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

This paper presents a data-driven framework using Koopman operator analysis and dynamic mode decomposition to reconstruct band dispersion, spectral functions, and quantum geometric properties directly from spatiotemporal data, offering a unified approach for analyzing wave propagation and topological phases in condensed matter and photonics without requiring an explicit Hamiltonian.

The Gist

Imagine you are trying to understand how a complex machine works, like a giant, invisible orchestra playing a symphony. Usually, to understand the music, you need the sheet music (the equations) and the conductor's score (the Hamiltonian). You have to know every instrument's position and every note before the music starts to predict how it will sound.

This paper proposes a different way. Instead of needing the sheet music, the authors suggest we can figure out the entire song just by listening to the recording of the orchestra playing.

Here is a breakdown of their idea using simple analogies:

1. The Old Way vs. The New Way

• The Old Way (Hamiltonian Floquet-Bloch): This is like trying to predict the weather by knowing the exact physics of every air molecule. You need a perfect model of the system first. If you don't know the exact rules (the equations) or if the system is messy (like a storm with disorder), this method gets stuck or becomes too hard to calculate.
• The New Way (Koopman-DMD): This is like analyzing a video of the storm. You don't need to know the physics of air pressure; you just look at the data (the video frames). The authors use a mathematical tool called Koopman-DMD to take a sequence of snapshots (like frames in a movie) and break them down into their "pure" moving parts.

2. The Magic Tool: DMD (Dynamic Mode Decomposition)

Think of a complex wave in a pond. It looks messy, with ripples going everywhere.

• DMD acts like a prism. When you shine white light through a prism, it splits into pure colors (red, blue, green).
• DMD splits the messy wave into pure "modes." Each mode is a simple, repeating pattern that has a specific speed (frequency) and a specific shape (spatial profile).
• Some of these patterns are extended waves (like a ripple traveling across the whole pond).
• Some are localized waves (like a splash that stays in one spot and fades away).

3. What They Found

The authors tested this "listening only" method on several types of "orchestras" (lattice models) used in physics:

• The Messy Orchestra (Disorder): In a system with random obstacles (like a forest with trees scattered randomly), the old method struggles because the "sheet music" is broken. The new method just looks at how the waves bounce around. It successfully identified that the waves were getting "stuck" in small spots (localization) rather than traveling freely.
• The Topological Orchestra (SSH Model): Some systems have special "edge states"—waves that only travel along the border of the material, like a train staying on a track. The new method found these special edge waves just by watching the data, even when the system was messy or being driven by an external rhythm.
• The 2D Orchestra (Graphene & Haldane): They looked at 2D materials (like a flat sheet of atoms). They could reconstruct the "shape" of the energy bands (the allowed notes the system can play) and even calculate "geometric" properties (how the waves twist and turn in space) without ever writing down the original equations.

4. The Big Picture: "Equation-Free" Physics

The most exciting part of this paper is that it bridges the gap between theory and experiment.

• Theory usually says: "If we build a perfect crystal, here is the math."
• Experiment often says: "Here is a messy, real-world sample. Here is the data we measured."

The authors show that you can take the messy experimental data, run it through their "prism" (Koopman-DMD), and get back the same answers you would get from the perfect math. It's like being able to read the sheet music just by listening to a slightly out-of-tune band playing in a noisy room.

Summary

The paper claims that you don't always need to know the underlying laws of physics (the equations) to understand how a system behaves. If you have enough data (snapshots of the system over time), you can use this data-driven method to:

1. Reconstruct the energy bands (what notes the system can play).
2. Find topological features (special edge states that are robust against noise).
3. Measure localization (where waves get stuck).
4. Calculate geometric properties (how the waves are shaped in space).

They demonstrated this on models of electrons in solids and light in crystals, showing that this "listen to the data" approach works just as well as the traditional "solve the equations" approach, especially when the system is messy, disordered, or too complex to model perfectly.

Technical Summary

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

Problem Statement
Conventional band-structure engineering relies on an equation-based paradigm where a Hamiltonian is constructed from crystal symmetry and material composition, followed by the solution of the Schrödinger (or Maxwell) eigenvalue problem. This approach faces significant limitations in systems where the exact Hamiltonian is unknown, partially constrained, or where translational symmetry is broken by disorder, defects, or strong nonlinearity. Furthermore, periodically driven (Floquet) systems and non-Hermitian systems introduce additional theoretical and numerical complexities, often necessitating computationally expensive real-space supercell calculations. There is a need for a methodology that can extract spectral functions, band dispersions, and topological properties directly from spatiotemporal data without prior knowledge of the governing equations.

