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REVIEW 3 major objections 5 minor 67 references

The even-odd effect in 2D electron scattering should sharpen a Corbino disk's resistance response to small magnetic fields.

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2026-08-02 00:40 UTC pith:PIIWYLT2

load-bearing objection Careful, honest numerics predicting a small-B enhancement of dR/d(B^2) from the even-odd effect; the new value is the all-regime solution and experimental parameter estimates, though the quantitative claim rests on an unmeasured relaxation spectrum and an exponent error in Eq. (28). the 3 major comments →

arxiv 2607.14929 v1 pith:PIIWYLT2 submitted 2026-07-16 cond-mat.mes-hall cond-mat.quant-gascond-mat.stat-mech

Tomographic flow regime vs even-odd effect for the magnetotransport in the Corbino geometry

Grigory A. Starkov This is my paper
classification cond-mat.mes-hall cond-mat.quant-gascond-mat.stat-mech PACS 72.10.-d72.20.My
keywords even-odd effecttomographic flow regimeCorbino geometrymagnetotransportlinearized Boltzmann equationelectron-electron scatteringlong-lived odd harmonicskinematic viscosity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

In two-dimensional electron systems, electron-electron collisions relax even angular harmonics of the distribution function far more efficiently than odd ones. This paper argues that this 'even-odd effect' has a distinctive transport signature: in a Corbino disk, the sensitivity of the resistance to the square of a small magnetic field, α = ∂R/∂(B²), is enhanced near zero field. The enhancement is strongest at the crossover from ballistic to tomographic flow, where only a few odd harmonics are long-lived, and fades as the system becomes more hydrodynamic. The paper also estimates that in the temperature range of a recent experiment the effect is small, so attributing the observed anomalous viscosity scaling to the even-odd effect may need re-examination. If correct, the prediction gives a way to directly probe the long-lived odd harmonics.

Core claim

The central claim is that the even-odd effect—the much slower relaxation of odd angular harmonics of the electron distribution compared with even ones—produces a specific, measurable signature in the magnetoresistance of a Corbino disk. Solving the linearized Boltzmann equation numerically across the ballistic, tomographic, and hydrodynamic regimes, the paper shows that the resistance sensitivity α = ∂R/∂(B²) is enhanced near B = 0 whenever long-lived odd harmonics are present, with the enhancement peaking at the crossover to the tomographic regime, where the even-harmonic scattering rate γ_e ≈ 1. In the standard approximation without such harmonics (the dual relaxation time model), this enh

What carries the argument

The key machinery is the harmonic decomposition of the non-equilibrium distribution function on a circular Fermi surface, together with the eigenvalue spectrum of the linearized electron-electron collision operator. The paper models the even-odd effect by taking even-harmonic relaxation rates ~ γ_e, odd-harmonic rates that grow as (2k+1)^4 γ_o with γ_o ~ γ_e²/a and saturate at the even rate, with a ≈ k_F r_b controlling how many odd harmonics are long-lived. The numerical solution of the resulting kinetic equation in the Corbino geometry—via direct discretization, cross-checked against an integral-equation reformulation based on the method of characteristics—carries the argument.

Load-bearing premise

The predictions rest on the assumed spectrum of odd-harmonic relaxation rates—that they grow as (2k+1)^4 and saturate at the even rate—taken from earlier theoretical work and parametrized by a single number a; if that spectrum is wrong at the experimentally relevant temperatures, the predicted enhancement and the estimate of its small size would both change.

What would settle it

A Corbino-disk transport measurement tuned to γ_e ≈ 1 that shows no low-field enhancement of ∂R/∂(B²) compared with the dual-relaxation-time prediction would contradict the paper's central claim; alternatively, a measurement of harmonic-resolved relaxation rates (e.g., via cyclotron resonance linewidths) that does not show the (2k+1)^4 growth of odd-harmonic rates would undermine the model.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A measurable enhancement of ∂R/∂(B²) near zero field in a Corbino sample with large k_F r_b would be direct evidence for long-lived odd harmonics and the even-odd effect.
  • The enhancement is strongest when γ_e ~ 1, the onset of tomographic flow, so experiments should tune temperature to this crossover rather than deeper into the hydrodynamic regime.
  • The magnetic field itself suppresses the effect by cutting off high odd harmonics, so the enhancement is confined to small B.
  • At the temperatures of a recent experiment (T/E_F ~ 10⁻²–10⁻¹), the paper estimates the enhancement is small, implying that the observed ν ∝ 1/T scaling of kinematic viscosity is probably not due to the even-odd effect.
  • The extension of the method of characteristics to multiple long-lived odd harmonics provides a new way to solve the linearized Boltzmann equation in confined geometries.

