	effective $\gamma(x)=\gamma^\dagger(x)$7 surface, exchange gap inversion	island edge
hcp Tl	mirror TCI surface state, $\gamma(x)=\gamma^\dagger(x)$8-wave SC, small Zeeman field	hinge
Dirac material $\gamma(x)=\gamma^\dagger(x)$9	$\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$0, odd-parity $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$1-wave + $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$2 mass	four $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$3-directed hinges
$\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$4 superconductor	boundary magnetization kink	surface domain wall
PDW Weyl semimetal	PDW phase, vortex winding, emergent $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$5	vortex line or ring

The QAHI–superconductor architecture remains the canonical 2D platform. In that setting the QAH edge channel fractionalizes into Majorana components at a superconducting interface, and the superconducting region enters trivial, $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$6, or $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$7 phases depending on model parameters (Osca et al., 2018, Yan et al., 2019). A closely related route uses the surface of a superconducting topological insulator. There the surface Dirac cone with induced $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$8-wave pairing behaves as an effective $\{\gamma(x),\gamma(x')\}=\delta(x-x')/2$9 superconductor, and a ferromagnetic overlayer or island adds an exchange field that can invert the superconducting gap. In the tight-binding study of a Bi$\psi_A=\gamma_1+i\gamma_2,$0Se$\psi_A=\gamma_1+i\gamma_2,$1-type STI with $\psi_A=\gamma_1+i\gamma_2,$2 eV and $\psi_A=\gamma_1+i\gamma_2,$3 meV, the gap closes at approximately $\psi_A=\gamma_1+i\gamma_2,$4 eV, beyond which the interface has $\psi_A=\gamma_1+i\gamma_2,$5 and the island edge hosts a single CMM. When the chemical potential is shifted to $\psi_A=\gamma_1+i\gamma_2,$6 eV, the same framework can yield two co-propagating CMMs (Chiu et al., 2016).

The FM/TI-thin-film/SC heterostructure reorganizes the same ingredients across two inequivalent TI surfaces: the bottom surface feels an exchange field and the top surface is proximitized by a superconductor. In the $\psi_A=\gamma_1+i\gamma_2,$7 limit and with $\psi_A=\gamma_1+i\gamma_2,$8-wave pairing, the phase with a single CMM occurs for $\psi_A=\gamma_1+i\gamma_2,$9, with $\Re(r_1r_2^\ast)$0. The paper emphasizes that, unlike earlier QAHI-based proposals, this platform does not require $\Re(r_1r_2^\ast)$1 and occupies “readily achievable parameter regions” (He et al., 2019).

The 2023 proposal for hcp thallium is notable because it integrates the topological precursor and superconductivity in a single element. The $\Re(r_1r_2^\ast)$2 mirror plane supports a large spin-orbit-driven inverted gap, the $\Re(r_1r_2^\ast)$3 surface hosts two inequivalent Dirac cones, and after surface relaxation one Dirac point lies about $\Re(r_1r_2^\ast)$4 meV from the Fermi level. Superconducting density functional theory gives a nearly momentum-independent $\Re(r_1r_2^\ast)$5-wave gap $\Re(r_1r_2^\ast)$6 meV and $\Re(r_1r_2^\ast)$7 K, consistent with experiment ($\Re(r_1r_2^\ast)$8 K). A small Zeeman field then drives the surface through a topological transition and produces a hinge-localized CMM on a structurally stable $\Re(r_1r_2^\ast)$9 hinge (Hirayama et al., 2023).

Beyond these platforms, twisted cuprate bilayers on Rashba substrates provide a ${\cal C}$0 route. In that proposal, a twisted cuprate bilayer near ${\cal C}$1 generates ${\cal C}$2 pairing, a one-sided Rashba substrate breaks the symmetries that otherwise force even Chern number, and an out-of-plane field yields a Chern-${\cal C}$3 topological superconductor with a single chiral Majorana edge mode and one Majorana zero mode per vortex (Margalit et al., 2022).

4. Boundary geometries: edges, hinges, domain walls, and vortices

The boundary manifestation of a CMM depends on how topology and symmetry-breaking masses terminate in real space. In 2D chiral topological superconductors the mode lives on the sample boundary. In the STI/FI construction, it propagates along the edge of a ferromagnetic island, with analytic interface solution

${\cal C}$4

for the localized CMM when the exchange field in the magnetized region exceeds the superconducting gap, ${\cal C}$5 (Chiu et al., 2016). In QAHI–superconductor junctions, the same edge localization appears as a mode on the superconducting side of a normal–superconductor interface (Osca et al., 2018).

