Floquet Non-Abelian Topological Insulators

• FNATIs are periodically driven three-band systems exhibiting non-Abelian topological phases characterized by quaternion charges in PT-symmetric settings.
• These systems incorporate an extra Floquet-replica gap that enables transitions and edge-state configurations unattainable in static models.
• They demonstrate multifold bulk-edge correspondence where ordered phase-band singularities determine diverse boundary-state patterns and interface modes.

Floquet non-Abelian topological insulators (FNATIs) are periodically driven topological phases in which the multi-gap topology is not captured by a single Abelian invariant, but by non-commuting charges associated with the global evolution of a real eigenframe, most commonly the quaternion group $Q_8$ in $\mathcal{PT}$-symmetric three-band settings. In the minimal construction, a one-dimensional three-band Floquet system acquires an additional quasienergy gap on the Floquet Brillouin zone, so that topological information is distributed across three intertwined gaps rather than two static band separations. This structure permits anomalous phases with edge states in all gaps despite trivial net bulk charge, multifold bulk-edge correspondence in which the same bulk charge supports several distinct boundary-state patterns, and interface modes generated purely by exchanging the order of driving steps (Li et al., 2023, Pan et al., 14 Mar 2025).

1. Minimal definition and Floquet setting

A Floquet system is specified by a time-periodic Hamiltonian $H(t)=H(t+T)$, with one-period evolution

$$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$

Its eigenvalues are $e^{-i\epsilon_n T}$, where the quasienergies $\epsilon_n$ are defined modulo $2\pi/T$. In FNATIs, the relevant setting is a three-band Floquet problem with $\mathcal{PT}$ symmetry, so that in an appropriate time frame the Floquet eigenvectors can be chosen real and orthonormal. As momentum winds around the one-dimensional Brillouin zone, the ordered eigenframe defines a loop in $M_3=O(3)/O(1)^3$, and the first homotopy group is the quaternion group $Q_8$ rather than $\mathcal{PT}$0 or $\mathcal{PT}$1 (Li et al., 2023, Lin et al., 8 Aug 2025).

The Floquet context adds an essential ingredient absent in static three-band non-Abelian insulators: the quasienergy axis is circular, and the lowest and highest bands become adjacent through the Floquet replica, or $\mathcal{PT}$2-gap. In the formulation of Li and Hu, the three gaps are the gap between bands 1 and 2, the gap between bands 2 and 3, and the third Floquet-replica gap crossing the quasienergy zone boundary at $\mathcal{PT}$3 (Li et al., 2023). This extra adjacency is what allows transitions and edge-state configurations with no static analog.

A central distinction from Abelian Floquet topological phases is that the relevant invariants do not commute. Abelian driven systems can assign independent $\mathcal{PT}$4 or $\mathcal{PT}$5 indices to separate gaps, but these invariants cannot encode frame-rotation braidings among three subspaces. FNATIs therefore sit outside the standard tenfold-way logic emphasized for two-band or single-gap problems (Li et al., 2023).

2. Quaternion charges, phase-band singularities, and gap-resolved topology

The quaternion group is

$\mathcal{PT}$6

with multiplication rules

$\mathcal{PT}$7

In FNATIs, these elements label non-Abelian topological charges of the real Floquet eigenframe. The global bulk charge may be computed from a lifted Wilson loop or holonomy in the real eigenstate frame, taking values in $\mathcal{PT}$8 (Li et al., 2023, Qiu et al., 9 Aug 2025).

For Floquet systems, the full micromotion matters. One therefore introduces continuous phase bands $\mathcal{PT}$9 associated with a $H(t)=H(t+T)$0-symmetric continuation $H(t)=H(t+T)$1. Isolated band-touching events in the two-dimensional $H(t)=H(t+T)$2 space appear as Dirac singularities, each carrying a quaternion charge $H(t)=H(t+T)$3. The bulk invariant is then the ordered product

$H(t)=H(t+T)$4

with the order fixed by the arrangement of singularities. This ordered structure is the source of the non-Abelian character: the same total product can arise from inequivalent factorizations (Li et al., 2023, Qiu et al., 9 Aug 2025).

