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Observation of returning Thouless pumping

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Pith's one-line read The paper reports the first experimental observation of returning Thouless pumping, realized in a chain of acoustic resonators with a synthetic extra dimension, and the first realization of a delicate topological insulator.

desk verdict A clever acoustic synthetic-dimension experiment reports the first observation of returning Thouless pumping with strong data-theory agreement, but the main-text Wilson loop formula is not actually closed and needs clarification. read the letter →

arxiv 2505.06808 v1 pith:PBS5SV3Z submitted 2025-05-11 cond-mat.mes-hall physics.class-ph

Zheyu Cheng , Sijie Yue , Yang Long , Wentao Xie , Zixuan Yu , Hau Tian Teo , Y. X. Zhao , Haoran Xue
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Baile Zhang
This is my paper · ORCID
classification cond-mat.mes-hallphysics.class-ph
keywords returningThoulesspumpdelicatetopologicalinsulatorsyntheticdimensionacousticcrystalsub-BrillouinzoneChernnumbermulticellularWannierfunctiongaplessedgemodesbulk-boundarycorrespondence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Returning Thouless pumping is a proposed variant of quantized transport in which a system pumps one quantum of polarization during the first half of an adiabatic cycle and gives it back during the second half, so the total Chern number (the integer that quantizes ordinary pumps) is zero and no net charge moves over a full cycle. This paper claims to observe that process directly, using a set of one-dimensional acoustic crystals in which a synthetic angle $\theta$ stands in for the missing second momentum dimension. From measured pressure fields the authors compute the bulk polarization through a discrete Wilson loop and find that it rises continuously from 0 to 1 and then falls back to 0 across the Brillouin zone. They also observe the predicted symmetric, multicellular Wannier functions, localized states that necessarily spread over several unit cells, and gapless edge modes, explained by Chern numbers $+1$ and $-1$ defined separately on the two halves of the Brillouin zone. The work turns delicate topological insulators, previously a purely theoretical classification, into a concrete experimental platform.

What carries the argument

The key machinery is the dimensional-reduction map $k_y \to \theta$, which converts a two-dimensional delicate topological insulator with imaginary and long-range hoppings into a one-dimensional chain with only nearest-neighbor, real hoppings, Eq. (2). Each acoustic unit cell contains two dipole resonators whose coupling sign and amplitude are adjusted geometrically, realizing hopping parameters $t_1 = 2t_{x1}\sin\theta$, $t_2 = 2t_{x2}\sin\theta$, and $\delta = 2t_y\cos\theta$. Bulk polarization is then extracted from the measured wavefunctions using the discrete Wilson loop of Eq. (4), and the Wannier functions are obtained by a gauge-fixed Fourier transform of the measured Bloch functions, Eq. (5). The second load-bearing element is the partition of the Brillouin zone at $\theta = 0$ and $\theta = \pi$ into two sub-Brillouin zones; mirror symmetry makes each half a closed manifold with Chern number $-1$ or $+1$, and Stokes' theorem, Eq. (6), links these sub-Brillouin-zone Chern numbers to both the $0 \to 1 \to 0$ polarization path and the gapless edge modes.

What would settle it

Take one of the twelve-sample chains and repeat the polarization extraction without the Lorentzian fitting step, or with a chain of more than twelve unit cells, and check whether the Wilson loop still runs from 0 to 1 to 0 exactly; any shortening or endpoint drift would indicate that the quantization is an artifact. Alternatively, drive $\theta$ in time through a full cycle while monitoring the pressure field, and look for one quantum of transported intensity during the first half-cycle and equal transport in the opposite direction during the second; if either half-cycle shows non-quantized or missing transport, the returning-pump interpretation fails.

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Extended reading notes

Core claim

In the authors' own terms, the discovery is that a one-dimensional acoustic crystal parameterized by a synthetic momentum $\theta$ faithfully realizes the two-dimensional tight-binding model of a delicate topological insulator, Eq. (1), and exhibits all of its predicted signatures. The measured lower-band polarization $p_x(\theta)$, computed from experimentally measured wavefunctions via Eq. (4), changes continuously from 0 to 1 as $\theta$ traverses the first half of the Brillouin zone and returns from 1 to 0 in the second half, directly displaying returning Thouless pumping. The Wannier functions of the lower band, built from the measured Bloch functions in a symmetry-preserving gauge, are symmetric and multicellular, extending beyond a single unit cell. In finite chains, the authors observe one pair of counterpropagating gapless edge modes per edge, and they interpret these as protected by sub-Brillouin-zone Chern numbers of $-1$ and $+1$ whose sum is zero. This establishes the bulk-boundary correspondence of a delicate topological insulator: gapless boundary modes coexist with a vanishing total Chern number.

