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A scanning-probe image is a projection of a transport operator, not a photograph of material coefficients; this paper shows how to extract quantitative electrical, thermoelectric, and viscous response channels from quantum Hall nanoscopy.

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2026-08-02 05:20 UTC pith:2LFNGR5P

load-bearing objection A serious framework paper: the thermoelectric zero is a dressed Mott identity, the Schur identifiability analysis is the real contribution, and the uncontrolled O(q²ℓ_B²) remainder is the main flaw. the 2 major comments →

arxiv 2607.16315 v1 pith:2LFNGR5P submitted 2026-07-15 cond-mat.mes-hall

Thermal and viscous contrast in quantum Hall scanning-probe images

P. Shubham Parashar This is my paper
classification cond-mat.mes-hall MSC 81V7082C7082D80 PACS 73.43.-f72.20.Pa72.10.-d
keywords quantum Hallscanning probethermoelectric contrastHall viscosityLandau-level projectionmagnetization currentSchur complementFisher information
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.

This paper develops a framework for interpreting scanning-probe images of quantum Hall systems, arguing that a probe measures a finite-resolution functional of a transport operator rather than a local material constant. In the strong-field regime, it derives a ratio between defect-induced thermoelectric and electrical Hall contrasts that is independent of defect shape and tip geometry, pinning the zero of thermoelectric contrast at the mobility edge. In the hydrodynamic regime, it shows that Hall-odd image shapes are not by themselves Hall-viscosity measurements, and it quantifies when a Hall-viscous coefficient is actually identifiable through a Fisher-information analysis. The central message is that visual features in nanoscopy images must be converted into a quantitative observability test before any transport coefficient is assigned.

Core claim

The paper's central claim is that under a local spectral-shift approximation, the defect-induced thermoelectric and electrical Hall contrasts of a smooth scalar defect satisfy δα_xy^tr(q)/δσ_xy(q) = (E_c−μ)/(eT). At the retained long-wavelength order, the orbital Landau-level form factor, the defect geometry, and the common tip transfer function all cancel after magnetization-current subtraction, so the zero of the thermoelectric contrast is fixed at E_c = μ rather than by defect shape. A second, complementary claim is that in the hydrodynamic regime the measurable q^2 tensor amplitudes mix Hall, longitudinal, transverse, boundary, electrothermal, and kinetic channels; the Hall-viscous lengt

What carries the argument

The core mechanism is the source–operator–probe decomposition M = ⟨W_P | L_R^{-1}(B) | S_src⟩, which separates what is measured from what is inferred. The microscopic vertex uses Landau-level projection to write the defect as F_N(q)U(q)ρ̄_{−q}, a particle-number Ward identity converts a scalar insertion into a spectral shift δΦ(ε,q) = −F_N(q)U(q)∂_εΦ(ε), and magnetization-current subtraction cures the bare heat-current bubble. In the hydrodynamic regime the key object is the Stokes–Ohm matrix A(q) and its q^2 expansion, whose inverse yields tensor amplitudes S_ij that mix d_L, d_⊥, d_H and non-viscous stiffness; tensor orthogonality at β=ω_c τ_mr=1 decouples even and Hall-odd channels before

Load-bearing premise

The ratio and its zero rely on the local spectral-shift approximation δΦ(ε,q) = −F_N(q)U(q)∂_εΦ(ε) + O(q^2ℓ_B^2), plus a sharp or symmetrically broadened mobility edge; if the heat-current vertex carries ε- or q-dependent structure beyond F_N(q) that the magnetization subtraction does not remove at finite q, or if disorder broadening is asymmetric, the zero shifts away from E_c = μ.

What would settle it

A direct experimental test would measure both the thermoelectric and electrical Hall contrast for the same defect in a quantum Hall sample, sweeping the gate voltage across the mobility edge; if the zero of δα_xy^tr/δσ_xy does not occur at the chemical potential where the mobility edge is independently known, the ratio fails. A sharper test uses a defect with a deliberately different shape (e.g., a triangular gate instead of a disk); if the zero moves with shape, the cancellation claimed in Eq. (30) is incomplete.

