Thouless Pumping in Topological Transport

• Thouless pumping is the quantized transport of particles in 1D systems via adiabatic modulation, where the transported charge equals the integer Chern number.
• It is typically demonstrated using the Rice–Mele model, with cyclic variation of Hamiltonian parameters producing measurable center-of-mass shifts in experiments.
• Robust in the adiabatic limit and adaptable to disorder, interactions, nonlinearities, and synthetic dimensions, Thouless pumping exemplifies versatile topological transport.

Searching arXiv for recent and foundational papers on Thouless pumping to ground the article in published work. Thouless pumping is the quantized transport of particles in a one-dimensional periodically modulated system with no net bias, obtained when Hamiltonian parameters are varied slowly and cyclically so that the many-body state returns to itself after one period while its polarization or center of mass shifts by an integer amount set by topology. In the standard formulation, the transported charge per cycle is the first Chern number of an occupied band over the torus formed by crystal momentum and time; in this sense Thouless pumping is a $(1+1)$-dimensional counterpart of the integer quantum Hall effect and one of the simplest manifestations of topological transport (Citro et al., 2022, Liu et al., 2024).

1. Topological formulation

In the adiabatic limit, Thouless pumping is defined by a family of Hamiltonians $H(k,t)$ that is periodic both in crystal momentum $k$ and in time $t\in[0,T]$. For an instantaneous Bloch eigenstate $\ket{u_n(k,t)}$, the Berry connection components are

$$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$

and the Berry curvature is

$F_n(k,t)=\partial_t A_k-\partial_k A_t.$

The pumped charge per cycle is

$$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$

namely the first Chern number of the occupied band on the $(k,t)$ torus (Citro et al., 2022).

This topological statement has several equivalent formulations in the literature. In band language, $Q$ is the Chern number; in real space, a Wannier state localized in band $H(k,t)$0 shifts by an integer number of unit cells; in current language, $H(k,t)$1; and in interacting settings the relevant invariant is the first Chern number of the many-body ground state over the $H(k,t)$2 torus, or equivalently the winding of a Berry phase in parameter space (Liu et al., 2024, Viebahn et al., 2023). The review literature also emphasizes a Floquet interpretation: for a translationally invariant pump, a filled Floquet band may wind across the Floquet Brillouin zone, and the pumped charge is then the winding number of the quasienergy band (Citro et al., 2022).

A common misconception is that quantization is merely geometric area accumulation in parameter space. The experimental verification with ultracold fermions instead showed that different loops with the same winding yield the same transported charge, while a back-and-forth loop with zero winding yields no pumping; the topological invariant is the winding around the gap-closing singularity, not the enclosed geometric area (Nakajima et al., 2015).

2. Minimal Hamiltonians and transport observables

The paradigmatic lattice realization is the Rice–Mele model, a dimerized chain with staggered on-site potential. In one standard form,

$H(k,t)$3

with a cyclic protocol such as

$H(k,t)$4

When $H(k,t)$5 encircles the gap-closing point once, one charge is pumped per cycle (Citro et al., 2022). A closely related hard-core-boson representation used in superconducting-qubit experiments is

$H(k,t)$6

with $H(k,t)$7 tracing a closed loop over a period $H(k,t)$8 (Liu et al., 2024).

Two observables recur across realizations. The first is the center-of-mass or Wannier-center displacement. For a localized Wannier state filling band $H(k,t)$9, the quantized displacement per cycle is

$k$0

where $k$1 is the unit-cell length (Liu et al., 2024). The second is the integrated current,

$k$2

which equals the Chern number for an ideal adiabatic pump (Liu et al., 2024). In the first ultracold-fermion realization, the pumped charge was extracted from the center-of-mass shift of the atomic cloud, and for two nontrivial pumping protocols the measured shift was $k$3 and $k$4 per cycle in the first six cycles, consistent with $k$5 (Nakajima et al., 2015).

The same formalism extends beyond single-particle band pumps. In the Rice–Mele–Hubbard model,

$k$6

the pumped charge remains topological, but the relevant invariant is many-body rather than single-particle (Viebahn et al., 2023).