Load-bearing premise

The entire extraction rests on the assumption that each acoustic resonator is accurately described by a single dipole mode, so the pressure measured at the two sublattice sites gives the full two-component Bloch wavefunction; any substantial contribution from higher modes, stray couplings, or boundary-induced phase shifts would make the Wilson-loop polarization a fitting artifact rather than a topological invariant.

Editorial extensions

If this is right

  • Delicate topological phases can be realized in wave systems with only nearest-neighbor, real-valued couplings, provided one momentum axis is replaced by a synthetic parameter; the imagined requirement of long-range complex hoppings is not an obstacle.
  • Gapless edge modes are a genuine signature of delicate topology: they appear on boundaries perpendicular to the synthetic dimension and are protected by sub-Brillouin-zone Chern numbers, even though the total Chern number is zero.
  • The polarization path $0 \to 1 \to 0$ is a measurable bulk invariant, extractable from wavefunction measurements through a discrete Wilson loop, making returning Thouless pumping a practical diagnostic.
  • Partitioning the Brillouin zone into more than two sub-zones should produce multiple pairs of gapless edge modes, offering a route to multimode waveguiding and signal multiplexing in a single platform.
  • The same gauge prescription that reveals symmetric multicellular Wannier functions provides an experimental method to identify other symmetry-protected phases that are Wannierizable but not atomically obstructed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: because the recipe uses only real nearest-neighbor couplings, the same delicate phase should be transferable to photonic, mechanical, or cold-atom lattices without any new design principle; the acoustic platform is one instance, not the only one.
  • Beyond the paper: a time-resolved version of this experiment, in which $\theta$ is swept continuously rather than sampled at twelve static values, would turn the inferred $0 \to 1 \to 0$ polarization into directly observed bidirectional transport, testing whether the returning pump survives non-adiabatic corrections.
  • Beyond the paper: the sub-Brillouin-zone Chern-number picture suggests a design rule, namely that any mirror-symmetric band structure whose Berry curvature integrates to an integer on each half Brillouin zone should exhibit the same returning-pump behavior; searching other symmetry classes for nonzero half-BZ Chern numbers could predict new delicate phases.
  • Beyond the paper: if the multicellular Wannier functions survive disorder and fabrication error at the level shown here, they could serve as a measurable fingerprint for distinguishing delicate topology from fragile or obstructed phases in noisy experimental data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript reports an acoustic experiment realizing a two-dimensional delicate topological insulator by promoting a momentum component to a synthetic parameter θ. Twelve one-dimensional acoustic crystals with 12 unit cells each are fabricated, with geometric parameters designed so that the extracted tight-binding parameters follow Eq. (3). The authors measure bulk dispersions and sublattice-resolved wavefunctions, compute the bulk polarization from Eq. (4), and observe a 0-to-1-to-0 variation of the polarization as θ traverses 2π. They also construct symmetric multicellular Wannier functions and observe edge modes whose frequencies match numerical dispersions. The results are interpreted as the first observation of returning Thouless pumping and of delicate topology, with sub-Brillouin-zone Chern numbers providing the bulk-boundary correspondence.

Significance. If correct, the experiment would be the first realization of a delicate topological insulator and the first observation of returning Thouless pumping. The main evidence, the measured Wilson-loop polarization and its agreement with tight-binding and full-wave simulations, is in principle compelling, and the manuscript includes error bars from five measurements. The synthetic-dimension approach is an elegant way to avoid implementing imaginary hoppings in a single 1D lattice. However, the key formula used for the central observable is not a closed Wilson loop as printed, and the experiment measures a static polarization rather than a dynamical pumping process, so the strength of the claim currently exceeds what is demonstrated in the text.