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

If this is right

  • An experimentalist can locate a mobility edge relative to the chemical potential by sweeping gate voltage and detecting where the defect-induced thermoelectric Hall contrast vanishes, without needing to know defect shape or tip calibration.
  • If the framework is correct, scanning-probe images of quantum Hall systems should be interpreted as functionals of transport operators, not as local maps of potential, temperature, or viscosity.
  • The derivation gives a direct check: sweep μ or T, compare electrical and thermoelectric Hall contrasts for the same defect, and look for sign reversal at E_c = μ.
  • The Hall-viscosity result implies that a Hall-odd image alone is insufficient evidence for Hall viscosity; any extraction must account for a specified nuisance library and report a Schur-marginalized error.
  • Leave-one-out analysis identifies boundary slip as the dominant competing direction for Hall viscometry in the representative geometry, suggesting that slip control is crucial for such measurements.

Where Pith is reading between the lines

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

  • If the zero of the thermoelectric contrast remains pinned at E_c = μ under symmetric or Gaussian broadening, the pinning may be robust to a wider class of disorder; the author's own symmetry argument suggests only asymmetric broadening would shift it, which is a testable prediction.
  • The operator-probe formulation could be extended to other probe channels, such as scanning SQUID or scanning NV thermometry, by replacing the tip kernel and keeping the same cancellation logic; this is a plausible generalization the paper does not work out.
  • One might test the thermoelectric ratio directly in a dual-gated graphene device with a known mobility edge, comparing photothermal vs electrical Hall images; the prediction is that the zero crossing stays at the same chemical potential regardless of the intrinsic defect geometry.
  • The observation that the residual Hall-odd Fisher weight is ~1% of the raw weight suggests a practical design rule: optimization of tip height and spot size cannot fully rescue the identifiability if boundary slip is mis-specified, so experiments should combine radial shape information with a frequency or temperature sweep.

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 / 4 minor

Summary. The manuscript develops a source–operator–probe framework for interpreting scanning-probe images of quantum Hall systems. In the strong-field, sharp-Landau-level regime, it uses Landau-level projection, a particle-number Ward identity, and magnetization-subtracted thermoelectric transport to derive Eq. (30), a ratio between defect-induced thermoelectric and electrical Hall contrasts that is claimed to vanish at E_c = μ, independent of defect shape and common probe kernel. In the hydrodynamic regime, it derives the q² tensor amplitudes of the Stokes–Ohm response and performs a Schur-complement Fisher analysis to quantify the identifiability of Hall viscosity d_H against a nuisance library, yielding a conditional sensitivity δd_H ≈ 68 nm² at SNR0 = 200. The paper emphasizes the distinction between measured contrast and transport coefficient, and it provides a detailed protocol for quantitative inversion of scanning-probe images.

Significance. If the central prediction of Eq. (30) survives corrections, it offers an attractive calibration-free method to locate a mobility edge relative to the chemical potential. The hydrodynamic analysis is a careful and conservative observability study that correctly warns against overinterpreting Hall-odd images as direct Hall-viscosity measurements. The algebraic derivations, including the tensor inversions in Section III and the Schur complement in Section IV, appear internally consistent, and the paper is commendably explicit about its assumptions, operating points, and the conditional nature of the quoted sensitivity. The principal weakness is the uncontrolled finite-q correction to the thermoelectric result, which is not negligible at the quoted experimental wave numbers and directly affects the headline zero.

