Non-Abelian Topological Charges (NATCs)

• Non-Abelian topological charges (NATCs) are group-valued invariants defined through the homotopy of real eigenstate frames, exemplified by quaternionic structures in PT-symmetric three-band systems.
• Experimental methods, including Wilson loop tomography in photonic and acoustic setups, have validated NATCs as predictors of edge-state patterns and nodal link configurations.
• NATCs underpin a refined bulk-boundary correspondence by encoding non-Abelian topological words, influencing gap-specific edge modes in static and Floquet systems.

Non-Abelian topological charges (NATCs) are group-valued invariants that classify multiband and multigap band structures when the transported object is the full eigenstate frame rather than a single isolated band. In parity-time-symmetric settings with a real Bloch basis, the relevant topology is the homotopy of ordered orthonormal frames modulo sign flips, and the resulting holonomies are non-commutative. In the three-band case this structure is quaternionic, with charges in $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$; in multigap insulators and Floquet systems it leads to bulk-boundary correspondences that are not captured by Abelian invariants such as Chern numbers, winding numbers, or single-gap Zak phases (Wu et al., 2018, Zhang et al., 22 Apr 2026, Li et al., 2023).

1. Homotopy classification and classifying spaces

In PT-symmetric three-band systems, the Bloch Hamiltonian can be chosen real, so the three real, orthonormal eigenstates form a moving orthonormal frame. The corresponding order-parameter space is

$M_3 = O(3)/O(1)^3,$

and its fundamental group is the quaternion group

$$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$

This is the basic homotopy statement behind quaternion NATCs in PT-symmetric three-band systems (Zhang et al., 22 Apr 2026).

More generally, for $N$ real bands with nondegenerate spectrum, the frame space is

$M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$

For $N\ge 3$, $\pi_1(M_N)$ is non-Abelian; in the many-band description it is the Salingaros vee group $\overline{\mathsf{P}_N}$, so quaternionic charges are the $N=3$ specialization of a broader real-frame topology (Wu et al., 2018).

A central subtlety is the distinction between homotopy with and without a fixed base point. With a fixed base point, loops are classified by elements of $\pi_1(M_3)$, hence by the full group $M_3 = O(3)/O(1)^3,$0. Without a fixed base point, the classification is by conjugacy classes

$M_3 = O(3)/O(1)^3,$1

so $M_3 = O(3)/O(1)^3,$2, $M_3 = O(3)/O(1)^3,$3, and $M_3 = O(3)/O(1)^3,$4 are merged. This distinction is operational in data-driven classification, where eight phases appear for based homotopy and five for free homotopy (He et al., 12 Apr 2025).

Even-band systems exhibit additional structure. In the four-band PT-symmetric case, the paper denotes the classifying space by $M_3 = O(3)/O(1)^3,$5, with

$M_3 = O(3)/O(1)^3,$6

the generalized quaternion group of order 16. This yields charge classes $M_3 = O(3)/O(1)^3,$7 together with the even-band-specific classes $M_3 = O(3)/O(1)^3,$8, which have no three-band analogue (Jiang et al., 2021).

2. Wilson loops, frame holonomy, and gauge structure

For a PT-symmetric three-band system with real orthonormal eigenstates $M_3 = O(3)/O(1)^3,$9, the Wilczek–Berry–Zak connection is

$$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$0

It is antisymmetric, so $$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$1. Using the Lie-algebra isomorphism $$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$2,

$$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$3

one lifts the connection to $$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$4 and defines the lifted Wilson loop

$$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$5

In the PT-symmetric three-band case,

$$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$6

which are identified with $$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$7. This gives a direct computational definition of NATCs from bulk eigenvectors and Wilson loops; the same construction extends to the PT-unbroken non-Hermitian regime with the biorthogonal connection $$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$8 (Zhang et al., 22 Apr 2026).

Gauge freedom does not change the physical rotation of the frame, but it does affect its representative in $$\pi_1(M_3)=\mathbb{Q}_8=\{\pm 1,\pm i,\pm j,\pm k\},\quad i^2=j^2=k^2=ijk=-1.$$9. Under a gauge transformation,

$N$0

and because $N$1 double-covers $N$2, one may also have the sign freedom $N$3. This sign freedom does not destroy the quaternion structure; rather, it makes a common reference point essential when one wants relative signs across multiple loops. The common reference point method transports the eigenbasis from a single $N$4 to each loop base point by maximizing overlaps and thereby fixes relative signs for non-Abelian multiplication such as $N$5 and $N$6 (Park et al., 2021).

