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arxiv: 2511.03894 · v2 · submitted 2025-11-05 · ❄️ cond-mat.mes-hall · cond-mat.quant-gas· physics.optics

Measuring non-Abelian quantum geometry and topology in a multi-gap photonic lattice

Martin Guillot , C\'edric Blanchard , Martina Morassi , Aristide Lema\^itre , Luc Le Gratiet , Abdelmounaim Harouri , Isabelle Sagnes , Robert-Jan Slager
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F. Nur \"Unal Jacqueline Bloch Sylvain Ravets
This is my paper

Pith reviewed 2026-05-18 00:29 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.quant-gasphysics.optics
keywords non-Abelian quantum geometric tensorphotonic latticemulti-gap systemsEuler curvaturequantum metricorbital-resolved polarimetrynon-Abelian topology
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The pith

Direct measurement of the non-Abelian quantum geometric tensor is achieved via orbital-resolved polarimetry in a six-band photonic lattice.

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

This paper establishes a direct experimental method for measuring the non-Abelian quantum geometric tensor in a multi-gap system. The authors implement an orbital-resolved polarimetry technique on a six-band two-dimensional synthetic photonic lattice to reconstruct the full Bloch Hamiltonian. This provides access to non-Abelian quaternion charges, Euler curvature, and the non-Abelian quantum metric. Such measurements matter because quantum geometry underlies phenomena like superconductivity in flat bands, optical responses, and quantum Hall physics, allowing study of non-Abelian topological phases that were previously inaccessible experimentally.

Core claim

By implementing a novel orbital-resolved polarimetry technique to probe the full Bloch Hamiltonian of a six-band two-dimensional synthetic lattice, we pioneer the direct measurement of the non-Abelian QGT, granting direct experimental access to non-Abelian quaternion charges, the Euler curvature, and the non-Abelian quantum metric associated with all bands.

What carries the argument

The orbital-resolved polarimetry technique for reconstructing the full Bloch Hamiltonian of the six-band lattice, which allows extraction of the non-Abelian quantum geometric tensor.

If this is right

  • Grants direct experimental access to non-Abelian quaternion charges associated with band nodes.
  • Enables measurement of the Euler curvature in multi-gap systems.
  • Provides the non-Abelian quantum metric for all bands in the lattice.
  • Unlocks experimental probing of topological phases at the confluence of topology, geometry, and non-Abelian physics.

Where Pith is reading between the lines

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

  • The same polarimetry approach could be adapted to probe non-Abelian geometry in other synthetic lattices or condensed-matter platforms.
  • Measured values of the non-Abelian metric may help predict optical or transport responses in multi-gap photonic devices.
  • This technique opens a route to test predictions about braiding properties of band singularities in laboratory settings.

Load-bearing premise

The orbital-resolved polarimetry technique accurately reconstructs the full Bloch Hamiltonian of the six-band lattice without significant systematic errors from fabrication imperfections, mode overlap, or calibration assumptions.

What would settle it

A direct comparison showing that the extracted non-Abelian quaternion charges or Euler curvature deviate from theoretical expectations for the known lattice parameters would falsify the reconstruction accuracy.

Figures

Figures reproduced from arXiv: 2511.03894 by Abdelmounaim Harouri, Aristide Lema\^itre, C\'edric Blanchard, F. Nur \"Unal, Isabelle Sagnes, Jacqueline Bloch, Luc Le Gratiet, Martina Morassi, Martin Guillot, Robert-Jan Slager, Sylvain Ravets.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of momentum space braiding in a multi-gap system (two gaps in between three [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Experimentally reconstructed Euler curvature Eu [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: a-d show in color scale the measured φn,n+1(k) for different pairs of bands, in 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

Recent discoveries in semi-metallic multi-gap systems featuring band singularities have galvanized enormous interest in particular due to the emergence of non-Abelian braiding properties of band nodes. This previously uncharted set of topological phases necessitates novel approaches to probe them in laboratories, a pursuit that intricately relates to evaluating non-Abelian generalizations of the Abelian quantum geometric tensor (QGT) that characterizes geometric responses. Here, we pioneer the direct measurement of the non-Abelian QGT. We achieve this by implementing a novel orbital-resolved polarimetry technique to probe the full Bloch Hamiltonian of a six-band two-dimensional (2D) synthetic lattice, which grants direct experimental access to non-Abelian quaternion charges, the Euler curvature, and the non-Abelian quantum metric associated with all bands. Quantum geometry has been highlighted to play a key role on macroscopic phenomena ranging from superconductivity in flat-bands, to optical responses, transport, metrology, and quantum Hall physics. Therefore, our work unlocks the experimental probing of a wide phenomenology of multi-gap systems, at the confluence of topology, geometry and non-Abelian physics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript reports an experimental protocol using orbital-resolved polarimetry in a six-band 2D synthetic photonic lattice to reconstruct the full Bloch Hamiltonian and thereby directly measure the non-Abelian quantum geometric tensor, granting access to non-Abelian quaternion charges, Euler curvature, and the non-Abelian quantum metric for all bands.

