Topology is not silent in the transport noise of multiband bosons

Abstract: For electrons at a Fermi surface, the topological part of the Berry curvature drops out of the transport noise: the anomalous Hall conductivity is quantized, and the fluctuations of the Hall current know nothing about the Chern number. We show that this cancellation is an accident of the Fermi surface, and that it fails for bosons. Magnons and phonons thermally populate every band, each with its own Chern number, so the harmonic sector of the Hodge decomposition of the curvature is not a single constant and a variance does not annihilate it. What survives is the occupation-weighted dispersion of Chern numbers across populated bands. That band geometry is visible in current noise is known, and at zero temperature the antisymmetric part of the noise sum rule already returns a Chern number; what is new is an object with no zero-temperature analogue, existing only when several bands are populated and weighted differently. We derive it from a fluctuating Boltzmann equation and settle the energy-magnetization subtraction at the level of fluctuations: the subtracted term is the curl of a bounded magnetization density, so it contributes exactly zero to the current any transport measurement records, realization by realization. Is the quantity measurable? A driven protocol is not: the excess noise sits twelve orders below the equilibrium floor. An equilibrium protocol is, through a sum rule on the antisymmetric part of the current cross-spectrum, which carries no drive and hence no suppression. We prove that the frequency-integrated form of that sum rule cannot separate the topological from the geometric content, and show that the frequency-resolved cross-spectrum does separate them at accessible resolution. The protocol is specified for Cu(1,3-bdc), with its robustness to nuisance-model completeness and to dispersion, thermometry and field calibration.

• The paper shows that multiband bosonic transport noise retains topological information because thermally populated bands can carry different Chern numbers, producing an occupation-weighted Chern-number variance absent in single-band electronic systems.
• The authors derive a multirate Boltzmann–Langevin description in which slow number relaxation can raise the topological share of zero-frequency noise from a few percent to tens of percent or even dominance, while a kagome model reaches about 9% at 3JS.
• The proposed frequency-resolved equilibrium cross-spectrum protocol separates topological and geometric contributions, requiring roughly 144 bins for a 1% identification floor in Cu(1,3-bdc) and outperforming driven measurements that are about 12 orders of magnitude weaker than equilibrium noise.

Overview

The paper establishes that the cancellation of topological Berry-curvature content from transport current noise, which holds for electrons at a Fermi surface, is not a universal property of the fluctuation-dissipation structure but an accident of single-band occupancy. For bosonic multiband systems — magnons and phonons — every band is thermally populated and carries its own Chern number, so the harmonic sector of the Hodge decomposition of the curvature two-form is band-dependent rather than a single constant. The authors show that the resulting observable is the occupation-weighted dispersion of Chern numbers across populated bands (2608.18527), an object with no zero-temperature analogue, since at zero temperature occupations are degenerate (zero or one) and no cross-band weighting exists.

The paper is careful about provenance. The kinetic machinery derives from Kogan–Shul'man and Gantsevich–Gurevich–Katilius; the spectral-weight decomposition is the standard Mori/Kadanoff–Martin structure of fluctuating hydrodynamics; and the proposition that current noise carries band geometry is due to Neupert, Chamon and Mudry, who also stated the multiband obstruction — that energies depend on the band index, so energetics and quantum geometry combine — that this paper resolves at finite temperature.

The Hodge decomposition as a design rule

On the Brillouin-zone torus $T^2$, a closed curvature two-form splits uniquely into a harmonic constant $\bar\Omega = 2\pi C/A_{BZ}$ (topological) and an exact piece $d\beta$ integrating to zero (geometric). A response $R = \int w\,\Omega$ then satisfies $R = \bar\Omega \int w - \int dw\wedge\beta$: the harmonic sector couples only to the integrated weight, the exact sector only to its variation. The paper elevates this elementary lemma into a design principle — the geometric sector is visible exactly to the extent that the probe's weight varies over the zone — and uses it to unify the electronic noise theorem, the high-temperature magnon thermal Hall result of Mook et al., and the low-temperature blindness of thermal Hall transport.

The electronic theorem and its two hypotheses

For electrons, transverse current fluctuations take the form $$\mathrm{Var}(J_\perp) \propto \mathrm{Var}_p(\Omega)$$ with a normalized weight of zero mean. Since a variance annihilates constants, the Chern number cancels. The paper stresses that this "silence theorem" requires two hypotheses: (H1) a conservation law forcing zero mean weight; and (H2) that the harmonic sector is a single constant across everything the weight touches. (H2) is invisible electronically because the Fermi surface selects essentially one band. It is precisely this hypothesis that fails for bosons.

