---
title: Majorana Corner Modes in 2D Superconductors
url: https://www.emergentmind.com/topics/majorana-corner-modes
type: topic
---

# Majorana Corner Modes in 2D Superconductors

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 [2012.00691, 1802.00270]. 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 [1901.04466, 1803.08545]. Closely related constructions also occur in fractionalized Majorana band structures, notably in a second-order Kitaev spin liquid on the Shastry–Sutherland lattice [1803.08922].

## 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 [2012.00691, 1802.00270]. 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 [1901.04466]. 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 [1803.08922].

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 [1901.04466, 1803.08545]. 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 [2012.00691, 1802.00270].

## 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 [2012.00691]. 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 [1802.00270]. 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 $s_\pm$ pairing symmetry of the proximate superconductor [1803.08545]. In the QSHI/$s$-wave/in-plane-Zeeman construction, the Zeeman field induces opposite effective Dirac masses between adjacent boundaries once $h_x>\sqrt{\mu^2+\Delta_0^2}$, leading to one Majorana mode at each corner [1905.08896].

Real-space mass engineering provides an alternative route. In the attractive Hubbard–Hofstadter model, a dark line soliton forces the $s$-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 [1901.04466]. This shows that momentum-space sign-changing pairing is not necessary: a conventional $s$-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 [1803.08545] | Pairing symmetry gives opposite induced edge masses | One MKP at each corner |
| QSHI + $s$-wave SC + in-plane Zeeman field [1905.08896] | Zeeman field drives adjacent edges to opposite effective Dirac masses | One Majorana mode at each corner |
| Noncentrosymmetric $(s+p)$-wave superconductor [2012.00691] | $m_i=V_Z\sin(\theta_Z-\gamma_i)$ tuned by field direction | Two MCMs on a square, movable between corner pairs |
| Attractive Hubbard–Hofstadter cold atoms [1901.04466] | Soliton-induced sign flip of $\Delta(\mathbf r)$ at soliton–edge intersections | Two MKPs at defect-defined corners |
| 2D TI + orbital-dependent exchange + $s$/$d$ pairing [2203.02879] | Orbital difference parameter $\alpha$ changes edge masses | One or two MCMs per corner |
| Altermagnet/TI/$s$-wave heterostructure [2311.05814] | Néel-vector-dependent altermagnetic mass competes with pairing | Two MCMs per corner at $\mu=0$; tunable patterns |
| Hexagonal lattices with $p_z$ or $p_{x,y}$ orbitals [2204.11178] | Nonuniform sublattice Zeeman field or pairing | MCMs at intersections between zigzag edges |
| 1T-PtSe$_2$ monolayer family [2308.12055] | ZZ edge becomes topological under Zeeman + $s$-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 [1802.00270, 1905.08896]. A second class uses unconventional pairing to imprint the sign structure directly onto the edge mass, as in the cuprate and iron-based heterostructures [1803.08545]. A third class engineers the sign structure in real space, either by a soliton in a cold-atom superfluid [1901.04466] or by orbital-, sublattice-, or altermagnetic anisotropies that make different edges effectively inequivalent [2203.02879, 2311.05814, 2204.11178].

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 [1803.08922].

## 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-$T_c$ QSHI heterostructures and the attractive Hubbard–Hofstadter superfluid with solitons [1803.08545, 1901.04466]. 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 [2012.00691, 1802.00270, 1905.08896].

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 $(s+p)$-wave and $p\pm ip$ models, the decisive quantity is the sign pattern of the edge masses, not a conventional bulk Chern number [2012.00691, 1802.00270]. In the QSHI/$s$-wave/in-plane-field heterostructure, Wilson-loop calculations yield a Majorana edge polarization, with $p_x^{\mathrm{edge},y}=1/2$ and $p_y^{\mathrm{edge},x}=0$ in the MCM phase, directly encoding that only one pair of edges is topological in the higher-order sense [1905.08896]. In the superconductor/topological-insulator/superconductor junction, a mirror winding number
\[
\nu_\pm=\frac{1}{i\pi}\log[\det W_{\pm,k_y}] \ \mathrm{mod}\ 2
\]
characterizes the mirror-resolved 1D sectors, and $\nu_+=\nu_-=1$ accompanies the Majorana-corner phase [2209.14885].

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 $r_{x,y}=1/2$ distinguish the second-order spin liquid from both the chiral spin liquid and a trivial insulator [1803.08922]. In the “perfectly localized” fermionic-lattice constructions, a nested Pfaffian invariant $\mathbf Q=(Q_x,Q_y)$ classifies whether the orthogonal edges are topological and hence which corners host MCMs [2303.01535]. In the 1T-PtSe$_2$ 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 [2308.12055]. In the Floquet second-order topological superconductor, the average quadrupolar motion distinguishes 0- and $\pi$-quasienergy MCMs dynamically [2208.05749].

## 5. Tunability, spectroscopy, and quantum operations

Field-angle control is a defining feature of several class-D platforms. In noncentrosymmetric $(s+p)$-wave superconductors, rotating the in-plane Zeeman field changes the sign of $V_\parallel=V_Z\sin(\theta_Z-\gamma)$ 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 $\theta_Z$ with period $\pi$, and plateaus at
\[
G(0)\approx 2e^2/h
\]
appear precisely in angular intervals where an MCM sits at the contacted corner [2012.00691]. The $p\pm ip$ field-induced second-order superconductor shows the same qualitative effect: a $2\pi$ field rotation moves the Majorana corner states around the sample boundary [1802.00270].

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 $2e^2/h$ for a single MCM per corner and $4e^2/h$ when two MCMs occupy the same corner, with the quantization robust to changes of the relevant system parameters [2203.02879]. 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 [2209.14885].

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 [1901.04466]. 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 $\pi$ 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 [1904.03161]. Floquet driving adds the further possibility of simultaneously engineering regular 0 and anomalous $\pi$ corner modes [2208.05749].

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

Majorana corner modes are robust, but their practical isolation depends strongly on geometry. In the noncentrosymmetric $(s+p)$-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 [2012.00691]. 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 [2012.00691]. In hexagonal-lattice constructions, the finite-size splitting of the corner modes decays much faster in the $p_x,p_y$ model than in the $p_z$ model, so the second model is less sensitive to finite-size effects [2204.11178].

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 $k_BT=0.01t$ still preserves the corner MKPs [1901.04466]. 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
\[
T_c \approx 0.012 J_z
\]
in the quantum Monte Carlo analysis [1803.08922]. 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/$s$-wave/in-plane-Zeeman framework, the 3D analogue is a topological-insulator heterostructure supporting Majorana hinge modes [1905.08896]. 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 [1907.02070]. 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 [2303.01535].

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 [2012.00691, 1901.04466]. Several materials proposals involve effectively 2D models for systems that are intrinsically 3D or require thin-film engineering [2012.00691]. Floquet implementations introduce anomalous $\pi$-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 [2208.05749]. 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.

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