Second-order realizations shift the locus from an edge to a hinge or nodal line. In superconducting Dirac materials with ${\cal C}$6 pairing, the chiral Majorana hinge dispersion is

${\cal C}$7

and the topological phase supports four chiral Majorana hinge modes along the hinges parallel to ${\cal C}$8, while the top and bottom ${\cal C}$9 surfaces remain gapless (Fu et al., 2020). In hcp Tl the relevant higher-order boundary is the hinge between the $\mathcal{N}$0 and $\mathcal{N}$1 surfaces, and the stability of that relaxed $\mathcal{N}$2 geometry is part of the physical argument for experimental viability (Hirayama et al., 2023). On a sphere decorated with magnetic impurities, the equator plays the role of a geometrically enforced hinge because the mass term $\mathcal{N}$3 changes sign there (Na et al., 2023).

Domain-wall realizations make the same sign-change mechanism explicit. In $\mathcal{N}$4 superconductors, surfaces become spontaneously spin-polarized, and a reversal of the sign of the $\mathcal{N}$5-wave component flips the surface magnetization. The resulting boundary kink is a magnetic domain wall that supports a single branch of dispersive CMMs, with velocity fixed by the chiral-symmetry index (Yang et al., 2017). In optically controlled chiral superconductors, locally switching the handedness of the condensate from $\mathcal{N}$6 to $\mathcal{N}$7 or from $\mathcal{N}$8 to its conjugate creates a domain wall hosting a pair of chiral Majorana edge modes; because the switched domain can persist after the pump is turned off, the domain-wall channel can also persist (Claassen et al., 2018).

Vortex defects provide a different one-dimensional geometry. In the PDW Weyl-semimetal proposal, a vortex line in one PDW component supports a dispersing chiral Majorana branch with

$\mathcal{N}$9

and a vortex ring carries discrete ring modes whose spectrum depends on the ring size and on whether the ring is threaded by an even or odd number of vortex lines (Chan et al., 2016).

5. Transport, interferometry, and external control

Transport signatures of CMMs are unusually subtle because a single CMM is charge neutral. In the scattering theory of chiral Majorana interferometry, a normal lead is decomposed into Majorana pairs, only one of which couples to the intrinsic CMM, and the current and noise are determined by Majorana interference rather than ordinary single-particle transmission. In Fabry–Perot geometries, loop resonances produce peaked Andreev differential conductances, anti-resonances suppress them, and the sign of current cross-correlations depends on vortex parity and on whether the loop is resonant (Li et al., 2012).

Concrete junction calculations show that conductance quantization does not map trivially onto “one mode equals half a conductance quantum.” In a normal–(QAH+SC) junction with a single outgoing CMM on the superconducting side, the incident electron is transmitted into the Majorana channel with $N$0, while $N$1 and $N$2, yet the conductance remains $N$3. For two CMMs ($N$4), the pair reconstructs an almost perfectly transmitted electron channel with $N$5, again giving $N$6. Orbital magnetic fields modify spatial overlap between the incident QAH edge state and the outgoing Majorana channel, producing oscillatory transmission and, at strong enough field, suppressing the chiral modes entirely (Osca et al., 2018).

Several proposals translate this interference physics into control protocols. In the Corbino QAHI–TSC–QAHI junction, coherent propagation of four chiral Majorana channels implements the same non-Abelian unitary as braiding four Majorana zero modes. In the odd-parity qubit sector the junction acts as $N$7, while a gate-defined phase shift

$N$8

produces a conductance readout

$N$9

The paper emphasizes that the oscillatory conductance, rather than a static plateau, is the signature of coherent gate action (Lian et al., 2017). Related QAHI–TSC devices use gate-induced dynamical phases to steer the outgoing CMM trajectory, yielding oscillatory transmission coefficients and conductances that are robust in period against disorder and gap fluctuations (Yan et al., 2019). An STM tip can regulate the phase of a chiral Majorana state between $N$0 and $N$1, toggling transport between perfect normal tunneling and perfect crossed Andreev reflection (Zhou et al., 2017).

