Second-Order Topological Superconductor
- Second-order topological superconductors are phases with a fully gapped bulk and conventional boundaries, yet they support (d-2)-dimensional Majorana modes at corners or hinges.
- They are realized through mechanisms such as edge-mass domain walls, sign-changing pairing, crystalline symmetries, and Floquet engineering, confirming their unique topological features.
- Experimental signatures include corner-localized zero-bias peaks and tunable Majorana braiding, offering robust platforms for quantum-information applications even under moderate disorder.
A second-order topological superconductor is a higher-order topological superconducting phase in which the bulk is fully gapped and the conventional -dimensional boundary is also gapped, while protected gapless modes appear on -dimensional boundaries: in two dimensions as Majorana corner modes and in three dimensions as hinge modes. In the literature summarized here, second-order topology is realized through several distinct mechanisms—edge-mass domain walls, sign-changing pairing on symmetry-related Fermi pockets, crystalline or effective chiral symmetries, and Floquet engineering—but the recurring physical content is the same: a gapped superconducting bulk supports lower-dimensional Majorana boundary modes that are not captured by ordinary first-order bulk–edge correspondence (Zhu, 2018, Pahomi et al., 2019, Volpez et al., 2018, Wang et al., 2 Dec 2025).
1. Definition and conceptual structure
In a conventional first-order topological superconductor in spatial dimensions, the bulk is gapped and the nontrivial topology appears through gapless boundary modes in dimension . A second-order topological superconductor instead has a gapped bulk and gapped -dimensional boundaries, but protected modes on boundaries of codimension two. In $2$D this means gapped edges and $0$D Majorana corner states; in $3$D it means gapped surfaces and $1$D hinge modes (Pahomi et al., 2019, Zhu, 2018).
This codimension-two structure is not tied to a single symmetry class or a single microscopic mechanism. Several works begin from a time-reversal-invariant parent superconductor in class DIII with helical Majorana edge modes and then gap those edges by perturbations whose sign depends on edge orientation; others formulate the phase directly in class D, or in crystalline subclasses diagnosed by mirror or rotation eigenvalues, real-space invariants, or quadrupole moments (Volpez et al., 2018, Wang et al., 2023, Wang et al., 2 Dec 2025).
A common misconception is that second-order topology is merely a corner version of ordinary edge topology. The cited works distinguish these cases sharply. Conventional chiral -wave superconductors carry one-dimensional Majorana edge modes, whereas the orbital-active 0 construction of a second-order phase yields gapped edges and anomalous zero-energy Majorana corner modes (Wang et al., 2023). Likewise, a 1D first-order topological superconductor may exhibit propagating edge Majoranas, while a 2D SOTSC is characterized by edges that are themselves massive and by corner-localized zero modes (Zhu, 2018, 1904.02437).
2. Symmetries and bulk diagnostics
The symmetry content of SOTSC models is diverse, but particle–hole symmetry is universal in the BdG description and is central to Majorana protection. In the square-lattice helical SOTSC of "Braiding Majorana corner modes in a second-order topological superconductor" (Pahomi et al., 2019), particle–hole symmetry is written as
3
and, together with an effective chiral symmetry, quantizes higher-order invariants. In that model the corner invariants 4 and 5, defined from nested Wilson loops, take values in 6, and the system is in a SOTSC phase whenever at least one of them equals 7 (Pahomi et al., 2019).
Nested Wilson loops are one important diagnostic, but not the only one. In heterostructures built from a QSHI, superconductivity, and noncollinear magnetic texture, the higher-order phase is characterized by a quantized quadrupole moment 8, with 9 in the topological phase and 0 in the trivial phase (Chatterjee et al., 2023). In the 2025 magnet/1-wave/QSHI proposal, the bulk topology is instead encoded in a quadrupolar winding number 2, with 3 corresponding to four Majorana corner modes and 4 corresponding to eight corner modes (Subhadarshini et al., 8 May 2025). In the intrinsic nonsymmorphic Dirac-semimetal construction, the second-order phase is tied to a quantized bulk quadrupole moment 5 in each mirror sector and to an obstructed atomic-limit description in real space (Wang et al., 2 Dec 2025).