major comments (3)
  1. [Observation of the RTP, Eq. (4)] Equation (4) as printed is not a closed Wilson loop: the product runs from ⟨u_{k_N}|u_{k_{N-1}}⟩ to ⟨u_{k_2}|u_{k_1}⟩ and omits the closing overlap ⟨u_{k_1}|u_{k_0}⟩. With N=12 and k_12≡k_0, the product contains N−1 links instead of N, so its argument is not the Berry phase of the band and is not gauge invariant under the U(1) phases of the measured states. The sentence setting |u_{k_12}⟩≡|u_{k_0}⟩ identifies the endpoints but does not replace the missing link. Please correct the formula, specify the periodic-gauge or parallel-transport convention used in the data analysis, and confirm that the 0-to-1-to-0 curve in Fig. 2b is obtained from the closed-loop expression. The analysis should also state how the measured Fourier amplitudes are phase-referenced across the k_i points.
  2. [Title, Abstract, and Observation of the RTP] The experiment measures the instantaneous bulk polarization at 12 discrete values of θ; it does not implement an adiabatic cycle in time or measure a transported charge (for example, a displacement of the center of mass). The phrase 'directly observe returning Thouless pumping' is therefore stronger than the data support. Please either temper the claim, for instance by writing 'observe the polarization signature of returning Thouless pumping', or add a dynamic measurement that sweeps θ and records the pumped quantity. This distinction matters because quantized transported charge is the defining feature of Thouless pumping.
  3. [Observation of symmetric multicellular WFs, Eq. (5)] The Wannier function construction in Eq. (5) requires an integral over the full two-dimensional Brillouin zone, including the synthetic momentum θ, but the experiment provides only 12 discrete θ samples. The main text does not state how the θ continuum is approximated (interpolation scheme, discrete Fourier transform, or a model-based filling), nor how the 'special gauge' that makes the Wannier functions symmetric is chosen and verified. Since the observation of symmetric multicellular Wannier functions is one of the headline claims, this information should be given in the main text or Methods rather than only in the Supplementary Information.
minor comments (6)
  1. [Observation of the RTP, Fig. 2b] The error bars on the measured p_x are described as standard deviations from five independent measurements, but the number of independent measurements and how the Wilson loop is averaged (per measurement versus pooled data) should be stated explicitly.
  2. [Methods, Data analysis] The sentence 'For each k, we fit the field to a Lorentzian line shape around the local maxima' should specify the fitting window, the number of k points, and the normalization convention for the extracted eigenfunctions, since these details directly affect the Wilson loop in Eq. (4).
  3. [Eq. (2)] Equation (2) uses t1, t2, and δ before these are defined in Eq. (3); please reorder the definitions or add a pointer to avoid confusion.
  4. [Data availability] The data availability statement contains a placeholder URL; for review, access to the experimental data and simulation files should be provided.
  5. [Fig. 3 caption and text] The caption mentions 'measured acoustic intensities' while the text describes pressure amplitude spectra; please clarify which quantity is plotted.
  6. [Observation of gapless edge modes] The statement that the chiral edge modes 'always span the entire bulk bandgap regardless of system parameters' is a strong claim; please specify the parameter range over which this is established or refer to the specific proof in the Supplementary Information.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the measured polarization, Wannier functions, and edge modes are independent data products compared against, not fitted from, the model.

full rationale

The paper's central claims are experimental observations of a previously proposed theoretical effect. The bulk polarization p_x(theta) in Eq. (4) is evaluated from measured sublattice-resolved wavefunctions, not from the tight-binding parameters t1, t2, delta that were extracted in Fig. 1d; the comparison in Fig. 2b is therefore between an independently measured quantity and the model, not a fitted parameter renamed as a prediction. The multicellular Wannier functions in Eq. (5) are Fourier transforms of measured Bloch functions with an explicitly gauge-selected phase, and the edge-mode spectra in Fig. 3 are direct pressure measurements; both are compared with, not generated by, the tight-binding and full-wave simulations. The only author-associated citations (Refs. 11, 44, 46) appear in technical contexts such as previous acoustic Thouless pumping studies or hopping-sign control, and none carries a load-bearing premise of the derivation. A reviewer concern that Eq. (4) as printed is an open Wilson loop, because it omits the closing overlap <u_{k1}|u_{k0}>, would be a correctness issue about the Berry-phase interpretation rather than a circularity issue: the quantity is still measured rather than derived from the claim. Accordingly, no circular step can be exhibited by reduction of an equation to its own inputs, and the honest finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central experiment rests on standard synthetic-dimension mapping, on a single-mode tight-binding description of the acoustic unit cell, and on chosen model parameters tx1=0.05 kHz, tx2=ty=0.1 kHz. The acoustic samples were reverse-engineered through a parameter scan to match Eq. (3), so the returning Thouless pump is designed into the fabrication rather than emerging from an independent parameter-free prediction. No new particles, forces, or hypothetical entities are introduced.