Other transport diagnostics exploit quantities beyond electrical conductance. In Shiba islands, the low-bias differential charge conductance measured by an STM tip obeys

$N$2

while the direction of the persistent edge supercurrent tracks $N$3 (Rachel et al., 2017). For an $N$4 chiral topological superconductor obtained from a QAH insulator proximitized by a topological $N$5-wave superconductor, random SO(3) mixing of three chiral Majorana edge modes leads to an averaged two-terminal conductance

$N$6

and thermal Hall conductance $N$7, with $N$8 (Wang et al., 2018). In the Fu–Kane heterostructure, a supercurrent along the magnetic boundary can invert the chirality of a Majorana edge mode once the Cooper-pair momentum exceeds $N$9, while the bulk gap remains open until $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$00. The inversion is marked by $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$01 and by a finite electrical conductance carried by a Dirac mode that emerges when the edge switches chirality (Vela et al., 2021). In a QAHI–CTSC–QAHI Laughlin pump, unit charge requires one flux quantum when transport is mediated by a chiral Dirac mode or by an $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$02 CTSC edge, but an $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$03 CMM-mediated path requires fractional flux quanta set by device geometry and by the parity of enclosed vortices (Cao et al., 18 Sep 2025).

6. Conceptual distinctions, limitations, and active issues

Several recurrent misconceptions are addressed directly in the literature. First, a half-quantized electrical conductance is not a universal local signature of a single CMM. In the one-contact QAH–superconductor junction, a single outgoing CMM still gives $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$04, not $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$05, because the setup lacks the counterpropagating channels present in a two-terminal NSN geometry (Osca et al., 2018). Second, plateau values by themselves do not establish coherent quantum logic. In the Corbino proposal, incoherent dynamics can pin the conductance at $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$06; only oscillation with the gate-induced phase demonstrates coherent CMM propagation (Lian et al., 2017). Third, STM observation of a gapless island edge on an STI surface is not sufficient by itself to distinguish a CMM from a chiral electron mode; transport information is required (Chiu et al., 2016).

A more recent conceptual caution concerns the relation between Majorana self-conjugacy and chirality. In a QAHI–superconductor heterostructure at $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$07, the edge state may retain Majorana character, with equal electron and hole components at the edge, while its dispersion becomes nonlinear or braid-like. In the regime $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$08 within the $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$09 phase, the band can intersect the Fermi level three times, with both positive and negative group velocities on the same edge. The result is propagation in both directions, so the mode is Majorana-like but no longer strictly chiral (Yue et al., 6 Mar 2026). This separates “Majorana-ness” from “chirality” in a way absent from the simplest $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$10 picture.

Practical realizations are also constrained by geometry, stability, and finite-size effects. In hcp Tl, the claim of an experimentally actionable hinge CMM is tied to surface and hinge relaxation calculations showing that the $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$11 surface remains mirror-symmetric, the surface Dirac point stays near the Fermi level, and the $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$12 hinge is structurally stable (Hirayama et al., 2023). In STI/FI islands with realistic $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$13 meV, opposite-edge hybridization produces a finite-size Majorana gap that decays exponentially and oscillates with ribbon width; for $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$14, the residual gap is about $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$15 meV (Chiu et al., 2016). In QAHI-based devices, orbital magnetic effects can shift and compress edge wavefunctions enough to weaken coupling or eliminate the chiral channels entirely (Osca et al., 2018).

Current directions in the field therefore extend beyond simple edge-state detection. One direction seeks intrinsic or nearly intrinsic platforms, such as hcp Tl and interaction-driven $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$16 Dirac materials, to reduce heterostructure complexity (Hirayama et al., 2023, Fu et al., 2020). Another uses nonequilibrium control to write and erase domain walls that carry CMMs, as in ultrafast optical switching of chiral superconductors (Claassen et al., 2018). A third explores odd-Chern chiral superconductors beyond the $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$17 case, including twisted cuprate bilayers on Rashba substrates and topological $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$18-wave/QAH hybrids with $H_M(x)=-i\hbar v_F\,\gamma(x)\partial_x\gamma(x),$19 (Margalit et al., 2022, Wang et al., 2018). Taken together, these developments frame CMMs not as a single phenomenology tied to one material family, but as a boundary manifestation of several distinct topological mechanisms whose common feature is a propagating, self-conjugate, one-dimensional mode protected by bulk or defect topology.