Closed-Space Thouless Pump

• Closed-space Thouless pump is a quantized adiabatic transport mechanism defined in a compact system where transport is encoded in bulk topology rather than edge states.
• It relies on geometric invariants such as the Chern number, with variations like returning pumps that exhibit quantized half-cycle transport despite zero full-cycle net movement.
• Experimental implementations span acoustics, ultracold atoms, and photonics, demonstrating quantized center-of-mass shifts, synthetic dimensional effects, and even non-Abelian holonomies.

A closed-space Thouless pump is an adiabatic pumping process in which the transported quantity is defined without open transport channels and is instead encoded by bulk current, polarization, center-of-mass motion, or an equivalent winding in a compact synthetic or internal space. In recent arXiv literature, the phrase appears in several closely related senses: pumping on a periodic ring or isolated finite system with no leads, autonomous pumping generated by an internal dynamical degree of freedom, returning pumps with zero full-cycle transport but quantized half-cycle transport, and pumps in synthetic spaces such as relative coordinate, spin space, or atomic configuration space (Bohm et al., 8 May 2026, Viebahn et al., 2023, Cheng et al., 11 May 2025, Kopaei et al., 2024, Mumford, 2022).

1. Formal structure and closed-geometry observables

The canonical starting point remains the adiabatic 1D pump introduced by David Thouless. For a gapped Bloch Hamiltonian $H(k,t)$ periodic in crystal momentum and in the pump parameter, the transported charge per cycle is the first Chern number of the occupied bundle on the $(k,t)$ torus:

$$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$

with $F_{kt}=\partial_k A_t-\partial_t A_k$ and $A_\mu=i\langle u|\partial_\mu u\rangle$. In 1D language, the same result is the change of polarization,

$$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$

and, for an isolated band, the Wannier center shifts by $\Delta x=C a$ over one cycle (Cheng et al., 11 May 2025).

What distinguishes the closed-space setting is not the disappearance of topology but the absence of leads and edge accumulation as the primary definition of transport. In a finite isolated ring with periodic boundary conditions, pumped charge can be measured by current integration, $Q=\int_0^T dt\,J(t)$, by the Zak-phase or polarization shift, or by the center-of-mass shift; for open chains one may instead observe edge accumulation during a cycle (Bohm et al., 8 May 2026). In trapped cold-atom realizations, the same bulk quantity is read out as a quantized center-of-mass deflection of a finite Mott-insulating cloud rather than as a transport current through contacts (Lohse et al., 2015).

For interacting systems, the closed-space formulation is naturally many-body. The polarization can be written through Resta’s formula,

$$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$

and the pumped charge is the many-body Chern number on the torus of boundary twist $\theta$ and pump parameter $(k,t)$0,

$(k,t)$1

This formulation makes explicit that quantized pumping in closed geometry is a bulk property and does not rely on edge states (Viebahn et al., 2023).

A related design viewpoint replaces time by a synthetic momentum or cyclic phase and treats the pump as an effective 2D band problem. In the generalized Creutz model, the Chern number is represented graphically as a linking number in Bloch space: the trajectory of the origin relative to the $(k,t)$2-circle determines the integer pump, and the closed-space pumped charge is again $(k,t)$3 for a filled band (Lv et al., 2024).

2. Returning pumps, sub-Brillouin-zone topology, and zero-net transport

A particularly important recent use of the term denotes the returning, or closed-space, Thouless pump. Here the first half of the cycle pumps an integer amount of polarization and the second half exactly undoes it, so the full-cycle Chern number vanishes even though each half-cycle is topological. In the synthetic torus $(k,t)$4 one defines

$(k,t)$5

with mirror symmetry enforcing $(k,t)$6 and therefore $(k,t)$7. The bulk polarization winds by $(k,t)$8 on $(k,t)$9 and by $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$0 on $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$1, so the full cycle is a pair of consecutive but opposite Thouless pumps (Cheng et al., 11 May 2025).

