{"id":"683ae5fa-526c-4d0f-8c59-72e157b79bde","arxiv_id":"2507.04068","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Bulk momentum-band topology is experimentally demonstrated in a PT-symmetric acoustic Floquet lattice via band inversion and a quantized Berry phase, with temporal interface states.","lead":"This paper reports acoustic experiments that reconstruct the effective Hamiltonian of a PT-symmetric Floquet lattice and extract a quantized Berry phase in the energy Brillouin zone, together with time-localized interface states. It is the first claimed bulk experimental evidence of momentum-band topology, extending topological physics to time-varying systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The measured quantized Berry phase rests on Eq. (4), whose derivation is deferred to the Supplemental Material and which presumes exact PT balance; neither the equivalence nor the gain/loss balance is quantified, so the central topological claim is not yet secured.","rationale":"The reader's conditional verdict is sound, but I want to sharpen the weakest link. The strongest claim is not merely that a phase winds by 2π, but that this winding equals a quantized EBZ Berry phase. That link is Eq. (4). Its proof is not in the main text, and non-Hermitian Berry phases are subtle: standard definitions use left and right biorthogonal eigenstates, whereas Eq. (2) as written uses ψ†∂_Eψ. The reader correctly identified PT balance as a load-bearing experimental condition; the additional point here is that even with perfect balance, the measured invariant is only the claimed Berry phase if the Supplemental Material establishes Eq. (4) rigorously for this two-band model. The paper itself provides independent support for its experimental method: reconstructing H_eff from two response measurements is a clean and well-posed approach, the temporal interface-state control with γ = −7 to −3 Hz is appropriately chosen, and the boundary-signature data are qualitatively consistent with the bulk calculation. The self-reported deviation in Fig. 5(b) after about 25 ms, attributed to sensitivity of the synthesized momentum to the accuracy of the unidirectional couplings, reinforces the need for quantitative error analysis on the reconstructed eigenstates before the quantized-invariant claim can be accepted. A biorthogonal recomputation of θ_E from the same reconstructed data is a decisive and practical test: if Eq. (4) holds numerically for the experimental parameters and for small imbalances, the central claim survives; if not, the measured phase winding is not a topological invariant. This is a check the authors can run with data they already possess, so it does not demand new experiments.","tokens_in":10605,"tokens_out":14133,"duration_ms":176864,"concrete_test":"Independently re-derive Eq. (4) from the measured 2×2 effective Hamiltonian: for the experimental parameters (w = 36 Hz, γs = 11 Hz, γ = ±7 Hz, Ω = 125 Hz), compute the EBZ Berry phase using the biorthogonal connection θ_E = i∮ dE ⟨ψ_L|∂_E ψ_R⟩ / ⟨ψ_L|ψ_R⟩ over the same contour used in Fig. 4(c), and compare it with half the winding of φ(E) = arg⟨ψ_R|PT|ψ_R⟩. If the two disagree by more than numerical precision, the experimentally quoted θ_E is not the standard non-Hermitian Berry phase and the central topological claim is unsupported. If they agree, repeat the comparison with a few percent imbalance in ±(γs+γd(t)) to establish the tolerance within which the quantization survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the phase winding in Fig. 4(c) yields the EBZ Berry phase θ_E = π for γ > 0 and θ_E = 0 for γ < 0. That step uses Eq. (4), θ_E = (1/2)∮∂_E φ(E)dE with e^{iφ} = ⟨ψ|PT|ψ⟩, whose derivation is entirely deferred to the Supplemental Material. For a non-Hermitian effective Hamiltonian, the standard Berry phase is built from biorthogonal left and right eigenstates; the object ψ†∂_Eψ appearing in Eq. (2) is not the usual connection unless a PT-symmetric normalization is imposed. Thus Eq. (4) is a nontrivial identity, not a definition, and the measured 2π winding of φ only counts as a quantized Berry phase if that identity holds for the experimental parameters. The identity also requires exact PT symmetry, i.e., gain +i(γs+γd(t)) and loss −i(γs+γd(t)) balanced to within a tolerance set by the gap. The paper reports no error bars on the eigenstate phases and no measured value of the imbalance; the only statement is the design intent of the positive and negative feedback circuits. Without those numbers, the first bulk evidence claim is conditional on an