Experimentally, the same Wilson-loop logic has been implemented in distinct ways. In a three-band photonic quantum walk, the bulk Floquet operator $N$7 was reconstructed tomographically from one-step dynamics, the real eigenvectors $N$8 were tracked through the Brillouin zone, and the quaternion charge was identified from their sign-flip pattern or from the lifted Wilson loop. In a dielectric double-diamond photonic crystal, overlap matrices along closed $N$9-space loops yielded $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$0, $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$1, and their products as loop charges (Lin et al., 8 Aug 2025, Park et al., 2021).

3. Nodes, links, and braiding of degeneracies

The earliest condensed-matter formulation of NATCs emphasized nodal lines in PT-symmetric metals with weak spin-orbit coupling. In that setting, nodal lines in consecutive gaps anticommute, loops around them carry non-commuting frame charges, and the resulting algebra constrains which nodal-line crossings, reconnections, and chains are allowed. A key consequence is that nodal-line rings must satisfy linking constraints with adjacent-gap lines, and the monopole charge of a nodal-line ring is tied to this linking structure; the monopole charge can flip sign when braided along a path with non-trivial Berry phase (Tiwari et al., 2019).

In dielectric photonic crystals, a direct realization of quaternionic nodal links was proposed in a double-diamond structure. There, an “orange” ring between the 3rd and 4th bands and a “cyan” ring between the 4th and 5th bands generate distinct frame rotations: loops encircling the orange ring carry charge $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$2, loops around the cyan ring carry charge $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$3, and a loop that encloses both rings yields $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$4 or $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$5 depending on the order of encirclement. The non-Abelian character is explicit in

$M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$6

so reversing the order of the two encirclements changes the charge (Park et al., 2021).

A more dynamical picture appears in kagome non-Abelian topological semimetals. There the relevant objects are frame charges, Dirac strings, and the Euler class of two-band patches. Braiding a node in gap I around a node in gap II acts by quaternion conjugation and flips its charge, while the parity $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$7 controls whether nodal pairs are removable. The experimentally observed phase diagram included charge transfer between gaps, symmetry-enforced triply-degenerate intermediate nodes, and stable double nodes with frame charge $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$8 and quadratic dispersion (Jiang et al., 2021).

The same principle was isolated in a minimal three-band acoustic metamaterial with one nodal pair per gap. There, mirror eigenvalues at high-symmetry points diagnosed whether a colliding pair could annihilate. After braiding, a gap-II pair acquired identical charge $M_N = O(N)/O(1)^N \cong SO(N)/\mathsf{P}_N.$9, so its total charge became

$N\ge 3$0

and the nodes failed to annihilate, instead bouncing to the other diagonal. This “creation, braiding, collision, and repulsion” sequence is the canonical minimal example of non-Abelian nodal braiding (Qiu et al., 2022).

A related photonic viewpoint interprets these objects as non-Abelian frame-charge flow in momentum space. In biaxial photonic crystals, the $N\ge 3$1 point acts as a source or sink because the zero-frequency longitudinal mode induces hidden braiding. Frame-charge arrows then obey a Kirchhoff-like rule at nodal junctions, while braiding with adjacent nodal lines flips their direction (Wang et al., 2022).

4. Multigap insulators and the “topological word”

For gapped multigap insulators, the main refinement beyond the global quaternion charge is the “topological word.” In this framework the total NATC is written as an ordered product

$N\ge 3$2

where each letter $N\ge 3$3 encodes the $N\ge 3$4 topology of a single adjacent gap, and the ordering records the sequence of adjacent-gap inversions. In static three-band systems with nearest-neighbor and next-nearest-neighbor hoppings, $N\ge 3$5, with $N\ge 3$6 attached to the upper gap and $N\ge 3$7 to the lower gap. The non-adjacent charge $N\ge 3$8 is not fundamental at the gap level but is realized as a word such as $N\ge 3$9 or, after a Hurwitz move, $\pi_1(M_N)$0 (Zhang et al., 22 Apr 2026).