In the acoustic realization of the mirror-protected returning pump, a 2D delicate topological insulator is reduced to a 1D family with synthetic momentum $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$2. Theory, full-wave simulation, and experiment all yield

$$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$3

The experimentally reconstructed Wilson-loop polarization increases from $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$4 to $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$5 over $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$6 and returns from $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$7 to $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$8 over $$Q=\frac{1}{2\pi}\int_0^T dt\int_{\mathrm{BZ}} dk\,F_{kt}(k,t)=C,$$9. The same experiment directly visualizes symmetry-respecting multicellular Wannier functions, showing that the occupied band is Wannierizable but cannot be represented by Wannier functions confined to a single primitive cell while preserving mirror symmetry (Cheng et al., 11 May 2025).

This zero-net pump is not topologically trivial. The same sub-BZ invariants enforce a bulk–boundary correspondence: on a boundary open in $F_{kt}=\partial_k A_t-\partial_t A_k$0, there is one chiral mode propagating in $F_{kt}=\partial_k A_t-\partial_t A_k$1 for $F_{kt}=\partial_k A_t-\partial_t A_k$2 and an oppositely propagating mode for $F_{kt}=\partial_k A_t-\partial_t A_k$3, producing a counterpropagating pair that spans the full band gap even though the total Chern number is zero (Cheng et al., 11 May 2025).

A second mechanism for returning pumping is Berry-dipole mediated. In a 1D three-sublattice acoustic model with adiabatic loop $F_{kt}=\partial_k A_t-\partial_t A_k$4, $F_{kt}=\partial_k A_t-\partial_t A_k$5, the synthetic space $F_{kt}=\partial_k A_t-\partial_t A_k$6 contains a Berry dipole rather than a Berry monopole. The total flux through the closed torus vanishes,

$F_{kt}=\partial_k A_t-\partial_t A_k$7

but the two hemi-tori carry quantized opposite charges,

$F_{kt}=\partial_k A_t-\partial_t A_k$8

Accordingly, the Wannier center advances by $F_{kt}=\partial_k A_t-\partial_t A_k$9 during the first half-cycle and retreats by $A_\mu=i\langle u|\partial_\mu u\rangle$0 during the second. In the finite acoustic array, an initial edge mode delocalizes into the bulk and returns to the same edge with its pseudospin flipped, a feature absent from previously proposed mirror-protected returning pumps (Mo et al., 13 May 2025).

A common misconception is that vanishing total Chern number implies trivial dynamics. Returning pumps show the opposite: $A_\mu=i\langle u|\partial_\mu u\rangle$1 can coexist with quantized half-cycle transport, enforced delocalization and relocalization, symmetry-protected multicellular Wannier functions, and gapless boundary modes defined on submanifolds of the synthetic torus (Cheng et al., 11 May 2025, Mo et al., 13 May 2025).

3. Autonomous operation, rapid cycles, and finite-frequency limits

Closed-space pumping also includes strictly isolated pumps whose full Hamiltonian is time-independent. In the autonomous topological pump for fermions on a 1D lattice, the external control parameters of a Rice–Mele pump are replaced by a quantum spin in a static magnetic field. The total Hamiltonian

$A_\mu=i\langle u|\partial_\mu u\rangle$2

is time-independent, and Larmor precession of the spin generates the control loop in the Rice–Mele plane. In the large-$A_\mu=i\langle u|\partial_\mu u\rangle$3, weak-back-action limit, the pumped charge per cycle is again the Chern number,

$A_\mu=i\langle u|\partial_\mu u\rangle$4

and for a loop encircling the origin once one finds $A_\mu=i\langle u|\partial_\mu u\rangle$5. Numerically, for $A_\mu=i\langle u|\partial_\mu u\rangle$6, $A_\mu=i\langle u|\partial_\mu u\rangle$7, $A_\mu=i\langle u|\partial_\mu u\rangle$8, and $A_\mu=i\langle u|\partial_\mu u\rangle$9, there is a quantized plateau $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$0 between a lower threshold set by back-action and an upper adiabatic breakdown near $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$1; the transport remains robust on average up to static on-site disorder $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$2 (Bohm et al., 8 May 2026).