major comments (2)
  1. [II.B/II.C, Eqs. (21) and (30)] The central result relies on the local spectral-shift approximation Eq. (21), whose O(q²ℓ_B²) remainder is never computed or bounded. At the stated operating point q ≈ 0.033 nm⁻¹ and ℓ_B = 10 nm, qℓ_B ≈ 0.33 and (qℓ_B)² ≈ 0.11. If this remainder contributes to δα_xy a term that does not vanish at E_c = μ, the zero will shift by an amount of order the spectral width times (qℓ_B)². The sharp-edge model (Eq. 22) and the symmetrically broadened model (Supplement S2) make the remainder absent by construction, so they cannot reveal this bias. The abstract's 'pinned' language and the proposed experimental protocol (sweeping gate voltage until the contrast vanishes) treat the leading-order zero as exact. The author should compute the next-order term for a concrete broadening/disorder model or provide a bound under which the zero is robust within a stated tolerance.
  2. [II.C vs. IV.A, Eq. (30)] The claim that the 'common tip kernel cancels' in Eq. (30) is not actually derived: Eqs. (23)–(29) are written for responses at fixed wave vector q and contain no tip transfer function; the tip kernel Th,s(q) first appears in Eq. (75). The cancellation is valid only if the same tip, with identical transfer function, is used to measure both the electrical and thermoelectric channels. This is a reasonable scenario but should be stated explicitly; if different probes or channel-dependent nonlinearities are involved, the ratio is not probe-independent. This is a presentational gap rather than a mathematical error, but it should be clarified.
minor comments (4)
  1. [Notation, Eq. (22)] The symbol T is used both for temperature (Eq. 22) and for the tip transfer function (Eq. 75). This is a potential source of confusion; consider using T_tip or a different symbol for the transfer function.
  2. [Fig. 2 caption] The caption states 'the sharp edge gives tg(t)' but does not specify the normalization. For a reader, the vertical axis is clear only after reading the text; a brief definition in the caption would help.
  3. [IV.B, leave-one-out] The leave-one-out results for removing T_c, T_th, T_η, T_ζ, T_4 are quoted only verbally; a table giving the residual fraction and resulting δd_H for each removal would make the limiting-nuisance claim more reproducible.
  4. [Appendix A, Eq. (A3)–(A5)] The definitions of Δ and the response-regime-dependent spectral weights are terse. Since this appendix is used to separate regimes, one or two sentences defining the transport coefficients σ_R^(0) and σ_R^(2) in the diffusive/hydrodynamic/kinetic cases would improve clarity.

Circularity Check

0 steps flagged

No significant circularity: Eq. (30) is a derived consequence of stated spectral-shift and sharp-edge assumptions; the only self-citation is not load-bearing.

full rationale

I walked the derivation chain: Eq. (21) follows from the particle-number Ward identity plus an explicit local spectral-shift approximation with an admitted O(q^2 l_B^2) remainder; Eqs. (23) and (26) are direct evaluations using the sharp mobility-edge ansatz Eq. (22); Eq. (30) is their algebraic quotient. The cancellation of F_N(q), U(q), and a common probe kernel is a within-model bookkeeping statement, not a circular reuse of the conclusion. The zero at E_c = mu is a moment identity for a step in Hall spectral weight under the (epsilon - mu) weighting in Eq. (24): with dPhi/depsilon concentrated at E_c and the weighting odd about mu, the first energy moment vanishes at E_c = mu. That is a legitimate consequence of the stated model, not a restatement of the output as an input: E_c is an input mobility-edge label, and the electrical contrast Eq. (23) provides an independent observable peaked at the same E_c, so the coincidence of the electrical peak and thermal zero is a real cross-check. The paper itself flags the finite-q limitation ('at the retained long-wavelength order', 'the O(q^2 l_B^2) term collects nonuniform-response corrections'); whether the uncontrolled remainder shifts the zero at the quoted q l_B ~ 0.33 is a quantitative correctness issue, not circularity. The only self-citation ([23]) appears in the acknowledgments and is not load-bearing for any derivation. The Fisher/Schur sensitivity for d_H is a forward-model observability bound, not a fitted parameter renamed as a prediction. No step meets the required standard of exhibiting a circular reduction, so the score is 0.

Axiom & Free-Parameter Ledger

6 free parameters · 8 axioms · 0 invented entities

The derivation is mostly self-contained given standard linear-response and hydrodynamics; the heavy lifting is in the model choices: local spectral shift, sharp/symmetric mobility edge, magnetization-subtracted thermoelectric coefficient, 2×2 Stokes–Ohm truncation, and an ad hoc boundary-slip template that controls the headline viscometry error bar. No new physical entities are introduced.