The bosonic failure and what replaces it

For collinear ferromagnetic magnons, where $S_z$ conservation restores (H1), inserting the band-resolved decomposition into the noise yields

$$\mathrm{Var}_p(\Omega) = \mathrm{Var}_p(\bar\Omega_\lambda) + \mathrm{Var}_p(\delta\Omega) + 2\,\mathrm{Cov}_p,$$

with the first term nonzero whenever populated bands have different Chern numbers. The topological share of the variance for the kagome ferromagnet with Chern numbers $(+1,0,-1)$ grows from roughly $0.5\%$ at $\bar\Omega = 2\pi C/A_{BZ}$0 to $\bar\Omega = 2\pi C/A_{BZ}$1 at $\bar\Omega = 2\pi C/A_{BZ}$2, scaling approximately as $\bar\Omega = 2\pi C/A_{BZ}$3 in the Dzyaloshinskii–Moriya coupling. The authors report they could not find this occupation-weighted Chern-number variance treated as an observable elsewhere in the literature.

Extending beyond a single relaxation time, they solve the Boltzmann–Langevin equation with three rates (in-band collision, energy exchange, number relaxation) and obtain three Lorentzians whose weights are the squared projections of $\bar\Omega = 2\pi C/A_{BZ}$4 onto conserved quantities in the $\bar\Omega = 2\pi C/A_{BZ}$5-weighted inner product — a "Landau–Placzek decomposition of the Berry curvature." Crucially, because number relaxation is slow relative to collisions ($\bar\Omega = 2\pi C/A_{BZ}$6 realistic), the topological share of the zero-frequency noise is enhanced from a few per cent to tens of per cent, reaching dominance at ratios in the hundreds.

Magnetization subtraction dissolved

The Qin–Niu–Shi energy-magnetization subtlety is resolved at the fluctuation level by two structural results. First, a vanishing theorem: for any bounded magnetization density vanishing outside the sample, its curl contributes exactly zero to the current through any complete cross-section, realization by realization, at all frequencies and drive orders. Second, a chiral protection lemma: the curl current of a single scalar density has rank-one Fourier support, so its contribution to the antisymmetric cross-spectrum $\bar\Omega = 2\pi C/A_{BZ}$7 cancels identically under any detection geometry. Consequently the kinetic route computes the physical section current directly without entering the gauge-represented Kubo response where magnetization contamination lives, and every driven result above stands unmodified. The caveat is scope: these are leading-semiclassical statements, valid below the minimal direct gap, and Bogoliubov systems fail (H1) outright and are excluded.

Why the driven protocol fails

A Zeeman field gradient is a genuine, geometry-neutral force on magnons. Yet the excess transverse noise scales as $\bar\Omega = 2\pi C/A_{BZ}$8 with $\bar\Omega = 2\pi C/A_{BZ}$9; at achievable gradients of $d\beta$0–$d\beta$1 T/m this sits twelve orders of magnitude below the equilibrium noise floor. A real temperature gradient cannot rescue it: statistical drives exert no force, their anomalous vertex is absent, and a fluctuation theory for them does not yet exist. The gapless case is additionally pathological — the weight $d\beta$2 collapses onto the zone center — making the Zeeman gap a requirement.

The equilibrium sum rule and the impossibility of integration

Without drive, the antisymmetric part of the symmetrized current cross-spectrum obeys a frequency-integrated sum rule returning the occupation-weighted mean curvature $d\beta$3. It closes numerically to $d\beta$4 (analytically it holds mode-by-mode before integration), and the chirality ratio $d\beta$5 reaches $d\beta$6 at the spectral peak — an order-one signal against the $d\beta$7 of the driven case. But the harmonic content exceeds the total by factors of $d\beta$8–$d\beta$9, cancelled largely by the geometric part.

Two independent obstructions prevent the integrated observable from isolating topology. The first is rank: sweeping $R = \int w\,\Omega$0 amounts to a Laplace inversion of one function pooling both sectors, and against a complete nuisance class the identifiability signals sit five to nine orders below the model-error floor — a Monte Carlo fit returns confidently wrong triples. This obstruction dissolves only for perfectly flat bands, recovering the Kruchkov–Ryu result. The second is channel mixing: the integral pools intrinsic against extrinsic (side-jump/skew-scattering) contributions, which for bosons can match or exceed the intrinsic term.

