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REVIEW 2 major objections 5 minor 76 references

A critical magnetic field kills one of two tomographic collective modes in 2D Fermi liquids, with which mode survives set by Landau parameters.

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2026-07-15 13:44 UTC pith:EFUJ4N63

load-bearing objection Clean, controlled calculation showing one of two zero-field tomographic poles dies at ω_c/γ_o ~ O(1), with F1 selecting the survivor; solid incremental theory for the electron-hydrodynamics crowd. the 2 major comments →

arxiv 2603.06518 v2 pith:EFUJ4N63 submitted 2026-03-06 cond-mat.str-el cond-mat.mes-hallcond-mat.quant-gas

Tomographic collective modes in a magnetic field

Jeff Maki , Johannes Hofmann This is my paper
classification cond-mat.str-el cond-mat.mes-hallcond-mat.quant-gas
keywords tomographic transportodd-even effectcollective modes2D Fermi liquidmagnetic fieldLandau parameterstransverse conductivityBoltzmann equation
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.

Two-dimensional Fermi liquids at low temperature relax even-parity Fermi-surface deformations much faster than odd-parity ones. That hierarchy produces a tomographic transport window that mixes hydrodynamic and collisionless physics and, in particular, two distinct diffusive poles in the transverse conductivity. This paper shows how those poles evolve under an out-of-plane magnetic field. Using an exact numerical solution of the linearized Boltzmann equation with separate even- and odd-parity relaxation rates, the authors find that at a critical cyclotron frequency of order the odd-parity rate one of the two tomographic modes disappears. Which mode survives is controlled by the p-wave Landau parameter; the remaining mode gradually loses its higher-odd-angular-momentum content and becomes ordinary hydrodynamic diffusion at high field. A variational ansatz for the angular shape of the deformation recovers both the critical field and the branch-selection rule. The result supplies a concrete magnetic-field fingerprint that can distinguish tomographic transport from other mechanisms in clean two-dimensional electron systems.

Core claim

In zero field the transverse conductivity of a 2D Fermi liquid with odd-even relaxation hierarchy supports two diffusive tomographic collective modes. At a critical magnetic field of order ω_c/γ_o ∼ O(1) one of those modes vanishes; which one persists is fixed by the Landau parameter F_1, and the survivor becomes increasingly dominated by the hydrodynamic current modes as the field is raised further.

What carries the argument

Numerically exact solution of the linearized Boltzmann equation in a generalized two-rate (even/odd) relaxation-time approximation, reduced via continued fractions for the higher harmonics and corroborated by a variational Gaussian or few-mode ansatz for the odd-parity Fermi-surface deformation that estimates the critical field from the angular-momentum variance.

Load-bearing premise

The full collision integral can be replaced by only two (or a simple angular-momentum-dependent) relaxation rates for even- and odd-parity modes, with higher Landau parameters set to zero.

What would settle it

Measure the damping poles of the transverse (or magnetically mixed longitudinal) current response while sweeping magnetic field through ω_c ~ γ_o; observe whether exactly one of the two zero-field tomographic poles disappears at a critical field whose value and which-branch selection track the predicted F_1 dependence.

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

If this is right

  • Tomographic transport signatures should be suppressed once the cyclotron radius falls below the dominant odd-parity mean free path.
  • Which of the two zero-field tomographic poles survives is a direct diagnostic of the Landau parameter F_1.
  • At still higher fields the surviving mode continuously evolves into ordinary hydrodynamic diffusion, recovering conventional magneto-transport.
  • The same mode structure is predicted for two-dimensional fermionic atomic gases with synthetic gauge fields, offering a cold-atom route to the same critical-field test.

Where Pith is reading between the lines

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

  • Because magnetic fields mix longitudinal and transverse responses, the tomographic poles may become visible in ordinary density or longitudinal probes at intermediate fields before they are fully suppressed.
  • Mapping the critical field versus density or temperature (which tune γ_o and F_1) would give a quantitative check of the odd-even hierarchy beyond static conductivity scaling.
  • If residual impurity scattering is present, the critical field window may shrink; the theory therefore also predicts how clean the sample must be for the branch-selection effect to be resolved.