A distinct route to finite-frequency exactness appears in the rapid-cycle Rice–Mele construction based on the zero-curvature representation of the Euclidean sinh-Gordon equation. For a family of complex-hopping protocols, the Floquet unitary over one cycle has an integer invariant

$$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$3

and the protocol is arranged so that the initial ground state is a Floquet eigenstate. For the class of orderly paths $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$4, the theorem proved in the paper gives $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$5, so the pump is exactly quantized at finite frequency and exhibits no end-of-cycle excitations (Malikis et al., 2021).

These constructive examples do not imply that exact finite-frequency quantization is generic. In the disordered Rice–Mele chain with periodic boundary conditions, a separate analysis found that for any finite disorder and any finite drive frequency the long-time pump rate decays to zero because non-adiabatic transitions between instantaneous localized states gradually equalize populations of the two counter-pumping bands. The decay is slow, with characteristic time exponentially large in the period, and in the adiabatic limit pumping survives only below the disorder threshold

$$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$6

For the parameters $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$7 and $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$8, this gives $$P(t)=\frac{e}{2\pi}\sum_{n\in\mathrm{occ}}\int_{\mathrm{BZ}} dk\,A_k^{(n)}(k,t),\qquad Q=\Delta P,$$9 (Vuina et al., 2024).

Taken together, these results sharpen the notion of closed-space pumping at finite frequency. Autonomous and integrable protocols show that exact quantization can survive without external time dependence or without the adiabatic limit in special models, whereas disorder at finite frequency generically destroys asymptotic pumping even though the decay can be parametrically slow (Bohm et al., 8 May 2026, Malikis et al., 2021, Vuina et al., 2024).

4. Synthetic, internal, configuration, and non-Abelian closed spaces

Recent work has expanded the closed-space idea far beyond transport of a particle’s center of mass in real space. One example pumps the interatomic separation of two atoms moving on a 1D ring. After separating center-of-mass and relative coordinates, the driven contact interaction produces an effective Hamiltonian in the relative coordinate,

$\Delta x=C a$0

with lattice constant $\Delta x=C a$1. The adiabatic cycle in $\Delta x=C a$2 pumps the mean relative distance by

$\Delta x=C a$3

and the simulations reported $\Delta x=C a$4 per cycle for $\Delta x=C a$5. Here the “space” being pumped is the compact relative-coordinate space of the two-body problem rather than laboratory position (Kopaei et al., 2024).

An analogous compression of space into an internal Hilbert space occurs in the two-particle Thouless spin pump. A giant spin-$\Delta x=C a$6 coupled to a spin-$\Delta x=C a$7 maps onto a synthetic SSH chain whose “sites” are $\Delta x=C a$8 eigenstates and whose two “sublattices” are the spin-$\Delta x=C a$9 states. With the chiral-symmetry-breaking term $Q=\int_0^T dt\,J(t)$0, the model becomes a Rice–Mele analogue in synthetic spin space; an adiabatic loop enclosing the point $Q=\int_0^T dt\,J(t)$1 pumps one quantum of giant spin per cycle, and the pumped quantity is again characterized by a Chern number on the synthetic $Q=\int_0^T dt\,J(t)$2 manifold (Mumford, 2022).

Closed-space pumping can also be genuinely non-Abelian. In the acoustic waveguide implementation of a non-Abelian Thouless pump, closed loops in a four-parameter space act on a three-band subspace through Wilson loops

$Q=\int_0^T dt\,J(t)$3

so the output depends on the order of loops rather than only on their net homotopy class. The experimentally realized $Q=\int_0^T dt\,J(t)$4 holonomies permute local states, and the sequences $Q=\int_0^T dt\,J(t)$5 and $Q=\int_0^T dt\,J(t)$6 produce different outcomes—rightward pumping in one case and trapping in the other—thereby directly exhibiting non-commutativity in a closed-space pump (You et al., 2021).