Floquet SOTSCs require separate diagnostics because the topology lives in quasienergy space. In periodically driven QSHI-based models, strong and weak Floquet SOTSC phases are distinguished by the Floquet quadrupole moment and by Floquet Wannier spectra. The strong phase hosts one zero-energy Majorana per corner and has quantized Floquet quadrupole moment, whereas the weak phase hosts two zero-energy Majoranas per corner and is distinguished by a different Wannier-center multiplicity (Ghosh et al., 2020, Ghosh et al., 2020).
These diagnostics are not interchangeable in a formal sense, but they encode related physics: a higher-order bulk obstruction forces a mismatch of edge masses or edge topological indices, and that mismatch reappears as corner or hinge Majorana modes.
3. Boundary mass domain walls and Majorana corner modes
The boundary theory of a 6D SOTSC is typically a collection of 7D Dirac or Majorana edge Hamiltonians whose mass terms depend on edge orientation and on external or induced perturbations. This is the most direct route to understanding why corner Majoranas appear.
In the magnetic-field-induced SOTSC of (Zhu, 2018), the parent state is a time-reversal-invariant 8 superconductor with helical Majorana edge modes. An in-plane Zeeman field gaps the edge modes and produces edge masses
9
so that opposite edges have opposite mass signs. A corner where adjacent masses satisfy 0 realizes a Jackiw–Rebbi mass kink and binds a Majorana zero mode (Zhu, 2018).
The same boundary-domain-wall logic appears in the Rashba bilayer 1-junction proposal, where the low-energy edge Hamiltonian takes the form
2
with edge mass
3
Changing the Josephson phase deviation 4 or the in-plane field direction 5 changes the signs of 6 on different edges, and the resulting domain walls localize Majorana corner states (Volpez et al., 2018).
The helical SOTSC model of (Pahomi et al., 2019) makes the boundary picture especially explicit. A 7D helical 8-wave parent state in class DIII has counterpropagating Majorana edge modes. An in-plane magnetic field gaps one set of edges, and proximity-induced spin-singlet pairing gaps the remaining ones. The bulk remains gapped while different edges acquire masses of opposite sign, so the corners become mass-domain-wall defects. Numerical diagonalization of a finite square yields exactly two zero-energy eigenstates for representative parameters, with energies approaching zero exponentially in system size (Pahomi et al., 2019).
Once such zero modes are found, particle–hole symmetry promotes them to bona fide Majorana operators. In (Pahomi et al., 2019), two zero-mode wavefunctions 9 and 0 can be chosen PHS self-conjugate, and the corresponding operators satisfy
1
This operator algebra is the minimal algebraic signature of Majorana corner modes (Pahomi et al., 2019).
A broader lesson from these models is that the protection of the zero mode does not require every edge to be topological in the same way. What matters is the relative topology of neighboring edges, or equivalently the sign structure of their effective masses.
4. Microscopic realizations and model classes
The current literature contains several distinct routes to second-order topological superconductivity. Some start from time-reversal-invariant helical superconductors and add symmetry-breaking masses; others generate effective 2-wave structure from SOC, Josephson phases, or magnetic texture; still others realize second-order topology intrinsically through sign-changing pairing on Dirac pockets or through orbital-active pairing. The following representative classes recur across the cited works.
