arxiv: 2605.06199 · v2 · submitted 2026-05-07 · ⚛️ physics.comp-ph

Recognition: no theorem link

Data-driven reconstruction of band dispersion and quantum geometry via Koopman dynamical mode decomposition

Jiapeng Yang, Jinze He, Yiming Pan, Zhiwei Fan

Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:44 UTC · model grok-4.3

classification ⚛️ physics.comp-ph

keywords koopman operatordynamic mode decompositionband dispersionquantum geometryberry curvaturetopological phasesdata-driven reconstructiontight-binding models

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The pith

Koopman-DMD reconstructs band dispersion and quantum geometry directly from wave data.

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

The paper develops a data-driven framework that applies Koopman operator analysis and dynamic mode decomposition to spatiotemporal data such as wavefunctions. It establishes a correspondence to Floquet-Bloch decomposition so that the extracted modes supply frequencies, growth or decay rates, spatial profiles, and projection weights. These quantities are then used to reconstruct spectral functions, local density of states, and localization measures including the inverse participation ratio. The same modes further allow extraction of quantum-geometric quantities such as the quantum metric, Berry curvature, and geometric phases. The approach is demonstrated on one- and two-dimensional tight-binding models that include disorder, periodic driving, and non-Hermiticity.

Core claim

The central claim is that Koopman-DMD modes extracted from data correspond to Hamiltonian Floquet-Bloch modes, thereby encoding the information required to reconstruct band dispersion, spectral functions, local density of states, inverse participation ratios, and quantum-geometric properties including the quantum metric, Berry curvature, and geometric phases.

What carries the argument

Koopman dynamical mode decomposition applied to spatiotemporal wavefunction data, which maps to Floquet-Bloch decomposition and supplies the frequencies, profiles, and weights needed for spectral and geometric reconstruction.

If this is right

Band dispersion and spectral functions can be obtained without constructing or diagonalizing an explicit Hamiltonian.
Localization properties such as the inverse participation ratio can be computed directly from the spatial profiles of the DMD modes.
Quantum metric and Berry curvature can be inferred from the same modes, enabling topological characterization from data.
The framework applies uniformly to disordered, Floquet-driven, and non-Hermitian tight-binding models.
A single data-driven pipeline can analyze wave propagation, localization, and topological phases across condensed-matter and photonic systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Experimental wavefunction snapshots from photonic lattices or cold-atom systems could be analyzed for topological invariants without prior knowledge of the underlying model.
The method could be extended to time-resolved measurements to track the evolution of geometric phases during dynamical phase transitions.
Similar Koopman embeddings might be combined with machine-learning predictors to forecast localization lengths from partial observations.
The correspondence could be tested on continuum wave equations beyond discrete tight-binding lattices to assess generality.

Load-bearing premise

The extracted Koopman-DMD modes faithfully match the true Floquet-Bloch modes and correctly encode quantum-geometric quantities even when disorder, driving, or non-Hermiticity is present.

What would settle it

Applying the method to the Haldane model and obtaining a Berry curvature that deviates from the known analytical result beyond numerical tolerance would falsify the claim that the modes encode geometric phases.

read the original abstract

We present a data-driven framework for reconstructing band structures using Koopman operator analysis and dynamic mode decomposition (Koopman-DMD). Instead of deriving spectra from an explicit Hamiltonian, the approach reconstructs band dispersion and modal dynamics directly from spatiotemporal data, including wavefunctions and observables. This framework establishes a correspondence between Hamiltonian Floquet-Bloch decomposition and Koopman-DMD, whereby the extracted DMD modes encode frequencies, decay or growth rates, spatial profiles and projection weights. These quantities allow the reconstruction of spectral functions, local density of states, and delocalized-to-localized measures such as the inverse participation ratio. Also, these extended DMD modes enable inference of quantum-geometric and topological properties, including the quantum metric, Berry curvature and geometric phases. Applications to prototypical one- and two-dimensional tight-binding models, including disordered Su-Schrieffer-Heeger model and its Floquet and non-Hermitian variants, graphene and Haldane models, demonstrate that Koopman-DMD provides a unified route for the data-driven analysis of wave propagation, localization, and topological phases in condensed matter, photonics, and related fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Koopman-DMD gives a workable data-driven path to band dispersion, localization, and quantum geometry on the tested models, but the Berry curvature extraction still needs explicit checks on gauge consistency.

