Fractionally Quantized Pumping

Updated 9 November 2025

Fractionally quantized pumping is the controlled transport of rational fractions of particle attributes per cycle, achieved by partitioning parameter space through discrete symmetries or interaction-driven degeneracies.
Experimental realizations in cold atoms, photonic systems, and nanowires utilize engineered lattice modulation, many-body interactions, and nonadiabatic protocols to manifest fractional responses.
This phenomenon extends traditional Thouless pumping by incorporating fractional Chern numbers and robust topological invariants, offering new avenues in quantum technology and spintronics applications.

Fractionally quantized pumping refers to the transport of well-defined rational fractions of a particle (charge, spin, or soliton) per cycle in a periodically driven quantum system. Unlike the conventional quantized Thouless pump, where the pumped quantity is integer-valued and determined by a topological invariant such as a Chern number, fractionally quantized pumping occurs due to discrete symmetries, degeneracies, or interaction-induced ground-state structure, producing a robust rational fraction of the relevant topological invariant. This phenomenon arises in a variety of systems, including superlattices, cold atomic gases, driven Josephson chains, synthetic dimensions, parafermion zero modes, nonlinear soliton pumps, and ultrafast nonadiabatic protocols near topological phase transitions. The underlying mechanisms typically involve either a division of parameter space (fractionalization by symmetry), many-body ground-state degeneracy (interaction-induced), or engineered non-Abelian band mixing.

1. Topological Foundations and Basic Principles

Quantized charge (or spin) pumping was established by Thouless as the transfer of an integer number $C$ of particles per adiabatic cycle of a periodic parameter, where $C$ is the first Chern number over the $(k,\varphi)$ parameter space: $Q = \frac{1}{2\pi} \int_0^{2\pi} d\varphi \int_{\rm BZ} dk\, \Omega(k,\varphi) = C$ with $\Omega(k,\varphi)$ the Berry curvature of an occupied band. Fractional quantization arises when, for symmetry or topological reasons, the relevant parameter space can be partitioned into $q$ equivalent sectors, producing a quantized fractional pumped charge in $1/q$ of the full period: $Q \left(\frac{\tau}{q}\right) = \frac{C}{q}$ provided the system exhibits the required symmetry—commensurability of lattice and modulation or multi-band/winding degeneracy (Marra et al., 2014). In interacting systems, fractionalization may result from ground-state degeneracy and many-body Berry phases, as in strongly correlated cold gases or nonlinear photonic pumps.

2. Symmetry-Protected Fractional Quantization

The Harper–Hofstadter superlattice and its generalizations exemplify symmetry-induced fractional pumping (Marra et al., 2014, Marra et al., 2017). Discrete parameter-space symmetries, such as the periodicity of the modulation phase $\varphi\to\varphi+2\pi/q$ , partition the $(k,\varphi)$ torus into $q$ congruent sectors, each enclosing $\frac{1}{q}$ of the total Berry curvature. The general result is that if the modulation period is commensurate with the lattice ( $\alpha=p/q$ ), then: $Q\left(2\pi \frac{m}{q}\right) = \frac{m}{q} C_j$ where $C_j$ is the Chern number of the relevant band. Robustness to open boundary conditions follows from the continuity equation and the vanishing of edge contributions at quantized fractional shifts. Experimental proposals in cold atomic gases use bichromatic optical lattices and phase sweeps to observe fractional plateaux in center-of-mass transport at accessible temperature and system sizes.

In the spinful Harper–Hofstadter model with a modulated Zeeman field, the same symmetry ensures fractional quantization of both charge and spin pumped during fractions of a cycle, with the spin-pumped fraction given by $C_s m/q$ where $C_s$ is the spin Chern number (Marra et al., 2017). This indicates a broader class of symmetry-protected fractional pumping in spintronics and solid-state realizations.

3. Interaction-Induced and Many-Body Fractionalization

Fractional quantization also emerges due to strong correlations and many-body ground-state degeneracy. In 1D lattices with interacting bosons or fermions, multi-body interactions at fractional filling can open topological gaps leading to quasi-degenerate ground-state manifolds, each characterized by a fractional Chern number $C_\alpha$ (Zeng et al., 2016). The pumped charge per cycle in a given ground state is then

$Q_\alpha = C_\alpha$

with the total response always summing to an integer over the manifold, but individual polarizations differing by quantized fractions that can be identified with elementary quasiparticle charges.

In synthetic dimension setups, such as spin-orbit coupled fermion gases mapped to $2$D ladders with flux, the system emulates a fractional quantum Hall (FQH) state. At filling $\nu=1/3$ and with large spin-component number, the many-body Chern number per ground state is $1/3$ and the pumped charge (measured as center-of-mass drift in the synthetic direction) per $\theta=2\pi$ cycle is quantized as $Q=1/3$ (Zeng et al., 2015).

Multi-band fractional Thouless pumps represent another route, where repulsive interactions stabilize crystals of multi-band Wannier states with fractional winding per cycle. Explicit numerical studies in off-diagonal Aubry–André–Harper chains with strong nearest- and next-nearest-neighbor repulsions show a sharp transition from integer- to fractional-quantized transport as interactions increase, with the latter characterized by polarization that returns to itself only after multiple cycles, shifting by an integer number of unit cells over $n$ cycles, i.e., $Q=p/n$ per cycle (Jürgensen et al., 12 Apr 2025).

