Non-Abelian Second-Order Topological Insulator

• Non-Abelian second-order topological insulators are quantum phases defined by codimension-two boundary states controlled by non-Abelian invariants such as quaternion charges or the second Stiefel–Whitney number.
• They employ hybrid models—like coupled-wire constructions combining a non-Abelian trimer chain with an SSH chain or Takagi formulations—to yield robust corner and hinge modes.
• Superconducting extensions in these systems enable Majorana excitations, offering practical platforms for realizing non-Abelian quasiparticle physics in higher-order topological settings.

to=arxiv_search 夫妻性生活影片 北京赛车能json {"query":"\"non-Abelian second-order topological insulator\" OR \"higher-order topological\" non-Abelian arXiv", "max_results": 10} to=arxiv 乐亚json {"query":"non-Abelian second-order topological insulator", "max_results": 10} A non-Abelian second-order topological insulator denotes a second-order topological phase in which the corner or hinge phenomenology is controlled by non-Abelian topological data rather than by Abelian invariants alone, or, in a closely related usage, a second-order topological insulator that becomes a platform for non-Abelian Majorana excitations after superconducting proximity coupling. In current arXiv usage, the term spans at least two technically distinct constructions. One is a genuinely non-Abelian higher-order phase built from non-Abelian topological charges with quaternion algebra and an Abelian winding number, yielding hybridized corner modes and weak boundary states (Pan et al., 24 Dec 2025). Another is a real, $\mathcal{PT}$-symmetric second-order phase whose bulk invariant is the second Stiefel–Whitney number $w_2$, equivalently formulated through Takagi factorization, and whose finite samples exhibit odd $\mathcal{PT}$-related pairs of corner zero modes (Liu et al., 2022). A further related direction replaces the first-order topological insulator in standard TI/SC proposals by a time-reversal-symmetry-broken second-order topological insulator, producing a second-order topological superconductor with Majorana corner or hinge modes, thereby importing non-Abelian quasiparticle physics into a higher-order setting (Yan, 2019).

1. Conceptual scope and defining structures

Second-order topological phases are characterized by boundary states localized at codimension-two boundaries, such as corners in two dimensions or hinges in three dimensions. The non-Abelian qualifier modifies this framework in different ways. In the coupled-wire construction of non-Abelian higher-order topological phases, the distinction is explicit: higher-order topological phases had largely been characterized by Abelian invariants such as winding and Chern numbers, whereas the relevant bulk data are non-Abelian topological charges whose algebra is noncommutative (Pan et al., 24 Dec 2025). In that setting, the minimal non-Abelian second-order topological insulator is a two-dimensional model whose corner physics depends jointly on a quaternion-valued charge and an SSH winding number.

A different but related usage appears in real-band $\mathcal{PT}$-symmetric systems with sublattice symmetry. There, the phase is second order, but the bulk topology is encoded not by a scalar Berry phase but by a non-Abelian Berry connection and an $O(N)$-type real bundle structure. The resulting invariant is the second Stiefel–Whitney number $w_2$, and the topology can be reformulated through Takagi’s factorization of a symmetric unitary matrix (Liu et al., 2022). This suggests that “non-Abelian” in the literature can refer either to noncommutative bulk charges in multigap topology or to non-Abelian gauge structure in real-band higher-order phases.

The superconducting extension adds a third meaning relevant to applications. A time-reversal-symmetry-broken second-order topological insulator, when proximity coupled to a superconductor, can realize a second-order topological superconductor hosting Majorana corner modes in two dimensions and chiral Majorana hinge modes in three dimensions. Because Majorana modes satisfy $\gamma^\dagger=\gamma$ and obey non-Abelian exchange statistics, the heterostructure functions as a non-Abelian descendant of second-order topological insulating physics (Yan, 2019).

