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Magnetic field suppression of tomographic electron transport

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A small magnetic field suppresses tomographic electron transport at a field scale set by the odd-parity mean free path, much below the scale for hydrodynamic suppression.

desk verdict A clean, internally consistent calculation showing that a small magnetic field suppresses tomographic transport at a scale set by the odd-parity mean free path; the main caveat is the assumed B-independence of collision rates, which is standard semiclassical input and not a loading flaw. read the letter →

arxiv 2411.08102 v2 pith:EFVUDM5S submitted 2024-11-12 cond-mat.mes-hall cond-mat.quant-gascond-mat.str-el

Habib Rostami , Nitay Ben-Shachar , Sergej Moroz , Johannes Hofmann This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.quant-gascond-mat.str-el
keywords magneticfieldtomographicelectronfreemeanregimetransport
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

In very clean two-dimensional metals, electron-electron collisions relax different distortions of the Fermi surface at vastly different rates. Even-parity distortions, like the quadrupole shear, relax quickly and give rise to hydrodynamic flow. Odd-parity distortions, which carry the electric current, have a much longer lifetime and remain nearly ballistic. This separation is called the tomographic regime, and it has been predicted to show up as anomalous scaling of conductance with channel width or temperature.

The problem is that testing these predictions requires fabricating samples of different sizes or tuning temperature, which makes the signatures hard to isolate. This paper shows that a magnetic field provides an in-situ control knob. Because time-reversal symmetry protects the parity effect, a magnetic field couples the odd modes to the faster even-mode relaxation. In their kinetic model, the tomographic power law k^2 sigma_T ~ (k xi)^(1/3) disappears once the cyclotron frequency omega_c exceeds gamma' (k xi), where gamma' is the small odd-mode damping rate and xi is a length scale set by the geometric mean of even and odd damping. Since gamma' is much smaller than the even-mode rate gamma, this happens at a magnetic field far below the one needed to suppress ordinary hydrodynamic flow.

The authors verify the suppression with exact numerical solution of the linearized Boltzmann equation, a derivative expansion, and a variational bound on the conductivity. They also propose that a Corbino-disk magnetoresistance measurement should show a resistance minimum at intermediate fields, which could be used to extract the odd-parity mean free path.

Extended reading notes

Core claim

The tomographic scaling window (k^2 sigma_T ~ (k xi)^(1/3)) is suppressed by a magnetic field at a critical cyclotron frequency omega_c^supp ~ gamma' (k xi) << gamma, corresponding to a cyclotron radius comparable to the dominant odd-parity mean free path. Quoting the paper, 'the tomographic region is restricted to a wedge that is terminated by the (small) critical magnetic field' (Sec. II, Eq. 8).

Load-bearing premise

The paper assumes that the magnetic field only adds the Lorentz streaming term omega_c d/dtheta to the kinetic equation and does not alter the collision rates gamma_m themselves: 'We follow Fermi liquid conventions and do not assume a strong dependence of the relaxation rates on the magnetic field' (Sec. I, near Eq. 3). If orbital effects, such as Landau quantization or field-dependent screening, change the odd-parity collision integral at fields omega_c << gamma, the predicted suppression scale would shift. This is a load-bearing input for the central claim.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies a minimal kinetic model of a two-dimensional Fermi liquid with parity-dependent collisional relaxation (the tomographic regime), in which even-parity Fermi-surface deformations relax at a rate γ and odd-parity deformations relax at the much smaller rate γ′m^4. A magnetic field is introduced through the Lorentz streaming term in the Boltzmann equation, and the static transverse conductivity σ_T(k,B) is computed by four complementary methods: exact numerical solution of the resulting tight-binding equation via continued fractions, a derivative expansion at long wavelengths cross-checked by a Hilbert expansion, and variational lower bounds. The central result is that the intermediate tomographic scaling window k^2σ_T ∼ (kξ)^{1/3} is suppressed at a small cyclotron frequency ω_c^supp ∼ γ′(kξ) ≪ γ, far below the field scale that suppresses hydrodynamic transport. The authors propose this as an in-situ experimental probe of tomographic transport, for example through the magnetoresistance of a Corbino device.