free parameters (6)
  • E_c (mobility-edge energy) = model input; not numerically specified
    Location of the sharp step in Hall spectral weight (Eq. 22). The zero-pinning statement is about the detuning E_c−μ; E_c is an input label, so the 'prediction' is relative to an input.
  • Φ0 (jump in Hall spectral weight) = cancels in ratio
    Sets the magnitude of both δσ_xy and δα_xy^tr; cancels in Eq. (30).
  • Default geometry and operating point (ℓ_B, a, h, s, ζ, β, SNR0) = (10,80,30,20,50) nm, 1, 200
    Chosen representative values (Eq. 4); the headline δdH≈68 nm² is conditional on them.
  • Nuisance template amplitudes (λ_c, λ_th, λ_η, λ_ζ, λ_4) = marginalized in Schur complement
    The five nuisance templates (Eqs. 87–91) are fitted to the image; the reported error bar is the conditional variance after this marginalization.
  • ℓ_th (electrothermal relaxation length) = set equal to a=80 nm
    Width of the T_th template; chosen, not measured.
  • w (mobility-edge broadening) = varied in Fig. 2 (0.5–2 k_B T)
    Controls Fig. 2; zero pinning holds only for symmetric broadening.
axioms (8)
  • standard math Particle-number Ward identity: at q=0 a uniform scalar insertion effectively shifts μ (⟨X⟩_μ derivative, Eq. 19)
    Grounds Eq. (21); exact at q=0; the finite-q extension is the local spectral-shift assumption.
  • domain assumption Local spectral-shift approximation: δΦ(ε,q) = −F_N(q)U(q)∂_εΦ(ε) + O(q²ℓ_B²)
    Stated in Sec. II B; the geometry/kernel cancellation in Eq. (30) holds only at the retained order.
  • domain assumption Sharp or symmetrically broadened mobility edge (Φ=Φ0Θ(ε−E_c); Gaussian broadening in Fig. 2)
    Eq. (22) and Fig. 2; asymmetric broadening would shift the zero of the thermoelectric contrast.
  • domain assumption Magnetization-subtracted α^tr is the transport thermoelectric response measured by the probe
    Sec. II C removes the f(t) circulating-current term; the paper gives no experimental protocol isolating the magnetization-subtracted α^tr from the raw heat-current response.
  • domain assumption 2×2 Stokes–Ohm truncation with Hall viscosity entering only as C(q)=Ω+η_H q²
    Sec. III A; all q² channels (longitudinal, transverse, boundary, electrothermal, kinetic) enter through the chosen templates.
  • domain assumption The electrical and thermal probes share the same tip transfer function T_{h,s}(q)
    Required for the 'common tip kernel' cancellation claimed after Eq. (30); the kernel never appears in Eqs. (23)–(29).
  • ad hoc to paper Boundary-slip template T_ζ = q²(1+qζ)^{−1}v_H correctly represents slip leakage
    Sec. IV B calls it 'a minimal representation of that crossover'; its 0.98 overlap with the target sets the 68 nm² bound and the 'limiting nuisance' conclusion.
  • domain assumption White, band-limited image noise with spectral density S_V
    Eq. (78); underpins the Fisher metric.

pith-pipeline@v1.3.0-alltime-deepseek · 13359 in / 28342 out tokens · 282083 ms · 2026-08-02T05:20:48.074620+00:00 · methodology

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read the original abstract

Quantum Hall scanning images are often read as maps of a local potential, temperature, or viscosity, whereas a probe records a finite-resolution functional of a transport operator. We formulate this functional using Landau-level projection, a particle-number Ward identity, magnetization-subtracted thermoelectric transport, a hydrodynamic Stokes-Ohm inversion, and finite-tip Fisher information. Two results follow in complementary transport regimes. In the strong-field, sharp-Landau-level regime, the defect-induced thermoelectric and electrical Hall contrasts of a smooth scalar defect obey $\delta\alpha_{xy}^{tr}/\delta\sigma_{xy}=(E_c-\mu)/(eT)$. At the retained long-wavelength order, the orbital form factor, defect geometry, and common tip kernel cancel after the heat-magnetization current is removed, so the zero of the thermoelectric contrast is pinned by energy weighting at $E_c=\mu$ rather than by defect shape. In the hydrodynamic regime, the measurable $q^2$ tensor amplitudes mix Hall, longitudinal, transverse, boundary, electrothermal, and kinetic channels, so a Hall-odd image is not by itself a Hall-viscosity measurement. For a representative graphene geometry, a Schur-complement fit against the stated nuisance library yields a conditional one-standard-deviation sensitivity of approximately 68 square nanometers at SNR0 = 200, with boundary slip the limiting nuisance. The framework turns visual interpretation of quantum Hall nanoscopy into a quantitative observability test for electrical, thermoelectric, and viscous response channels.

Figures

Figures reproduced from arXiv: 2607.16315 by P. Shubham Parashar.