unquantified experimental symmetry and an unshown theoretical equivalence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reports an experimental study of momentum-band topology in a PT-symmetric acoustic Floquet lattice. The authors implement the lattice with coupled acoustic cavities and active feedback circuits, reconstruct the Floquet effective Hamiltonian from two measured time-domain wavefunctions, and extract quasi-energy spectra and eigenstates. They observe a dynamic gain/loss induced momentum gap, momentum-band inversion signaled by the exchange of Floquet exceptional-point momenta when the sign of the gain/loss amplitude γ is reversed, and a quantized Berry phase θ_E = π (γ>0) versus 0 (γ<0) obtained from the winding of the PT eigenvalue phase. They also observe temporally localized interface states at a temporal domain wall, as a boundary signature. The central claim is that this constitutes the first direct bulk experimental evidence of momentum-band topology.","tokens_in":10857,"tokens_out":9351,"duration_ms":108849,"significance":"If fully substantiated, this experiment would provide a genuine advance: direct bulk observation of a topological invariant in the energy Brillouin zone of a Floquet system, complementing earlier observations of temporal interface states. The technique of reconstructing the Floquet operator from two injected wavefunctions is elegant and transferable to other non-equilibrium systems. The theory is standard Floquet theory, clearly presented, and the bulk and boundary evidence are presented together, with a trivial-control experiment included. The paper would be strengthened by explicit error quantification and by making the derivation of the central identity used for the Berry-phase measurement fully transparent. The claim of 'first bulk evidence' is conditional on those points, but the experimental platform and reconstruction method are significant even apart from that claim.","major_comments":[{"comment":"The central topological measurement is based on the identity θ_E = (1/2)∮ ∂_E φ(E)dE with e^{iφ}=⟨ψ|PT|ψ⟩, which is stated in the main text and its derivation is entirely deferred to the Supplemental Material. Since the directly measured quantity is the winding of φ and not the integral in Eq. (2), the identification of a 2π winding with a quantized Berry phase θ_E=π depends on this nontrivial identity. The authors should provide the derivation in the main text or an extended appendix, specify the normalization of ψ, and state the conditions under which the identity holds (including the branch of the Floquet logarithm and the behavior near exceptional points).","section":"§4, Eq. (4)"},{"comment":"The experimental phase evolution φ(E) is plotted as circles with no error bars and no statistical analysis. The central quantitative claim—that the winding is exactly 2π for γ>0 and 0 for γ<0—needs an uncertainty estimate to be a convincing measurement of a topological invariant. The authors should report the number of independent measurements, the reproducibility of the winding, and the sensitivity of the extracted phase to noise in the measured wavefunctions.","section":"§4, Fig. 4(c)"},{"comment":"The reconstruction of the Floquet operator in Eq. (3) requires that the two injected wavefunctions ψ1(t0) and ψ2(t0) are linearly independent and that the measured responses over one period faithfully represent the free Floquet evolution. The 2×2 matrix inversion in Eq. (3) has a condition number that depends on the angle between the two states and on the measurement noise. The authors should report the condition number (or an equivalent measure) and discuss how errors propagate from the measured time traces into the reconstructed eigenstates and phases. This directly affects the reliability of all subsequent eigenstate-based conclusions.","section":"Experimental setup, §3"},{"comment":"The theoretical model and the Berry-phase quantization rely on exact PT symmetry, i.e., gain +i(γ_s+γ_d(t)) on cavity 1 balanced by loss −i(γ_s+γ_d(t)) on cavity 2. The manuscript states the design intent of the positive and negative feedback circuits but does not report any measured value of the gain/loss imbalance or the calibration procedure. Since an unquantified imbalance can smear the PT eigenvalue phase and invalidate Eq. (4), the authors should provide a measured balance (e.g., from transmission or reflection calibration) and state the tolerance required by the size of the momentum gap.","section":"§2 and §3, PT balance"}],"minor_comments":[{"comment":"The caption refers to 'Floquet-EP