Significance. If the Hamiltonian reconstruction is shown to be accurate, the work would be significant for topological photonics and multi-gap systems, as it provides the first claimed direct experimental probe of non-Abelian quantum geometry quantities that govern braiding of band nodes and related responses in superconductivity, optics, and transport. The approach addresses a clear experimental gap in non-Abelian topology.

major comments (2)
  1. [Orbital-resolved polarimetry technique and Hamiltonian reconstruction] The central claim of 'direct' measurement of the non-Abelian QGT rests on the orbital-resolved polarimetry inverting to the exact six-band Bloch Hamiltonian. Potential systematic errors from mode overlap, fabrication imperfections, or calibration assumptions could introduce non-unitary perturbations to the eigenvectors; because non-Abelian quantities are defined via the full Berry connection matrix, such errors can alter quaternion charges or Euler curvature by O(1) amounts. A quantitative fidelity analysis (e.g., via simulated data with known perturbations or cross-validation against Abelian limits) is needed to support the reconstruction accuracy.
  2. [Results and extracted non-Abelian quantities] The manuscript should include explicit comparisons of the extracted non-Abelian quantities (quaternion charges, Euler curvature, quantum metric) to independent theoretical calculations for the specific lattice parameters, including error propagation from the polarimetry data, to demonstrate that the measured values are not dominated by reconstruction artifacts.
minor comments (2)
  1. [Theoretical background] Clarify the precise definition and normalization used for the non-Abelian quantum metric in the multi-band context, and ensure all derived quantities are accompanied by units or dimensionless scaling where appropriate.
  2. [Figures and data presentation] Add error bars or uncertainty estimates to all experimental plots of the reconstructed Hamiltonian components and derived geometric quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the robustness of our Hamiltonian reconstruction and the validation of the extracted non-Abelian quantities. We address each major comment below and have revised the manuscript accordingly to strengthen the supporting evidence.

read point-by-point responses

  1. Referee: [Orbital-resolved polarimetry technique and Hamiltonian reconstruction] The central claim of 'direct' measurement of the non-Abelian QGT rests on the orbital-resolved polarimetry inverting to the exact six-band Bloch Hamiltonian. Potential systematic errors from mode overlap, fabrication imperfections, or calibration assumptions could introduce non-unitary perturbations to the eigenvectors; because non-Abelian quantities are defined via the full Berry connection matrix, such errors can alter quaternion charges or Euler curvature by O(1) amounts. A quantitative fidelity analysis (e.g., via simulated data with known perturbations or cross-validation against Abelian limits) is needed to support the reconstruction accuracy.

    Authors: We agree that a quantitative assessment of reconstruction fidelity is important for supporting the accuracy of the non-Abelian quantities. In the original manuscript we already verified unitarity of the reconstructed eigenvectors and consistency with the designed band dispersions, but we acknowledge that these checks do not fully quantify the impact of realistic perturbations. We have therefore added a new supplementary section containing simulated polarimetry data that incorporates controlled levels of mode overlap, fabrication disorder, and calibration offsets. The reconstructed Hamiltonians are then used to compute the quaternion charges and Euler curvature; the deviations remain below 12% even for perturbation strengths exceeding those estimated from our experimental calibration, and the topological invariants are preserved. We have also included a cross-validation against the Abelian limit by reducing the system to an effective two-band subspace. These results are now presented in the revised manuscript and demonstrate that the non-Abelian features are robust against the cited error sources. revision: yes

  2. Referee: [Results and extracted non-Abelian quantities] The manuscript should include explicit comparisons of the extracted non-Abelian quantities (quaternion charges, Euler curvature, quantum metric) to independent theoretical calculations for the specific lattice parameters, including error propagation from the polarimetry data, to demonstrate that the measured values are not dominated by reconstruction artifacts.

    Authors: We concur that direct, quantitative comparison to theory with propagated uncertainties is necessary to rule out reconstruction artifacts. We have added new panels to Figure 4 (and corresponding supplementary figures) that overlay the experimentally extracted quaternion charges, Euler curvature, and non-Abelian quantum metric against independent tight-binding calculations performed with the exact lattice parameters used in the experiment. Error bars are obtained by propagating the measured intensity uncertainties through the polarimetry inversion and subsequent Berry-connection matrix construction. The measured values agree with theory to within one standard deviation across the Brillouin zone, and the topological features (e.g., the locations and magnitudes of the quaternion charges) remain clearly distinguishable from zero even after accounting for experimental noise. These comparisons are now explicitly discussed in the revised text. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in experimental reconstruction protocol

full rationale

The paper describes an experimental measurement of the non-Abelian quantum geometric tensor via orbital-resolved polarimetry in a synthetic photonic lattice, reconstructing the six-band Bloch Hamiltonian directly from measured data. No derivation chain reduces a claimed prediction or first-principles result to its own inputs by construction, nor does any load-bearing step rely on self-citation of an unverified uniqueness theorem or ansatz. The central result is presented as direct experimental access rather than a fitted or self-referential computation, making the protocol self-contained against external benchmarks of the physical system.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the photonic lattice as a faithful simulator of the target Hamiltonian and on the polarimetry method extracting the geometric tensor without additional modeling assumptions.

axioms (1)
  • domain assumption The synthetic photonic lattice accurately realizes the desired six-band Bloch Hamiltonian.
    Invoked when claiming direct access to the QGT from measured polarimetry data.

pith-pipeline@v0.9.0 · 5775 in / 1136 out tokens · 28155 ms · 2026-05-18T00:29:09.922749+00:00 · methodology

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Forward citations

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  2. Interband Berry connection measurement in the optical honeycomb lattice

    cond-mat.quant-gas 2026-05 unverdicted novelty 6.0

    Ultracold fermionic atoms in a shaken optical honeycomb lattice map the interband Berry connection via polarization-dependent resonant excitations, revealing transparency lines and Dirac strings between K and K' points.

  3. Euler Topology in Superconducting Honeycomb Lattices

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    Superconducting pairings in IST-symmetric honeycomb lattices realize valley-Euler and Euler superconductors with mirror-protected helical domain-wall modes and non-Abelian Dirac-node braiding.

  4. Magnetononlinear Hall effect from multigap topology in metal-organic frameworks

    cond-mat.mes-hall 2026-04 unverdicted novelty 6.0

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