Non-Abelian Thouless Pumping

• Non-Abelian Thouless pumping is a topological phenomenon where adiabatic evolution in degenerate bands produces both real-space displacements and internal state rotations via matrix holonomies.
• Its mechanism exploits matrix-valued gauge potentials and noncommutative cycle ordering, distinguishing it from traditional Abelian Thouless pumps.
• Experimental and theoretical studies across photonic, acoustic, and Rydberg platforms demonstrate control over topological charge, internal state mixing, and disorder-induced transitions.

Non-Abelian Thouless pumping is an adiabatic cyclic transport phenomenon in a one-dimensional periodically modulated system whose relevant adiabatic subspace is degenerate, or effectively multiband, rather than an isolated nondegenerate band. In that regime the geometric evolution is governed not by a scalar Berry phase but by a matrix-valued Wilczek–Zee holonomy, so a pumping cycle can simultaneously induce real-space transport and an internal unitary transformation within the degenerate manifold. The defining consequence is noncommutativity: two closed pumping cycles can yield different final states when applied in opposite orders. This framework has been formulated and realized across photonic, acoustic, flat-band, ladder, disorder-induced, nonlinear, and Rydberg-synthetic settings (Brosco et al., 2020, You et al., 2021, Huang et al., 2024, Danieli et al., 2024, Wu et al., 10 Jun 2025, Guo et al., 8 Jul 2026).

1. Conceptual definition and distinction from the Abelian pump

The conventional Thouless pump is a one-dimensional dynamical topological effect that stems from the same topological mechanism as two-dimensional Chern insulators, with one momentum dimension replaced by a cyclic evolution parameter. In that Abelian setting the system adiabatically follows an isolated band, the Berry connection is scalar-valued, and the transported quantity over one cycle is controlled by a Chern number defined on a two-dimensional parameter space such as $(k,\lambda)$ or $(k,t)$ (You et al., 2021).

Non-Abelian Thouless pumping arises when adiabatic evolution occurs inside a degenerate subspace. The Berry connection then becomes matrix-valued,

$$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$

and the adiabatic evolution over a loop is described by a path-ordered holonomy

$$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$

Because the relevant geometric objects are matrices, different loops need not commute: $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$ This failure of commutativity is the characteristic diagnostic of non-Abelian pumping (You et al., 2021).

The physical content is correspondingly broader than in the Abelian case. Standard Thouless pumping produces a quantized displacement associated with scalar geometric data. Non-Abelian pumping instead produces displacement together with a holonomic transformation among states in the degenerate manifold. In the photonic Lieb-lattice formulation, the pump is explicitly described as “displacement plus a holonomic transformation among the different bands,” while the Rice–Mele ladder formulation emphasizes simultaneous control over position and internal pseudo-spin composition (Brosco et al., 2020, Danieli et al., 2024).

A recurrent misconception is that any multiband, coherent, or pseudospin-dependent pump is therefore non-Abelian. The literature instead draws a sharper distinction: non-Abelianity requires a degenerate pumping subspace and matrix-valued adiabatic transport. Multiband participation by itself is insufficient unless the geometry is that of a degenerate manifold or an effectively braided set of bands carrying a non-Abelian connection (Huang et al., 2024, Wu et al., 10 Jun 2025).

2. Gauge structure, field strength, and topological observables

The gauge-theoretic formulation generalizes the Berry-phase description of ordinary pumping. In a degenerate subspace, the Wilczek–Zee connection has matrix elements

$$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$

with $\mu$ denoting an adiabatic parameter such as $k$, $t$, or the propagation coordinate $z$. The associated non-Abelian curvature contains the commutator term that has no Abelian counterpart: $(k,t)$0 In the photonic Lieb chain this structure appears as

$(k,t)$1

and the commutator term is identified as the hallmark of the non-Abelian geometry (Brosco et al., 2020).

Transport observables are correspondingly matrix-valued before projection onto an initial state. For light initially prepared in a degenerate flat-band manifold, the displacement is written as

$(k,t)$2

with

$(k,t)$3

Although the full response depends on the initial superposition coefficients $(k,t)$4, the trace of the displacement matrix is topological: $(k,t)$5 where $(k,t)$6 is the first Chern number of the degenerate band (Brosco et al., 2020).