states' rendered as thick arrows, but at an exceptional point the two eigenstates coalesce. Please clarify whether each arrow represents the single coalesced eigenvector at the EP momentum, or one of the two eigenstates on either side of the EP.","section":"§4, Fig. 4(b)"},{"comment":"The notation ψ† in Eq. (2) is the usual conjugate transpose, but for non-Hermitian systems one often needs biorthogonal left and right states. Please state explicitly the normalization adopted for the eigenstates used in Eq. (2) and in the experimental extraction.","section":"§3, Eq. (3)"},{"comment":"The deviation between experiment and simulation for t > 25 ms is attributed to sensitivity to the synthesized momentum. Please provide a quantitative estimate of the momentum uncertainty and show, if possible, a comparison at a slightly different k to confirm this interpretation.","section":"Fig. 5(b)"},{"comment":"Reference [43] is given as 'See Supplemental Material [url]'; in the arXiv version the URL is a placeholder. Please ensure the supplemental material is accessible and that all equations cited there are available to the reader.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents an appealing and potentially important experiment, and the reconstruction technique is a genuine strength. The main risk is that the headline 'first bulk evidence' rests on the unproven-in-main-text identity in Eq. (4) and on an unquantified PT balance, with no error bars on the measured winding. These are fixable with additional analysis and calibration data, so I recommend major revision rather than rejection. I would also gently note that the phrase 'unambiguous observation' in the abstract is stronger than the data in Fig. 5(b) support, given the noted deviation for later times; toning this down would improve accuracy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers something real: the first direct bulk measurement of momentum-band topology. The authors reconstruct the effective Hamiltonian from two injected wavefunctions, resolve the band structure and Floquet exceptional-point momenta, and read a 0/pi Berry phase from the PT-eigenvalue winding across the energy Brillouin zone. That is a genuine step beyond Ren et al. and Feis et al., which observed temporal interface states but not the bulk invariant. The reconstruction method is elegant, the control experiment with a trivial temporal interface is a good check, and the data in Fig. 4(c) look consistent with theory.\n\nThe soft spots are real but addressable. The stress-test is right that Eq. (4) is not derived in the main text. For a non-Hermitian effective Hamiltonian, the equivalence between the standard Berry connection and half the PT-eigenvalue winding is not a definition; it requires the PT-symmetric normalization and exact balance between gain and loss. The paper gives no error bars on the measured eigenstates or phase angles, and no measured value of the PT imbalance. The wording \"first bulk evidence\" and \"conclusive verification\" is stronger than the presented uncertainty justifies. Also, no raw data are released. These are not fatal, but they are exactly what a referee should probe.\n\nThe central claim is probably right. The band inversion, the 2pi winding for one sign of gamma and zero for the other, and the temporal TIM all hang together. What is missing is a quantitative statement of how precisely the PT symmetry is realized and how robust the quantized Berry phase is to the residual imbalance.\n\nThis deserves a serious referee. It is a solid experimental advance in a fast-moving area. The revision should move the derivation of Eq. (4) into the main text or at least state its assumptions explicitly, and it should include error bars on the eigenstate phases and the Berry phase. I would take it to reading group either way, and I would cite it if I worked on momentum-gap physics.","headline":"A credible first bulk measurement of momentum-band topology, but the quantized Berry-phase claim needs the missing error analysis and the deferred Eq. (4) derivation.","tokens_in":638,"tokens_out":2668,"would_cite":true,"duration_ms":38964,"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":"This paper reports the first bulk experimental evidence of momentum-band topology, measured in a PT-symmetric acoustic Floquet lattice by reconstructing its effective Hamiltonian and detecting a quantized energy-Brillouin-zone Berry phase.","keywords":["momentum-band topology","PT-symmetric Floquet lattice","energy Brillouin zone","Berry phase","exceptional points","temporal interface