Majorana Corner Modes in 2D Superconductors

• Majorana corner modes are zero-energy excitations localized at 0D boundaries in 2D second-order topological superconductors, appearing at geometric corners or soliton intersections.
• They emerge from edge-mass domain wall physics where sign changes in the Dirac mass of helical edge modes bind localized Majorana states tunable by external fields.
• Various platforms such as QSHI-based heterostructures, noncentrosymmetric superconductors, and cold-atom systems realize these modes with distinct symmetry classes and experimental signatures.

Majorana corner modes are zero-energy Majorana excitations localized at codimension-two boundaries of two-dimensional systems: geometric corners in finite samples, or defect-defined intersections such as the meeting points of a soliton line and an edge. In the higher-order-topology language, they are the characteristic boundary modes of second-order topological superconductors, where the 2D bulk and 1D edges are both gapped but 0D corner states remain gapless (Ikegaya et al., 2020, Zhu, 2018). In time-reversal-invariant realizations they occur as Majorana-Kramers pairs (MKPs), while in class-D realizations they typically appear as single Majorana zero modes (Zeng et al., 2019, Yan et al., 2018). Closely related constructions also occur in fractionalized Majorana band structures, notably in a second-order Kitaev spin liquid on the Shastry–Sutherland lattice (Dwivedi et al., 2018).

1. Boundary-of-boundary Majorana modes

Majorana corner modes differ from the Majorana zero modes of 1D Kitaev-chain-like setups and from gapless Majorana edge modes of first-order 2D topological superconductors. In 1D, Majorana zero modes appear at the ends of a topological segment; in a 2D second-order topological superconductor, the bulk is 2D, the edges are 1D and gapped, and only selected 0D corners host zero-energy bound states (Ikegaya et al., 2020, Zhu, 2018). This is the bulk–edge–corner correspondence emphasized across higher-order superconducting proposals.

The word “corner” is not purely geometric. In the attractive Hubbard–Hofstadter cold-atom proposal, the relevant corners are the intersection points between a line soliton of the pairing order parameter and the physical edges of the sample; the bulk and edges remain gapped, but the soliton-induced sign reversal produces MKPs at those intersections (Zeng et al., 2019). In crystalline implementations, the corners are literal sample corners, often selected by mirror or rotational symmetry. In the second-order Kitaev spin liquid, for example, four exponentially localized zero-energy Majorana modes appear at the corners of a square-shaped cluster, while the edges are gapped and the bulk Chern number vanishes (Dwivedi et al., 2018).

A further distinction concerns symmetry class. In class DIII, where time-reversal symmetry with $\mathcal{T}^2=-1$ is preserved, a corner zero mode necessarily comes with its Kramers partner, yielding an MKP (Zeng et al., 2019, Yan et al., 2018). When time-reversal symmetry is broken, most commonly by an in-plane Zeeman field, the system falls into class D and individual Majorana corner modes appear (Ikegaya et al., 2020, Zhu, 2018).

2. Edge-mass domain walls as the universal mechanism

A recurring low-energy description is a 1D Dirac theory for helical edge modes with a position-dependent mass. In the magnetized noncentrosymmetric $(s+p)$-wave model, projection onto the helical edge subspace gives

$${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$

where $\gamma$ is the edge-normal angle and $\theta_Z$ is the in-plane Zeeman-field direction (Ikegaya et al., 2020). In the time-reversal-invariant $p\pm ip$ model driven into a second-order phase by an in-plane field, the corresponding edge mass is

$m_{\mathrm{edge}}=-V\sin(\theta+\phi+2\varphi_n),$

with $\varphi_n$ labeling the four edges (Zhu, 2018). In both cases, adjacent edges can acquire opposite mass signs, so that a corner becomes a Jackiw–Rebbi mass kink binding a Majorana zero mode.