Significance. If correct, the result is significant: it offers a magnetic-field-based protocol for identifying the tomographic regime without requiring multiple samples or temperature sweeps. The central prediction is expressed directly in terms of the model's damping rates, and it is supported by three independent calculations: exact continued-fraction solutions, a derivative expansion that is independently reproduced by a Hilbert expansion, and variational bounds that match the numerical conductivity within 9.6–20.2%. The paper also makes a concrete, falsifiable prediction for the magnetoresistance factor α(B) in a Corbino geometry, including a resistance minimum at intermediate fields whose location is controlled by γ′. The main assumption, that the collision rates γ_m are independent of B, is standard semiclassical input and is explicitly stated in the manuscript. The remaining issues are local presentation and proof-detail problems rather than errors in the central suppression mechanism.

major comments (3)
  1. [Abstract and Sec. II C] The statement that suppression occurs when the cyclotron radius is comparable to the ballistic mean free path of the dominant odd-parity mode is inconsistent with the derivation leading to Eq. (8). The balance γ′\bar m^4 ≃ ω_c \bar m with \bar m ∼ (kξ)^{1/3} gives ω_c^supp ≃ γ′(kξ). The ratio of the cyclotron radius to the mean free path of that mode is r_c/l_{\bar m} = \bar m, not 1; for γ′/γ = 10^{-4} and kξ = 100 it is about 4.6, and it grows as (kξ)^{1/3}. Please correct the geometric interpretation in the abstract and in the caption of Fig. 3, e.g., by stating that suppression occurs when the cyclotron frequency matches the dominant odd-mode damping rate divided by its angular momentum, or equivalently when r_c ≈ \bar m l_{\bar m}.
  2. [Sec. III B and Appendix C] The variational lower bound (23) is derived from a Cauchy-Schwarz inequality applied to the 'nonnegative norm' ⟨f|G^{-1}|f⟩. However, G^{-1} defined in Eq. (21) contains the anti-Hermitian streaming terms i k·v(θ) + ω_c ∂/∂θ, so this quadratic form is complex for a general trial function and is not a norm. The proof as written therefore does not establish a rigorous bound for arbitrary h̃. The bound may be valid for the specific parity-symmetric trial functions used, because their streaming expectation vanishes, but the general claim and the word 'rigorous' need to be justified, for example by restricting the argument to the relevant subspace or by using only the Hermitian part of G^{-1}. This does not affect the central suppression result, which is confirmed by the exact continued-fraction solution, but the mathematical status of Eq. (26) is overstated.
  3. [Abstract and Sec. I] The statement that the magnetic field 'breaks time-reversal invariance, which is a prerequisite for the odd-even parity effect in the collisional relaxation' is not what the calculation implements. In the model, the relaxation rates γ_m in Eq. (3) are independent of B, and the odd-even structure of the collision integral is preserved at all fields; the suppression arises from the parity-mixing Lorentz streaming term −i m ω_c in Eq. (4). Please rephrase to avoid implying that B modifies the collision rates themselves.
minor comments (3)
  1. [Sec. I near Eq. (3)] The B-independence of γ_m is load-bearing but standard, and the paper states it explicitly. Since the proposed protocol extracts γ′ from the suppression field, a brief remark that quantum corrections to γ_m(B) are suppressed by powers of ω_c/T (in the relevant window ω_c^supp/T ≲ (T/T_F)^{3/2}) would strengthen the experimental discussion.
  2. [Sec. II A, Eq. (9)] In Eq. (9), the notation γ_2 and γ_3 is used before these quantities are defined; please define them at first use and check the typesetting of the denominators, which is difficult to parse in the current version.
  3. [Fig. 1 and Sec. IV] The impurity scattering rate γ_i is set to 10^{-7}γ in Fig. 1 and Fig. 6 but is omitted in the analytic results of Secs. III B and III C; a sentence clarifying the role of impurities in the phase diagram and in the Stokes-Ohm modeling would improve readability.
Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the odd-even relaxation hierarchy (gamma' << gamma), imported from prior Fermi-liquid theory, and on the assumption that the magnetic field only modifies the streaming term. No new entities are introduced. The model parameters gamma' and gamma_i are inputs, not fitted to the magnetic-field response.