The same topological logic extends to atomic configuration space under periodic boundary conditions. In electronically gapped materials, a closed adiabatic loop $Q=\int_0^T dt\,J(t)$7 in atomic configuration space yields a quantized dipole displacement

$Q=\int_0^T dt\,J(t)$8

When strong adiabaticity holds, the integers $Q=\int_0^T dt\,J(t)$9 reduce to additive atomic topological charges that coincide with oxidation states; when it fails, zero-winding loops can still pump charge, producing non-convective transport in insulating non-stoichiometric electrolytes. This is a closed-space Thouless pump in the space of nuclear coordinates rather than in momentum-time space (Pegolo et al., 2020).

A further generalization is possible even in a single qubit. In the NV-center experiment on a generalized Thouless pump, the total transported quantity over $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$0 cycles is

$$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$1

where $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$2 is an interband-coherence contribution controlled by the switching-on rate of the loop. In that setting the conventional topological part is supplemented by a continuously tunable coherent term, and the strongest response appears near a band-touching point (Ma et al., 2017).

5. Representative realizations and diagnostics

The experimental and theoretical platforms already span acoustics, ultracold atoms, semiconductors, photonics, and synthetic internal spaces.

Platform	Closed-space aspect	Primary observable
1D acoustic crystals with synthetic $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$3	returning pump with sub-BZ topology	Wilson-loop polarization $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$4, Berry curvature, multicellular Wannier functions (Cheng et al., 11 May 2025)
Berry-dipole acoustic waveguide array	adiabatic loop in $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$5 over $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$6	edge $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$7 bulk $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$8 same edge evolution and pseudospin flip at $$P=\frac{e}{2\pi}\,\mathrm{Im}\ln\left\langle\Psi\left|e^{i\frac{2\pi}{L}\hat X}\right|\Psi\right\rangle,$$9 (Mo et al., 13 May 2025)
Trapped ultracold bosons in an optical superlattice	finite closed cloud without leads	center-of-mass shift $\theta$0; reversed deflection in the first excited band (Lohse et al., 2015)
Ring of Kerr resonators with periodic boundary conditions	bona-fide closed-space pump of Bogoliubov excitations	phase winding, Wannier-center shift, barycenter motion (Ravets et al., 2024)
Serpentine semiconducting narrow channel	ring or wire pumped by a rotating magnetic field, with no local ac gates	weak-field phase with $\theta$1 per cycle and $\theta$2 for $\theta$3 (Pandey et al., 2017)
Two $\theta$4 atoms on a ring	pump in relative-coordinate space	quantized change of mean separation, $\theta$5 per cycle for $\theta$6 (Kopaei et al., 2024)

The acoustic returning-pump experiment implemented twelve 1D samples with $\theta$7 discretized into twelve values; each sample contained $\theta$8 resonators, the working band was $\theta$9–$(k,t)$00 around $(k,t)$01, and the lattice constant was $(k,t)$02. From the measured Bloch functions the authors reconstructed the Berry curvature and extracted $(k,t)$03 and $(k,t)$04 (Cheng et al., 11 May 2025).

In the Berry-dipole acoustic device, the unit-cell lattice constant was $(k,t)$05 and the adiabatic loop was implemented by structural misalignments $(k,t)$06 and $(k,t)$07. The full $(k,t)$08 cycle was distributed over a $(k,t)$09 propagation section, and measurements at $(k,t)$10, $(k,t)$11, and $(k,t)$12 directly imaged the returning evolution and pseudospin inversion (Mo et al., 13 May 2025).