This refinement is necessary because the same global charge can support different edge-state patterns. In the canonical static three-band examples, $\pi_1(M_N)$1 corresponds to the word $\pi_1(M_N)$2 and one pair of edge states in the upper gap; $\pi_1(M_N)$3 corresponds to $\pi_1(M_N)$4 and one pair in the lower gap; $\pi_1(M_N)$5 corresponds to $\pi_1(M_N)$6 and one pair in each adjacent gap. More strikingly, the same total charge $\pi_1(M_N)$7 can be represented by $\pi_1(M_N)$8, $\pi_1(M_N)$9, or $\overline{\mathsf{P}_N}$0, giving, respectively, two lower-gap pairs, two upper-gap pairs, or pairs distributed across both gaps. The complete non-Abelian bulk-boundary correspondence is therefore not determined by the global quaternion element alone (Zhang et al., 22 Apr 2026).

The topological word is constructed from a minimal interpolation $\overline{\mathsf{P}_N}$1 between bulks, whose Dirac singularities carry local charges $\overline{\mathsf{P}_N}$2. Under open boundaries, the interpolation from the trivial phase yields the canonical word; the number of letters attached to a given gap equals the number of edge-state pairs in that gap, bounded by the maximal hopping range $\overline{\mathsf{P}_N}$3. Hurwitz moves,

$\overline{\mathsf{P}_N}$4

together with base-point conjugation and stabilization by trivial pairs, generate gauge-equivalent words with the same edge predictions (Zhang et al., 22 Apr 2026).

This non-Abelian bulk-boundary principle has experimental and model-specific counterparts. In a PT-symmetric three-band transmission-line system, hard boundaries and domain walls were classified by a quotient such as $\overline{\mathsf{P}_N}$5, and the observed edge spectra followed the predicted conjugacy classes. In a reciprocal Hermitian acoustic realization, single-phase endpoint states identified only the quotient classes $\overline{\mathsf{P}_N}$6, $\overline{\mathsf{P}_N}$7, and $\overline{\mathsf{P}_N}$8, while interfaces implemented the non-commutative product $\overline{\mathsf{P}_N}$9. In an ultracold-atom proposal, the same logic appears as $N=3$0, with interface modes determined by the nonconservation multiplication relation of quaternion charges (Guo et al., 2020, Sun et al., 2023, Wang et al., 2024).

5. Floquet NATCs, phase-band singularities, and anomalous phases

Floquet driving introduces an additional quasienergy gap at the Floquet zone edge $N=3$1, so a three-band Floquet system has three gaps rather than two. For a time-periodic Hamiltonian,

$N=3$2

the effective Hamiltonian $N=3$3 may be PT-symmetric even when the micromotion is essential. The relevant topology is encoded not only in $N=3$4 but in the full phase-band structure of $N=3$5, whose Dirac singularities carry local quaternion charges $N=3$6 and multiply in order to give the bulk charge

$N=3$7

Because the highest and lowest Floquet bands are adjacent by periodicity, the letter set enlarges to $N=3$8, and words with trivial total charge can still host edge states in every gap (Li et al., 2023, Pan et al., 14 Mar 2025).

This produces a multifold bulk-edge correspondence. The same global quaternion charge can arise from distinct singularity patterns and hence from distinct gap-by-gap edge configurations. In one-dimensional three-band Floquet non-Abelian topological insulators, $N=3$9 can come either from two singularities $\pi_1(M_3)$0, giving edge modes in the first and second gaps, or from a single $\pi_1(M_3)$1, giving an edge mode only in the third gap. The most distinctive case is the anomalous non-Abelian phase with trivial bulk charge $\pi_1(M_3)$2 but edge states in all three gaps, coming from an ordered product such as

$\pi_1(M_3)$3

A further Floquet-only signature is the swap-interface effect: two drives related by a time-frame shift have identical bulk quasienergy spectra, yet an interface between them can host robust modes because the ordered singularity data, and hence the non-Abelian composition, differ (Li et al., 2023).

These phenomena have been realized experimentally in a photonic quantum walk. There, bulk-dynamic tomography reconstructed $\pi_1(M_3)$4 and identified quaternion charges $\pi_1(M_3)$5 and $\pi_1(M_3)$6, while spatially resolved injection spectroscopy resolved the gap structure of domain-wall states. Two inequivalent configurations with the same domain-wall charge $\pi_1(M_3)$7 were observed: one with states only in the second gap, another with states in the first and third gaps. An anomalous configuration with $\pi_1(M_3)$8 nevertheless exhibited domain-wall states in all three gaps, directly confirming the multifold non-Abelian bulk-boundary correspondence (Lin et al., 8 Aug 2025).