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

2 major / 5 minor

Summary. The paper studies the magnetic-field evolution of tomographic collective modes in a two-dimensional Fermi liquid. Using a numerically exact continued-fraction solution of the linearized Boltzmann equation within a generalized relaxation-time model that encodes the odd-even hierarchy of quasiparticle lifetimes (γ_e ≫ γ_o), the authors show that the two zero-field diffusive tomographic poles of the transverse conductivity persist only up to a critical cyclotron frequency of order ω_c/γ_o = O(1). At that field one branch disappears; which branch survives is controlled by the p-wave Landau parameter F_1, with a critical F*_1 at which the branches coalesce. A variational ansatz for the Fermi-surface deformation recovers the same critical scale via √⟨m²⟩ ω_c ∼ γ_o and captures the angular structure of the modes. At still larger fields the surviving mode becomes increasingly dominated by hydrodynamic (m=±1) components. Results are consistent for both constant and m-dependent odd-parity rates.

Significance. This is a technically solid and timely contribution to the theory of tomographic transport. A magnetic-field scale that extinguishes one of the two characteristic poles is a clear, falsifiable diagnostic of the odd-even lifetime hierarchy. Strengths include recovery of the known zero-field analytic poles (Eq. 31), consistent results for two distinct models of γ_o(m), an independent variational bound that becomes equality for the true deformation and yields a transparent estimate of the critical field (Eq. 49), and explicit microscopic Fermi-surface deformations (Fig. 5). The work is of clear interest to electronic hydrodynamics and to ultracold-atom platforms with synthetic gauge fields.

major comments (2)
  1. Abstract and Sec. V: The abstract states that the modes “can in principle be observed by examining the damping of longitudinal and transverse current responses in finite magnetic fields,” yet Sec. V reports that “the residue of the tomographic modes in the longitudinal response is negligible.” This tension should be resolved more carefully—either by quantifying the residue in σ_L (or in the mixed response at finite B and finite k) or by tempering the observability claim and focusing on transverse probes and the atomic-gas proposals already mentioned.
  2. Sec. IV, Eq. (49) and Fig. 3: The critical-field estimate √⟨m²⟩ ω_c ∼ γ_o is physically transparent and agrees qualitatively with the numerical termination points (vertical lines in Fig. 3). A short quantitative comparison—e.g., the ratio of the numerical ω_c^crit to the variational estimate across the F_1 range shown—would make the corroboration more precise and would clarify how much of the F_1-dependent branch selection is captured by the variance alone.
minor comments (5)
  1. Sec. III.A: typo “qunatity” → “quantity”.
  2. Sec. IV: encoding artifacts in “ans¨ atze” and similar places; also “them-dependent” → “the m-dependent” in several figure captions and the text around Fig. 3.
  3. Fig. 1 and Fig. 2 captions: special-character rendering is broken in places (e.g., “Re[σ L]”, “kξ”); clean for production.
  4. After Eq. (17): the choice F_m≥2 = 0 is stated without comment. A one-sentence justification (or a note that finite F_2, F_3 would only quantitatively shift the critical field) would help readers unfamiliar with the prior literature.
  5. Fig. 4: the stability diagram is useful; adding the corresponding diagram (or a sentence) for the m-dependent rate model of Eq. (13) would make the F*_1 shift fully transparent.

Circularity Check

1 steps flagged

No significant circularity: B-field spectra and critical-field branch selection are computed from the kinetic equation; prior self-citations supply only the established zero-field odd-even inputs.

specific steps
  1. self citation load bearing [Sec. III A 1, Eq. (31) and surrounding text]
    "For the staggered odd-even relaxation rates in Eq. (12), our results agree with an analytical expression for the dispersion of these diffusive poles [54] ω_tom,±(k)=−iγ_o (1+F1)/8F1 [4+(kξ)2(1+F1)∓(kξ)2√((1+F1+4/(kξ)2)2−16F1/(kξ)2)]."

    The zero-field tomographic poles that serve as the starting point of the B-field analysis are taken from the authors’ own prior work [54]. This is a minor self-citation: the poles are independently re-derived numerically in the present paper and are used only as a known baseline, not as the new claim being established.