A recent refinement is the “topological word” framework, which replaces a single global quaternion by an ordered sequence

$H(t)=H(t+T)$5

In Floquet three-band systems the enlarged alphabet reflects the circular quasienergy axis: $H(t)=H(t+T)$6 label the Floquet $H(t)=H(t+T)$7-gap topology, while $H(t)=H(t+T)$8 and $H(t)=H(t+T)$9 encode the other two gaps. The ordered product reproduces the global quaternion charge, but the sequence retains band-adjacency information that a lone quaternion loses. In this formulation, the parity of edge-state pairs in a given gap equals the number of appearances of the corresponding letter (Zhang et al., 22 Apr 2026).

This gap-resolved viewpoint clarifies why global homotopy classification and observable edge-state patterns are not identical pieces of information. A single quaternion describes the total frame rotation, whereas the topological word records how that rotation is built from elementary singularities associated with particular gaps. That distinction is decisive for Floquet boundary physics (Zhang et al., 22 Apr 2026).

3. Anomalous phases and multifold bulk-edge correspondence

The hallmark of FNATIs is multifold bulk-edge correspondence. In ordinary Abelian settings, one bulk invariant typically predicts a unique boundary-state count in a given gap. In FNATIs, the same bulk quaternion charge can correspond to several distinct edge-state configurations because different ordered factorizations of the same quaternion correspond to different phase-band singularity patterns (Li et al., 2023).

Each nontrivial singularity charge signals a gapwise phase-band closing and, by a Jackiw-Rebbi argument, binds one topological edge mode in that specific gap. Since quaternions do not commute, the product is not sufficient to determine how many singularities occurred in each gap or in which order. The same total charge can therefore support distinct combinations of edge modes across the three gaps (Li et al., 2023, Qiu et al., 9 Aug 2025).

The anomalous non-Abelian phase is the sharpest example. In the one-dimensional three-band Floquet construction of Qiu et al., three singularities with charges $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$0, $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$1, and $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$2 occur in successive gaps, so that

$$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$3

The net quaternion charge is trivial, yet there are exactly three topological edge modes, one in each gap. From the viewpoint of Abelian invariants, the Floquet operator over the full period carries vanishing total charge in each gap; in a static three-band problem, this would imply no edge states. FNATIs violate that expectation because the topology resides in the ordered factorization of the evolution rather than in the net bulk charge alone (Qiu et al., 9 Aug 2025).

This is the main correction to a common misconception: a trivial global quaternion charge does not imply the absence of protected boundary states in periodically driven non-Abelian systems. Another misconception is that the quaternion charge uniquely determines the boundary spectrum. The 2025 analysis of Floquet non-Abelian charges and edge states showed that a given conjugacy class such as $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$4 or $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$5 can correspond to distinct edge-mode patterns, depending on whether the relevant transition involved ordinary gaps or the anomalous Floquet $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$6-gap (Pan et al., 14 Mar 2025).

A second signature is the swap effect. If one half of a system is driven by $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$7 and the other by the swapped sequence $$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$8, the two bulks can have the same spectrum and even trivial net quaternion charge, yet their phase-band singularities are ordered differently. In the acoustic realization, this produces a nontrivial interface charge in the third gap,

$$U(T)=\mathcal T \exp\!\Bigl(-i\!\int_0^T H(\tau)\,d\tau\Bigr).$$9

and hence a protected topological interface mode. Theoretical analyses identify this phenomenon as a genuine signature of non-Abelian Floquet braiding and absent in Abelian Floquet insulators (Li et al., 2023, Qiu et al., 9 Aug 2025).

4. One-dimensional model constructions and experimental realizations

By 2025, FNATIs had moved from theory to two complementary experimental platforms: acoustics and photonic quantum walks. Both platforms targeted the minimal one-dimensional three-band setting, but they emphasized different observables.