| Platform or mechanism | Key ingredients | Representative paper |
|---|---|---|
| Magnetic-field-induced helical SOTSC | 3 or helical 4-wave parent state, in-plane Zeeman field, edge-mass sign changes | (Zhu, 2018) |
| Rashba bilayer 5-junction | Two tunnel-coupled Rashba 2DEGs, phase difference 6, weak in-plane Zeeman field | (Volpez et al., 2018) |
| QSHI plus unconventional superconductivity | QSHI edge states proximitized by 7-wave or 8-wave pairing | (Zhang et al., 2019) |
| Phase-biased nanowire array | Array of Majorana nanowires with Josephson-phase-controlled interwire Majorana coupling | (1904.02437) |
| Floquet FM/2DEG/SC heterostructure | Driven ferromagnet, spin–orbit-coupled 2DEG, induced 9-wave pairing, oscillating Zeeman field | (Plekhanov et al., 2019) |
| Magnetic-texture heterostructure | QSHI, $2$0-wave superconductor, noncollinear magnetic texture inducing effective Zeeman and SOC terms | (Chatterjee et al., 2023) |
| Magnet/$2$1-wave/QSHI hybrid | $2$2-wave superconductor, QSHI, noncollinear magnetic texture, tunable one or two MCMs per corner | (Subhadarshini et al., 8 May 2025) |
| Intrinsic nonsymmorphic Dirac semimetal | Two Dirac Fermi pockets at $2$3 and $2$4, even-parity spin-singlet $2$5-wave pairing with opposite signs on the pockets | (Wang et al., 2 Dec 2025) |
| Orbital-active $2$6 superconductor | Two-orbital model with spin–orbit couplings and orbital-dependent $2$7 pairing, no external field | (Wang et al., 2023) |
These constructions are not merely different microscopic realizations of the same mean-field ansatz. They implement higher-order topology through genuinely different organizing principles. The Rashba bilayer and QSHI-based models are proximity platforms built from conventional ingredients; the magnetic-texture proposals generate effective in-plane Zeeman fields and spin–orbit coupling by unitary transformation of a noncollinear exchange pattern (Chatterjee et al., 2023, Subhadarshini et al., 8 May 2025); the nonsymmorphic Dirac-semimetal proposal is an intrinsic second-order superconductor in which the sign of the even-parity $2$8-wave gap differs on the two Dirac Fermi pockets (Wang et al., 2 Dec 2025); and the orbital-active $2$9 construction realizes a time-reversal-breaking second-order phase without any external magnetic field, in contrast to more conventional chiral $0$0-wave states with one-dimensional Majorana edge modes (Wang et al., 2023).
Floquet constructions add a separate axis of tunability. In one line of work, a resonantly driven ferromagnet above a proximitized spin–orbit-coupled $0$1DEG produces a Floquet helical topological superconductor and, with an in-plane drive component, a Floquet higher-order phase with zero-energy Majorana corner states (Plekhanov et al., 2019). In another, periodic kicks of a mass term generate weak or strong Floquet SOTSC phases from a QSHI with unconventional pairing, again distinguished by the number of corner Majoranas and by Floquet multipole diagnostics (Ghosh et al., 2020).
5. Manipulation, braiding, and quantum-information protocols
One of the strongest motivations for SOTSCs is that corner Majoranas can be moved by tuning global parameters rather than by moving vortices or wire ends. The clearest example is the square helical SOTSC of (Pahomi et al., 2019), where the spatial positions of the two Majorana corner modes are controlled by rotating an in-plane magnetic field $0$2 and tuning proximity-induced singlet pairing $0$3. During an adiabatic cycle the corner wavefunctions trace worldlines around the square, and after one full cycle each Majorana acquires a Berry phase
$0$4
so that
$0$5
The paper emphasizes that the legitimate braid is the full closed loop in parameter space; the halfway point at $0$6 swaps positions but does not return the Hamiltonian to itself (Pahomi et al., 2019).
The earlier magnetic-field-induced proposal already showed that rotating the in-plane magnetic field causes the Majorana corner states to hop from one corner to the next and complete a full circuit when the field rotates by $0$7 (Zhu, 2018). This suggests a planar alternative to nanowire- or vortex-based exchange protocols.
A crucial qualification is that a single isolated pair of Majoranas realizes only the minimal $0$8 sector of the braid group. In that case a double exchange reduces to a phase, not a noncommuting matrix action on a larger degenerate Hilbert space. The explicit non-Abelian gate structure appears only when more than one Majorana pair is available (Pahomi et al., 2019). This is precisely the direction taken in "Topological and holonomic quantum computation based on second-order topological superconductors" (Zhang et al., 2020), which develops fusion rules, trijunction braiding schemes, and shamrock-like holonomic architectures built from multiple SOTS islands. There the effective couplings between Majoranas are controlled by geometry, superconducting phase differences, and the direction and magnitude of in-plane magnetic fields, enabling topological exchanges and holonomic gates in a Majorana qubit space (Zhang et al., 2020).
