{"id":"a6ebbcd4-4166-4222-95c2-d692e450cebb","arxiv_id":"2505.06808","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Acoustic metamaterials with a synthetic dimension directly show the predicted returning Thouless pump, multicellular Wannier functions, and gapless edge modes of a delicate topological insulator.","lead":"This paper reports the first experimental realization of a delicate topological insulator using acoustic crystals, observing a returning Thouless pump where a quantized bulk polarization rises over half a synthetic cycle and returns to zero over the other half. It matters because it turns a recent theoretical prediction about a new class of fragile topological matter into a tabletop acoustic experiment.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4) as written is an open path, not a closed Wilson loop; the 0-to-1-to-0 polarization in Fig. 2b may not be a Berry phase.","rationale":"The reader's weakest assumption concerns whether the measured pressure field is faithfully represented by a two-component spinor so that Eq. (4) yields the true Berry phase. I agree that this is important, but the more immediate and concrete flaw is that Eq. (4) as written is not a closed Wilson loop: it omits the link between k_1 and k_0=k_N. Even if this is a typesetting typo, the main text does not specify how the measured U(1) gauges are made periodic, and without that the computed polarization is not gauge invariant. This is directly load-bearing for the headline claim of observing returning Thouless pumping, because Fig. 2b is the quantitative evidence. The paper has supporting full-wave simulations and edge-mode observations, so the concern may be resolved by the SI or a small correction, but the preprint as written does not establish the central invariant. I therefore recommend UNVERDICTED rather than a flat rejection: the claim cannot be assessed until Eq. (4) is corrected and the gauge procedure is specified and verified.","tokens_in":9798,"tokens_out":13815,"duration_ms":138228,"concrete_test":"Recompute p_x(θ) from the raw measured (or full-wave) sublattice fields using the closed Wilson loop W = ∏_{i=1}^{N} ⟨u_{k_i}|u_{k_{i-1}}⟩ with k_0=k_N, after applying an explicit parallel-transport gauge (fix each u_{k_i} up to a phase by requiring ⟨u_{k_i}|u_{k_{i-1}}⟩ real positive), and compare to Fig. 2b. Also repeat with N=24. If the 0-to-1-to-0 curve changes by more than the error bars or fails to close, the RTP observation is not supported.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central observation of returning Thouless pumping is the measured polarization p_x(θ) of Fig. 2b, computed via Eq. (4). As printed, Eq. (4) is the product ⟨u_{k_N}|u_{k_{N-1}}⟩⋯⟨u_{k_2}|u_{k_1}⟩ with k_N≡k_0. This contains N−1 overlaps and omits the closing link ⟨u_{k_1}|u_{k_0}⟩, so it is an open path whose argument is not the Berry phase of the band. The sentence 'we set |u_{k_12}⟩≡|u_{k_0}⟩' identifies endpoints but does not fix the U(1) gauges of the measured states at intermediate k_i. The measured field amplitudes inherit k-dependent source-coupling phases; these cancel only in a genuinely closed loop with a periodic gauge. Without an explicit parallel-transport or smooth-gauge construction, the logarithm in Eq. (4) can return a value that depends on the data-analysis convention, and the 0-to-1-to-0 curve could be an artifact. The SI is said to contain the Wannier-gauge procedure, but the main text's key observable should stand alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10026,"tokens_out":8915,"duration_ms":87750,"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":[{"comment":"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.","section":"Observation of the RTP, Eq. (4)"},{"comment":"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.","section":"Title, Abstract, and Observation of the RTP"},{"comment":"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.","section":"Observation of symmetric multicellular WFs, Eq. (5)"}],"minor_comments":[{"comment":"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.","section":"Observation of the RTP, Fig. 2b"},{"comment":"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).","section":"Methods, Data analysis"},{"comment":"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.","section":"Eq. (2)"},{"comment":"The data availability statement contains a placeholder URL; for review, access to the experimental data and simulation files should be provided.","section":"Data availability"},{"comment":"The caption mentions 'measured acoustic intensities' while the text describes pressure amplitude spectra; please clarify which quantity is plotted.","section":"Fig. 3 caption and text"},{"comment":"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.","section":"Observation of gapless edge modes"}],"recommendation":"major_revision","confidential_remarks":"I see no ethical issues or citation problems. The main concern is that the central quantitative claim is currently supported by a misprinted Wilson-loop formula and by an indirect static measurement; both are fixable within a revision. The experimental effort is substantial and the topic is well suited to the journal, but the claims need to be made commensurate with the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline is that this is a genuinely new experiment: a 1D acoustic crystal with a synthetic momentum parameter that realizes a 2D delicate topological insulator and returning Thouless pumping. The synthetic-dimension mapping from k_y to theta is a neat trick that avoids imaginary hoppings, and the measured bulk polarization, Wannier functions, and edge modes agree with both tight-binding and full-wave simulations across all 12 theta steps. That agreement, with error bars from five independent measurements, is the paper's core strength. The claim of a delicate topological phase is credible, and the Wannier functions clearly show multicellular structure.