Where Pith is reading between the lines

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

  • One could test the prediction by measuring α(T) at a fixed small magnetic field across samples with different k_F r_b; the enhancement should scale with the sample size and density, not just temperature.
  • A similar enhancement may appear in other axisymmetric geometries (e.g., a disk with a small central contact) and could be used to extract the phenomenological parameter a from the shape of α(B).
  • Since the even-odd effect relies on inversion symmetry ε_k = ε_{-k}, samples with strong trigonal warping should show a suppressed enhancement; comparing warped and unwarped materials could isolate the effect.
  • The paper's result suggests that re-analysis of existing Corbino-disk data at low fields—not just the high-field plateau—could reveal whether the even-odd enhancement was present but unnoticed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies magnetotransport in a Corbino disk using the linearized Boltzmann equation with an angular-harmonic-resolved electron-electron collision operator that implements the even-odd effect. The central claim is that the resistance sensitivity α=∂R/∂(B^2) is enhanced at small B when long-lived odd harmonics are present, with the enhancement most pronounced near the ballistic-to-tomographic crossover γ_e^(ee)~1 and vanishing deeper into the hydrodynamic regime. The authors further estimate that for the parameters of the recent experiment [33] the effect is small, and therefore the attribution of the reported ν∝1/T scaling to the even-odd effect requires reconsideration. A side result is an extension of the method of characteristics to multiple long-lived odd harmonics, turning the Boltzmann equation into a system of integral equations, used as a cross-check of a direct discretization method.

Significance. If the prediction holds, it provides a concrete, falsifiable transport signature of the even-odd effect in a geometry already used experimentally, and it directly challenges the interpretation of a recent high-profile experiment. The paper is careful in describing two independent numerical approaches, gives a detailed account of the discretization and integral-equation methods, and makes explicit which quantities are model-dependent. However, the quantitative conclusions rest on a phenomenological collision spectrum and on parameter estimates, and the manuscript does not currently supply the convergence data or sensitivity analysis needed to fully support the numerical predictions. With those additions, the paper would be a solid contribution to the tomographic-transport literature.

major comments (3)
  1. [§II, Eq. (28)] Equation (28) is internally inconsistent. From Eqs. (1), (2) and (23), γ_e/γ_o ∝ (E_F/T)^2, so √(E_F/T) = (γ_e/γ_o)^{1/4}, not (γ_e/γ_o)^{1/2}. The correct estimate in the tomographic regime is k' ~ ½(γ_e/γ_o)^{1/4} ~ ½a^{1/6}, not ½a^{1/3}. For the estimated a≈10^3–10^4 this changes k' from ~10–20 to ~3–5. This should be corrected; the corrected count also makes the asymptotic (2k+1)^4/saturation spectrum apply at even lower harmonic numbers, where it is least controlled.
  2. [§III.B, Fig. 4] The central quantitative claim—the enhancement of α at B→0 and its disappearance with increasing γ_e^(ee)—is presented without convergence metrics or error bars. The direct method uses N_ρ=1000, N_θ=1600 and m'=5 in a magnetic field; the integral method uses N=400 and m'=5–8. No data show how α(B→0), or the height of the peak at r_b/R_L≈2.67, converges with N_ρ, N_θ, or m', nor how the two methods agree for the 'selected set' of parameters. Since the comparison with experiment in §IV is quantitative, these convergence checks are essential.
  3. [§II, Eqs. (24)-(27); §IV] The predicted signature and the reinterpretation of Ref. [33] both depend on the phenomenological interpolation for the odd-harmonic relaxation spectrum, Eq. (27), with the parameter a estimated from screening parameters rather than measured. In the experimentally relevant T/E_F≈10^-2–10^-1, only a few odd harmonics are long-lived, so the asymptotic large-m form is used precisely for the modes (m=3,5) where logarithmic-accuracy results are least controlled. The paper varies a over a wide range, but does not test sensitivity to the low-m rates themselves; e.g., changing the m=3 and m=5 rates by factors of order unity can materially shift the predicted α-enhancement. A sensitivity analysis, or a microscopic calculation of the low-m lifetimes, is needed to support the quantitative conclusions.
minor comments (5)
  1. [Appendix A.2] The quantity in Eq. (A36) is described as 'Standard Error', but it is actually the relative deviation of ρη_1^(s)(ρ) from a constant. Please rename to 'relative deviation' or 'relative error'.
  2. [§II, Table I] The definition of k' in Eq. (22) is √(E_F/T)/2, and Table I is consistent with it. Consider adding a short comment that this estimate is asymptotic and that the exact number of long-lived modes is sensitive to constants of order unity, especially for T/E_F=10^-1.
  3. [§III.A] The text states that 'the results seem to be qualitatively the same' for γ_mr=0.5 and 1.0, but the corresponding figures are only in the Supplementary Material. It would help to show one representative comparison in the main text or to state explicitly that the qualitative statements rely on the supplementary plots.
  4. [Appendix A.1] The notation m' for the number of retained long-lived harmonics in the integral method is close to the notation k' used for the number of long-lived odd harmonics in Section II. Please distinguish them (e.g., use M for the truncation order) to avoid confusion.
  5. [General] The paper would benefit from a brief note on the sign convention in Eq. (12), since the Larmor-radius term appears with a minus sign depending on the definition of θ and the direction of B; a reader may otherwise question the magnetic-field dependence in later figures.

