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REVIEW 5 major objections 6 minor 43 references

Direct Measurement of Zak Phase and Higher Winding Numbers in an Electroacoustic Cavity System

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An electroacoustic cavity experiment measures the Zak phase directly, reporting a quantized $\pi$ for winding number 1 and a $2\pi$ phase for winding number 2, by canceling dynamical phases along two symmetric evolution paths.

desk verdict Real W=2 electroacoustic Zak-phase data, but the phase-extraction derivation needs a fix before this is solid. read the letter →

arxiv 2505.21131 v1 pith:D4TC3WQN submitted 2025-05-27 quant-ph cond-mat.other

Guang-Chen He , Zhao-Xian Chen , Xiao-Meng Zhang , Ze-Guo Chen , Ming-Hui Lu This is my paper
classification quant-phcond-mat.other
keywords ZakphaseSu-Schrieffer-HeegermodelwindingnumbertopologicalinvariantsadiabaticevolutionelectroacousticcavityBerrynext-nearest-neighborcoupling
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

The paper reports an experimental method to read the Zak phase—the 1D Berry phase that labels topological phases—directly from the bulk dynamics of an electroacoustic cavity system, without relying on boundary states. The key move is a dimensional extension that maps the complex two-band SSH Hamiltonian onto a real four-level time-dependent Hamiltonian realized by voltage-controlled acoustic cavities. The authors show that the phase difference accumulated along two symmetric adiabatic paths cancels the dynamical phase and leaves the geometric Zak phase. They measure a quantized $\pi$ phase in the standard SSH model and a $2\pi$ phase in the extended model with next-nearest-neighbor coupling, corresponding to winding numbers 1 and 2. If correct, the approach provides a direct, platform-generic way to probe bulk topological invariants in classical wave systems.

What carries the argument

The load-bearing object is the 'dimensional extension' map that sends the complex $2\times2$ Hamiltonian $H(t) = d_x(t)\sigma_x + d_y(t)\sigma_y$ to a real $4\times4$ matrix by replacing the imaginary unit with a real antisymmetric block, so that the real and imaginary parts of the complex eigenstate become independent real components. This conversion turns the complex phase factor $e^{i\theta(k)}$ of the off-diagonal coupling into real coupling amplitudes $\pm d_x, \pm d_y$ that can be implemented with voltage-controlled amplifiers in the cavity circuit. The second mechanism is two-path interference: evolving the same initial state along two symmetric paths (Path A and Path B) in momentum–time space makes the path-dependent dynamical phases equal, so their difference isolates the geometric phase. Together these allow the Zak phase accumulated over the Brillouin zone to be read out as a phase difference between cavity signals.

What would settle it

Numerically integrate the full time-dependent Schrödinger equation for the $4\times4$ real Hamiltonian used in the experiment and compute the exact phase difference between Path A and Path B for a range of evolution times $T$ and loss rates. If the difference deviates from the Zak phase as $T$ is shortened or as path asymmetry is introduced, the cancellation of dynamical phases is not exact and the measured phase contains dynamical contamination. Alternatively, run the experiment with deliberately unequal path durations or unequal loss on the two paths and check whether the extracted phase shifts away from the quantized values.

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Extended reading notes

Core claim

The central claim is that the relative phase difference between two symmetric adiabatic evolutions in a four-level electroacoustic cavity directly equals the Zak phase of the corresponding SSH Hamiltonian. The experiment implements the complex coupling $w + v e^{-ik}$ as real, time-dependent coupling amplitudes $d_x(t) = w + v\cos(k(t))$ and $d_y(t) = v\sin(k(t))$ in a real $4\times4$ matrix, whose eigenstates are the real and imaginary parts of the original complex eigenstates. By preparing identical initial states and evolving them along two mirror-symmetric paths in parameter space, the dynamical phases cancel and the measured phase difference accumulates the geometric phase. In the standard SSH model the authors observe a phase difference of $\pi$ in the topologically nontrivial regime and zero in the trivial regime; in the extended SSH model with next-nearest-neighbor coupling they observe phase accumulation up to $2\pi$, which they associate with winding number 2. The paper argues that this constitutes a direct measurement of bulk topological invariants rather than an inference from boundary states.

Load-bearing premise

The load-bearing premise is that the two symmetric evolution paths acquire exactly equal dynamical phases, so that subtracting them isolates the geometric phase; the paper asserts this cancellation without a rigorous derivation or an error budget for path-dependent dynamical contributions.

