---
title: Majorana Corner States
url: https://www.emergentmind.com/topics/majorana-corner-states
type: topic
---

# Majorana Corner States

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Majorana corner states, also called Majorana corner modes, are zero-energy Majorana bound states localized at the corners of a two-dimensional system whose bulk and one-dimensional edges are gapped. They are the codimension-2 boundary manifestation of second-order topological superconductivity. In the principal constructions developed so far, the corner mode appears where adjacent edges acquire Dirac or boundary masses of opposite sign, so that a Jackiw–Rebbi domain wall forms at the corner. Depending on symmetry class and crystalline structure, a corner can host a single Majorana zero mode, a Majorana Kramers pair, or multiple protected zero modes [1802.00270] [1804.04711] [1806.07002] [2409.11791].

## 1. Boundary mass inversion and the corner-mode mechanism

The standard field-theoretic description starts from a two-dimensional topological superconductor with gapless helical or chiral boundary modes and introduces perturbations that gap the edges in an orientation-dependent way. In the in-plane-field construction for a time-reversal-invariant \(p\pm ip\) superconductor, the low-energy edge Hamiltonian takes the form
\[
\mathcal{H}^{\text{Edge}}(k_{\bar r})=\nu \Delta k_{\bar r}\eta_z - V\sin(\theta+\phi+2\varphi_n)\eta_y,
\]
so the Zeeman term acts as an edge mass
\[
m_{\text{Edge}}=-V\sin(\theta+\phi+2\varphi_n).
\]
A Majorana corner state appears when two adjacent edges meeting at a corner have opposite mass signs [1802.00270].

This mass-domain-wall picture reappears in magnetic topological-insulator/high-\(T_c\) heterostructures. There, proximity-induced \(d\)-wave or \(s_{\pm}\)-wave pairing gaps the edge modes, and the sign structure of the effective pairing produces opposite Dirac masses on neighboring edges. The corner then binds a zero mode by the Jackiw–Rebbi mechanism; when mirror symmetry is present, the Hamiltonian decomposes into mirror sectors and each sector contributes a corner state, yielding two Majorana bound states per corner [1806.07002].

A closely related but more geometry-explicit formulation is provided by the phase-biased altermagnetic Josephson junction. Near \(\phi=\pi\), the projected edge theory is
\[
H_{\rm edge}^{(0)} = v k_\parallel \chi_z,
\]
while altermagnetic spin splitting generates a boundary mass
\[
m_\vartheta = m_0 \sin(\varphi - \vartheta) \left[ \cos\beta \cos(2\vartheta) + \frac{1}{2}\sin\beta \sin(2\vartheta) \right].
\]
Here the mass depends simultaneously on the boundary orientation \(\vartheta\) and the Néel-vector azimuth \(\varphi\), and the corner criterion is \(m_{\vartheta_1}m_{\vartheta_2}<0\) [2606.16931].

This recurring structure indicates that Majorana corner states are most naturally understood as defects of boundary mass textures rather than as isolated bulk objects. The bulk topology constrains the existence of the edge theory, while the corner localization is fixed by how competing masses are distributed along the boundary.

## 2. Symmetry classes, topological indices, and mode multiplicity

Majorana corner states occur in both time-reversal-invariant and time-reversal-broken settings. In class DIII, the corner excitation can be a Majorana Kramers pair. A square-shaped two-dimensional topological insulator proximitized by an \(s_{\pm}\)-wave superconductor realizes such a phase, and its bulk \(\mathbb{Z}_2\) invariant is defined as the parity of the number of occupied Kramers doublets with \(\mathcal{C}_4\) eigenvalues \(e^{\pm i\pi/4}\) and inversion eigenvalue \(-1\) at the \(\mathcal{C}_4\)-invariant momenta \((\pi,0)\) and \((0,\pi)\) [1804.04711].

A different DIII realization is the tunnel-coupled topological-insulator bilayer with a \(\pi\)-phase difference between top and bottom superconductors. In the absence of time-reversal breaking, the system supports a Kramers pair of Majorana corner states at each corner when the interlayer tunneling dominates the induced pairing, \( |\Gamma|>|\Delta_{sc}| \). A weak in-plane Zeeman field breaks time-reversal symmetry and generates a richer phase diagram with phases containing two Majorana corner states per corner and an intermediate phase with one Majorana corner state per corner [2007.13579].