2. Coupled-wire non-Abelian SOTI and quaternion topology

The coupled-wire construction starts from a Kronecker-sum decomposition

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$

A corner mode appears when each constituent subsystem contributes a boundary-localized state. In the minimal two-dimensional model, the $x$-direction is a one-dimensional non-Abelian trimer chain $H_x$, and the $w_2$0-direction is a one-dimensional SSH chain $w_2$1. The lattice Hamiltonian is

$w_2$2

with

$w_2$3

The Hamiltonian has the form

$w_2$4

If

$w_2$5

then

$w_2$6

This factorized structure underlies the corner-state construction.

The non-Abelian character originates in the $w_2$7-symmetric trimer subsystem $w_2$8. Because $w_2$9 symmetry forces the Bloch eigenvectors to be real, the occupied eigenvectors form a real three-dimensional eigenframe whose rotation across the Brillouin zone is encoded by a quaternion-valued charge

$\mathcal{PT}$0

with

$\mathcal{PT}$1

and $\mathcal{PT}$2 the affine BWZ connection. The allowed values are

$\mathcal{PT}$3

identified with the quaternion group

$\mathcal{PT}$4

The paper interprets $\mathcal{PT}$5 as $\mathcal{PT}$6-rotations of the real eigenframe about different axes, $\mathcal{PT}$7 as a $\mathcal{PT}$8 rotation that is topologically nontrivial but invisible to ordinary Zak-phase diagnostics, and $\mathcal{PT}$9 as the trivial phase (Pan et al., 24 Dec 2025).

The $\mathcal{PT}$0-subsystem is the SSH chain with chiral symmetry $\mathcal{PT}$1 and Bloch Hamiltonian

$\mathcal{PT}$2

Its winding number is

$\mathcal{PT}$3

3. Hybrid invariant, boundary correspondence, and phase transitions

The coupled-wire model is classified by the hybrid topological vector

$\mathcal{PT}$4

with multiplication rule

$\mathcal{PT}$5

Its components have distinct origins: $\mathcal{PT}$6 is the non-Abelian quaternion charge of the $\mathcal{PT}$7-symmetric trimer chain, and $\mathcal{PT}$8 is the Abelian SSH winding number. Because quaternions do not commute, $\mathcal{PT}$9 is genuinely non-Abelian when $O(N)$0 (Pan et al., 24 Dec 2025).

Corner states appear if and only if both components are nontrivial,

$O(N)$1

The counting rule is

$O(N)$2

For the minimal model with $O(N)$3, this yields $O(N)$4 for $O(N)$5, $O(N)$6 for $O(N)$7, and no corner states if $O(N)$8. The corner states are tensor products of subsystem edge states,

$O(N)$9

Since the SSH edge modes are pinned at zero energy in the topological phase, $w_2$0.

The same construction yields weak topological edge phases when only one component of $w_2$1 is nontrivial. For

$w_2$2

the system supports weak non-Abelian edge states localized along $w_2$3, with the number of edge bands

$w_2$4

For

$w_2$5

the system supports weak Abelian edge states localized along $w_2$6, with

$w_2$7

A notable consequence is that weak boundary states can themselves be non-Abelian in origin.

The phase diagram contains both Abelian and non-Abelian topological transitions. The Abelian transition occurs at

$w_2$8

where the SSH winding changes between $w_2$9 and $\gamma^\dagger=\gamma$0. The non-Abelian transition occurs when the trimer subsystem changes quaternion charge, for example among $\gamma^\dagger=\gamma$1, $\gamma^\dagger=\gamma$2, $\gamma^\dagger=\gamma$3, $\gamma^\dagger=\gamma$4, and $\gamma^\dagger=\gamma$5, and requires a gap closing in the relevant band structure. The charge $\gamma^\dagger=\gamma$6 is especially significant because the Zak phases of both gaps can be trivial even though protected edge states remain; this is the paper’s central example of topology beyond Abelian phase diagnostics (Pan et al., 24 Dec 2025).

4. Takagi topological insulator and the Stiefel–Whitney formulation

A second major realization of a non-Abelian second-order topological insulator is the Takagi topological insulator on the honeycomb lattice. The symmetry setting is

$\gamma^\dagger=\gamma$7

so the Bloch Hamiltonian can be made real in an appropriate basis. The model also has sublattice symmetry

$\gamma^\dagger=\gamma$8

and inversion exchanges sublattices, so

$\gamma^\dagger=\gamma$9

This anticommutation makes the Hamiltonian block off-diagonal and enables the Takagi formulation (Liu et al., 2022).