Editorial extensions

If this is right

  • Bulk topological invariants can be measured directly in classical wave systems without preparing or probing boundary states.
  • The measured geometric phase scales with the winding number: $\pi$ for winding 1 and $2\pi$ for winding 2 in the extended SSH model.
  • The dimensional-extension construction applies to any two-band Hamiltonian whose off-diagonal coupling traces a closed curve in the complex plane, so the scheme is not restricted to nearest-neighbor couplings.
  • The electroacoustic platform, with gain feedback and voltage-controlled modulation, provides enough coherence to track adiabatic geometric phase accumulation over the full evolution.
  • The same framework can be extended to higher-dimensional, non-Hermitian, or long-range coupled topological models, as the paper states.

Reading between the lines

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

  • The symmetry-based cancellation of dynamical phases is the step most worth stress-testing: a direct numerical integration of the full four-level Schrödinger equation, comparing the exact phase difference with the Zak phase as a function of evolution time $T$, would reveal whether residual dynamical contamination appears at finite $T$.
  • Since the measured phase in the $W=2$ case accumulates continuously through $2\pi$ rather than wrapping modulo $2\pi$, the readout tracks the argument of the complex off-diagonal coupling $q(k)$; a natural test is whether the final phase continues to grow if the path winds twice in the opposite direction.
  • The real-matrix embedding doubles the eigenvalue multiplicity, so the experiment effectively probes a symmetrized four-level system; one could exploit the same embedding to measure non-Abelian Berry phases or Wilson loops in multi-band models.
  • The scheme could be transferred to other classical platforms such as mechanical or circuit lattices by substituting the electroacoustic cavities with the corresponding tunable resonators.
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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

5 major / 6 minor

Summary. The manuscript reports an experimental method for direct measurement of the Zak phase in an electroacoustic cavity system. The SSH Hamiltonian is implemented through a time-dependent real 4x4 matrix obtained by a dimensional-extension mapping from the complex 2x2 Hamiltonian. Initial states are evolved along two symmetric paths in parameter space, and the phase difference between the paths is claimed to isolate the geometric phase because dynamical phases cancel by symmetry. Measurements for the standard SSH model yield phase differences near 0 or pi for the trivial and nontrivial regimes, and for the extended SSH model with next-nearest-neighbor coupling the measured accumulated phase reaches approximately 0, pi, and 2pi for winding numbers W=0,1,2. The authors conclude that this constitutes a direct measurement of bulk topological invariants without relying on boundary states.

Significance. If the central cancellation claim is correct, the experiment provides a clean platform for direct bulk topological-invariant measurement in classical wave systems, with the advantage of not requiring boundary-state inference. The paper's strengths include the absence of fitted parameters in the phase comparisons, the use of three distinct winding regimes predicted from the implemented Hamiltonian, and the clear extension to a next-nearest-neighbor model. However, the theoretical derivation of the two-path dynamical-phase cancellation is not rigorously presented, and Eq. (5) contains a state-evolution error that is load-bearing for the phase-extraction logic. The significance is potentially high, but the manuscript needs a substantial revision to make the derivation sound and the claims precise.