When time-reversal symmetry is broken, many constructions fall into class D. In the semi-Dirac proposal, each edge is mapped to an effective Kitaev chain, and the relevant one-dimensional invariant is the Pfaffian \(\mathbb{Z}_2\) index
\[
\nu = \mathrm{sgn}\left( \frac{\mathrm{Pf}[A(0)]}{\mathrm{Pf}[A(\pi)]} \right),
\]
with \(A(k)=\mathcal{H}(k)\tau_x\). In the topological regime, finite geometries host four zero-energy corner modes [2604.22553].

Multiplicity beyond a single mode per corner requires additional protection. For a quantum spin Hall insulator with two pairs of helical edge states, robust twofold Majorana corner modes can appear when the edges are gapped by a combined magnetic exchange field and \(s\)-wave pairing, or by \(s+p\) mixed-parity pairing. Their stability is attributed to chiral symmetry, which forbids hybridization between same-chirality Majorana zero modes. In that setting, the integer multipole chiral number \(N_{xy}\) equals the number of protected zero modes per corner [2409.11791].

A common misconception is that corner-mode multiplicity is always fixed by Kramers degeneracy or by a simple \(\mathbb{Z}_2\) classification. The existing proposals show instead that multiplicity can arise from mirror-sector doubling, from Kramers structure, or from chiral-symmetry-protected multichannel edge physics, and these mechanisms are not equivalent.

## 3. Platform diversity

The literature spans a wide range of microscopic settings, from proximitized band insulators to synthetic and strongly correlated systems. The shared theme is the conversion of edge-localized low-energy channels into one-dimensional topological superconductors whose endpoints coincide with sample corners.

| Platform | Core mechanism | Corner-state outcome |
|---|---|---|
| 2D TI + \(s_{\pm}\)-wave superconductor [1804.04711] | Sign reversal of effective edge pairing on orthogonal edges | One Majorana Kramers pair at each corner |
| Magnetic TI + \(d\)-wave or \(s_{\pm}\)-wave superconductor [1806.07002] | Pairing-symmetry-induced Dirac masses with mirror-sector doubling | Two Majorana bound states per corner |
| Attractive QSHI with opposite in-plane Zeeman energy [2202.06254] | Opposite Dirac masses on adjacent edges and Majorana edge polarization | Second-order topological superfluid with corner Majoranas |
| TI bilayer / SC-TI-SC / Josephson-junction platforms [2007.13579] [2209.14885] [2509.03949] | \(\pi\)-phase bias, Zeeman gapping, or phase-controlled edge orthogonality | Kramers-pair, single, or phase-steered corner modes |
| Semi-Dirac system + Rashba SOC + Zeeman + \(s\)-wave proximity [2604.22553] | Edge-selective channels acquire effective \(p\)-wave pairing | Four corner Majorana zero modes |
| 1T-PtSe\(_2\) unconventional monolayer [2308.12055] | SI-free unconventionality, anisotropic obstructed edge states, Rashba splitting, SC + magnetic field | Majorana corner modes in an anisotropic two-BHZ description |
| Dice lattice with \(C=\pm 2\) flat bands [2210.09610] | Mixed singlet-triplet pairing on topological flat bands | Second-order TSC with corner Majorana bound states |

Additional realizations emphasize that Majorana corner physics is not confined to conventional proximitized semiconducting heterostructures. Square and Kagome quantum spin liquids can localize either fermions or Majorana particles at corners depending on coupling values, with the square-lattice model analytically tractable via Lieb’s theorem [2101.01180]. Two coupled sheets of bilayer graphene support a second-order topological superconducting phase with four Majorana corner states, and strong interactions elevate the construction to a fractional counterpart with zero-energy \(\mathbb{Z}_{2m}\) parafermion corner states [1912.10931]. Electric circuits simulate Majorana edge and corner states through circuit Laplacians and detect them through impedance rather than charge transport [1902.03716].

This range of platforms suggests two recurrent design strategies. One starts from helical or chiral edge states and gaps them with competing masses. The other exploits intrinsically anisotropic or obstructed boundary channels that already behave as separated one-dimensional wires and then induces effective \(p\)-wave pairing on those channels.

## 4. Dynamic steering and braiding

A distinctive feature of Majorana corner states, compared with more static corner bound states in ordinary higher-order insulators, is that several proposals permit deterministic relocation within a fixed sample geometry. In the in-plane-field proposal, rotating the magnetic field changes the edge-mass pattern, so the Majorana corner states hop from one corner to a neighboring one and complete a full revolution around the boundary when the field rotates by \(2\pi\) [1802.00270].

