Fractional Pumping Regimes Explained

Updated 17 December 2025

Fractional pumping regimes are defined by transport quantized to rational multiples through the interplay of symmetry, nonlinearity, and interactions.
They rely on topological invariants, including Abelian and non-Abelian Chern numbers, to explain experimental fractional plateaux in diverse systems.
Practical implementations in photonics, cold atoms, and electronic devices demonstrate controlled fractional charge pumping under varied parameter regimes.

Fractional Pumping Regimes

Fractional pumping refers to quantum transport processes in which the charge or physical quantity pumped per cycle is quantized to a rational (fractional) multiple rather than an integer. Such regimes arise in diverse contexts—single-particle, many-body, topological, interacting, and nonlinear systems—and are underpinned by various physical mechanisms, including generalized topological invariants, symmetry-protected band structure, many-body Berry phases, and dynamical effects. Recent advances, particularly in nonlinear Thouless pumps and interacting quantum systems, have clarified the conditions and invariants governing fractional pumping and established robust experimental realizations in photonics, cold atoms, and electronic platforms.

1. Topological Invariants Unifying Integer and Fractional Pumping

In generalized Thouless pumps, transport is governed by topological invariants defined over the parameter space of the system’s Hamiltonian. For nonlinear and interacting systems, as established by Wu et al. (Wu et al., 10 Jun 2025), the relevant invariant interpolates between the Abelian (single-band) Chern number and the non-Abelian (multi-band) Chern number. The displaced charge per cycle is given by

$D = C_1$ (Abelian Chern number) for a non-degenerate, well-isolated band.
$D = C^{NA}/N$ where $C^{NA}$ is the non-Abelian Chern number of an $N$ -fold degenerate band manifold, producing fractional pumping per cycle when $N>1$ .

Fractional plateaux arise when the nonlinearity or interaction induces band coalescence or braiding, such that multiple nonlinear bands (or collective modes) are intertwined and the effective topological response is shared among them. This framework encompasses the entire crossover from integer to fractional pumping and sets precise conditions: strong enough nonlinearity to close gaps between the $N$ bands, but with the $N$ -band subspace still separated from all higher bands (Wu et al., 10 Jun 2025, Jürgensen et al., 2022, Bohm et al., 30 May 2025).

2. Mechanisms and Physical Realizations

Nonlinear and Many-Body Systems

Nonlinear Lattice Solitons / Photonic Systems: Fractional transport of solitons is realized in Kerr nonlinear lattices (photonic waveguides or cold atoms). Under weak nonlinearity, solitons bifurcate from a single band and show integer pumping. Increasing nonlinearity induces band mixing, and the soliton braids between multiple bands, producing plateaux at rational values such as $1/2$, $1/3$, or $1/4$ of a unit cell per cycle (Wu et al., 10 Jun 2025, Jürgensen et al., 2022, Tao et al., 10 Feb 2025). Experimental observations in coupled waveguide arrays confirm these fractional plateaux, which are linked to underlying topological invariants (Abelian or non-Abelian Chern numbers, Berry matrices over degenerate bands).
Quantum Many-Body Pumps: In systems of interacting bosons or fermions at fractional fillings ( $\nu = p/q$ ), the charge transported per cycle is quantized to $p/q$ , the ratio of the many-body Chern number to the ground-state degeneracy. This appears in bosonic superlattices (Zeng et al., 2016), fermionic synthetic dimensions (Zeng et al., 2015), and alkaline-earth(-like) atom platforms (Taddia et al., 2016). The precise mapping to topological invariants is achieved through Wilczek-Zee non-Abelian curvature integrals, and fractionalization requires stabilized ground-state degeneracy, strong interactions, and gapped spectral structures.

Single-Particle and Coherent Regimes

Quantum Dot and Electronic Pumps: In quantum dot turnstiles, at high driving frequencies comparable to or exceeding the dwell time, charge transferred per cycle becomes fractional due to incomplete loading/unloading and quantum coherence between dot states (Lin et al., 2011). In adiabatic almost-topological quantum dot pumps, the average pumped charge per cycle is a fraction $f$ of an electron, with $f$ set by the reservoir band structure (Lamb shift to level broadening ratio) (Hasegawa et al., 2019).
Symmetry-Protected Fractionalization: For 1D charge pumps with rational superlattice modulation (Harper–Hofstadter models), symmetry in parameter space (magnetic translation) ensures that fractions of the pumping cycle result in pumped charges exactly quantized to $r\,C/q$ , where $C$ is the Chern number, and $q$ is the period (Marra et al., 2014). This quantization is robust to boundary conditions and applies both in open and closed geometries.
Topological Phase Transition Regimes (Type-II/Geometric Pumping): In systems driven through topological phase transitions in the adiabatic ( $\omega \to 0$ ) and gapless ( $\Delta \to 0$ ) regime, pumping yields fractional occupation per cycle controlled by geometric (Berry curvature) factors, with pumped charge per cycle $P_G = 1/2$ at the critical point—interpreted as geometric rather than energetic pumping (Song et al., 2 Aug 2024).