Platform	Core Floquet construction	Main demonstrated signature
Acoustic lattice	$e^{-i\epsilon_n T}$0	Edge modes in all three gaps and swapped-drive interface mode
Photonic quantum walk	$e^{-i\epsilon_n T}$1	Quaternion-charge tomography and gap-resolved boundary spectroscopy

In the acoustic experiment of Qiu et al., the starting point was a one-dimensional three-site unit cell with $e^{-i\epsilon_n T}$2 symmetry and Bloch Hamiltonian

$e^{-i\epsilon_n T}$3

where time periodicity entered solely through the complex coupling $e^{-i\epsilon_n T}$4 in $e^{-i\epsilon_n T}$5. The authors chose a piecewise-constant drive with $e^{-i\epsilon_n T}$6 for $e^{-i\epsilon_n T}$7 and $e^{-i\epsilon_n T}$8 for $e^{-i\epsilon_n T}$9, so that the one-period operator factorized as $\epsilon_n$0. The experimental system comprised ten unit cells of three cylindrical cavities with resonant frequencies $\epsilon_n$1 Hz, $\epsilon_n$2 Hz, and $\epsilon_n$3 Hz; static couplings $\epsilon_n$4 Hz, $\epsilon_n$5 Hz, $\epsilon_n$6 Hz, $\epsilon_n$7 Hz, and $\epsilon_n$8 Hz; and time-periodic $\epsilon_n$9 Hz realized by unidirectional active feedback circuits. A 50% duty cycle and phase shift $2\pi/T$0 or $2\pi/T$1 implemented the two-step drive. By changing the period from $2\pi/T$2 ms to $2\pi/T$3 ms, the experiment moved from phase E, with edge modes only in gaps 1 and 2, to the anomalous phase G, with edge modes in all three gaps. Spatial Fourier transforms of the $2\pi/T$4th and $2\pi/T$5st harmonic responses mapped the bulk quasienergy bands, and a domain wall between two phase-G bulks with swapped driving sequence revealed the predicted counterintuitive interface mode (Qiu et al., 9 Aug 2025).

The photonic quantum-walk realization used a time-multiplexed walk with a three-dimensional coin and Floquet step

$2\pi/T$6

The coin states were encoded as $2\pi/T$7, $2\pi/T$8, and $2\pi/T$9, where $\mathcal{PT}$0 denote two fiber modes and $\mathcal{PT}$1 denote polarization. The architecture combined fiber loops, electro-optic modulators, triple-PBS interferometers, and waveplates to implement the conditional shift $\mathcal{PT}$2 and the two rotations $\mathcal{PT}$3 and $\mathcal{PT}$4. Momentum-space tomography reconstructed the $\mathcal{PT}$5 matrix $\mathcal{PT}$6 from single-step bulk dynamics, and diagonalization yielded real orthonormal eigenvectors whose sign-flip patterns over the Brillouin zone identified the quaternion charge. A second diagnostic, spatially resolved injection spectroscopy at a domain wall, measured the quasienergies of localized boundary states and determined which of the three Floquet gaps contained them. This combination of direct bulk-dynamic detection and edge spectroscopy established the multifold bulk-boundary correspondence and identified the anomalous case $\mathcal{PT}$7 with edge modes in all three gaps (Lin et al., 8 Aug 2025).

Taken together, these experiments did not merely observe isolated in-gap modes. They resolved the specific FNATI claim that non-Abelian Floquet topology is encoded jointly in bulk micromotion and gap-resolved boundary structure.

5. Two-dimensional extensions: helicity, flat bands, and higher-order modes

The non-Abelian Floquet program has already extended beyond the minimal one-dimensional setting in two distinct directions.