major comments (5)
  1. [Zak Phase Measurement of the SSH Model, Eq. (5)] Equation (5) is not the correct adiabatic evolution formula. For an initial eigenstate |Psi_n(0)> = |u_n(k(0))>, the standard adiabatic theorem gives |Psi_n(t)> = e^{i phi_g(t)} e^{-i integral E_n(t') dt'} |u_n(k(t))>, where |u_n(k(t))> is the instantaneous eigenstate. The manuscript writes |Psi_n(0)> instead of |u_n(k(t))>. This is not a typographical issue alone: the subsequent two-path phase-difference argument compares phases at a common final time, and without the instantaneous-eigenstate factor the comparison is ill-defined. The authors should correct Eq. (5), define the reference state at each time, and explain how the experimental signal is related to that reference.
  2. [Eq. (5) and Fig. 1(d)] The claim that 'symmetry ensures the cancellation of dynamical phases' is asserted rather than proven for the actual Hamiltonian used in the experiment. The dimensional extension produces a real 4x4 Hamiltonian whose eigenvalues are duplicated, so the state evolves in degenerate eigenspaces. In a degenerate subspace, adiabatic evolution generically involves non-Abelian holonomies, and the simple U(1) phase in Eq. (5) is not well-defined without an additional argument. The authors need to provide explicit Path A and Path B definitions, show that the dynamical-phase integrals along the two paths are exactly equal (or bound their difference), and prove that the degenerate subspace evolution reduces to an Abelian phase for the initial states and parameters actually used. A reference to the Supplementary Material is insufficient unless that derivation is included in the review material.
  3. [Experimental Observation of the Zak Phase, initial state preparation] The manuscript states that the initial state [1 1 1 1]^T is prepared and evolved. This vector must be shown to be an eigenstate of the real 4x4 Hamiltonian at t=0, or at least to lie entirely within one degenerate eigenspace corresponding to a single band of the original SSH Hamiltonian. If [1 1 1 1]^T is a superposition of different energy eigenstates, the system does not undergo adiabatic evolution of a single eigenstate, and the phase measured from the time-domain signal would mix bands. Please provide the explicit eigenvalue decomposition of H(0) for the experimental parameters and verify that the initial state satisfies the adiabatic-evolution premise.
  4. [Zak Phase Measurement of the SSH Model with Next-Nearest-Neighbor Coupling] The Zak phase is conventionally defined modulo 2pi, so a W=2 loop gives an accumulated geometric phase of 2pi, which is equivalent to zero as a Zak phase. The paper presents the observation of 2pi as 'phase doubling' and as evidence for higher winding numbers. This is valid only if the claim is about the continuously accumulated phase along the prescribed time-dependent loop in parameter space, not about a distinct quantized value of the Zak phase modulo 2pi. The text should state explicitly that the measured quantity is the continuous geometric phase 2pi W accumulated along the chosen loop, and should clarify in what sense this goes beyond the conventional mod-2pi Zak phase.
  5. [Fig. 3(b) and Fig. 4(e-g)] The agreement between theory and experiment is presented visually without error bars, numbers of repetitions, or quantitative residuals. Since the central conclusion relies on the phase difference being close to 0, pi, or 2pi, the manuscript should provide statistical measures (e.g., standard errors over repeated runs, confidence intervals for the final phase, or a chi-square comparison) to support the statements 'robust' and 'excellent agreement'. Without such quantification, the claim that the observed values are quantized cannot be fully assessed.
minor comments (6)
  1. [Eq. (3) and Eq. (7)] The matrix displays for H(t) and H'(t) are garbled in the typeset version; please ensure that the 4x4 block form is unambiguous and matches the block representation in Eq. (13).
  2. [Methods, Eq. (10)] The notation sigma_y' = i sigma_y is confusing because sigma_y' is claimed to be a real matrix; please rewrite this step with an explicit real matrix definition and avoid mixing i into the real-formalism derivation.
  3. [Methods, Eq. (14)] The basis transformation P is displayed in a malformed way; please specify the 4x4 permutation matrix explicitly, for example in a standard bracket format, so that the relation between H(t) and H'(t) can be checked.
  4. [Throughout] There are several OCR-like artifacts in the formulas, such as 'c c J s' in Eq. (6) and inconsistent use of bold or calligraphic symbols for H(t). A careful proofreading pass is needed.
  5. [Supplementary Material] The paper repeatedly refers to the Supplementary Material for the derivation of dynamical-phase cancellation. Since this derivation is load-bearing, it should be either included in the main text or made available as a clearly referenced appendix in the review version.
  6. [Data and code availability] The code availability statement says code is available upon request; for a measurement-based claim with no fitting parameters, depositing the analysis code and raw data in a public repository would strengthen reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the measured phase is compared against an unfitted analytic prediction, and the W=2 measurement is an independent extension.

full rationale

The derivation chain is self-contained. Equations (1)-(4) and the Methods (Eqs. (8)-(14)) give an exact realification of the complex SSH Hamiltonian into the 4-level real Hamiltonian; the eigenstate and eigenvalue correspondence is proven there, so the Berry phase of the complex model is faithfully represented without fitting. The experimental observable is the relative phase between two symmetric adiabatic paths, and the paper compares it to the analytic Zak phase (Eq. (2)) and winding number without fitting any parameter to the phase data. The W=2 result (Eqs. (6)-(7)) is a distinct prediction of the same construction, and observing a continuous 2pi accumulation is a nontrivial, falsifiable outcome. The assertion that dynamical phases cancel between the two paths is an adiabaticity/symmetry assumption; if wrong it would be a systematic error, but it is not a circular reduction because the paths are not defined to reproduce the target phase and the measured values are checked against theory rather than imposed by construction. The self-citation to Ref. 21 is prior peer-reviewed experimental work and, even if the measurement scheme is related, it does not supply the W=2 claim or the specific electroacoustic data reported here.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; the auxiliary levels in the dimensional extension are a mathematical and experimental construction rather than a postulated particle or force. The central claim rests on control parameters (couplings, evolution time, initial state) and on physical assumptions about adiabaticity, phase cancellation, and the dominance of the designed four-level subspace.