The momentum-space Hamiltonian is a nearest-neighbor dimerized honeycomb model with six sites per unit cell,

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$0

Its bulk phase boundary follows from

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$1

so the gap closes when

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$2

The topological regime is

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$3

for which the second Stiefel–Whitney number is $$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$4; the opposite inequality is trivial.

The invariant can be diagnosed through the Wilson loop

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$5

where $$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$6 is the non-Abelian Berry connection of the valence bands. Writing the Wilson-loop eigenvalues as

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$7

the $$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$8 invariant is

$$H=H_{x_1}\oplus H_{x_2}\oplus\cdots\oplus H_{x_d}, \qquad A\oplus B=A\otimes \mathbb{I}_B+\mathbb{I}_A\otimes B.$$9

where $x$0 is the number of crossings at $x$1. In the topological phase of the honeycomb model, the Wilson loop has a single crossing, giving $x$2.

The same topology admits a Takagi-factorization formulation. The flattened Hamiltonian can be written as

$x$3

Thus $x$4 is a unitary symmetric matrix, and Takagi’s factorization gives

$x$5

The classifying space is

$x$6

On the overlap $x$7 of north and south hemispheres, the Takagi factors differ by an orthogonal transition function

$x$8

and the obstruction to a global Takagi factorization is precisely the same $x$9 data as $H_x$0. In this sense, the second-order phase is “non-Abelian” because its topology is controlled by matrix-valued real-bundle data rather than by an Abelian scalar phase (Liu et al., 2022).

5. Boundary masses, corner zero modes, and geometry dependence

The boundary mechanism of the Takagi topological insulator is formulated in terms of effective edge masses. For edges parallel to $H_x$1, the mass is

$H_x$2

When $H_x$3, the corresponding edge is gapless and hosts helical edge modes. When $H_x$4, the edge modes are gapped. A corner between two edges with opposite mass signs behaves as a Jackiw–Rebbi domain wall and traps a localized zero mode (Liu et al., 2022).

The finite-sample consequence is sharper than the bare existence of corner states. In $H_x$5-symmetric samples, corner zero modes come in $H_x$6-related pairs, and the number of such pairs is odd when $H_x$7. If sublattice symmetry is present, these corner modes are pinned exactly at zero energy; finite-size splitting is exponentially small with system size. The model demonstrates this structure for hexagonal and octagonal geometries. The boundary phenomenology is not one-to-one with the bulk invariant: the same $H_x$8 can be realized by gapped edges with corner zero modes, by helical edge states at boundary criticality, or by different corner-localization patterns depending on geometry. This is described as a bulk-boundary criticality or one-to-many correspondence.

A related but distinct boundary logic governs the coupled-wire non-Abelian SOTI. There, corners are not generated by a single Dirac-mass sign change but by the coexistence of boundary-localized states from both constituent subsystems. The corner modes are tensor products of an $H_x$9 edge state and an $w_2$00 edge state, and their existence requires both the quaternion charge and the SSH winding to be nontrivial. This produces a hybridized bulk-edge-corner correspondence in which corner physics is jointly controlled by non-Abelian and Abelian invariants (Pan et al., 24 Dec 2025).

Taken together, these constructions show that the second-order boundary signature can arise from different microscopic mechanisms. One is a mass-domain-wall mechanism tied to real-bundle topology and $w_2$01-related zero modes. Another is a tensor-product mechanism tied to a Kronecker-sum Hamiltonian and a hybrid quaternion-winding invariant. The shared feature is codimension-two boundary localization enforced by bulk topology, but the topological data and protection principles are different.