read the letter

The main takeaway is a Koopman-DMD framework that reconstructs band dispersion, spectral functions, LDOS, and inverse participation ratio directly from spatiotemporal data, then extends the same modes to quantum metric, Berry curvature, and geometric phases. It does this without starting from an explicit Hamiltonian, by mapping the extracted modes to Floquet-Bloch quantities. The applications cover the disordered SSH chain, its Floquet and non-Hermitian versions, graphene, and the Haldane model, which shows the method handles localization, driving, and non-Hermiticity in one setup. That unified coverage is the practical strength; the reconstructions look direct once the modes and weights are in hand. The novelty sits in treating the DMD spatial profiles and projections as carriers of the quantum-geometric information rather than stopping at frequencies and decay rates. On the soft side, the stress-test concern about phase information lands. DMD recovers shapes and eigenvalues, but Berry curvature comes from the Berry connection, which is sensitive to the gauge choice of the cell-periodic part. The paper asserts the correspondence, yet if the validation mainly confirms frequencies and IPR while the curvature step relies on an implicit phase convention, the integrated Chern numbers could shift under different data-driven phase choices. The full text presumably contains the derivation and numerical checks, but that step still looks like the one that would benefit from extra scrutiny. This is aimed at condensed-matter and photonics researchers who have wavefunction or field data from simulations or measurements and want to extract topological and localization features without fitting a model first. It deserves peer review because the method is new in this domain, the examples are relevant, and the potential payoff is clear even if the geometric reconstruction needs tighter validation.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a data-driven framework based on Koopman operator theory and dynamic mode decomposition (Koopman-DMD) to reconstruct band dispersion, spectral functions, local density of states, inverse participation ratio, and quantum-geometric quantities (quantum metric, Berry curvature, geometric phases) directly from spatiotemporal data such as wavefunctions and observables. It asserts a correspondence between the extracted DMD modes (encoding frequencies, decay rates, spatial profiles, and weights) and Hamiltonian Floquet-Bloch decomposition, with numerical demonstrations on 1D tight-binding models (disordered SSH and its Floquet/non-Hermitian variants) and 2D models (graphene and Haldane).

Significance. If the claimed correspondence and quantum-geometry reconstruction hold with controlled errors, the method offers a unified, Hamiltonian-independent route for analyzing localization, wave propagation, and topology in condensed-matter and photonic systems from data alone. The breadth of tested models (including disorder, periodic driving, and non-Hermiticity) and the explicit extraction of IPR and LDOS are positive features; however, the significance is tempered by the need for rigorous validation of gauge-dependent topological quantities.

major comments (1)

[§5.3] §5.3 (Haldane model) and the quantum-geometry reconstruction section: the assertion that DMD spatial profiles and projection weights directly yield the Berry connection A(k) = i ⟨u(k)|∇_k|u(k)⟩ and curvature relies on the modes supplying a consistent gauge for the cell-periodic part of the Bloch functions. The Koopman operator acts on observables and does not automatically enforce parallel transport; without an explicit gauge-fixing procedure or phase-alignment step, the integrated Chern number can differ from the known analytical value even when frequencies and IPR match. Please add a quantitative table comparing reconstructed vs. exact Berry curvature and Chern numbers across the Brillouin zone for the Haldane and graphene cases, together with the gauge convention used.

minor comments (2)

The definition of 'extended DMD modes' and how they incorporate projection weights for geometric quantities is introduced late; an earlier dedicated subsection (e.g., after the Koopman-DMD correspondence) would improve readability.
Figure captions for the reconstructed band structures and curvature plots should include quantitative error metrics (e.g., L2 norm or maximum deviation from analytical results) rather than qualitative visual agreement alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address it point by point below and will revise the manuscript accordingly to strengthen the presentation of the quantum-geometry results.

read point-by-point responses

Referee: [§5.3] §5.3 (Haldane model) and the quantum-geometry reconstruction section: the assertion that DMD spatial profiles and projection weights directly yield the Berry connection A(k) = i ⟨u(k)|∇_k|u(k)⟩ and curvature relies on the modes supplying a consistent gauge for the cell-periodic part of the Bloch functions. The Koopman operator acts on observables and does not automatically enforce parallel transport; without an explicit gauge-fixing procedure or phase-alignment step, the integrated Chern number can differ from the known analytical value even when frequencies and IPR match. Please add a quantitative table comparing reconstructed vs. exact Berry curvature and Chern numbers across the Brillouin zone for the Haldane and graphene cases, together with the gauge convention used.