states","acoustic circuits","Hamiltonian reconstruction"],"falsifier":"Detune the positive and negative feedback gains by about 10% while reconstructing the effective Hamiltonian; if the measured energy-Brillouin-zone Berry phase leaves the quantized set $\\{0,\\pi\\}$ or the Floquet exceptional-point momenta stop swapping exactly with the sign of $\\gamma$, the PT-balance assumption is the load-bearing condition that fails.","tokens_in":10393,"feed_emoji":"🔊","tokens_out":7927,"duration_ms":78648,"temperature":0.7,"pith_summary":"The paper sets out to prove that momentum-band topology—a band structure defined over energy rather than over crystal momentum—can be observed directly in the bulk of a laboratory system, not merely inferred from boundary states. It builds a PT-symmetric Floquet lattice from coupled acoustic cavities and external feedback circuits that supply balanced gain and loss, reconstructs the effective Hamiltonian from measured sound responses, and finds a Floquet momentum gap whose bulk invariant, an energy-Brillouin-zone Berry phase, equals $\\pi$ for one sign of the dynamic gain/loss and $0$ for the other. It also creates a temporal interface between the two phases and observes a time-localized interface state, the temporal counterpart of an edge state. If correct, the result establishes the energy-Brillouin-zone Berry phase as a measurable bulk invariant for momentum gaps and supplies a general reconstruction method for Floquet systems.","feed_headline":"First bulk proof of momentum-band topology in acoustic lattice","feed_subtitle":"Reconstructing the Floquet Hamiltonian yields a quantized Berry phase and time-localized interface states.","key_machinery":"The load-bearing object is the effective Hamiltonian $H_{\\mathrm{eff}}=(i2\\pi T)^{-1}\\ln U$ obtained from the Floquet operator $U$, with $U$ reconstructed experimentally from two measured wavefunctions via $U=[\\psi_1(t_0+T),\\psi_2(t_0+T)][\\psi_1(t_0),\\psi_2(t_0)]^{-1}$. The topological invariant is the energy-Brillouin-zone Berry phase $\\theta_E=i\\oint \\langle\\psi(E)|\\partial_E|\\psi(E)\\rangle\\,dE$, which the paper rewrites, in the PT-symmetric real-energy sector, as half the winding of the PT eigenvalue phase $\\varphi(E)$ around the origin of the complex plane; that winding is what the experiment actually records. The momentum-band inversion appears as an exchange of the two Floquet exceptional-point momenta $k_1$ and $k_2$ as $\\gamma$ crosses zero, and the temporal interface state is captured by a momentum-space Dirac model $M(\\delta E)=v_D^{-1}(m\\sigma_y-\\delta E\\,\\sigma_z)$, with Dirac mass $m$ proportional to $\\gamma$.","core_discovery":"The central claim is that in a PT-symmetric Floquet lattice with balanced static and dynamic gain/loss, periodically driving the gain/loss opens a momentum gap at the Floquet Dirac point, and the topology of that gap is carried by the eigenstates across the energy Brillouin zone. The paper reports direct experimental reconstruction of the $2\\times 2$ effective Hamiltonian $H_{\\mathrm{eff}}=(i2\\pi T)^{-1}\\ln U$ from two independent sound responses, extracts the eigenstates, and shows that the two Floquet exceptional-point states exchange their momenta when the sign of the dynamic gain/loss parameter $\\gamma$ is reversed. It then evaluates the energy-Brillouin-zone Berry phase through the winding of the PT eigenvalue phase and finds $\\theta_E=\\pi$ for $\\gamma>0$ and $\\theta_E=0$ for $\\gamma<0$, which it identifies as nontrivial versus trivial momentum-band topology. Finally, it creates a temporal interface between the two signs and observes sound intensity that first grows and then decays, localized at the interface in time, which is the temporal counterpart of an edge state.","pith_inferences":["The Hamiltonian-reconstruction method should transfer to any Floquet system with a small Hilbert space and controllable inputs, such as photonic, mechanical, or electrical circuits, as long as two linearly independent initial states can be prepared and read out.","The quantized Berry phase suggests a temporal Thouless pumping invariant; because time evolution is causal, temporal pumps may behave differently from spatial ones, a direction the paper flags but does not develop.","A natural test is to replace the square-wave drive with a sinusoidal modulation and check whether $\\theta_E$ stays quantized or a different invariant is needed.","Because the degeneracies are exceptional points in the complex quasi-energy spectrum, the platform opens a route to observing exceptional-point