\n\nThe main soft spot is in the main-text Wilson loop formula. Eq. (4) as printed is not a closed loop: it includes overlaps from k_N down to k_1 but omits the closing link ⟨u_k1|u_k0⟩. With k_N ≡ k_0, the product gives a path from k_0 to k_1, and its argument is not gauge-invariant unless the SI's Wannier-gauge procedure explicitly fixes the phases at every intermediate k. The data likely are fine—the agreement with the two independent simulations would be hard to fake—but the main text should state the closed-loop form or the gauge construction, otherwise the central observable is under-specified. This is fixable, but the SI needs to be checked.\n\nThe other concern is semantic: the paper says \"observe returning Thouless pumping,\" but the measurement is a static set of 12 theta samples, not a time-resolved pumping cycle. That is a fair way to probe the polarization winding, but \"direct observation\" overreaches slightly. Also, the claim of gapless edge modes would benefit from showing an actual dispersion (frequency vs k_x) rather than peak positions at a single edge site, though the evidence is suggestive.\n\nThe citation pattern is honest, giving adequate credit to the theory papers and prior experiments. No circularity: the measured polarization is computed directly from measured wavefunctions, not from fitted parameters. This is a serious experimental paper, with real new results and a clever platform. I'd send it to a good referee, with a specific request to scrutinize the Wilson loop evaluation and the SI. If that holds, it's a milestone for delicate topology.\n\nFor a reading group, yes—it's a good example of experiment meeting tricky band topology. I'd cite it if I were writing about synthetic dimensions or delicate phases.\n\nRecommendation: engage with it; it deserves peer review, and the Eq. (4) issue should be fixed but doesn't sink the paper.","headline":"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.","tokens_in":10602,"tokens_out":4006,"would_cite":true,"duration_ms":36781,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["returning Thouless pump","delicate topological insulator","synthetic dimension","acoustic crystal","sub-Brillouin zone Chern number","multicellular Wannier function","gapless edge modes","bulk-boundary correspondence"],"falsifier":"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.","tokens_in":9579,"feed_emoji":"🔊","tokens_out":8969,"duration_ms":86853,"temperature":0.7,"pith_summary":"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.","feed_headline":"Acoustic crystals show a pump that returns its charge","feed_subtitle":"Measured bulk polarization sweeps 0 to 1 and back, the signature of returning Thouless pumping and delicate topology.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the quantized particle pump whose returning variant is the subject of the paper.","marker":"[1]"},{"why":"Defines delicate topological insulators and predicts multicellular Wannier functions, the target signatures observed experimentally.","marker":"[23]"},{"why":"Provides the crystalline-symmetry theory underlying returning Thouless pumping in delicate topological bands.","marker":"[24]"},{"why":"Supplies the two-dimensional tight-binding model of a delicate topological insulator, Eq. (1), that the acoustic chain realizes.","marker":"[33]"},{"why":"Provides the sub-Brillouin-zone Chern-number framework used to explain the gapless edge modes.","marker":"[38]"},{"why":"Basis of the Wilson-loop formula, Eq. (4), used to extract polarization from measured wavefunctions.","marker":"[48]"}],"fun_headline_variants":["Returning Thouless pumping observed in acoustic crystals","Pump that pushes charge out and pulls it back seen in acoustic crystals","Delicate topological insulator shows charge pump with a round trip","Acoustic lattice revisits Thouless pump: charge goes 0 to 1 and back","Charge shuttle that returns: direct evidence of returning Thouless pumping"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Returning Thouless pumping observed in acoustic crystals","Pump that pushes charge out and pulls it back seen in acoustic crystals","Delicate topological insulator shows charge pump with a round trip","Acoustic lattice revisits Thouless pump: charge goes 0 to 1 and back","Charge shuttle that returns: direct evidence of returning Thouless pumping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000227,"raw_usage":{"total_tokens":1476,"prompt_tokens":954,"completion_tokens":522,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":432}},"tokens_in":570,"tokens_out":522,"duration_ms":5422,"temperature":1.0,"reasoning_tokens":432,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:32:14.101405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Nelson, T","cited_arxiv_id":null,"evidence_quote":"Defines delicate topological insulators and predicts multicellular Wannier functions, the target signatures observed experimentally."},{"cited_title":"Nelson, T","cited_arxiv_id":null,"evidence_quote":"Provides the crystalline-symmetry theory underlying returning Thouless pumping in delicate topological bands."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional tight-binding model of a delicate topological insulator, Eq. (1), that the acoustic chain realizes."},{"cited_title":"Chen, Y .-P","cited_arxiv_id":null,"evidence_quote":"Provides the sub-Brillouin-zone Chern-number framework used to explain the gapless edge modes."}],"review_version":1}