---
title: Non-Abelian Second-Order Topological Insulator
url: https://www.emergentmind.com/topics/non-abelian-second-order-topological-insulator
type: topic
---

# Non-Abelian Second-Order Topological Insulator

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{"query":"\"non-Abelian second-order topological insulator\" OR \"higher-order topological\" non-Abelian arXiv", "max_results": 10}
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{"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 [2512.21179]. 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 [2205.05873]. 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 [1907.02070].

## 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 [2512.21179]. 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 [2205.05873]. 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 [1907.02070].

## 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 \(y\)-direction is a one-dimensional SSH chain \(H_y\). The lattice Hamiltonian is
\[
\begin{split}
H&=\sum_{n,m=1}^{N,M/2}\sum_{\alpha,\beta=A,B,C}\big[ (J_{1}\ket{n,2m,\alpha}\bra{n,2m-1,\alpha} \\
 &+J_{2}\ket{n,2m+1,\alpha}\bra{n,2m,\alpha} +\text{h.c.}) \\
 &+S_{\alpha}\ket{n,m,\alpha}\bra{n,m,\alpha} \\
 &+(J_{\alpha\beta}\ket{n+1,m,\beta}\bra{n,m,\alpha}+\text{h.c.})\big ],
\end{split}
\]
with
\[
J_{AB}=J_{BA}=iu,\qquad J_{BC}=J_{CB}=iv,\qquad J_{AC}=J_{CA}=0.
\]
The Hamiltonian has the form
\[
H=H_x\oplus H_y = H_x\otimes{\mathbb I}_y+{\mathbb I}_x\otimes H_y.
\]
If
\[
H_x|\Psi_x\rangle=E_x|\Psi_x\rangle,\qquad H_y|\Psi_y\rangle=E_y|\Psi_y\rangle,
\]
then
\[
|\Psi\rangle = |\Psi_x\rangle\otimes|\Psi_y\rangle,\qquad E=E_x+E_y.
\]
This factorized structure underlies the corner-state construction.

The non-Abelian character originates in the \(\mathcal{PT}\)-symmetric trimer subsystem \(H_x\). Because \(\mathcal{PT}\) 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
\[
q_x=\hat{\mathsf{P}\exp\left[\oint_{-\pi}^{\pi}\bar{A}(k_x)\,dk_x\right],
\]
with
\[
[A(k_x)]_{q,q'}=\braket{\psi_q(k_x)|\partial_{k_x}|\psi_{q'}(k_x)},
\]
and \(\bar A(k_x)\) the affine BWZ connection. The allowed values are
\[
q_x=\{\sigma_0,\ \pm i\sigma_x,\ \pm i\sigma_y,\ \pm i\sigma_z,\ -\sigma_0\},
\]
identified with the quaternion group
\[
Q_x=\{1,\pm i,\pm j,\pm k,-1\}.
\]
The paper interprets \(i,j,k\) as \(\pi\)-rotations of the real eigenframe about different axes, \(-1\) as a \(2\pi\) rotation that is topologically nontrivial but invisible to ordinary Zak-phase diagnostics, and \(1\) as the trivial phase [2512.21179].

The \(y\)-subsystem is the SSH chain with chiral symmetry \(\Gamma=\sigma_3\) and Bloch Hamiltonian
\[
H_y(k_y)=(J_1+J_2\cos k_y)\sigma_1+J_2\sin k_y\,\sigma_2
= d_1(k_y)\sigma_1+d_2(k_y)\sigma_2.
\]
Its winding number is
\[
w=\frac{1}{2\pi}\int_{-\pi}^{\pi} \frac{d_1(k_y)\partial_{k_y}d_2(k_y)-d_2(k_y)\partial_{k_y}d_1(k_y)}
{|\mathbf d(k_y)|^2}\,dk_y.
\]

## 3. Hybrid invariant, boundary correspondence, and phase transitions

The coupled-wire model is classified by the hybrid topological vector
\[
\nu=(Q_x,w)\in \mathbb{Q}_8\times\mathbb{Z},
\]
with multiplication rule
\[
\nu_1\circ \nu_2\equiv (Q_{x1}Q_{x2},\, w_1+w_2).
\]
Its components have distinct origins: \(Q_x\) is the non-Abelian quaternion charge of the \(\mathcal{PT}\)-symmetric trimer chain, and \(w\) is the Abelian SSH winding number. Because quaternions do not commute, \(\nu\) is genuinely non-Abelian when \(Q_x\neq 1\) [2512.21179].

Corner states appear if and only if both components are nontrivial,
\[
Q_x\neq 1,\qquad w\neq 0.
\]
The counting rule is
\[
N_c=
\begin{cases}
4|w||Q_x^2|, & Q_x=\pm i,\pm k,\\
8|w||Q_x^2|, & Q_x=\pm j,-1,\\
0, & Q_x=1.
\end{cases}
\]
For the minimal model with \(w=1\), this yields \(N_c=4\) for \(Q_x=\pm i,\pm k\), \(N_c=8\) for \(Q_x=\pm j,-1\), and no corner states if \(Q_x=1\). The corner states are tensor products of subsystem edge states,
\[
|\Psi_c\rangle=|\psi_{xe}\rangle\otimes|\psi_{ye}\rangle,
\qquad
E_c=E_{xe}+E_{ye}.
\]
Since the SSH edge modes are pinned at zero energy in the topological phase, \(E_c=E_{xe}\).