3. Mathematical Structure and Chern Number Fractionalization

The universal feature across these systems is the emergence of quantized, rational values in transport as a consequence of the partitioning of parameter space—be it Brillouin zone, pumping parameter, or adiabatic cycle—via symmetry, interaction-induced degeneracy, or nonlinearity. Explicitly, for a state returning to itself after $m$ pump cycles with displacement $n$ unit cells, the mean pumping per cycle is $\langle \Delta n \rangle = n/m$ , and it is directly related to the Chern number computed in the appropriate (possibly enlarged or degenerate) Hilbert space (Tao et al., 10 Feb 2025, Bohm et al., 30 May 2025).

The non-Abelian generalization, essential in the presence of multi-band degeneracy, operates through the Berry-Wilczek–Zee connection and curvature over parameter manifolds, or via time-ordered Wilson loops in momentum space. As interaction or nonlinearity merges bands, the effective transport per cycle is given by $C^{NA}/N$ , where $N$ is the dimension of the degenerate manifold (Wu et al., 10 Jun 2025, Bohm et al., 30 May 2025).

4. Experimental Realizations and Parameter Regimes

Several platforms have achieved or are poised to realize fractional pumping:

System Type	Regime	Observed Fraction(s)
Photonic nonlinear waveguides (Jürgensen et al., 2022, Tao et al., 10 Feb 2025)	Moderate to strong Kerr nonlinearity	$1$, $1/2$, $1/3$, $1/4$ unit cells/cycle
Bose-Einstein condensates in superlattices (Hu et al., 17 Jan 2024, Fu et al., 2021)	Commensurate lattice periods and amplitude tuning	Integer or fractional steps, e.g., $1/3$
Synthetic dimension Fermi gases (Zeng et al., 2015, Taddia et al., 2016)	Strong interaction and rational filling	$1/3$ per cycle (Laughlin-like)
Quantum dot turnstile (Lin et al., 2011, Hasegawa et al., 2019)	High-frequency (non-adiabatic) drive; nontrivial reservoir structure	$<1$ electron per cycle
Topological Cooper-pair pumping (Weisbrich et al., 2022)	Non-adiabatic regime + flux offset	$1/\nu$ fraction in transconductance

Tuning key parameters—e.g., nonlinearity ( $g$ ), interaction ( $U/t$ ), synthetic dimension size, driving frequency—controls the transition between integer, fractional, and non-quantized pumping or breakdown regimes (Bohm et al., 30 May 2025, Fu et al., 2021, Hu et al., 17 Jan 2024).

5. Physical Interpretation and Generalizations

Fractional pumping regimes reflect the redistribution of quantized topological response across physically or spectrally degenerate subspaces, with the fractionalization mechanism differing according to context:

In nonlinear photonic and atomic systems, the self-induced soliton potential or interaction-induced band merging fractionalizes the response by sharing the topological invariant over multiple bands or states (Wu et al., 10 Jun 2025, Tao et al., 10 Feb 2025, Bohm et al., 30 May 2025).
For symmetry-enforced fractionalization, the quantization arises from the splitting of parameter or momentum space by magnetic translation or related symmetries, partitioning the Berry curvature into $q$ congruent domains (Marra et al., 2014).
In geometric/Type-II regimes, the relevant structure becomes the statistical occupation difference imposed by the geometric path through phase space, modified by criticality, rather than energetic resonance (Song et al., 2 Aug 2024).
The non-Abelian and Wilson-loop formalism provides a unifying mathematical description for all such mechanisms in adiabatic and interacting regimes.

6. Outlook and Design Principles

Fractional pumping regimes are broadly accessible in current experimental platforms, with realization guided by:

Control of nonlinear or interaction strength to induce targeted merging of bands.
Engineering unit-cell sizes and symmetries that commensurate the desired fractional quantization.
Leveraging auxiliary parameters (flux, synthetic dimensions) to partition Hilbert or parameter space.

Designing pumps capable of arbitrary rational fractions $C/N$ is feasible within the constraints of adiabaticity and band isolation (Wu et al., 10 Jun 2025, Hu et al., 17 Jan 2024, Marra et al., 2014), and parallels exist with the fractional quantum Hall effect, where interaction and degeneracy underlie topological response fractionalization. The robustness and universality of these regimes, beyond linear and single-particle paradigms, support future investigation into non-Hermitian, quantum-correlated, and higher-dimensional generalizations.