The ultracold-boson realization used a superlattice with $(k,t)$13, so $(k,t)$14 and $(k,t)$15. The ground band displayed a center-of-mass shift by one long-lattice spacing per cycle, while atoms prepared in the first excited band moved in the opposite direction, consistent with $(k,t)$16 and $(k,t)$17 (Lohse et al., 2015).

Other closed-space implementations emphasize wave or quasiparticle transport rather than particle number. In the nonlinear photonic Thouless pump, numerical propagation with periodic boundary conditions showed a pumped soliton regime in which the center of mass shifts by exactly one unit cell per cycle, whereas a pitchfork bifurcation at higher power produces a trapped regime with $(k,t)$18 (Jürgensen et al., 2021). In arrays of coupled spin chains, ring-like connections and network motifs inherit the same bulk Chern quantization while using XY edge couplers as coherent splitters and routers for pumped spin excitations (Bastidas, 2022). In an interaction-enabled nonsliding optical lattice, repulsive Hubbard interactions split the noninteracting singularity and create a closed-space pump with one atom transferred per cycle in the first period for $(k,t)$19, while the $(k,t)$20 and strong-repulsion limits remain trivial (Viebahn et al., 2023).

6. Robustness, limitations, and broader significance

Across these realizations, the conditions for closed-space quantization are stringent but conceptually uniform: a nonvanishing gap along the cycle, adiabatic evolution with respect to the relevant gap, and preservation of the symmetry that protects the pump when such a symmetry is essential. In the mirror-protected returning pump, breaking $(k,t)$21 spoils the equality-and-opposite relation between half-BZ Chern numbers and can gap out the boundary modes across the cycle (Cheng et al., 11 May 2025). In the driven-dissipative Kerr array, one must satisfy $(k,t)$22 and keep the loss rate smaller than the relevant gaps so that the instantaneous Bogoliubov bands remain well defined (Ravets et al., 2024).

Robustness is equally model dependent. The autonomous pump requires the Larmor frequency to exceed a back-action scale while remaining below the fermionic gap scale (Bohm et al., 8 May 2026). The interaction-enabled nonsliding lattice pump exists only in a finite interaction window, because at small $(k,t)$23 the path encloses no singularity and at large $(k,t)$24 the singularity exits the cycle again (Viebahn et al., 2023). In disordered finite-frequency Rice–Mele chains, quantization is only preasymptotic: the long-time pump rate vanishes for any $(k,t)$25 and $(k,t)$26, even though the decay time can be exponentially large in $(k,t)$27 (Vuina et al., 2024).

Another frequent misconception is that closed-space pumping is synonymous with ordinary charge pumping on a ring. The literature now makes clear that it also encompasses pumps of spin, mirror or orbital quantum numbers, relative distance, acoustic energy, and matrix-valued holonomies. In enlarged three-parameter spaces of competing mass terms, distinct closed loops can pump different conserved quantities because the allowed parameter space is multiply connected by removed gapless lines (Lopes et al., 2016). In the generalized Creutz model, the same idea is recast graphically: different closed trajectories have different linking numbers with the $(k,t)$28-circle, and those linking numbers are the Chern numbers of the pump (Lv et al., 2024).

This suggests a broad unifying view. A closed-space Thouless pump is best understood as quantized adiabatic transport defined on a compact manifold—real-space ring, synthetic torus, internal coordinate, atomic configuration space, or multi-band parameter space—where the relevant observable is fixed by geometric data of the occupied subspace. In some cases the result is a conventional nonzero-Chern pump; in others it is a returning pump with $(k,t)$29 but quantized half-cycle topology; in still others it is an autonomous, non-Abelian, interaction-enabled, or internal-coordinate pump. What remains invariant across these versions is the replacement of open transport by a closed geometric cycle and the encoding of transport in bulk topology rather than in dissipative conduction channels (Cheng et al., 11 May 2025, Bohm et al., 8 May 2026, Kopaei et al., 2024, You et al., 2021).