Josephson phase control provides a complementary route. In the phase-tunable nanowire-array proposal, the couplings between Majoranas on adjacent wires depend on phase bias, allowing the device to move between trivial, weak-TSC, and second-order regimes. The array realizes an extrinsic SOTSC in which the edges behave as effective Kitaev chains and the corners carry unpaired Majoranas when the interwire couplings are dimerized appropriately (1904.02437).
6. Experimental signatures, robustness, and unsettled issues
The most direct experimental signature across essentially all proposals is a corner-localized zero-bias anomaly. The magnetic-field-induced SOTSC paper proposes local tunneling spectroscopy showing zero-bias conductance peaks at corners that move as the in-plane field is rotated (Zhu, 2018). The Rashba bilayer $0$9-junction work likewise points to local tunneling spectroscopy, corner-sensitive transport, and phase-controlled relocation of corner states as the main observables (Volpez et al., 2018). In the Josephson-junction analysis of SOTS leads, the higher-order topology manifests as a doping-controlled $3$0-$3$1 transition driven by the sign change of the effective edge pairing, offering a transport signature that is specific to the second-order setting (Zhang et al., 2019).
Robustness is a recurrent theme. In the Rashba bilayer model, Majorana corner states remain stable against moderate potential and magnetic disorder and disappear only when disorder closes the bulk gap (Volpez et al., 2018). The ferromagnetic-resonance Floquet proposal finds stability against disorder, detuning, and static perturbations so long as the bulk and edge gaps remain open (Plekhanov et al., 2019). The Floquet SOTSC generated from unconventional pairing retains quantized Floquet invariants and corner modes under moderate disorder (Ghosh et al., 2020). The nanowire-array platform is robust against disorder in both the wires and the Josephson junctions, and even supports a second-order topological Anderson phase in which disorder converts an initially trivial system into a topological one (1904.02437).
Several objective distinctions in the literature are worth keeping separate. First, “strong” and “weak” SOTSC are used in more than one sense. In Floquet and magnetic-texture settings, the distinction often refers to one versus two Majoranas per corner or to different multipole/Wannier diagnostics (Ghosh et al., 2020, Subhadarshini et al., 8 May 2025). Second, not every proposal relies on exact crystalline symmetry to the same degree. Some models require mirror, rotation, or nonsymmorphic symmetry for quantization of bulk invariants (Wang et al., 2 Dec 2025, Wang et al., 2023), while others emphasize that the essential ingredients are the persistence of particle–hole symmetry and the existence of a robust edge-mass sign pattern, even without exact spatial symmetry (Plekhanov et al., 2019). Third, not every corner zero mode implies the same computational utility: a pair of well-separated Majoranas confirms fractional statistics under an adiabatic cycle, but scalable non-Abelian operations require larger Majorana networks and controlled couplings (Pahomi et al., 2019, Zhang et al., 2020).
Taken together, the literature presents second-order topological superconductivity as a broad organizing principle rather than a single model family. It encompasses magnetic-field-induced helical superconductors, Josephson $3$2-junction bilayers, QSHI-based proximity structures, phase-biased nanowire arrays, Floquet platforms, noncollinear magnetic-texture heterostructures, intrinsic sign-changing $3$3-wave superconductors on Dirac pockets, and orbital-active $3$4 states. Across these realizations, the defining feature remains constant: a gapped superconducting bulk and gapped edges conspire to produce Majorana modes on corners or hinges, with bulk diagnostics, boundary mass theories, and transport signatures all pointing to the same higher-order topological origin (Zhu, 2018, Pahomi et al., 2019, Volpez et al., 2018, Chatterjee et al., 2023, Wang et al., 2 Dec 2025).