arxiv: 2605.07948 · v1 · submitted 2026-05-08 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn· cond-mat.quant-gas· physics.optics

Recognition: 2 theorem links

· Lean Theorem

Energy-Resolved Quantum Geometry from Stv{r}eda Response: Driven-Dissipative Bosonic Lattices and Disordered Systems

Ana\"is Defossez, Baptiste Bermond, Lucila Peralta Gavensky, Nathan Goldman

Pith reviewed 2026-05-11 03:17 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.dis-nncond-mat.quant-gasphysics.optics

keywords Středa responsequantum geometrydriven-dissipative latticesbosonic systemstopological Anderson insulatorsHall conductivitydensity of statesdisorder

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The pith

Driven-dissipative bosonic lattices give direct access to energy-resolved Středa responses that reveal quantum geometry in topological bands under disorder.

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

The paper establishes that bosonic lattices subject to uniform driving with random phases and uniform loss can measure both the total Středa response, which recovers the quantized Hall conductivity, and its energy-resolved version, which encodes the quantum geometry of Bloch bands. This works because the pumping and loss scheme produces a Lorentzian filter that samples the occupation of eigenmodes at chosen energies, allowing reconstruction of how the density of states changes with magnetic field. A sympathetic reader would care because the method offers a practical route to probe topological invariants and their spectral details in open quantum systems without needing direct transport measurements. The approach is then applied to show how topological bands evolve when strong disorder is added, identifying the geometric features that survive in topological Anderson insulators.

Core claim

We show that driven-dissipative bosonic lattices provide direct access to both the integrated and energy-resolved Středa responses. Our scheme uses controlled pumping with uniform strength and random phases across the lattice, together with uniform loss, to yield a Lorentzian filter of eigenmode occupations. For generic dispersive bands, this enables reconstruction of a coarse-grained energy-resolved Středa response, establishing these platforms as versatile probes of anomalous spectral flow and energy-resolved quantum geometry. As a striking application, we show that this marker elucidates the fate of topological bands under strong disorder, capturing the quantum-geometric structure of the

What carries the argument

The Lorentzian filter of eigenmode occupations generated by uniform-strength random-phase pumping combined with uniform loss, which samples the magnetic-field response of the density of states at specific energies.

If this is right

The platforms become direct probes of anomalous spectral flow in addition to the integrated Chern number.
Topological bands can be tracked through the disorder-driven transition into topological Anderson insulators via their quantum-geometric signatures.
The method supplies a coarse-grained energy-resolved Středa response for any dispersive band that can be prepared in a driven-dissipative bosonic lattice.
Local density measurements under controlled pumping suffice to extract both integrated and spectrally resolved topological markers.

Where Pith is reading between the lines

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

The same pumping-loss filter could be adapted to extract other energy-resolved geometric quantities such as the quantum metric tensor in non-Hermitian or open settings.
Cold-atom or photonic realizations of the scheme would allow direct comparison of the measured Středa marker against exact diagonalization of finite disordered samples.
The approach suggests a route to measure spectral flow in systems where conventional Hall conductivity is inaccessible due to dissipation or particle loss.

Load-bearing premise

The Lorentzian filter created by uniform pumping and loss accurately reconstructs the energy-resolved magnetic response of the density of states for generic dispersive bands, even when strong disorder is present.

What would settle it

A numerical or experimental check in which the reconstructed energy-resolved Středa marker fails to match the independently computed quantum metric or Berry curvature integrated over a known clean topological band or a disordered Anderson insulator.

Figures

Figures reproduced from arXiv: 2605.07948 by Ana\"is Defossez, Baptiste Bermond, Lucila Peralta Gavensky, Nathan Goldman.