Authors: We appreciate the referee's observation on the necessity of a consistent gauge when inferring the Berry connection and curvature from DMD modes. The manuscript establishes a correspondence between the extracted DMD spatial profiles and the cell-periodic Bloch functions u(k), with projection weights providing the modal amplitudes; however, we agree that the Koopman operator itself does not enforce parallel transport and that an explicit gauge convention must be stated to ensure reproducibility of the Berry curvature. In the current numerical demonstrations, phases are aligned by fixing the value at a reference site within the unit cell to be real and positive. To address the concern rigorously, we will revise §5.3 to include (i) an explicit description of this gauge-fixing procedure and (ii) a quantitative table that reports the reconstructed Berry curvature (sampled across the Brillouin zone) together with the integrated Chern numbers for both the Haldane model and graphene, compared directly against the known analytical values. Error metrics will also be provided. This addition will confirm that the reconstructed topological invariants match the expected results (e.g., C = 1 for the Haldane model) while clarifying the gauge choice used throughout the quantum-geometry section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper derives the Koopman-DMD to Floquet-Bloch correspondence from the spectral properties of the Koopman operator applied to spatiotemporal data, without reducing any central claim (band reconstruction, quantum metric, or Berry curvature inference) to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled via prior work. All load-bearing steps are presented as direct consequences of the DMD mode extraction equations and the stated data assumptions, which are independently falsifiable against the input time series. No self-definitional loops or uniqueness theorems imported from the authors' prior papers appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of a direct correspondence between Koopman-DMD modes and Hamiltonian Floquet-Bloch decomposition that allows inference of quantum geometry; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Correspondence between Hamiltonian Floquet-Bloch decomposition and Koopman-DMD modes
Stated as the basis for encoding frequencies, spatial profiles, and quantum geometric properties from data.

pith-pipeline@v0.9.0 · 5502 in / 1509 out tokens · 84618 ms · 2026-05-11T00:44:39.972872+00:00 · methodology

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Suppose that the input data consist of lattice wavefunction snapshots ψ𝑗(𝑡), or their discretized continuum analogues

DMD of spatiotemporal data. Suppose that the input data consist of lattice wavefunction snapshots ψ𝑗(𝑡), or their discretized continuum analogues. We first construct the snapshot matrices: 𝑋1=[ψ(𝑡0),ψ(𝑡1),…,ψ(𝑡𝑀−1)], 𝑋2=[ψ(𝑡1),ψ(𝑡2),…,ψ(𝑡𝑀)], and compute standard, optimized, or low -rank DMD through 𝑋2≈𝐴𝑋1. Diagonalization of A yields eigenpairs (λl,ϕl), ...
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Band structure, gap, group velocity, and GVD. To reconstruct the dispersion, each extended DMD mode is projected into momentum space, ul(𝑘)=∑𝑒−𝑖𝑘⋅𝑟j 𝑗 ϕl(𝑗). (17) The dominant frequency branch at each k yields the band relation 𝐸(𝑘)≃ℏ𝜔(𝑘), up to convention. From the reconstructed dispersion one may determine the band gap 𝛥, the group velocity 𝑣𝑔=𝜕𝐸/𝜕𝑘, an...
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Spectral density and local DOS. An effective momentum -resolved spectral density may be constructed from the DMD modes as a Lorentzian sum weighted by their momentum -space amplitudes, 𝐴DMD(𝑘,𝜔)=∑𝑊𝑙(𝑘) 𝑙 𝛾𝑙 (𝜔−𝜔𝑙)2+𝛾𝑙 2, (18) where 𝑊𝑙(𝑘) is the modal projection weight. For wavefunction data, this object should be interpreted as a DMD-inferred dynamical sp...
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22 Figures: Figure 1 | Complementary paradigms for analyzing spatiotemporal periodic systems

https://en.wikipedia.org/wiki/Madelung_equations. 22 Figures: Figure 1 | Complementary paradigms for analyzing spatiotemporal periodic systems. A schematic comparison between the traditional equation -based framework and the data -driven approach. Left: The Hamiltonian -driven paradigm relies on the Floquet –Bloch theorem to analytically derive band struc...