3. Disorder, localization, and breakdown of quantization

The canonical robustness statement is conditional: weak disorder does not change the pumped charge so long as the instantaneous many-body gap remains open during the cycle (Citro et al., 2022). The review literature further notes that, although static eigenstates may be Anderson-localized, Floquet modes can hybridize and delocalize across a ring, thereby enabling transport in the disordered adiabatic pump (Citro et al., 2022).

Recent experiments on a 41-qubit superconducting quantum processor made this interplay explicit in a time-dependent Rice–Mele model with both diagonal and off-diagonal disorder,

$k$7

There, the clean single-loop pump gave a center-of-mass shift $k$8 for $k$9, in agreement with $t\in[0,T]$0 and the expected displacement of about two lattice sites. With on-site disorder, $t\in[0,T]$1 remained quantized for $t\in[0,T]$2, decayed rapidly once $t\in[0,T]$3, and vanished for $t\in[0,T]$4. The interpretation given is gap closing by random potential fluctuations and the onset of Landau–Zener transitions (Liu et al., 2024).

The same experiment also exhibited disorder-induced topological pumping. In a double-loop protocol that is trivial in the clean limit, moderate on-site disorder $t\in[0,T]$5 produced a net displacement $t\in[0,T]$6 per cycle. For quasiperiodic hopping disorder,

$t\in[0,T]$7

a loop that is topologically trivial in the clean limit became topological, with $t\in[0,T]$8 emerging above $t\in[0,T]$9; this was identified as an intrinsic topological Anderson pump (Liu et al., 2024).

A separate theoretical development sharpens the limits of disorder protection at finite frequency. The result termed “absence of disordered Thouless pumps at finite frequency” states that for any finite disorder $\ket{u_n(k,t)}$0 and finite drive frequency $\ket{u_n(k,t)}$1, the pump rate ultimately decays to zero because of non-adiabatic transitions between instantaneous eigenstates, although the decay time is exponentially large in the period of the drive. In the strict adiabatic limit, the instantaneous gap closes at a critical disorder strength $\ket{u_n(k,t)}$2, and above this point pumping ceases (Vuina et al., 2024). This clarifies a frequent overstatement: robustness to weak disorder is an adiabatic statement, not a guarantee of indefinitely stable quantized transport at arbitrary drive frequency.

4. Interaction-driven, nonlinear, and dissipative pumps

Interactions can do more than perturb an existing pump; they can create one. In a dynamical superlattice realizing the Rice–Mele–Hubbard Hamiltonian, a “boomerang” loop that is trivial at $\ket{u_n(k,t)}$3 becomes topological for sufficiently large repulsive Hubbard interaction because the noninteracting singularity at $\ket{u_n(k,t)}$4 splits into two interaction-driven singularities near $\ket{u_n(k,t)}$5, one of which enters the loop. The measured response displayed three regimes: $\ket{u_n(k,t)}$6 for $\ket{u_n(k,t)}$7, a plateau $\ket{u_n(k,t)}$8 for $\ket{u_n(k,t)}$9, and a re-entrant trivial regime for $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$0. The experiment was interpreted as a genuinely interaction-induced Thouless pump, not adiabatically connected to the noninteracting limit (Viebahn et al., 2023).

Nonlinear and driven-dissipative settings support further variants. In a chain of coupled Kerr resonators, a space-time modulation of onsite Kerr interaction energies generates $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$1-dimensional topological bands in the Bogoliubov spectrum. For $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$2, the five lower-energy bands were reported to carry Chern numbers $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$3, identical to the Harper–Hofstadter model with flux $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$4, and the corresponding Wannier centers move by exactly $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$5 lattice sites per cycle. Increasing pump power toward the nonlinear bistability threshold leads to a gap closing between the third and fourth normal bands and a reopened phase with inverted Chern numbers, constituting an interaction-induced topological transition (Ravets et al., 2024).