Phase bias provides a more directly superconducting control parameter. In a vertical superconductor–insulator–superconductor Josephson junction, the position of the corner modes is determined by the boundary angle satisfying an edge-state orthogonality condition, and varying the superconducting phase difference \(\varphi\) continuously moves the modes along the edge. The same work proposes a braiding protocol based on three circular vertical Josephson junctions with independently controlled phase biases [2509.03949].

The altermagnetic Josephson junction goes further by identifying two independent macroscopic knobs: the superconducting phase difference \(\phi\) and the Néel-vector orientation \(\boldsymbol n\). For a square sample with \(d_{x^2-y^2}\)-type form factor, \(\boldsymbol n\) along \([1,1]\) places the Majorana corner modes at the top-left and bottom-right corners, while \(\boldsymbol n\) along \([1,-1]\) moves them to the top-right and bottom-left corners. Phase tuning away from ideal \(\pi\) also reshapes the localization pattern, enabling deterministic steering without changing sample geometry [2606.16931].

Braiding constructions are not restricted to electronic matter. In electric circuits simulating a BHZ-type model with a particle-hole-symmetry-preserving effective Zeeman term, adiabatic variation of an angle parameter \(\theta\) exchanges the corner states. The Berry phase accumulated over a double exchange is \(\Gamma(2\pi)=\pi\), giving \(\sigma^2=-1\), the characteristic Ising-anyon relation [1902.03716].

A complementary route uses the internal spin structure of the corner states. In a class of higher-order topological superconductors built from two-dimensional topological insulators with \(s\)-wave pairing and in-plane magnetic fields, the corner-mode spin polarization is perpendicular to the applied field and opposite on the intrinsic orbitals, producing a ferrimagnetic texture. Because the existence of the modes does not depend on the in-plane field direction, this geometry supports all-electronic braiding networks controlled by gates and Josephson couplings rather than by field reorientation [2111.12359].

## 5. Detection, spectroscopy, and discrimination from trivial states

A recurring methodological point is that a zero-bias peak alone is not an unambiguous signature of a Majorana corner state. The altermagnetic Josephson-junction proposal makes this explicit: when a normal-metal lead probes a corner occupied by a Majorana corner mode, resonant Andreev reflection yields a quantized zero-bias conductance peak,
\[
G(0)=2e^2/h.
\]
When the mode is relocated away from that corner, the peak disappears. As \(\phi\) or the Néel-vector azimuth is tuned, the zero-bias peak turns on at the new corner and off at the old one in a phase-locked, spatially correlated fashion. Because trivial Andreev bound states are fixed by local disorder or impurity potentials, they cannot reproduce this macroscopic control-correlated switching across distant corners [2606.16931].

In time-reversal-invariant proposals, the conductance can reflect Kramers degeneracy. The square 2D topological-insulator/\(s_{\pm}\)-wave-superconductor platform predicts quantized zero-bias tunneling conductance of \(4e^2/h\) per corner because the corner excitation is a Majorana Kramers pair [1804.04711].

Local density of states remains a standard numerical and experimental diagnostic. In the semi-Dirac construction, the zero modes are well localized at the corners, and their Majorana character is checked by a Majorana polarization that is close to unity and uniform at each corner for the zero modes [2604.22553]. In partial-corner-state realizations, the LDOS immediately reveals whether only selected corners participate, which is essential for distinguishing genuine symmetry-broken higher-order behavior from a symmetric four-corner pattern [2007.07525].

Synthetic platforms broaden the diagnostic repertoire. In electric circuits, the impedance between nodes is given by \(Z_{ab}=G_{ab}=J^{-1}_{ab}\), and corner zero modes appear as impedance peaks at the corresponding nodes [1902.03716]. In spin-texture-based proposals, the ferrimagnetic structure of the corner mode yields transverse spin-selective Andreev reflection and a gate-tunable \(4\pi\)-periodic \(\phi_0\) Josephson current, features that differ qualitatively from topologically trivial \(\phi_0\)-junction behavior under rotation of the in-plane field [2111.12359].

An objective conclusion from these results is that spectroscopy is strongest when it is combined with a nonlocal control parameter. Zero-energy localization, by itself, is necessary but often not sufficient.

## 6. Variants, departures from the canonical picture, and broader generalizations

The simplest textbook image

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