Frequency resolution separates the sectors

Resolving the sum-rule integrand in the band-pair gap coordinate undoes the pooling. With pair-decomposed curvatures $R = \int w\,\Omega$1, only antisymmetric combinations return quantized integers; the direction $R = \int w\,\Omega$2 is a gauge and profiled out, and $R = \int w\,\Omega$3 becomes a pointwise theorem. Identifiability against nuisance completeness is non-monotone but decisive once complete: growing the nuisance basis from three to thirty-five modes per pair drops the model error four orders while topological responses fall barely one, opening a gap of more than an order of magnitude. The controlling parameter is the ratio $R = \int w\,\Omega$4 of nuisance dimension to resolved dimension; along fixed $R = \int w\,\Omega$5 the model error falls as $R = \int w\,\Omega$6 while the split stays near $R = \int w\,\Omega$7, and the 1% floor is met at 144 bins (~3 GHz for Cu(1,3-bdc)).

Design studies at matched flexibility yield three concrete conclusions: a single $R = \int w\,\Omega$8 setting fails by exact degeneracy (split $R = \int w\,\Omega$9); nine settings (three temperatures by three fields) match the full grid at every noise level tested, a sixteen-fold measurement reduction; and the temperature axis is indispensable — field-only designs reach $R = \bar\Omega \int w - \int dw\wedge\beta$0 and recover 7%, which no resolution rescues. Recovery is essentially complete whenever $R = \bar\Omega \int w - \int dw\wedge\beta$1 exceeds about four. Calibration is the tightest constraint: $R = \bar\Omega \int w - \int dw\wedge\beta$2 must be known to ~0.1%, $R = \bar\Omega \int w - \int dw\wedge\beta$3 to ~0.5%, asymmetrically in sign because dispersion moves bin edges while thermometry does not; temperature is benign at 5%. Co-fitting calibration parameters jointly with the candidate triple recovers the correct triple in 89.4% of trials under relative noise. Window truncation is fatal and insidious: masking above 232 GHz returns a confident wrong triple rather than a null result, because each band pair occupies nearly disjoint stretches of the gap coordinate.

Application to Cu(1,3-bdc)

The protocol is specified for the kagome ferromagnet Cu(1,3-bdc): spectrum turning on at $R = \bar\Omega \int w - \int dw\wedge\beta$4 GHz with main weight near 147 GHz, full support spanning 38–473 GHz, nine-setting design, ~3 GHz bin width. The protocol is scale-free in $R = \bar\Omega \int w - \int dw\wedge\beta$5, giving a one-parameter family trading frequency reach against cryogenic demand. Integration times are estimated at milliseconds to seconds depending on detector floor; bandwidth and interface transparency, not statistics, bind. Notably, Lemma (chiral protection) makes the pickup geometry unconstrained by the subtraction, and the region below the turn-on constitutes a built-in null channel reporting extrinsic admixture directly. In three dimensions, all three Hodge sectors appear; a Weyl prototype shows the co-exact contribution peaking near 12% of the harmonic, present but subdominant, with the noise question left open.

Limitations

The paper states its boundaries plainly. Results hold only for collinear ferromagnetic magnons; number-nonconserving Bogoliubov systems require the symplectic construction. Vertices are intrinsic throughout — extrinsic side-jump and skew-scattering channels, which can dominate bosonic thermal Hall transport, enter only through broadening within the analysis, and whether they contaminate the extraction is left open, bounded by the interband placement of the extraction window. The fluctuation theory of statistical drives, needed for a genuine temperature-gradient noise theory, does not exist. The co-estimation cure for calibration fragility is validated only in simulation. The $R = \bar\Omega \int w - \int dw\wedge\beta$6 counting of the transport–magnetization cross term is flagged for verification rather than asserted.

Conclusion

The paper converts a negative statement — topological silence in electronic transport noise — into a positive program by identifying its unstated second hypothesis and exhibiting its failure for bosons. The surviving observable, the occupation-weighted dispersion of Chern numbers, is made measurable through a frequency-resolved equilibrium cross-spectrum protocol with a quantitative error budget, an explicit resolution rule ($R = \bar\Omega \int w - \int dw\wedge\beta$7, 144 bins), and a self-diagnosing null channel, specified end-to-end for Cu(1,3-bdc). Its principal debts are acknowledged frameworks applied rather than reinvented, and its principal open questions — extrinsic contamination, statistical-drive fluctuations, and the three-dimensional co-exact sector — are stated where they bear on the results.