4. Fractional Pumping in Soliton and Nonlinear Systems

Fractionally quantized transport is not confined to linear single-particle systems. Nonlinearity, such as Kerr or mean-field interactions, supports soliton-like excitations whose motion under adiabatic parameter cycling can become fractionally quantized. In photonic waveguide arrays implementing a nonlinear Aubry–André–Harper pump, solitons undergo pitchfork bifurcations as the nonlinearity increases, leading to orbits where the soliton returns to itself only after $n$ cycles and net displacement $p$ unit cells, so the observed plateau is $f=p/n$ (Jürgensen et al., 2022, Bohm et al., 30 May 2025). The quantization is governed by the many-body (non-Abelian) Wilson loop of the soliton bands rather than a single-band Chern number. This fractional transport survives disorder, is observable in photonic and atomic systems, and extends to dark solitons and synthetic nonlinearities realized by engineering coupling cutoffs in the AAH model (Tao et al., 9 Aug 2025, Maisuriya et al., 4 Nov 2025).

5. Parafermion and Fractional Quasiparticle Pumps

Fractionally quantized adiabatic pumping can be engineered in hybrid topological structures involving parafermion zero modes at domain walls between alternating superconducting and ferromagnetic segments at fractional quantum Hall edges (Herasymenko et al., 2018). In these architectures, fine-tuned sequential tunneling protocols with quantum antidots ensure that the average pumped charge per cycle is $(d-1)e^*/2$ , where $e^*=\nu e$ is the fractional quasiparticle charge and $d=2/\nu$ is the topological degeneracy. The transport statistics are universal: the current noise reaches a parameter-independent Fano factor $F=(d+1)/6$ in the adiabatic limit, robust to microscopic details and set exclusively by the system’s topological properties. The same protocol yields a universal $1/2$ Fano factor for Majorana-based pumps.

In quantum dot systems with driven dot-reservoir coupling, adiabatic almost-topological fractional pumping is achievable in the absence of interactions or fractional particles (Hasegawa et al., 2019). Here, the pumped charge fraction is determined not by a Chern number, but by the ratio of the Lamb shift to the level broadening set by reservoir band-structure. For uniformly wide bands, the quantization is $Q=e/2$ per cycle. Topological protection holds asymptotically but is lost with increasing nonadiabaticity.

6. Fractional Pumping via Nonadiabatic Protocols and Topological Phase Transitions

Recent work has established a new class of fractional quantization, termed geometric (type-II) pumping, as distinct from resonance-driven (type-I) mechanisms (Song et al., 2 Aug 2024). In the adiabatic limit near a topological phase transition (TPT), where both drive frequency $\omega$ and the gap $\Delta$ vanish ( $\omega, \Delta\to0$ ) simultaneously, conventional Fermi golden rule arguments break down. Instead, the population change (e.g., of the upper band in a two-band model) after many cycles converges to $1/2$ per mode for each $k$ at which the path in parameter space encircles the gap-closing point. This fractional pumping is non-directional, has an entropy increase (probabilistic irreversibility over ensemble or many cycles), and is governed by a geometric winding number—the TPT invariant. The experimental signature is a long-time plateau in the pumped quantity, observable in ultrafast coherent phonon driving experiments in Dirac semimetals such as ZrTe $_5$ .

7. Experimental Realizations and Prospects

Fractionally quantized pumping has been implemented or proposed in a range of platforms:

Cold atom systems in optical superlattices using bichromatic potentials and phase sweeps to realize and measure fractional Chern responses (Marra et al., 2014, Zeng et al., 2016).
Synthetic dimensions in spin-orbit coupled Fermi or bosonic gases, leveraging Raman-like couplings to achieve lattice models with a magnetic flux, ground-state degeneracy, and fractionally quantized pumping equivalent to the Laughlin state (Zeng et al., 2015).
Photonic waveguide arrays and nonlinear optical systems, where Kerr nonlinearity or synthetic nonlinearities stabilize fractionally pumped soliton transport, observable by center-of-mass imaging after one or more cycles (Jürgensen et al., 2022, Bohm et al., 30 May 2025, Maisuriya et al., 4 Nov 2025, Tao et al., 9 Aug 2025).
Superconducting Josephson-junction chains supporting engineered ground-state manifolds and nonadiabatic Landau–Zener transitions, leading to robust fractional transconductance plateaux (Weisbrich et al., 2022).
Nanowires with parafermion or Majorana zero modes, with pumping protocols designed to engineer universal fractionally quantized current and noise (Herasymenko et al., 2018).
Ultrafast pump–probe experiments in topological Dirac/Weyl materials, detecting fractional geometric pumping at topological phase transitions (Song et al., 2 Aug 2024).

The robustness of fractional quantization is dictated by the relevant symmetry or topological invariants: for symmetry-protected cases by commensurability (parameter-space symmetry), for interaction-driven cases by the stability of ground-state degeneracy and topological gaps, for many-body soliton systems and synthetic nonlinearities by the non-Abelian Berry phase structure, and for almost-topological dot pumps by the shape and location of Berry curvature peaks in parameter space.

Fractionally quantized pumping presents a rich experimental and theoretical platform to paper the interplay of topology, symmetry, interactions, and nonlinearity, and generalizes fundamental concepts from the quantum Hall effect and topological quantum matter to driven, finite, and strongly correlated systems.