The same construction yields weak topological edge phases when only one component of \(\nu\) is nontrivial. For
\[
\nu=(Q_x,0),\qquad Q_x\neq 1,
\]
the system supports weak non-Abelian edge states localized along \(x\), with the number of edge bands
\[
N_{ex}=
\begin{cases}
4|Q_x^2|, & Q_x=\pm i,\pm k,\\
8|Q_x^2|, & Q_x=\pm j,-1.
\end{cases}
\]
For
\[
\nu=(1,w),\qquad w\neq 0,
\]
the system supports weak Abelian edge states localized along \(y\), with
\[
N_{ey}=6|w|,\qquad Q_x\in\mathbb{Q}_8.
\]
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
\[
|J_1|=|J_2|,
\]
where the SSH winding changes between \(w=1\) and \(w=0\). The non-Abelian transition occurs when the trimer subsystem changes quaternion charge, for example among \(i\), \(j\), \(k\), \(-1\), and \(1\), and requires a gap closing in the relevant band structure. The charge \(Q_x=-1\) 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 [2512.21179].

## 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
\[
(\mathcal{PT})^2=1,
\]
so the Bloch Hamiltonian can be made real in an appropriate basis. The model also has sublattice symmetry
\[
\hat S=I_3\otimes \sigma_3,
\]
and inversion exchanges sublattices, so
\[
\{\hat P,\hat S\}=\{\mathcal{PT},\hat S\}=0.
\]
This anticommutation makes the Hamiltonian block off-diagonal and enables the Takagi formulation [2205.05873].

The momentum-space Hamiltonian is a nearest-neighbor dimerized honeycomb model with six sites per unit cell,
\[
\H(\k)=
\begin{bmatrix}
0&t_3&0&\chi^{(2)}_{\k} &0&t_1\\
t_3&0&t_2&0&\bar{\chi}^{(1)}_{\k} &0\\
0&t_2&0&t_1&0&\chi^{(3)}_{\k}\\
\bar{\chi}^{(2)}_{\k}&0&t_1&0&t_3&0\\
0&\chi^{(1)}_{\k}&0&t_3&0&t_2\\
t_1&0&\bar{\chi}^{(3)}_{\k}&0&t_2&0
\end{bmatrix},
\qquad
\chi^{(i)}_{\k}=t_i^\prime e^{-i\k\cdot \mathbf{a}_i}.
\]
Its bulk phase boundary follows from
\[
\det[\H(\Gamma)] = -\big(t_1^2t_1' + t_2^2t_2' + t_3^2t_3' - 2t_1t_2t_3 - t_1't_2't_3'\big)^2,
\]
so the gap closes when
\[
t_1^2t'_1+t_2^2t'_2+t_3^2t'_3 = 2t_1t_2t_3+t'_1t'_2t'_3.
\]
The topological regime is
\[
t_1^2t'_1+t_2^2t'_2+t_3^2t'_3 < 2t_1t_2t_3+t'_1t'_2t'_3,
\]
for which the second Stiefel–Whitney number is \(w_2=1\); the opposite inequality is trivial.

The invariant can be diagnosed through the Wilson loop
\[
W(k_y)=\mathcal P \exp\!\left(-i\int_{C_{k_y}} dk_x\, \mathcal A(k_x,k_y)\right),
\]
where \(\mathcal A\) is the non-Abelian Berry connection of the valence bands. Writing the Wilson-loop eigenvalues as
\[
\lambda_m(k_y)=e^{i\theta_m(k_y)},\qquad \theta_m(k_y)=\Im[\log \lambda_m(k_y)],
\]
the \(\mathbb Z_2\) invariant is
\[
w_2=\zeta \bmod 2,
\]
where \(\zeta\) is the number of crossings at \(\theta=\pi\). In the topological phase of the honeycomb model, the Wilson loop has a single crossing, giving \(w_2=1\).

The same topology admits a Takagi-factorization formulation. The flattened Hamiltonian can be written as
\[
\tilde\H(\k)=
\begin{bmatrix}
0 & Q(\k) \\
Q^\dagger(\k) & 0
\end{bmatrix},
\qquad
Q=Q^T,\qquad QQ^\dagger=I_M.
\]
Thus \(Q(\k)\) is a unitary symmetric matrix, and Takagi’s factorization gives
\[
Q(\k)=U(\k)U^T(\k),\qquad U(\k)\in U(M).
\]
The classifying space is
\[
US(M)=U(M)/O(M),
\qquad
\pi_2[US(M)]=\mathbb Z_2.
\]
On the overlap \(S^1\) of north and south hemispheres, the Takagi factors differ by an orthogonal transition function
\[
\mathcal O_{S^1}=U_N^\dagger|_{S^1}\,U_S|_{S^1}\in O(M),
\]
and the obstruction to a global Takagi factorization is precisely the same \(\mathbb Z_2\) data as \(w_2\). 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 [2205.05873].