The same structure appears in QSHI-based heterostructures. In the high-$T_c$ proposals, the induced pairing on the helical edge serves as a Dirac mass whose sign changes between edges because of the $d$-wave or $(s+p)$0 pairing symmetry of the proximate superconductor (Yan et al., 2018). In the QSHI/$(s+p)$1-wave/in-plane-Zeeman construction, the Zeeman field induces opposite effective Dirac masses between adjacent boundaries once $(s+p)$2, leading to one Majorana mode at each corner (Wu et al., 2019).

Real-space mass engineering provides an alternative route. In the attractive Hubbard–Hofstadter model, a dark line soliton forces the $(s+p)$3-wave order parameter to vanish on a line and change sign across it. Where that soliton intersects an edge, the edge pairing gap changes sign, producing a Majorana-Kramers pair by the same mass-domain-wall mechanism (Zeng et al., 2019). This shows that momentum-space sign-changing pairing is not necessary: a conventional $(s+p)$4-wave order parameter can also generate corner Majoranas if its sign is made spatially inhomogeneous.

3. Principal realization routes

Several distinct microscopic routes realize the same edge-mass-domain-wall physics.

Platform	Mass-sign control	Corner content
QSHI + cuprate or iron-based superconductor (Yan et al., 2018)	Pairing symmetry gives opposite induced edge masses	One MKP at each corner
QSHI + $(s+p)$5-wave SC + in-plane Zeeman field (Wu et al., 2019)	Zeeman field drives adjacent edges to opposite effective Dirac masses	One Majorana mode at each corner
Noncentrosymmetric $(s+p)$6-wave superconductor (Ikegaya et al., 2020)	$(s+p)$7 tuned by field direction	Two MCMs on a square, movable between corner pairs
Attractive Hubbard–Hofstadter cold atoms (Zeng et al., 2019)	Soliton-induced sign flip of $(s+p)$8 at soliton–edge intersections	Two MKPs at defect-defined corners
2D TI + orbital-dependent exchange + $(s+p)$9/$${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$0 pairing (Lu et al., 2022)	Orbital difference parameter $${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$1 changes edge masses	One or two MCMs per corner
Altermagnet/TI/$${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$2-wave heterostructure (Li et al., 2023)	Néel-vector-dependent altermagnetic mass competes with pairing	Two MCMs per corner at $${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$3; tunable patterns
Hexagonal lattices with $${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$4 or $${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$5 orbitals (Luo, 2022)	Nonuniform sublattice Zeeman field or pairing	MCMs at intersections between zigzag edges
1T-PtSe$${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$6 monolayer family (Sheng et al., 2023)	ZZ edge becomes topological under Zeeman + $${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$7-wave pairing, AC edge remains trivial	Corner MCMs at ZZ–AC intersections

These platforms separate into several design classes. One class starts from helical edge modes of a QSHI or time-reversal-invariant topological superconductor and then gaps them by superconductivity plus a symmetry-breaking mass, so that neighboring edges carry opposite signs (Zhu, 2018, Wu et al., 2019). A second class uses unconventional pairing to imprint the sign structure directly onto the edge mass, as in the cuprate and iron-based heterostructures (Yan et al., 2018). A third class engineers the sign structure in real space, either by a soliton in a cold-atom superfluid (Zeng et al., 2019) or by orbital-, sublattice-, or altermagnetic anisotropies that make different edges effectively inequivalent (Lu et al., 2022, Li et al., 2023, Luo, 2022).

Beyond electronic superconductors, the second-order Kitaev spin liquid on the Shastry–Sutherland lattice realizes Majorana corner modes in an exactly solvable interacting spin model. There the emergent itinerant Majoranas form a gapped second-order phase protected by two mirror symmetries, and the corner modes are the codimension-two manifestation of that fractionalized bulk topology (Dwivedi et al., 2018).

4. Symmetry classes and topological diagnostics

The symmetry content determines whether a corner hosts a single Majorana or an MKP. DIII constructions retain time-reversal symmetry and therefore support Majorana-Kramers pairs; this is the case for the high-$${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$8 QSHI heterostructures and the attractive Hubbard–Hofstadter superfluid with solitons (Yan et al., 2018, Zeng et al., 2019). In contrast, once an in-plane Zeeman field or equivalent TRS-breaking perturbation is present, the system is generally in class D, and corner states are single Majorana zero modes (Ikegaya et al., 2020, Zhu, 2018, Wu et al., 2019).