free parameters (5)
  • tx1 (target tight-binding hopping) = 0.05 kHz
    Hand-chosen model parameter in Eq. (3). The acoustic geometry is designed so that the extracted t1 = 2 tx1 sin theta follows the target curve in Fig. 1d. The polarization curve depends on this scale, though the topological quantization is scale-independent.
  • tx2 (target tight-binding hopping) = 0.1 kHz
    Chosen model parameter in Eq. (3). Sets the kx-dependent hopping amplitudes and the size of the band gap. The acoustic structures are fitted to reproduce t2 = 2 tx2 sin theta.
  • ty (target tight-binding hopping) = 0.1 kHz
    Chosen model parameter in Eq. (3) that controls the on-site term delta = 2 ty cos theta and the mirror-symmetric mass term. Used to set the synthetic-dimension dispersion.
  • f0 (resonance frequency) = 5.7 kHz
    Operating frequency chosen for the acoustic dipole mode. It sets the frequency window for the measurements but does not enter the topological invariant.
  • Acoustic geometric parameters for 12 samples = 12 parameter sets listed in SI
    Per-sample values of d1, l1, l2, A, and edge-cavity corrections are chosen through an exhaustive parameter scan so that the extracted tight-binding parameters match Eq. (3). This is a design-fitting step, distinct from fitting the measured polarization.
assumptions (4)
  • domain assumption Each acoustic resonator supports only one relevant dipole mode in the 5.3 to 6.1 kHz range, and higher modes and long-range couplings are negligible.
    Invoked when reducing the unit cell to the two-band tight-binding model Eq. (2) and when identifying the measured pressure field with the tight-binding wavefunction. Residual higher modes would contaminate the Wilson loop.
  • domain assumption The synthetic-dimension mapping ky to theta is faithful: measuring 12 independent 1D crystals at discrete theta is equivalent to sampling the 2D band structure at discrete ky values.
    This is the standard synthetic-dimension assumption and is used throughout the design and the interpretation of Fig. 2b as a returning Thouless pump.
  • domain assumption The discrete Wilson loop with N=12 and the periodic gauge |u_k12> = |u_k0> gives a reliable estimate of the bulk polarization.
    Used in Eq. (4). Discretization errors and gauge misalignment are assumed small relative to the observed 0-to-1-to-0 variation, but no convergence study is shown.
  • domain assumption The mirror symmetry My = sigma_z and the time-reversal-like symmetry T = sigma_z K of the tight-binding model are sufficiently well realized in the acoustic structures to preserve the sub-Brillouin-zone Chern numbers.
    The sBZ Chern number picture in Fig. 3a relies on these crystalline symmetries. Fabrication disorder and geometric asymmetries could break them, though the measured spectra suggest the effects are small.

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Pith. "Pith review of Observation of returning Thouless pumping." pith.science (2026). https://pith.science/paper/PBS5SV3Z

@misc{pith2026250506808,
  author       = {Pith},
  title        = {Pith review of: Observation of returning Thouless pumping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBS5SV3Z}},
  note         = {Machine review of arXiv:2505.06808}
}
read the original abstract

Introduced by David Thouless in 1983, Thouless pumping exemplifies topological properties in topological systems, where the transported charge is quantized by the Chern number. Recently, returning Thouless pumping was theoretically proposed, in which quantized charge is pumped during the first half of the cycle but returns to zero in the second half. This mechanism leads to crystalline symmetry-protected delicate topological insulators. Unlike conventional topological bands, a delicate topological band is Wannierizable but not atomically obstructed, which features multicellular Wannier functions extending beyond a single unit cell. Here, by replacing the second dimension with a synthetic dimension, we realize a two-dimensional delicate topological insulator via a set of one-dimensional acoustic crystals with fine-tuned geometric parameters. Through acoustic bands and wavefunction measurements, we directly observe returning Thouless pumping and symmetric multicellular Wannier functions, followed by establishing the bulk-boundary correspondence between sub-Brillouin zone Chern numbers and gapless boundary modes. As enriched by crystalline symmetries, our experimental demonstration of returning Thouless pumping expands the current understanding of topological phases of matter.

Figures

Figures reproduced from arXiv: 2505.06808 by the authors.

Figure 1
Figure 1. FIG. 1. Design of a synthetic acoustic delicate topological insulator. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Observation of the returning Thouless pump and symmetric multicellular Wannier functions. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Observation of multiple gapless edge modes. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Delicate Wannier insulators

    cond-mat.mes-hall 2025-06 conditional novelty 7.0 of 10

    Delicate Wannier insulators are a new family of topological phases whose Wannier bands carry delicate invariants, yielding hinge and corner modes only at sharp boundaries.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.