Methodology
The paper proposes a data-driven framework utilizing Koopman operator theory and Dynamic Mode Decomposition (Koopman-DMD) to reconstruct band structures and quantum geometric properties. The core methodology involves:

1. Equivalence Formulation: The authors establish a theoretical correspondence between the Hamiltonian Floquet–Bloch decomposition (FBD) and Koopman-DMD. While FBD is equation-based, Koopman-DMD is equation-free and data-driven, linearly embedding nonlinear dynamics into an infinite-dimensional observable space.
2. Data Decomposition: Given spatiotemporal snapshots (e.g., wavefunctions $\psi(x,t)$ or electric fields $E(x,t)$), the method decomposes the data into spatial modes ($\phi_j$), projection weights ($w_j$), and temporal dynamics characterized by complex frequencies ($\omega_j$).
   • Extended Modes: For Bloch-like modes, the dominant momentum $k$ and oscillation frequency $\Re\{\omega\}$ reconstruct the band dispersion $\omega(k)$.
   • Localized Modes: For disordered systems, the method identifies localized modes characterized by decay rates and spatial confinement.
3. Observable vs. State Data: The framework distinguishes between state-level data (wavefunctions), where DMD directly recovers eigenenergies, and observable-level data (densities, currents), where DMD recovers transition frequencies. For the latter, delay-embedded Hankel matrices are used to handle memory effects and non-Markovian dynamics.
4. Reconstruction of Physical Quantities:
   • Spectral Functions & LDOS: Constructed as Lorentzian sums weighted by modal amplitudes.
   • Localization Metrics: The Inverse Participation Ratio (IPR) is calculated from DMD modes to distinguish extended from localized states.
   • Topological Invariants: Using a discrete, gauge-invariant formulation (Fukui–Hatsugai–Suzuki construction), the method computes Zak phases (1D) and Chern numbers (2D) directly from the overlap of extracted DMD bulk modes.
   • Quantum Geometry: The quantum metric and Berry curvature are reconstructed from the fidelity (inner product overlap) between adjacent DMD modes in momentum space.

Key Results
The framework was validated across several prototypical one- and two-dimensional tight-binding models, demonstrating high agreement with exact Hamiltonian FBD results:

• Anderson Localization: In 1D disordered models, Koopman-DMD successfully captured the crossover from extended to localized states. It reproduced the Wigner–Dyson to Poisson level-spacing statistics transition and extracted localization lengths that decreased monotonically with disorder strength, without requiring explicit diagonalization of a disordered Hamiltonian.
• Topological Transitions in SSH Models:
  • Static Disordered SSH: The method tracked the merging of topological zero modes into the Anderson regime as disorder increased, quantified by IPR changes.
  • Floquet SSH: Applied to periodically driven systems, it reconstructed quasienergy spectra and identified robust 0- and $\pi$-modes, capturing their eventual breakdown into Anderson localization.
• Non-Hermitian and Competing Effects: In non-Hermitian SSH models with disorder, the framework disentangled three regimes: robust topological phases, non-Hermitian skin effect (NHSE) dominance (characterized by boundary accumulation), and Anderson dominance (random bulk localization). It successfully reconstructed generalized Brillouin zone deformations.
• 2D Topological Phases:
  • Higher-Order Topological Insulators (2D SSH): The method distinguished bulk, edge, and corner states under open boundary conditions and extracted nontrivial fractional winding numbers and quantum metrics.
  • Graphene and Haldane Models: It reconstructed the Dirac-cone dispersion of pristine graphene and the gapped topological phase of the Haldane model. Crucially, it recovered the quantized Chern number ($C=1$) for the Haldane model and the vanishing total Berry curvature for graphene, while resolving singular enhancements in the quantum metric near Dirac points.

Significance and Claims
The paper claims that Koopman-DMD provides a unified, data-driven route for analyzing wave propagation, localization, and topological phases in condensed matter, photonics, and related fields. Its primary significance lies in:

1. Model Agnosticism: It operates without requiring an explicit Hamiltonian, making it applicable to systems with unknown parameters, strong disorder, or complex driving protocols where traditional analytical or numerical methods struggle.
2. Complementarity: The authors position Koopman-DMD not as a replacement for Hamiltonian FBD, but as a complementary framework. While FBD derives properties from governing equations (top-down), Koopman-DMD infers effective dynamics directly from observation (bottom-up).
3. Direct Extraction of Geometry: It enables the direct inference of quantum geometric properties (Berry curvature, quantum metric) and topological invariants from finite-size, finite-time data, bridging the gap between experimental observables and ideal Bloch-band theory.
4. Versatility: The framework is shown to be effective across diverse physical regimes, including equilibrium, non-equilibrium (Floquet), non-Hermitian, and disordered systems, offering a practical alternative for diagnosing many-body localization and topological transitions where exact diagonalization is computationally prohibitive.

The paper concludes that this approach extends band theory into a form naturally aligned with modern data-driven science, retaining direct physical interpretability while overcoming the limitations of equation-based modeling in complex, real-world systems.