Because many of these second-order phases are boundary-obstructed rather than first-order topological, their diagnosis often proceeds through edge rather than bulk invariants. In the noncentrosymmetric $${\cal H}= \begin{pmatrix} \gamma k_{\parallel} & V_{\parallel}\ V_{\parallel} & -\gamma k_{\parallel} \end{pmatrix}, \qquad V_{\parallel}=V_Z\sin(\theta_Z-\gamma),$$9-wave and $\gamma$0 models, the decisive quantity is the sign pattern of the edge masses, not a conventional bulk Chern number (Ikegaya et al., 2020, Zhu, 2018). In the QSHI/$\gamma$1-wave/in-plane-field heterostructure, Wilson-loop calculations yield a Majorana edge polarization, with $\gamma$2 and $\gamma$3 in the MCM phase, directly encoding that only one pair of edges is topological in the higher-order sense (Wu et al., 2019). In the superconductor/topological-insulator/superconductor junction, a mirror winding number

$\gamma$4

characterizes the mirror-resolved 1D sectors, and $\gamma$5 accompanies the Majorana-corner phase (Chen et al., 2022).

Other works use real-space or Wannier-based diagnostics. In the exactly solvable second-order Kitaev spin liquid, gapped Wannier bands with mid-gap Wannier centers at $\gamma$6 distinguish the second-order spin liquid from both the chiral spin liquid and a trivial insulator (Dwivedi et al., 2018). In the “perfectly localized” fermionic-lattice constructions, a nested Pfaffian invariant $\gamma$7 classifies whether the orthogonal edges are topological and hence which corners host MCMs (Poduval et al., 2023). In the 1T-PtSe$\gamma$8 family, the underlying “SI-free” unconventionality is diagnosed not by symmetry indicators but by Wannier charge centers extracted from a one-dimensional Wilson loop, and the corner Majoranas arise after superconducting proximity and an external magnetic field act on those obstructed edge states (Sheng et al., 2023). In the Floquet second-order topological superconductor, the average quadrupolar motion distinguishes 0- and $\gamma$9-quasienergy MCMs dynamically (Ghosh et al., 2022).

5. Tunability, spectroscopy, and quantum operations

Field-angle control is a defining feature of several class-D platforms. In noncentrosymmetric $\theta_Z$0-wave superconductors, rotating the in-plane Zeeman field changes the sign of $\theta_Z$1 on each edge, so the corner modes move from one corner pair to another. In a square island, the zero-bias differential conductance measured by a lead attached to a corner is a periodic function of $\theta_Z$2 with period $\theta_Z$3, and plateaus at

$\theta_Z$4

appear precisely in angular intervals where an MCM sits at the contacted corner (Ikegaya et al., 2020). The $\theta_Z$5 field-induced second-order superconductor shows the same qualitative effect: a $\theta_Z$6 field rotation moves the Majorana corner states around the sample boundary (Zhu, 2018).

Transport probes can also resolve the number of corner modes. In the orbital-dependent-exchange model, a normal probe terminal attached to a corner gives a quantized zero-bias conductance of $\theta_Z$7 for a single MCM per corner and $\theta_Z$8 when two MCMs occupy the same corner, with the quantization robust to changes of the relevant system parameters (Lu et al., 2022). In the superconductor/topological-insulator/superconductor junction, the corner phase and the nodal phase are sharply distinguished: the former exhibits two zero-energy Majorana corner modes in a fully gapped system, while the latter exhibits flat-band Majorana edge modes connecting bulk nodes (Chen et al., 2022).