Circularity Check

2 steps flagged

The predicted α(B→0) enhancement is a real transport calculation but its magnitude and the small-effect conclusion are contingent on an imported odd-harmonic spectrum with an estimated parameter a; no step is circular by construction, yet the key signature is strongly shaped by the input spectrum.

specific steps
  1. ansatz smuggled in via citation [Section II, Eqs. (20), (21), (24), (27)]
    "1/τ(ee)_{2k+1} = (2k+1)^4/τ(ee)_o ... For large k, they saturate at the level of even relaxation rates [14–16], which can be modelled by [19] 1/τ(ee)_{2k+1} = 1/τ(ee)_e [1 + (τ(ee)_o/τ(ee)_e) 1/(2k+1)^4]^{-1} ... γ(ee)_o/γ(ee)_e = a^{-1} γ(ee)_e ... γ_{2k+1} = γ(mr) + [(2k+1)^4 a^{-1} γ(ee)_e] / [1 + (2k+1)^4 a^{-1} γ(ee)_e] γ(ee)_e"

    Not circular by construction: the odd-harmonic spectrum is an input, and the transport coefficient is computed from the kinetic equation. However, the principal prediction (α enhancement at small B) is a direct consequence of injecting long relaxation times for odd harmonics into the collision operator. The spectrum is imported from Refs. [14-16] and [19] and condensed into the phenomenological parameter a, so the central signature is, at most, a demonstration of that assumed spectrum rather than an independent test of it.

  2. fitted input called prediction [Section II, Eqs. (24), (29) and Section IV; Supplementary Eq. (S13)]
    "Using Eq. (1) ... γ(ee)_o/γ(ee)_e = a^{-1} γ(ee)_e, where a is the phenomenological parameter that governs how quickly odd scattering rates catch up with the even scattering rate ... a ∼ k_F r_b ... More accurate calculations provided in the supplementary [43] suggest that the values of a in the experiment [33] were device-dependent and varied in the range 1000−5000."

    The parameter a is not fitted to the prediction target α(B), but it is estimated from the same even/odd scattering rates used to construct the model. The paper's qualitative conclusion that the effect is small for the experiment is obtained by combining this estimated a, the estimated γ(ee)_e, and the model spectrum. Thus the quantitative punchline depends on input parameters that are not measured from the target data; this is a fragility/risk rather than a by-construction circularity.

full rationale

The paper's core result—the α = ∂R/∂(B^2) enhancement at small B—is computed by numerically solving the linearized Boltzmann equation with a collision operator in which odd harmonics are given long relaxation times (Eqs. (20)-(27)). In that sense the result is not circular: the kinetic equation could in principle have produced transport coefficients insensitive to the odd-harmonic lifetimes, and the calculation is a genuine solution of the model. However, the model itself builds the even-odd effect into the input spectrum via Eqs. (20)-(27), importing the (2k+1)^4 growth and saturation from Refs. [14-16] and [19] and compressing it into a single phenomenological parameter a. The predicted α-enhancement and the quantitative claim that the experimental effect is small (Section IV) are therefore contingent on that assumed spectrum and on the estimated a (Supplementary Eq. (S13)), not on an independent measurement or a derivation from first principles. This is not the strongest kind of circularity: a is not fitted to the predicted α curves, the transport calculation is a real numerical experiment, and the paper includes an internal check (method-of-characteristics integral equations vs. direct discretization). I rate it 3 rather than 0-2 because the central observable is heavily shaped by an organic input spectrum that is itself the physical claim being tested, and the small-effect conclusion is parameter-estimate-dependent. No self-citation chain is load-bearing, no prediction reduces by construction to its own fit, and the arXiv [27] note actually shows an independent near-simultaneous calculation, which supports the external content of the result.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