free parameters (2)
  • gamma' (odd-parity damping amplitude) = gamma'/gamma = 10^-4 in main figures; 10^-6 to 10^-2 in Fig. 5 scan
    Sets the hierarchy gamma' << gamma that defines the tomographic regime; the predicted suppression field scales linearly with gamma'. It is an input from prior Fermi-liquid theory, not fitted to the magnetic-field response.
  • gamma_i (impurity scattering rate) = gamma_i/gamma = 10^-7
    Included for the Ohmic cutoff in Figs. 1 and 6; it is small and does not affect the central suppression scale.
assumptions (5)
  • domain assumption Binary collisions conserve the angular momentum index m on a circular Fermi surface
    This makes the collision integral diagonal in angular harmonics, used in Eq. (B1) and the tight-binding model (4). It is standard for circular Fermi surfaces but would fail for anisotropic or warped Fermi surfaces.
  • domain assumption Odd-parity modes relax with rate gamma_m = 1/(1/gamma + 1/(gamma' m^4))
    This Matthiessen interpolation (Eq. 3) captures the low-m gamma' m^4 scaling and the crossover to gamma at large m, motivated by exact diagonalization studies [18,19]. The specific m^4 power law is load-bearing for the tomographic scaling.
  • ad hoc to paper Collision rates do not depend on the magnetic field
    Stated in Sec. I: 'We follow Fermi liquid conventions and do not assume a strong dependence of the relaxation rates on the magnetic field.' This ensures B enters only through omega_c d/dtheta; if violated, the suppression scale would shift.
  • domain assumption Landau parameters are set to zero (alpha_m = 1)
    Neglected in the main text for simplicity; they would renormalize velocities and add factors (1+F1) to currents without changing the scaling structure.
  • domain assumption Low-temperature rigid Fermi-surface deformation with (-df0/depsilon) = delta(epsilon - mu)
    Used in Eq. (1) and (A11) to project the kinetic equation onto angular modes at the Fermi surface; assumes T << T_F.

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Pith. "Pith review of Magnetic field suppression of tomographic electron transport." pith.science (2026). https://pith.science/paper/EFVUDM5S

@misc{pith2026241108102,
  author       = {Pith},
  title        = {Pith review of: Magnetic field suppression of tomographic electron transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFVUDM5S}},
  note         = {Machine review of arXiv:2411.08102}
}
read the original abstract

Degenerate two-dimensional electron liquids are theoretically established to possess two vastly distinct collisional electron mean free paths, where even-parity deformations of the Fermi surface are hydrodynamic with a short collisional mean free path but odd-parity deformations remain near ballistic (known as the "tomographic" transport regime). Predicted signatures of this regime rely on the scaling of observables with temperature or device dimension, both of which are difficult to establish with certainty. Here, we consider magnetotransport in a minimal model of tomographic electrons and show that even a small magnetic field suppresses tomographic transport signatures and thus acts as a sensitive and unique probe of this regime. Fundamentally, the magnetic field breaks time-reversal invariance, which is a prerequisite for the odd-even parity effect in the collisional relaxation. We analyze in detail the scaling of the transverse conductivity, which has been linked to small-channel conductance of interaction-dominated electrons, and show that a tomographic scaling regime at intermediate wave numbers is quickly suppressed with magnetic field to a hydrodynamic or collisionless form. We confirm that the suppression occurs at relatively small magnetic fields when the cyclotron radius is comparable to the ballistic mean free path of the dominant odd-parity mode. This occurs at a much smaller magnetic field than the magnetic field strength required to suppress hydrodynamic electron transport, which suggests an experimental protocol to extract the odd-parity mean free path.

Figures

Figures reproduced from arXiv: 2411.08102 by the authors.

Figure 1
Figure 1. FIG. 1. Density plot of the logarithmic derivative of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Transverse conductivity [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Inverse of the transverse conductivity [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Fermi surface deformations in response to a static perturbation [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Transverse conductivity at zero magnetic [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Magnetoresistance in a Corbino device (inset sketch) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Illustration of the tight-binding structure of the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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