In the quasiperiodically disordered Lieb chain, the non-Abelian Zak phase and Chern number are formulated using a twisted boundary phase $(k,t)$7. The center-of-mass shift satisfies

$(k,t)$8

with

$(k,t)$9

The numerical implementation uses a discrete Wilson-loop or Fukui–Hatsugai–Suzuki-style construction on the discretized $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$0 torus (Huang et al., 2024).

The nonlinear extension preserves this logic while changing the nature of the eigenstates. In nonlinear optical waveguide lattices with Kerr nonlinearity, a unified invariant is proposed that reduces to the Abelian Chern number when the nonlinear band is isolated and becomes the trace of the non-Abelian curvature when strong nonlinearity causes band braiding. In that framework the pumped displacement after one cycle is the nonlinear topological invariant divided by the number of interacting bands, which can produce fractional quantization of the transported charge (Wu et al., 10 Jun 2025). This suggests that non-Abelian pumping is not restricted to linear degenerate bands; it can also emerge when nonlinear bands intertwine so that an Abelian band-by-band classification becomes ill-defined.

3. Mechanisms and canonical model families

Two distinct mechanisms are prominent in the literature. One is continuous adiabatic evolution inside an exactly or effectively degenerate manifold, as in photonic Lieb lattices, non-Abelian Lieb chains, Rice–Mele ladders, and Rydberg synthetic lattices (Brosco et al., 2020, Huang et al., 2024, Danieli et al., 2024, Guo et al., 8 Jul 2026). The other is permutation of eigenstates through designed crossings in parameter space, as in the acoustic three-band $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$1 pump (You et al., 2021).

In the acoustic realization, the non-Abelian effect originates from time evolution of eigenstates in $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$2 pump cycles that traverse several band degeneracies in a three-band system. At the common base point the three eigenstates are $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$3, $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$4, and $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$5, and a single loop $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$6 performs the cyclic permutation

$$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$7

The companion loop $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$8 implements the opposite permutation direction and involves a unit-cell shift in the $$\mathcal{A}_{mn}(\lambda)= i\langle u_m(\lambda)|\partial_\lambda u_n(\lambda)\rangle,$$9 sector. Because

$$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$0

a single elementary loop does not by itself generate a net lattice shift after three repetitions, but composite sequences of $$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$1 and $$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$2 can move the state left, right, or trap it depending on the order of composition. The experimentally tested sequences

$$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$3

produce different final transport outcomes for the same initial state, providing a direct realization of noncommutative pumping paths (You et al., 2021).

In the photonic Lieb chain, the crucial ingredient is the presence of two degenerate flat bands. Their flatness suppresses ordinary dispersive spreading, while their degeneracy makes the connection matrix-valued. Slow modulation of the waveguide couplings along the propagation direction therefore pumps light across the lattice and rotates it inside the degenerate subspace. The model explicitly exhibits cycles that primarily generate displacement and others that primarily generate internal rotation, and the order of those cycles matters because the corresponding holonomies do not commute (Brosco et al., 2020).

In the Rice–Mele ladder, each Bloch band is doubly degenerate,

$$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$4

and the adiabatic cycle acts by a $$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$5 Wilczek–Zee matrix. Two model cycles illustrate the separation between translation and internal rotation. The cycle $$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$6 yields a diagonal Wilson loop,

$$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$7

which implements pure displacement in opposite directions for the two basis states, whereas the cycle $$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$8 yields

$$\mathcal{U}=\mathcal{P}\exp\!\left(i\oint \mathcal{A}\cdot d\lambda\right).$$9

which swaps the internal basis without translating the state. Since these matrices do not commute, the ladder provides a transparent realization of non-Abelian cycle ordering. The same paper interprets the ladder through an $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$0 structure connected to a Yang monopole model, placing the pump within a higher-dimensional non-Abelian topological geometry (Danieli et al., 2024).