The first is a two-dimensional Floquet topological photonic insulator on a Lieb lattice of coupled microring resonators. Over one period, the system executes four successive nearest-neighbor coupling steps with Bloch Hamiltonians $\mathcal{PT}$8, and the reduced Floquet operator is

$\mathcal{PT}$9

At perfect coupling $M_3=O(3)/O(1)^3$0, all three quasienergy bands are perfectly flat with $M_3=O(3)/O(1)^3$1, yet the Floquet modes undergo nontrivial micromotion during the cycle. By tracking the evolution of the sublattice-resolved wavepackets, the authors showed that the corresponding world lines braid around each other within each unit cell. The relevant invariant is a quantized non-Abelian helicity

$M_3=O(3)/O(1)^3$2

which equals $M_3=O(3)/O(1)^3$3 for the three-band Floquet-Lieb model at $M_3=O(3)/O(1)^3$4. Equivalently, the braid winding satisfies $M_3=O(3)/O(1)^3$5 with $M_3=O(3)/O(1)^3$6. This topological structure coexists with vanishing band Chern numbers $M_3=O(3)/O(1)^3$7: the nontriviality is not a 2D band invariant but a $M_3=O(3)/O(1)^3$8-dimensional Floquet invariant of the micromotion. The paper further proposed measurement through a synthetic magnetic field and showed in full-wave simulations of a $M_3=O(3)/O(1)^3$9 lattice that the three flat-band resonances shift rigidly in a manner that reproduces $Q_8$0 (Leng et al., 26 Jan 2026).

The second direction is a two-dimensional higher-order Floquet non-Abelian topological insulator on a square lattice with a two-step periodic drive between $Q_8$1 and $Q_8$2. In the model of Zhou et al., the Floquet spectrum on an open lattice supports corner and edge states in all energy gaps despite trivial quaternion charge $Q_8$3. At the same time, the system carries a non-zero composite Chern number, and the configuration of these boundary states is determined by quadruple-degenerate phase-band singularities in the time evolution. Spatially exchanging the driving sequence across a boundary generates interface modes, again as a direct consequence of non-commutativity of the driving protocol. This suggests that the anomalous non-Abelian logic of the one-dimensional FNATI survives in higher-order form when both corner and edge channels are present (Zhou et al., 18 Aug 2025).

These two-dimensional results broaden the concept of FNATIs from a three-gap 1D boundary problem to a more general dynamical topology of micromotion, braiding, and higher-order boundary localization.

6. Classification refinements, limitations, and current directions

Recent work indicates that the global quaternion charge is indispensable but not always sufficient. The topological word framework was proposed precisely because a single $Q_8$4 captures homotopy but discards gap-adjacency information that controls the edge-state pattern across multiple gaps. In this formulation, phase-band singularities, braiding representations, and boundary-state counting become different descriptions of the same ordered gap-resolved topology (Zhang et al., 22 Apr 2026).

Another important refinement concerns symmetry. The non-Abelian quaternion classification relies on real eigenframes and therefore on $Q_8$5 symmetry in the constructions discussed here. When $Q_8$6 symmetry is broken, some gaps may close or the eigenstates may become complex, rendering the global quaternion charge ill-defined. The topological word proposal is notable because it is stated to remain informative for the surviving open gaps even when the global non-Abelian topology is no longer well defined (Zhang et al., 22 Apr 2026).

At the level of phase diagrams, Floquet driving does more than reproduce static non-Abelian phases. It reshuffles the non-driven phase diagram, produces both gapped and gapless Floquet band structures with non-Abelian charges, and allows transitions driven solely by the anomalous $Q_8$7-gap $Q_8$8, a process inaccessible to static three-band models because the lowest and highest static bands cannot meet without passing through the middle band. The same work showed that one can restore a one-to-one bulk-edge map by fixing a base point in parameter space and tracking which direct gap closes and reopens along a chosen path (Pan et al., 14 Mar 2025).

The immediate research directions named in the literature include Floquet-induced non-Abelian braidings, temporal higher-order modes, non-Hermitian skin effects under time modulation, and photonic flat-band platforms for strongly correlated phenomena (Qiu et al., 9 Aug 2025, Leng et al., 26 Jan 2026). A plausible implication is that FNATIs are becoming less a single model class than a broader framework for organizing dynamical, multigap, and non-commuting topology across acoustics, photonics, and engineered lattice systems.