free parameters (3)
  • Coupling parameters (w, v, J) = e.g. (1,5), (5,1); (4,1,1), (1,4,1), (1,1,4)
    Chosen by hand to place the system in winding-number phases W=0,1,2. They define the implemented Hamiltonian and therefore the expected phase, but they are not fitted to the phase data.
  • Total evolution time T = 0.5 s
    Chosen as a compromise between adiabatic following and acoustic losses. The validity of the adiabatic phase extraction depends on this choice.
  • Initial state preparation = [1,1,1,1]^T
    The prepared initial state is chosen to be an eigenstate of the initial Hamiltonian; the phase measurement depends on this preparation.
assumptions (7)
  • domain assumption The adiabatic theorem applies: a slowly varying Hamiltonian keeps the state in the instantaneous eigenstate of the time-dependent Hamiltonian.
    The phase extraction in Eq. (5) and Fig. 3(a) depends on adiabatic following; the paper chooses T=0.5 s to balance this against losses but does not quantify diabatic errors.
  • standard math The complex-to-real dimensional extension preserves the Zak phase accumulated by the corresponding eigenstates.
    The Methods (Eqs. 8-15) asserts an isomorphism between the complex two-band Hamiltonian and the real four-level matrix, and states that phase information is preserved, but the proof is sketched rather than complete.
  • domain assumption The two symmetric paths have identical dynamical-phase contributions, so the relative phase difference isolates the geometric phase.
    This is the core of the AB-inspired cancellation in Fig. 1(d); the paper does not derive the equality of dynamical phases for the full four-level system.
  • domain assumption The prepared initial state [1,1,1,1]^T is an eigenstate of the initial Hamiltonian.
    Adiabatic following requires starting in an eigenstate; the paper states the preparation but does not show the eigenvalue check or population analysis.
  • domain assumption The active gain circuit and phase shifter do not introduce uncontrolled phase offsets that could be mistaken for geometric phase.
    The Methods describes compensation of phase mismatches, but no independent calibration or stability data are provided.
  • domain assumption The four-cavity system is accurately described by the designed four-level real Hamiltonian; other acoustic modes are negligible.
    The experiment uses first-order cavity modes near f0=1955 Hz and ignores higher modes; no modal purity measurement is presented.
  • standard math The winding number for the extended SSH model is computed from q(k)=w+v e^{ik}+J e^{i2k} using the standard argument principle.
    Used to define W=0,1,2 in Fig. 4(c); follows from cited references 42 and 43, but the phase diagram realized in hardware is not independently verified.

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Pith. "Pith review of Direct Measurement of Zak Phase and Higher Winding Numbers in an Electroacoustic Cavity System." pith.science (2026). https://pith.science/paper/D4TC3WQN

@misc{pith2026250521131,
  author       = {Pith},
  title        = {Pith review of: Direct Measurement of Zak Phase and Higher Winding Numbers in an Electroacoustic Cavity System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4TC3WQN}},
  note         = {Machine review of arXiv:2505.21131}
}
read the original abstract

Topological phases are states of matter defined by global topological invariants that remain invariant under adiabatic parameter variations, provided no topological phase transition occurs. This endows them with intrinsic robustness against local perturbations. Experimentally, these phases are often identified indirectly by observing robust boundary states, protected by the bulk-boundary correspondence. Here, we propose an experimental method for the direct measurement of topological invariants via adiabatic state evolution in electroacoustic coupled resonators, where time-dependent cavity modes effectively emulate the bulk wavefunction of a periodic system. Under varying external driving fields, specially prepared initial states evolve along distinct parameter-space paths. By tracking the relative phase differences among states along these trajectories, we successfully observe the quantized Zak phase in both the conventional Su-Schrieffer-Heeger (SSH) model and its extension incorporating with next-nearest-neighbor coupling. This approach provides compelling experimental evidence for the precise identification of topological invariants and can be extended to more complex topological systems.

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