The disorder-induced case shows that non-Abelian pumping need not be present in the clean limit. In the disorder-tunable Lieb chain with two degenerate flat bands, quasiperiodic disorder renormalizes the effective hopping so that a singularity described as a monopole is dragged into the region enclosed by the pumping loop. The system then undergoes a transition from a trivial phase with $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$1 to a nontrivial non-Abelian pump with $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$2. At larger disorder strength the monopole broadens into a nodal line extending beyond the topological region, and the pump disappears again, returning to $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$3 (Huang et al., 2024).

4. Platforms, control protocols, and observables

The first experimental observation reported explicitly as a non-Abelian Thouless pump was carried out in a one-dimensional periodic acoustic waveguide array made from 3D-printed rigid acoustic structures (You et al., 2021). Each unit cell contains three rectangular waveguides, $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$4, $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$5, and $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$6, coupled by interconnected narrow T-shaped tubes. The system uses the $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$7-mode of the rectangular waveguide, whose antisymmetric field profile along the long axis allows the sign and magnitude of the coupling coefficient to be controlled by the offset of the connecting tube from the waveguide center. The effective coupling configuration is encoded by four normalized offsets $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$8. Excitation is implemented by a pair of out-of-phase armature drivers to selectively launch the $$\mathcal{U}(C_1)\mathcal{U}(C_2)\neq \mathcal{U}(C_2)\mathcal{U}(C_1).$$9-mode, and the pressure field is measured by drilling small holes along the propagation direction.

To make long cycles experimentally feasible, that work introduced a cut-and-join construction. Equivalent field-distribution points along the full pumping path are identified and joined, allowing the pump to be compressed into two short building blocks $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$0 and $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$1. Their reversed order directly reveals noncommutativity: simulations and measurements show that $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$2 pumps the wave packet to the right, whereas $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$3 leaves it trapped. The central observable is the acoustic pressure distribution as a function of propagation distance and lattice site, and agreement between full-wave simulations and experiments is the decisive empirical confirmation (You et al., 2021).

The photonic implementation is formulated in a modulated waveguide array governed by coupled-mode theory and a Schrödinger-like propagation equation,

$$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$4

The proposed observables are beam displacement, population transfer within the degenerate flat-band sector, and noncommutativity under different cycle orderings. The paper also specifies representative photonic parameters, including waveguide size of order $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$5–$$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$6, separation of order $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$7–$$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$8, coupling $$[\Gamma^\mu]_{ab}=\langle \psi_a|i\partial_\mu|\psi_b\rangle,$$9, and modulation wavelength $\mu$0, with $\mu$1 sufficient for adiabatic pumping over several cycles (Brosco et al., 2020).

The Rydberg-synthetic implementation encodes a three-cell Lieb lattice into twelve selected microwave-coupled $\mu$2 Rydberg levels, with four synthetic sites per cell and six zero-energy states forming the working subspace (Guo et al., 8 Jul 2026). The Hamiltonian is

$\mu$3

and two elementary cycles $\mu$4 and $\mu$5 differ only by the temporal order of the $\mu$6 and $\mu$7 pulses. The order dependence of the resulting projected population maps is exactly consistent with noncommuting matrix-valued adiabatic operations in the zero-energy subspace. To select pulse timing, the authors reduce an active transfer step to a local $\mu$8 system and define a global adiabatic criterion based on the mean and variance of a nonadiabatic factor,

$\mu$9

For the Gaussian pulse family studied, a delay parameter around $k$0 is chosen. Lindblad simulations include state-dependent Rydberg loss, detuning gradients, amplitude errors, static on-site disorder, and relative coupling disorder, and the GAC-selected schedule yields a higher target-state population than two literature-adapted Gaussian schedules over the simulated parameter ranges (Guo et al., 8 Jul 2026).

The Rice–Mele ladder has not yet been presented as an experimental realization in the supplied material, but it is explicitly proposed for cold atoms in optical lattices, where control of $k$1, $k$2, $k$3, and $k$4 and site-resolved readout would allow direct observation of both unit-cell transport and internal-state swapping (Danieli et al., 2024).

5. Relation to adjacent pumping phenomena and common misconceptions

Several nearby topics are often conflated with non-Abelian Thouless pumping but are conceptually distinct. The generalized Thouless pump observed with a single NV-center spin in diamond is not a non-Abelian pump. Its novelty is the contribution of initial-state interband coherence in a nondegenerate two-band system, which adds a tunable, nontopological transport term on top of the standard Thouless contribution. The formalism does not involve degenerate adiabatic subspaces, matrix-valued gauge potentials, Wilczek–Zee holonomies, or non-Abelian curvature (Ma et al., 2017).

