A reciprocal mechanical lattice can show Chern topology in its fluctuations even when its deterministic dynamics and noise spectrum are both topologically trivial, a theoretical preprint reports. The finding concerns a correlation spectrum: a frequency- and wavevector-resolved description of displacement fluctuations, whose eigenvalues indicate fluctuation intensity. It is not a claim that the underlying mechanics has become topological.
That distinction matters because the model tracks correlations, not just the equations of motion. Its displacement spectrum is built from the system’s response and force spectrum; the spectrum is Hermitian positive-semidefinite, and its eigenvalues quantify how strongly each mode fluctuates. The reported Chern numbers therefore describe the geometry of fluctuation bands, not direct evidence of protected one-way mechanical transport.
A lattice driven by rotating noise
The main calculation used a periodic two-dimensional honeycomb lattice with two sites in each repeating cell. Every site received its own independent random force, modeled as a chiral Ornstein–Uhlenbeck process — a colored-noise process with memory whose fluctuations rotate with a preferred sense.
The study compared chiral forcing with an unpolarized case and examined how the calculated spectrum changed across model parameters, frequencies and geometries. It used analytical matrix calculations and numerical spectra for bulk lattices, ribbons and interfaces. Chern numbers were evaluated by summing Berry flux — the geometric twisting of the calculated eigenvectors — over a momentum mesh whose two dimensions each contained 81 points.
Topology appears when a gap reopens
The clearest transition occurred in the middle gap between fluctuation bands. A Chern number is a label for how a band twists across the modeled momentum space. In the unpolarized system, that gap had C2 = 0. After the gap reopened in the polarized cases, the reported value was C2 = +1 for positive spectral polarization and C2 = −1 for negative polarization.
The two signs of rotating noise were therefore associated with opposite topological phases in the model. The result is conditional on the relevant gap remaining open, and it applies to the modeled correlation bands. It does not establish that the reciprocal lattice transmits force preferentially in one direction.
The phase was not fixed once and for all for a given steady state. The same modeled steady state could display different Chern numbers when examined at different observation frequencies. In one reported frequency sweep, the middle gap changed from C2 = 0 to a nonzero value and later returned to C2 = 0 as the magnitude of the rotation rate increased.
The calculation also reported topology across all three correlation gaps: at infinitesimal positive spectral polarization, the gap-Chern tuple was (C1, C2, C3) = (+1, 0, +1). The displayed reciprocal-lattice examples included several other zero, positive and negative Chern-number patterns.
Edges and activity walls
The author then examined how the bulk topology appeared near boundaries. For a bulk–boundary comparison, the two sides of an interface must share a usable indirect gap — a frequency window without bulk states on either side — as well as the direct gap needed to define the Chern number. Within such a common gap, the difference between the two sides’ Chern numbers characterizes branches connecting the bulk bands below and above it.
In the displayed activity-wall example, the Chern mismatch across the interface had magnitudes (2, 0, 2) in the lower, middle and upper gaps. The calculation showed topological branches in the lower and upper gaps, where the mismatch was nonzero. These branches belong to the fluctuation spectrum and should not be read as proof of protected, unidirectional mechanical transport.
The boundary modes also became less confined as the system approached a topological transition. The reported localization length — the distance over which a mode’s intensity fades from the boundary — scaled inversely with the distance from the transition, written as ξ ∼ (∆χ)−1 near the specified transitions.
What remains untested
The evidence is limited to analytical and numerical results within the stated linear reciprocal elastic-lattice models and prescribed colored noise. There is no experimental validation or biological data, and the study does not show that every reciprocal lattice with chiral forcing is topological. Its conclusions depend on the gaps used in the calculation remaining open.
Unpinned lattices can contain floppy modes and nonstationary displacement processes, so finite-frequency spectra in that setting are treated as transfer spectra or in a weak-pinning limit. The slope of a correlation-spectrum edge is not a mechanical group velocity, and the reciprocal mechanical propagator remains Chern-trivial. The exact number of states around a finite circular interface can also depend on interface details.
The document identifies the work as a submission to SciPost Physics and says publication information will appear upon publication. It also states that the source code used to generate the figures is available at github.com/Syrocco/topology. Whether the proposed fluctuation-band topology can be measured in a physical platform, and how it responds to nonlinearities, disorder or more general noise correlations, remains open.
Paper data and sources
Original title: Topology of Fluctuation Bands in Chiral Active Matter
Authors: Raphaël Maire
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text