No new physical entities are introduced. The paper's central claim rests on a standard kinetic model plus a phenomenological scattering-rate spectrum whose key parameter a is estimated, not directly measured. The free parameters are the usual dimensionless scattering rates, the aspect ratio, and the harmonic truncation; the main load-bearing choice is the odd-rate interpolation of Eq. (27).

free parameters (5)
  • a (odd/even relaxation hierarchy parameter) = estimated 1000-5000 for experiment [33]; varied from 0 to infinity in numerics
    Controls the ratio gamma_o/gamma_e = a^{-1} gamma_e in Eq. (24); the central enhancement exists only for finite positive a.
  • gamma_mr (momentum-relaxing scattering rate) = 0.2 in main text; 0.5 and 1.0 in SI
    Fixed by hand in the numerical runs; the main qualitative conclusions are checked against alternative values in the supplementary.
  • gamma_e (even electron-electron scattering rate) = swept across regimes; estimated to be of order tens at T=100 K in SI
    Dimensionless even e-e scattering rate acts as the temperature-like knob in Figs. 3-4 and defines the ballistic/tomographic/hydrodynamic regimes.
  • r_b/r_a (Corbino aspect ratio) = 4
    Kept fixed at the experimental value of Ref. [33] throughout all computations.
  • m' (number of long-lived harmonics retained in integral method) = 5 with magnetic field; 8 without
    Numerical truncation adopted to keep computation time manageable; affects accuracy in the ballistic regime.
axioms (6)
  • domain assumption Degenerate Fermi gas limit T << E_F; energy dependence of delta-f is neglected and velocities are pinned to the Fermi level.
    Invoked around Eq. (11) to reduce the linearized Boltzmann equation to angular-only dynamics.
  • domain assumption The Fermi surface is circular and the collision operator eigenmodes are angular harmonics with relaxation rates given by Eqs. (20)-(21).
    This is the mathematical content of the even-odd effect as borrowed from Refs. [14-16].
  • domain assumption Matthiessen's rule holds and the momentum-relaxing rate is mode-independent, while charge conservation makes tau_0 infinite and momentum conservation makes tau_1 infinite.
    Used in Eq. (19) and in the definitions of gamma_0 and gamma_1 in Eq. (25).
  • domain assumption Boundary conditions are of Fuchs type with fully diffusive leads, r_theta = 0.
    Eqs. (17)-(18) and Section III state this choice; the predicted alpha curves depend on it.
  • ad hoc to paper Odd-harmonic relaxation rates saturate to the even rate according to the interpolation formula Eq. (27), with ratio gamma_o/gamma_e = a^{-1} gamma_e.
    This phenomenological interpolation is essential for the central prediction; a is a free parameter rather than an independently measured quantity.
  • standard math In a magnetic field, complex angular harmonic rates are renormalized as gamma_m -> gamma_m - i m r_b/R_L (Eq. (33)).
    This is the standard representation of the Lorentz force in the angular harmonic basis, used to justify the magnetic-field cutoff of the long-lived harmonics.

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read the original abstract

In two dimensions, the geometric constraints due to Pauli blocking and conservation laws lead to the even-odd effect exhibited by the electron-electron scattering lengths: electron-electron collisions are more efficient at relaxing the even angular harmonics of the distribution function than the odd ones. Inspired by a recent experiment on the magnetotransport in the Corbino disk geometry, we numerically analyze the electron flows in this geometry across all the regimes. We predict a clear signature of the even-odd effect - enhancement of the resistance sensitivity $\partial R/\partial(B^2)$ at small magnetic fields $B\rightarrow 0$. This enhancement is most prominent at the crossover from the ballistic to the tomographic regime, and gradually disappears when the temperature is further increased. Our estimates suggest that in the temperature range of the experiment, the effect should be small. This implies that the attribution of the anomalous scaling of the kinematic viscosity, that was observed in the experiment, to the even-odd effect might need more careful consideration. As a side note, we show how the method of characteristics can be extended to treat the long-lived odd harmonics, which allows one to recast the linearized Boltzmann equation as a system of integral ones.

Figures

Figures reproduced from arXiv: 2607.14929 by Grigory A. Starkov.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic representation of a Corbino disk. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Normalized total conductance [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Normalized total conductance [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Normalized electrochemical potential ¯µ [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Normalized tangential current density [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Geometrical sketch of an electron trajectory in the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Triangle formed by the point [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

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Reference graph

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