6. Superconducting descendants and non-Abelian Majorana boundary modes

Second-order topological insulators without time-reversal symmetry acquire a different kind of non-Abelian significance when coupled to a superconductor. In two and three dimensions, a time-reversal-symmetry-broken SOTI/SC heterostructure realizes a second-order topological superconductor hosting Majorana corner modes in two dimensions and chiral Majorana hinge modes in three dimensions (Yan, 2019). This is a higher-order analog of the familiar TI/SC and QAHI/SC platforms: instead of starting from a first-order topological insulator or a quantum anomalous Hall insulator, the construction starts from a second-order topological insulator whose first-order boundary states are already gapped by a TRS-breaking term.

In two dimensions the BdG Hamiltonian is

$w_2$02

with

$w_2$03

$w_2$04

The pairing $w_2$05 may represent $w_2$06-wave, $w_2$07-wave, or $w_2$08-wave pairing, and the proposal does not require special pairing symmetry. The BdG Hamiltonian always has particle-hole symmetry,

$w_2$09

For $w_2$10 there is also chiral symmetry,

$w_2$11

but the Majorana corner modes remain protected by particle-hole symmetry even when chiral symmetry is broken by $w_2$12.

The low-energy edge theory is

$w_2$13

With $w_2$14, the superconducting edge theory splits into two sectors,

$w_2$15

In the weak pairing limit, each charged corner mode splits into two Majorana zero modes, one in each $w_2$16 sector, but these are not generically robust because they can couple through perturbations such as $w_2$17. A robust single Majorana per corner appears when only one sector changes mass sign at a corner. For square geometry with $w_2$18-wave pairing and

$w_2$19

the condition is

$w_2$20

In that regime the two-dimensional SOTI/$w_2$21-wave SC heterostructure realizes a second-order topological superconductor with four Majorana corner modes, one per corner.

The three-dimensional construction follows the same logic. The Hamiltonian is

$w_2$22

with

$w_2$23

Adding $w_2$24 gaps the lateral surface Dirac cones and leaves one chiral electronic mode per hinge. After superconducting proximity coupling, each chiral electronic hinge mode becomes, in the weak-pairing limit, two chiral Majorana modes. Increasing the pairing strength can drive a boundary topological phase transition from two chiral Majoranas per hinge to one chiral Majorana per hinge, again under

$w_2$25

for the geometry studied with $w_2$26 and $w_2$27. Because Majorana modes are non-Abelian, these SOTI/SC heterostructures constitute non-Abelian higher-order platforms without requiring special pairings or magnetic fields (Yan, 2019).

7. Relation to adjacent non-Abelian higher-order-like phases

Not every non-Abelian topological insulator with subdimensional topology is a second-order topological insulator in the strict corner-or-hinge sense. A useful boundary of the concept is provided by non-Abelian Hopf–Euler insulators. These are three-band, $w_2$28-symmetric non-Abelian topological insulators in three dimensions whose strong invariant is a Hopf number

$w_2$29

and whose weak topology is described by Euler classes

$w_2$30

If the occupied bands are also split, the relevant invariant becomes a Pontryagin index on the real flag manifold. The same work derives surface Euler topology, helical nodal structures in the Brillouin zone, a quantized optical bulk integrated circular shift effect, and quantum-geometric breathing in the real-space Wannier functions (Jankowski et al., 2024).

The crucial distinction is that these phases are not explicitly formulated as second-order topological insulators with protected corner states or hinge modes. Their boundary signatures are surface Euler topology and boundary Wilson-loop structure rather than canonical codimension-two corner or hinge states. The relation to second-order topology is therefore indirect: the systems are higher-order-like through subdimensional invariants and boundary topology, but they do not furnish the standard definition of a second-order topological insulator.

This distinction clarifies the scope of the term non-Abelian second-order topological insulator. The label is most precise when reserved for phases in which codimension-two boundary states are protected by non-Abelian bulk data, as in the quaternion-based coupled-wire construction, or by real non-Abelian gauge structure encoded in $w_2$31 and Takagi factorization, as in the honeycomb Takagi topological insulator. It remains closely related, but not identical, to broader classes of non-Abelian topological phases with subdimensional boundary structure (Pan et al., 24 Dec 2025, Liu et al., 2022, Jankowski et al., 2024).