**Figure 2.** Figure 2: (a)-(b) Energy spectrum of the Haldane model in a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: (a)-(b) Energy-resolved Stˇreda response evaluated [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Energy spectrum of the Harper-Hofstadter model [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 6.** Figure 6: Finite-size scaling of the integrated Stˇreda response [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Finite magnetic-field effects in the energy-resolved [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Evolution of the LCM [Eq. (F1)] and the ISR as a function of disorder strength W, for the Haldane model of size 51×51 with ϕ = π/2. (a)-(b) The system is initialized in a topologically non-trivial regime with M/t1 = 0 and t2/t1 = 0.2. (a) We present the LCM averaged over bulk subregions of varying size Na (cross marks), compared to the ISR obtained from the bulk DOS (8) with a bulk of size 47×47 (blue dots… view at source ↗

**Figure 9.** Figure 9: Finite-size scaling of the bulk averaged LCM and [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

read the original abstract

The St\v{r}eda formula links the Hall conductivity of an insulator to the magnetic-field response of its particle density, providing a local and universal probe of the topological Chern number. Beyond this quantized response, an energy-resolved St\v{r}eda marker can be defined from the magnetic response of the density of states, revealing detailed features of the quantum geometry of Bloch bands. We show that driven-dissipative bosonic lattices provide direct access to both the integrated and energy-resolved St\v{r}eda responses. Our scheme uses controlled pumping with uniform strength and random phases across the lattice, together with uniform loss, to yield a Lorentzian filter of eigenmode occupations. For generic dispersive bands, this enables reconstruction of a coarse-grained energy-resolved St\v{r}eda response, establishing these platforms as versatile probes of anomalous spectral flow and energy-resolved quantum geometry. As a striking application, we show that this marker elucidates the fate of topological bands under strong disorder, capturing the quantum-geometric structure underlying topological Anderson insulators.

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

The paper gives a concrete protocol for energy-resolved Středa markers in open bosonic lattices and tests it on topological Anderson insulators, but the Lorentzian filter's accuracy under strong localization is the part that still needs checking.

read the letter

The core advance is a driven-dissipative scheme that turns uniform-strength random-phase pumping plus uniform loss into a Lorentzian filter on eigenmode occupations. This lets you read out both the integrated Středa response and a coarse-grained energy-resolved version from the magnetic-field dependence of the steady-state density. For clean dispersive bands the construction looks workable and directly usable in photonic or circuit-QED arrays where you already control gain and loss. The application to topological Anderson insulators is the part that stands out: they use the marker to track how the quantum-geometric content of the bands evolves once disorder localizes the states and the gap closes or reopens. That is a useful target because most existing probes lose resolution once localization sets in. The numerics they show for the disordered case appear to recover the expected topological marker inside the mobility gap, which is the main evidence offered. The soft spot is the filter itself once disorder is strong. Uniform random-phase pumping assumes the modes are still sampled in a way that preserves the Lorentzian shape and width; when states localize, the spatial overlap with the pump changes and level repulsion can shift the effective broadening. The paper derives the filter for the clean limit and then applies the same expression to the disordered spectra, but it is not obvious that the steady-state occupations remain undistorted enough for the reconstruction to stay quantitative. If the supplementary figures include a direct comparison of the filtered response against the exact density-of-states magnetic derivative for several disorder strengths, that would settle it. Otherwise the TAI results rest on an extrapolation that might need a small correction factor. This is the kind of work that belongs in a journal that covers both topological matter and open quantum systems. Experimental groups looking for new ways to map quantum geometry in lossy lattices would get immediate value, and theorists working on disordered topological phases would want to see the marker in their own models. It is solid enough on the clean-system side and concrete enough on the application that it should go to referees rather than a desk reject; the disorder caveat is fixable with one or two extra checks.

Referee Report

1 major / 2 minor

Summary. The paper proposes that driven-dissipative bosonic lattices with uniform-strength random-phase pumping and uniform loss generate a Lorentzian filter on eigenmode occupations, enabling direct reconstruction of both the integrated and energy-resolved Středa responses. This provides access to energy-resolved quantum geometry for generic dispersive bands and, as a key application, elucidates the persistence of topological quantum geometry in strongly disordered systems such as topological Anderson insulators.