The returning Thouless pump mediated by a Berry dipole is likewise not non-Abelian. Although it exhibits topological pumping, delocalization and return to the same edge, and pseudospin flipping in a one-dimensional acoustic waveguide array, its theory is built around a single-band Abelian Berry connection and Berry curvature. The pseudospin there is an edge-mode symmetry label rather than a protected degenerate internal manifold carrying noncommuting holonomies (Mo et al., 13 May 2025).

Dispersion management in ordinary Thouless pumping also does not imply non-Abelianity. Re-localization echo protocols and high-order tunneling suppression were derived for isolated-band Abelian pumps, where the central issue is dynamical-phase-induced wave-packet spreading rather than matrix-valued geometric transport (Hu et al., 2019).

The same caution applies to broader multiband and nonequilibrium settings. A first-principles nonadiabatic Thouless pump in trans-polyacetylene uses Floquet theory, time-dependent maximally localized Wannier functions, and sums over occupied states, and the supplied material notes that this places it in a broader family of multiband pumping frameworks. However, the paper does not explicitly formulate a non-Abelian Wilson-loop description, so its relation to non-Abelian pumping should be treated as contextual rather than direct (Zhou et al., 2022).

A concise criterion follows from these contrasts: neither multiband occupation, nor interband coherence, nor pseudospin conversion, nor nonadiabaticity is by itself sufficient. The non-Abelian designation is appropriate when the adiabatic transport is defined on a degenerate subspace and the geometric evolution is intrinsically matrix-valued, with observable consequences such as internal holonomy, state mixing, or noncommuting cycle composition (Brosco et al., 2020, You et al., 2021).

6. Significance, extensions, and current directions

The significance of non-Abelian Thouless pumping lies in its fusion of topological transport and holonomic control. In the Rice–Mele ladder it is presented as a mechanism for simultaneous displacement and internal unitary transformation, with explicit relevance to quantum metrology and holonomic quantum computing (Danieli et al., 2024). In the Rydberg synthetic-lattice setting, the controllable zero-energy manifold and order-dependent cycle composition directly realize non-Abelian geometric transport in a finite quantum system, and the work identifies this as a path toward holonomic gates in Rydberg synthetic dimensions (Guo et al., 8 Jul 2026).

Classical-wave realizations have already demonstrated that the effect is not confined to quantum matter. The acoustic experiment established that path-order-dependent pumping can be observed directly through pressure maps in a 3D-printed waveguide array, showing that matrix-valued topological transport can be engineered and measured in a classical platform (You et al., 2021). The photonic Lieb-chain formulation similarly emphasizes that flat-band photonics provides a clean environment in which dispersive effects vanish and non-Abelian gauge structure becomes directly visible in beam displacement and mode rotation (Brosco et al., 2020).

Current extensions broaden both mechanism and phenomenology. Quasiperiodic disorder can create and later destroy a non-Abelian pump by moving a singularity into and then out of the loop-enclosed topological region, indicating that disorder may act as a control parameter rather than only as a source of degradation (Huang et al., 2024). Strong nonlinearity can force nonlinear bands to braid into an effectively non-Abelian manifold, in which the pumped charge is governed by a non-Abelian Chern number shared among several interacting bands and can become fractional (Wu et al., 10 Jun 2025). These developments enlarge the subject from a narrow degenerate-band construction into a broader framework for matrix-valued adiabatic transport in linear, disordered, and nonlinear media.

A plausible implication is that the field is converging on a common viewpoint: non-Abelian Thouless pumping is best understood as topological transport on a degenerate fiber bundle, where real-space displacement, internal-state holonomy, and cycle ordering are inseparable. Across the existing models, the most robust signatures remain the same—degenerate-subspace adiabatic evolution, matrix-valued gauge structure, and experimentally resolvable noncommutativity of composite pumping paths (You et al., 2021, Danieli et al., 2024, Guo et al., 8 Jul 2026).