## 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 \(\mathbf a_i\), the mass is
\[
m_i=t_it_i'-t_jt_k,\qquad i\neq j\neq k.
\]
When \(m_i=0\), the corresponding edge is gapless and hosts helical edge modes. When \(m_i\neq 0\), 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 [2205.05873].

The finite-sample consequence is sharper than the bare existence of corner states. In \(\mathcal{PT}\)-symmetric samples, corner zero modes come in \(\mathcal{PT}\)-related pairs, and the number of such pairs is odd when \(w_2=1\). 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 \(w_2\) 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 \(H_x\) edge state and an \(H_y\) 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 [2512.21179].

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 \(\mathcal{PT}\)-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 [1907.02070]. 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
\[
H(\mathbf{k})= \epsilon(\mathbf{k})\sigma_z\tau_z +\lambda_x \sin k_x\, \sigma_x s_z +\lambda_y \sin k_y\, \sigma_y \tau_z +\Lambda(\mathbf{k}) \sigma_x s_x \tau_z +\mu \tau_z +\Delta(\mathbf{k}) s_y \tau_y ,
\]
with
\[
\epsilon(\mathbf{k}) = m_0 - t_x \cos k_x - t_y \cos k_y,
\qquad
\Lambda(\mathbf{k}) = \Lambda_x \cos k_x - \Lambda_y \cos k_y,
\]
\[
\Delta(\mathbf{k}) = \Delta_0 + \Delta_x \cos k_x + \Delta_y \cos k_y.
\]
The pairing \(\Delta(\mathbf{k})\) may represent \(s\)-wave, \(s_\pm\)-wave, or \(d\)-wave pairing, and the proposal does not require special pairing symmetry. The BdG Hamiltonian always has particle-hole symmetry,
\[
P H(\mathbf{k}) P^{-1}=-H(-\mathbf{k}),\qquad P=\tau_x K.
\]
For \(\mu=0\) there is also chiral symmetry,
\[
\{C,H\}=0,\qquad C=\sigma_x s_y \tau_z,
\]
but the Majorana corner modes remain protected by particle-hole symmetry even when chiral symmetry is broken by \(\mu\).

The low-energy edge theory is
\[
H_{\rm Edge} = -i \lambda(l)\partial_l s_z + M_\Lambda(l) s_y + M_S(l) s_y \tau_y.
\]
With \(\Lambda_x=\Lambda_y\), the superconducting edge theory splits into two sectors,
\[
H_{\tau_y=\pm 1} = -i\lambda(l)\partial_l s_z + \big(M_\Lambda(l) \pm M_S(l)\big)s_y.
\]
In the weak pairing limit, each charged corner mode splits into two Majorana zero modes, one in each \(\tau_y\) sector, but these are not generically robust because they can couple through perturbations such as \(\mu\tau_z\). A robust single Majorana per corner appears when only one sector changes mass sign at a corner. For square geometry with \(s\)-wave pairing and
\[
m=t_x+t_y-m_0,
\qquad
\frac{m\Lambda_x}{t_x} < \frac{m\Lambda_y}{t_y},
\]
the condition is
\[
\frac{m\Lambda_x}{t_x} < \Delta_0 < \frac{m\Lambda_y}{t_y}.
\]
In that regime the two-dimensional SOTI/\(s\)-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
\[
H(\mathbf{k})= \xi(\mathbf{k})\sigma_z\tau_z +\sum_{i=x,z}\lambda_i \sin k_i \,\sigma_x s_i +\lambda_y \sin k_y\, \sigma_x s_y \tau_z +\Lambda(\mathbf{k})\sigma_y +\mu \tau_z +\Delta_0 s_y\tau_y,
\]
with
\[
\xi(\mathbf{k}) = m_0 - t_x\cos k_x - t_y\cos k_y - t_z\cos k_z.
\]
Adding \(\Lambda(\mathbf{k})\) 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
\[
\frac{m\Lambda_x}{t_x} < \Delta_0 < \frac{m\Lambda_y}{t_y},
\qquad
m=t_x+t_y+t_z-m_0>0,
\]
for the geometry studied with \(\mu=0\) and \(\Lambda_x=\Lambda_y\). Because Majorana modes are non-Abelian, these SOTI/SC heterostructures constitute non-Abelian higher-order platforms without requiring special pairings or magnetic fields [1907.02070].

## 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, \(\mathcal{PT}\)-symmetric non-Abelian topological insulators in three dimensions whose strong invariant is a Hopf number
\[
H\in \pi_3(S^2)\cong \mathbb Z,
\qquad
H=-\frac{1}{16\pi^2}\int a\wedge Eu,
\]
and whose weak topology is described by Euler classes
\[
\chi_i=\frac{1}{2\pi}\int_{T^2_i} Eu,\qquad i=x,y,z.
\]
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 [2405.17305].

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 \(w_2\) 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 [2512.21179; 2205.05873; 2405.17305].

Source: https://www.emergentmind.com/topics/non-abelian-second-order-topological-insulator