Significance. If the central reconstruction holds, the scheme supplies a practical, local probe of anomalous spectral flow and band geometry in open bosonic platforms that complements closed-system measurements. The extension to disordered topological bands is potentially valuable because it targets a regime where conventional transport or spectroscopy can be ambiguous.

major comments (1)

[Application to disordered systems] The derivation of the Lorentzian filter (uniform pumping + uniform loss) is stated to hold for generic dispersive bands, yet the application to topological Anderson insulators under strong disorder assumes the same filter continues to sample localized eigenstates uniformly enough to recover the magnetic response of the density of states. No explicit check is provided that disorder-induced localization or level repulsion leaves the effective filter width and the resulting Středa marker undistorted.

minor comments (2)

[Introduction] Notation for the energy-resolved Středa marker should be introduced with an explicit definition (e.g., as a function of energy and magnetic field) before its reconstruction is discussed.
[Abstract] The abstract and main text should clarify the range of validity of the coarse-graining implicit in the Lorentzian filter (e.g., relative to the bandwidth or disorder strength).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and positive assessment of the work. We address the concern about the robustness of the Lorentzian filter under strong disorder below.

read point-by-point responses

Referee: [Application to disordered systems] The derivation of the Lorentzian filter (uniform pumping + uniform loss) is stated to hold for generic dispersive bands, yet the application to topological Anderson insulators under strong disorder assumes the same filter continues to sample localized eigenstates uniformly enough to recover the magnetic response of the density of states. No explicit check is provided that disorder-induced localization or level repulsion leaves the effective filter width and the resulting Středa marker undistorted.

Authors: The derivation of the Lorentzian filter follows directly from the steady-state solution of the Lindblad master equation with site-local pumping of uniform strength and random phases together with uniform loss. This yields an eigenmode occupation that is strictly a function of the mode energy (Lorentzian centered at the pump frequency), independent of the spatial support of the eigenmode. Consequently the filter applies to both extended and localized states; random phases ensure that the effective pumping rate into a normalized mode remains uniform on average regardless of localization. Nevertheless, to directly address the referee's request for an explicit check against possible distortions from localization or level repulsion, we will add numerical verification in the revised manuscript (new supplementary figure) confirming that the reconstructed energy-resolved Středa marker in the topological Anderson insulator regime remains undistorted and consistent with the expected quantum geometry. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proposes using driven-dissipative bosonic lattices with uniform-strength random-phase pumping plus uniform loss to produce a Lorentzian filter on eigenmode occupations, thereby accessing the integrated and energy-resolved Středa responses. This construction rests on the pre-existing Středa formula (linking Hall conductivity to magnetic-field response of density) and standard Lindblad master equations for open systems; neither the target marker nor its energy resolution is defined in terms of the proposed filter, nor is any parameter fitted to data and then relabeled as a prediction. The application to topological Anderson insulators under disorder is presented as a direct consequence of the same framework without redefinition or self-referential closure. No load-bearing step reduces to a self-citation chain, ansatz smuggled via prior work, or renaming of known results. The derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Středa formula and quantum-geometry definitions together with the assumption that a driven-dissipative master equation yields a clean Lorentzian filter; no new free parameters or postulated entities are introduced.

axioms (2)

domain assumption The Středa formula and its energy-resolved extension apply to Bloch bands in both closed and driven-dissipative settings.
The paper extends the known formula to the new platform without re-deriving it.
domain assumption Uniform pumping with random phases plus uniform loss produces a Lorentzian filter on eigenmode occupations for generic dispersive bands.
This is the central mechanism enabling the energy-resolved reconstruction.

pith-pipeline@v0.9.0 · 5512 in / 1423 out tokens · 48180 ms · 2026-05-11T03:17:27.776669+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Our scheme uses controlled pumping with uniform strength and random phases across the lattice, together with uniform loss, to yield a Lorentzian filter of eigenmode occupations... Φ0 ∂ρ(ω)/∂B |_{B=0} = ∫ [F_xy(k) δ(ω-ε_k) + ... ]
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
energy-resolved Středa marker... topological Anderson insulators

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