Cold-atom proposals replace tunneling spectroscopy by spatially resolved probes of quasiparticle structure. In the attractive Hubbard–Hofstadter model, spatially resolved radio-frequency spectroscopy can probe zero-bias features at the soliton–edge intersections, and moving the soliton line shifts the MKPs along the boundary (Zeng et al., 2019). This suggests a controllable route to braiding-type operations in a clean optical-lattice environment.

A distinct line of work treats corner Majoranas as computational primitives. In a periodically modulated second-order topological superconductor, four Majorana zero modes and four Majorana $\theta_Z$9 modes can reside at the four corners, giving eight Majorana corner modes in one island. Two logical qubits and one ancilla qubit can then be encoded, and topologically protected gate operations can be implemented by sequences of Majorana parity measurements achieved through Mach–Zehnder-type interference in conductance (Bomantara et al., 2019). Floquet driving adds the further possibility of simultaneously engineering regular 0 and anomalous $p\pm ip$0 corner modes (Ghosh et al., 2022).

6. Geometry, disorder, dimensional extension, and open problems

Majorana corner modes are robust, but their practical isolation depends strongly on geometry. In the noncentrosymmetric $p\pm ip$1-wave model, a large cut in a square island creates additional corners and yields four distinct MCMs, whereas a smaller cut causes strong hybridization so that several would-be corner modes merge into a single extended zero mode (Ikegaya et al., 2020). Random edge roughness does not destroy the MCMs themselves, but it reduces the edge gap and introduces subgap edge states, especially near field angles where the clean edge mass is small; this complicates controlled braiding protocols (Ikegaya et al., 2020). In hexagonal-lattice constructions, the finite-size splitting of the corner modes decays much faster in the $p\pm ip$2 model than in the $p\pm ip$3 model, so the second model is less sensitive to finite-size effects (Luo, 2022).

Several proposals report explicit disorder or thermal stability within their modeling assumptions. In the attractive Hubbard–Hofstadter system, moderate thermal fluctuations and on-site disorder do not remove the zero modes, and the calculation at $p\pm ip$4 still preserves the corner MKPs (Zeng et al., 2019). In the second-order Kitaev spin liquid, the corner-mode phase remains stable in the presence of thermal fluctuations and is separated from the high-temperature regime by a finite-temperature phase transition at

$p\pm ip$5

in the quantum Monte Carlo analysis (Dwivedi et al., 2018). These results indicate robustness, but they do not eliminate the need for long edges, sizable boundary gaps, and careful control of symmetry-breaking perturbations.

The dimensional extension of the corner-mode paradigm leads to hinge physics in 3D. In the QSHI/$p\pm ip$6-wave/in-plane-Zeeman framework, the 3D analogue is a topological-insulator heterostructure supporting Majorana hinge modes (Wu et al., 2019). In SOTI/superconductor heterostructures, chiral electronic hinge modes of the normal second-order topological insulator convert into chiral Majorana hinge modes under uniform superconducting pairing, with one or two chiral Majorana channels per hinge depending on the boundary topological phase (Yan, 2019). In lattice-construction language, stacking lower-dimensional Majorana-corner architectures yields 3D cubes with exactly localized corner Majoranas and suggests a general hypercubic construction in arbitrary dimension (Poduval et al., 2023).

Open issues recur across the literature. Many analyses are BdG mean-field treatments; beyond-mean-field effects, realistic multi-band structures, orbital effects of magnetic fields, and self-consistent feedback of superconductivity are often not treated in full (Ikegaya et al., 2020, Zeng et al., 2019). Several materials proposals involve effectively 2D models for systems that are intrinsically 3D or require thin-film engineering (Ikegaya et al., 2020). Floquet implementations introduce anomalous $p\pm ip$7-Majorana corner modes and rich time dependence, but the strongest agreement between exact numerics and Floquet perturbation theory is obtained in the higher-frequency regime and mainly for the 0-quasi-energy sector (Ghosh et al., 2022). Across these realizations, the central technical problem remains the same: maintain a gapped bulk and gapped edges while engineering edge masses whose sign changes only where corner Majoranas are desired.