Optical solitons furnish a nonlinear transport regime in which topology and spectral occupancy are intertwined. In a nonlocal self-defocusing medium with two slowly sliding sublattices, broad low-power fundamental solitons do not exhibit transport because they excite only a small portion of the spectral band, whereas higher-power solitons with broader spectral projections show stable quantized transport governed by the space-time Chern index. Nonlocality shifts the lower pumping threshold to higher power and suppresses the high-power breakdown common in local Kerr media; dipole and tripole solitons can also be pumped stably, though only within finite power windows (Ye et al., 8 Jul 2025).

A qualitatively different mechanism appears in a mechanical Frenkel–Kontorova realization with topological kink solitons. There the pump is non-adiabatic, dissipation is necessary, and the quantized soliton displacement per cycle is not described by a Chern number. Instead, friction forces the kink into successive instantaneous minima of an effective time-periodic potential, producing integer plateaux in the displacement $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$6. With an added gradient, the experiment reported transport against the pumping sequence and a rich plateau structure (Jürgensen et al., 19 Feb 2025).

The term “generalized Thouless pump” has also been used for a distinct non-equilibrium setting in which interband coherence in the initial state adds a non-topological contribution $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$7 to the transported charge. In that case the response is continuously tunable by the switching-on rate, vanishes for a quadratic ramp with $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$8, and is most pronounced near a band-touching point (Ma et al., 2017). This usage does not refer to the conventional integer-quantized adiabatic pump of a filled band.

5. Experimental realizations and measurement strategies

The first direct demonstration of topological Thouless pumping with ultracold atoms used a dynamically controlled optical superlattice loaded with a degenerate Fermi gas of $$A_k=i\langle u_n|\partial_k u_n\rangle,\qquad A_t=i\langle u_n|\partial_t u_n\rangle,$$9. The time-dependent potential

$F_n(k,t)=\partial_t A_k-\partial_k A_t.$0

maps in the deep-lattice limit to the Rice–Mele model. The phase $F_n(k,t)=\partial_t A_k-\partial_k A_t.$1 was swept linearly to realize a pump cycle, and in-situ imaging of the atomic density yielded the cloud center of mass $F_n(k,t)=\partial_t A_k-\partial_k A_t.$2, from which the pumped shift per cycle was extracted (Nakajima et al., 2015).

Superconducting circuits provide a distinct route. On the 41-qubit processor used to study disorder, each qubit frequency was modulated as

$F_n(k,t)=\partial_t A_k-\partial_k A_t.$3

with $F_n(k,t)=\partial_t A_k-\partial_k A_t.$4 in the high-frequency limit. A rotating-wave expansion gives effective hoppings $F_n(k,t)=\partial_t A_k-\partial_k A_t.$5, while the slow offset $F_n(k,t)=\partial_t A_k-\partial_k A_t.$6 implements the staggered onsite potential, thereby Floquet-engineering the target time-dependent Rice–Mele Hamiltonian (Liu et al., 2024). Related superconducting proposals for Coulomb-blockaded Josephson-junction arrays formulate both Rice–Mele and Harper–Hofstadter pumps, with topological Cooper-pair transport $F_n(k,t)=\partial_t A_k-\partial_k A_t.$7 in the adiabatic infinite-chain limit (Athanasiou et al., 2023).

Acoustic implementations have made nonstandard topological pumps directly observable. In one returning Thouless-pump experiment, a two-dimensional delicate topological insulator was realized using one-dimensional acoustic crystals and a synthetic dimension; the measured bulk polarization $F_n(k,t)=\partial_t A_k-\partial_k A_t.$8 increased from $F_n(k,t)=\partial_t A_k-\partial_k A_t.$9 to $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$0 for $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$1 and then returned from $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$2 to $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$3 for $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$4, establishing forward pumping in the first half-cycle and backward pumping in the second (Cheng et al., 11 May 2025). In a second acoustic realization, a Berry-dipole-mediated returning pump in a three-sublattice waveguide array showed an edge-localized mode that delocalized into the bulk and returned to the original edge with a pseudospin flip, again with zero net displacement over a full cycle (Mo et al., 13 May 2025).