Floquet non-Abelian topology also extends to higher-order systems. In a two-dimensional three-band Floquet higher-order topological insulator, corner and edge states appeared in all quasienergy gaps despite a trivial quaternion charge $\pi_1(M_3)$9, while spatially exchanging the drive generated interface modes because $M_3 = O(3)/O(1)^3,$00. In that setting the full edge-state configuration was determined by quadruple degenerate phase-band singularities, and the composite Chern number of the full three-band multiplet was

$M_3 = O(3)/O(1)^3,$01

showing that the anomalous phase is globally non-trivial even when the net quaternion charge is trivial (Zhou et al., 18 Aug 2025).

6. Extensions, diagnostics, and open problems

The NATC framework has already moved beyond the minimal three-band, first-order setting. In four-band PT-symmetric systems, the generalized quaternion group $M_3 = O(3)/O(1)^3,$02 governs frame topology, and the classes $M_3 = O(3)/O(1)^3,$03 are distinguished by opposite four-dimensional rotation senses on stereographically projected Clifford tori. In a distinct direction, a coupled-wire construction of a non-Abelian second-order topological insulator combines a quaternion charge $M_3 = O(3)/O(1)^3,$04 from a PT-symmetric three-band wire with an Abelian winding number $M_3 = O(3)/O(1)^3,$05 from an SSH wire, forming a topological vector $M_3 = O(3)/O(1)^3,$06; corner states emerge only when both components are nontrivial, while weak topological edge states arise when the quaternion charge alone is nontrivial (Jiang et al., 2021, Pan et al., 24 Dec 2025).

Non-Hermitian systems realize a different non-Abelian topology, based not on quaternion frame rotations but on braid groups of complex eigenvalues. In a three-band non-Hermitian Hamiltonian implemented with an NV center in diamond, the NATC is the braid word $M_3 = O(3)/O(1)^3,$07 extracted from complex eigenvalue braids along a loop in parameter space. Across a non-Hermitian non-Abelian topological transition, the discriminant number remained zero while the braid invariant changed from the identity to

$M_3 = O(3)/O(1)^3,$08

and two second-order exceptional points of opposite Abelian charge merged into a third-order exceptional point. This shows that the term NATC now covers at least two distinct, rigorous non-Abelian structures: quaternionic frame charges in real-band topology and braid-group invariants in non-Hermitian spectral topology (Wang et al., 22 Mar 2025).

Algorithmic classification is also advancing. Diffusion maps applied to one-dimensional PT-symmetric three-band loops recover the eight based-homotopy phases of $M_3 = O(3)/O(1)^3,$09, the five free-homotopy conjugacy classes, and even the multiplication table of the quaternion group by concatenating samples and reclustering them. This addresses a practical difficulty of NATCs: the classification depends on multiband frame continuity, on based versus free homotopy, and on non-commuting loop composition rather than on a single scalar invariant (He et al., 12 Apr 2025).

Several caveats recur across the literature. The sign of a NATC is gauge-dependent unless a common reference is fixed; the global quaternion element does not, by itself, determine the gap-resolved edge pattern; the canonical topological word depends on a minimal interpolation and is only defined up to Hurwitz moves, conjugation, and stabilization; and when PT symmetry is broken the global quaternion invariant becomes ill-defined even though gapwise remnant topology and edge-state survival may still be tracked. Extending complete non-Abelian bulk-boundary correspondences to higher dimensions and $M_3 = O(3)/O(1)^3,$10 bands remains an open direction (Park et al., 2021, Zhang et al., 22 Apr 2026).

In a distinct but related usage, non-Abelian topological charges also label excitations in Kitaev quantum double models. There the topological sectors are pairs $M_3 = O(3)/O(1)^3,$11, with $M_3 = O(3)/O(1)^3,$12 a conjugacy class of a finite group $M_3 = O(3)/O(1)^3,$13 and $M_3 = O(3)/O(1)^3,$14 an irreducible representation of the centralizer of a representative of $M_3 = O(3)/O(1)^3,$15. Single-qudit perturbations reduce the gauge symmetry from $M_3 = O(3)/O(1)^3,$16 to $M_3 = O(3)/O(1)^3,$17, producing condensation and confinement of topological charges. This usage is not a frame-holonomy classification of Bloch bands, but it shows that “non-Abelian topological charge” already spans both band topology and topological order in lattice gauge systems (0712.0190).