braiding in the time domain."],"forward_implications":["The energy-Brillouin-zone Berry phase can serve as a measurable bulk invariant for PT-symmetric Floquet momentum gaps, so momentum-band topology no longer has to be inferred only from temporal boundary states.","Temporal bulk-boundary correspondence holds in this acoustic lattice: a time interface between opposite signs of $\\gamma$ hosts a time-localized interface state, whereas an interface between two trivial configurations does not.","Reversing the sign of the dynamic gain/loss parameter swaps the momenta of the two Floquet exceptional points, giving a clean experimental signature of momentum-band inversion.","Reconstructing $H_{\\mathrm{eff}}$ from two independent pulsed wavefunctions is a viable route to quasi-energy spectra and eigenstates in small Floquet systems.","The same tunable acoustic platform can be extended to temporal quasicrystals, time-domain Anderson effects, and temporal Moiré superlattices."],"supporting_citations":[{"why":"Defines momentum-band topology in photonic time crystals, the concept the experiment is designed to verify.","marker":"[25]"},{"why":"Establishes the Berry phase for energy bands, the prototype of the energy-Brillouin-zone Berry phase measured here.","marker":"[12]"},{"why":"Identifies band inversion as a topological phase-transition signature, the energy-band analogue the paper maps to momentum.","marker":"[42]"},{"why":"Observed momentum-gap topology at temporal interfaces in a time-synthetic lattice, the boundary-level evidence this work complements with bulk measurements.","marker":"[38]"},{"why":"Realized acoustic Floquet modes in a time-varying lattice, providing the circuit-controlled time-varying acoustic platform used here.","marker":"[50]"},{"why":"Demonstrated acoustic non-Hermitian behavior from feedback circuits, grounding the gain/loss implementation in the experiment.","marker":"[47]"}],"fun_headline_variants":["First bulk evidence of momentum-band topology in acoustic lattice","Acoustic experiment reveals momentum-gap Berry phase","Sound waves confirm momentum-band topology in Floquet lattice","Time-localized interface states sign momentum-band topology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the gain on cavity 1 and the loss on cavity 2 are exactly balanced and remain PT-symmetric throughout the modulation; if the feedback amplitudes drift apart, the phase winding that produces the quantized Berry phase can smear, and the claimed topological invariant is no longer protected.","fun_headline_variants_meta":{"raw":{"variants":["First bulk evidence of momentum-band topology in acoustic lattice","Acoustic experiment reveals momentum-gap Berry phase","Sound waves confirm momentum-band topology in Floquet lattice","Time-localized interface states sign momentum-band topology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000708,"raw_usage":{"total_tokens":3192,"prompt_tokens":954,"completion_tokens":2238,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":2176}},"tokens_in":570,"tokens_out":2238,"duration_ms":21398,"temperature":1.0,"reasoning_tokens":2176,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:57:10.154258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Detune the positive and negative feedback gains by about 10% while reconstructing the effective Hamiltonian; if the measured energy-Brillouin-zone Berry phase leaves the quantized set $\\{0,\\pi\\}$ or the Floquet exceptional-point momenta stop swapping exactly with the sign of $\\gamma$, the PT-balance assumption is the load-bearing condition that fails.","supporting_citations":[{"cited_title":"Lustig, Y","cited_arxiv_id":null,"evidence_quote":"Defines momentum-band topology in photonic time crystals, the concept the experiment is designed to verify."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies band inversion as a topological phase-transition signature, the energy-band analogue the paper maps to momentum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Observed momentum-gap topology at temporal interfaces in a time-synthetic lattice, the boundary-level evidence this work complements with bulk measurements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Realized acoustic Floquet modes in a time-varying lattice, providing the circuit-controlled time-varying acoustic platform used here."},{"cited_title":"Zhang, Y","cited_arxiv_id":null,"evidence_quote":"Demonstrated acoustic non-Hermitian behavior from feedback circuits, grounding the gain/loss implementation in the experiment."}],"review_version":1}