Other platforms discussed in the cited literature include single-spin quantum simulation with an NV center in diamond (Ma et al., 2017), optical and photonic settings (Citro et al., 2022, Ye et al., 8 Jul 2025), time-modulated spin-chain arrays with engineered edge couplings (Bastidas, 2022), and driven one-dimensional optical lattices that realize pumping in time-space crystalline structures (Braver et al., 2022). Across these platforms, the standard observables are center-of-mass shift, instantaneous or cycle-integrated current, Berry-phase or Wilson-loop polarization, Floquet quasienergy winding, and direct imaging of bulk or edge mode motion.

6. Synthetic dimensions, returning pumps, and fast protocols

Thouless pumping has become a general construction principle for topological transport rather than a property of the Rice–Mele model alone. Synthetic dimensions turn internal states, modulation phases, or propagation coordinates into additional coordinates on which topological invariants can be defined (Citro et al., 2022). A particularly explicit extension is the time-space crystal construction in which a resonantly driven quantum well yields an effective lattice in the angle variable $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$5, and adiabatic variation of a drive phase realizes pumping along this synthetic time coordinate. Extending to a driven one-dimensional optical lattice produces a genuine two-dimensional time-space crystalline structure with separate or simultaneous quantized pumping in space and time (Braver et al., 2022).

Returning Thouless pumps form a symmetry-enriched subclass. In the “delicate” topological version, the full-zone Chern number vanishes, but each half of a synthetic Brillouin zone carries a nonzero sub-Brillouin-zone Chern number, so polarization increases by one unit in the first half-cycle and decreases by one unit in the second. The associated Wannier functions are multicellular rather than atomically localized (Cheng et al., 11 May 2025). The Berry-dipole realization differs from conventional monopole-mediated pumps by having zero net flux through the full cycle yet nontrivial internal pseudospin dynamics (Mo et al., 13 May 2025).

A major current theme is the relaxation of the adiabatic constraint. Counter-diabatic constructions for the Rice–Mele model add an auxiliary Hamiltonian that cancels nonadiabatic transitions and locks the state to the instantaneous eigenstate for arbitrary driving speed. One formulation shows that the pumped charge remains the quantized Chern number under the shortcut Hamiltonian and reports a speed-up of up to $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\rm BZ} dk\,F_n(k,t)=C_n\in\mathbb Z,$$6 orders of magnitude together with suppressed wavepacket diffusion and robustness against moderate noise levels (Liu et al., 2024). A complementary analysis proves Chern-number quantization of the pumped charge across each bond under counter-diabatic driving, while emphasizing that the exact auxiliary term is generically long-ranged in real space, motivating special trajectories and optimized nearest-neighbor approximations (Chiel et al., 2024). An additional finite-frequency construction based on a zero-curvature mapping to the Euclidean sinh-Gordon equation yields a family of rapid-cycle protocols with exact quantization and no excitations at the end of each cycle (Malikis et al., 2021).

Dispersion control is a separate practical issue. In non-flat bands, dynamical phase differences between Bloch states broaden a pumped wavepacket even when the mean displacement remains quantized. Two suppression schemes were proposed: a re-localization echo protocol, in which the Hamiltonian is reversed during a second cycle to cancel dynamical-phase dispersion, and a high-order tunneling suppression protocol that flattens the band and eliminates the virtual processes responsible for spreading (Hu et al., 2019). A plausible implication is that future high-fidelity pumps will need to control both nonadiabatic leakage and intraband dispersion.

Further extensions in the cited literature include quantized pumping of the relative distance between two atoms by modulating the s-wave scattering length in time (Kopaei et al., 2024), graphical Chern-number constructions for generalized Creutz ladders with tunable Peierls phases and diagonal hoppings (Lv et al., 2024), and networked transport of single spin excitations in coupled spin-chain arrays with disorder-robust splitting at junctions (Bastidas, 2022). Taken together, these developments place Thouless pumping at the intersection of band topology, Floquet engineering, disorder physics, many-body interactions, nonlinear dynamics, and synthetic geometry.