full rationale

The paper’s load-bearing results—the disappearance of one of the two zero-field tomographic poles of σ_T at ω_c/γ_o = O(1), the F1-dependent selection of the survivor, the high-field hydrodynamic takeover, and the matching variational estimate √⟨m^{2}⟩ω_c ∼ γ_o—are obtained by direct numerical solution of the linearized Boltzmann equation (Eqs. 14–17 with continued-fraction x_± via Lentz) and by an independent variational deformation constructed in Sec. IV. The zero-field analytic poles (Eq. 31) and the odd-even hierarchy (Eqs. 11–13) are imported from prior literature (including the authors’ own works), but those results are independent derivations (exact diagonalization of the collision integral, analytic continued fractions) that function here as fixed inputs, not as the output being “predicted.” No quantity is fitted to data and then re-presented as a prediction, no uniqueness theorem is smuggled in to force the branch selection, and the variational ansatz is introduced only as a corroborative tool after the numerical spectra are already obtained. The single minor self-citation therefore does not render the central claim circular.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central claim is a numerical/analytic consequence of the linearized 2D Fermi-liquid Boltzmann equation under a two-rate (or m-dependent) relaxation-time model. Free parameters are the ratios and Landau parameters scanned in the figures; axioms are standard kinetic-theory and conservation-law statements plus the deliberate truncation F_m≥2=0. No new particles or forces are postulated; 'tomographic modes' are inherited from prior literature and re-derived here.

free parameters (4)
  • γ_o / γ_e
    Ratio of odd- to even-parity relaxation rates; set by hand to 10^{-2}–10^{-5} in figures to place the system deep in the tomographic window. Controls the scale ξ and the critical field in units of γ_o.
  • F1 (p-wave Landau parameter)
    Scanned over 0–10; selects which tomographic branch dies and sets F*_1 ≈ 2.5–2.6 for the chosen kξ. Not fitted to data but treated as a free material parameter.
  • kξ (or v_F k / γ_e)
    Dimensionless momentum fixed near the tomographic window (kξ≈1.58) for the critical-field scans; changes F*_1 quantitatively.
  • choice of γ_o(m) model
    Constant γ_o vs. the m^4 form (Eqs. 12–13) is a modeling choice that shifts F*_1 slightly but is not constrained by new data in this paper.
axioms (4)
  • domain assumption Linearized Boltzmann equation for a 2D Fermi liquid with Landau mean-field interactions and a homogeneous out-of-plane B field (Eqs. 1–5, 14).
    Standard semiclassical kinetic theory; Berry curvature neglected (stated after Eq. 3).
  • domain assumption Collision integral is diagonal in angular harmonics with rates γ_m of the staggered or m-dependent form (Eqs. 10–13), with γ_0 = γ_±1 = 0 by particle and momentum conservation.
    Generalized relaxation-time approximation justified by prior exact diagonalizations of the collision integral; load-bearing for the existence of a clean tomographic window.
  • ad hoc to paper Landau parameters F_m = 0 for all |m| ≥ 2.
    Explicit truncation after Eq. 17; simplifies the continued fraction but is not derived from a microscopic interaction.
  • standard math Collective modes are poles of the longitudinal/transverse conductivity obtained from the 0,±1 subspace closed by continued-fraction x_±2 (Eqs. 17–29).
    Standard linear-response identification of collective modes.

pith-pipeline@v1.1.0-grok45 · 25800 in / 3268 out tokens · 33788 ms · 2026-07-15T13:44:48.158991+00:00 · methodology

0 comments
read the original abstract

Two-dimensional Fermi liquids at low temperatures have been theoretically established to exhibit an odd-even effect in the collective quasiparticle relaxation rates where even-parity deformations of the Fermi surface decay at a much faster rate than odd-parity ones. A predicted consequence of this effect is a new tomographic transport regime that mixes hydrodynamic and collisionless transport. In the presence of a magnetic field, however, the tomographic regime is expected to evolve towards conventional transport regimes as soon as the cyclotron radius becomes smaller than the dominant odd-parity mean-free path. In this work, we examine this transition from the point of view of collective modes, using a numerically exact solution of the linearized Boltzmann equation within a generalized relaxation time approximation for the odd-parity and even-parity modes. In the absence of a magnetic field, the transverse conductivity exhibits two diffusive tomographic collective modes, and we find that at a critical magnetic field one of these two tomographic modes disappears. Which tomographic mode persists depends on the Landau parameters, and becomes increasingly dominated by hydrodynamic modes at high fields. We corroborate our analysis using a variational approach for the Fermi surface deformation that captures the angular structure of the deformation and the critical magnetic field strength. The collective modes discussed here can in principle be observed by examining the damping of longitudinal and transverse current responses in finite magnetic fields.

discussion (0)

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Reference graph

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