Figure 1
Figure 1. Figure 1: FIG. 1. Source–operator–probe structure of the paper. A scanning experiment measures a finite-resolution functional [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Thermoelectric Hall contrast for a mobility edge of finite spectral width. The horizontal axis is the dimensionless [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Parametric Schur-complement study for Hall-viscous identifiability. Left: marginalized error [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

23 extracted references · cited by 1 Pith paper

  1. [1]

    Four-terminal phase-coherent conductance,

    M. B¨ uttiker, “Four-terminal phase-coherent conductance,” Phys. Rev. Lett.57, 1761 (1986). 15

    1986

  2. [2]

    Partial densities of states, scattering matrices, and Green’s functions,

    V. Gasparian, T. Christen, and M. B¨ uttiker, “Partial densities of states, scattering matrices, and Green’s functions,” Phys. Rev. A54, 4022 (1996)

    1996

  3. [3]

    Nanoscopic tunneling contacts on mesoscopic multiprobe conductors,

    T. Gramespacher and M. B¨ uttiker, “Nanoscopic tunneling contacts on mesoscopic multiprobe conductors,” Phys. Rev. B56, 13026 (1997)

    1997

  4. [4]

    Shockley-Ramo theorem and long-range photocurrent response in gapless materials,

    J. C. W. Song and L. S. Levitov, “Shockley-Ramo theorem and long-range photocurrent response in gapless materials,” Phys. Rev. B90, 075415 (2014)

    2014

  5. [5]

    Visualizing Poiseuille flow of hydrodynamic electrons,

    J. A. Sulpizioet al., “Visualizing Poiseuille flow of hydrodynamic electrons,” Nature576, 75 (2019)

    2019

  6. [6]

    Imaging viscous flow of the Dirac fluid in graphene,

    M. J. H. Kuet al., “Imaging viscous flow of the Dirac fluid in graphene,” Nature583, 537 (2020)

    2020

  7. [7]

    Imaging the breakdown of ohmic transport in graphene,

    A. Jenkinset al., “Imaging the breakdown of ohmic transport in graphene,” Phys. Rev. Lett.129, 087701 (2022)

    2022

  8. [8]

    Nanoscale imaging of magnetotransport around a circular p–n junction in graphene,

    Z. J. Krebset al., “Nanoscale imaging of magnetotransport around a circular p–n junction in graphene,” Phys. Rev. Lett.136, 136301 (2026)

    2026

  9. [9]

    Viscosity of quantum Hall fluids,

    J. E. Avron, R. Seiler, and P. G. Zograf, “Viscosity of quantum Hall fluids,” Phys. Rev. Lett.75, 697 (1995)

    1995

  10. [10]

    Hall viscosity and electromagnetic response,

    C. Hoyos and D. T. Son, “Hall viscosity and electromagnetic response,” Phys. Rev. Lett.108, 066805 (2012)

    2012

  11. [11]

    Formalism for the quantum Hall effect: Hilbert space of analytic functions,

    S. M. Girvin and T. Jach, “Formalism for the quantum Hall effect: Hilbert space of analytic functions,” Phys. Rev. B29, 5617 (1984)

    1984

  12. [12]

    Magneto-roton theory of collective excitations in the fractional quantum Hall effect,

    S. M. Girvin, A. H. MacDonald, and P. M. Platzman, “Magneto-roton theory of collective excitations in the fractional quantum Hall effect,” Phys. Rev. B33, 2481 (1986)

    1986

  13. [13]

    Electronic properties of graphene in a strong magnetic field,

    M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys.83, 1193 (2011)

    2011

  14. [14]

    Theory of thermal transport coefficients,

    J. M. Luttinger, “Theory of thermal transport coefficients,” Phys. Rev.135, A1505 (1964)

    1964

  15. [15]

    Thermoelectric effect in a weakly disordered inversion layer subject to a quantizing magnetic field,

    M. Jonson and S. M. Girvin, “Thermoelectric effect in a weakly disordered inversion layer subject to a quantizing magnetic field,” Phys. Rev. B29, 1939 (1984)

    1939

  16. [16]

    Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field,

    N. R. Cooper, B. I. Halperin, and I. M. Ruzin, “Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field,” Phys. Rev. B55, 2344 (1997)

    1997

  17. [17]

    Energy magnetization and the thermal Hall effect,

    T. Qin, Q. Niu, and J. Shi, “Energy magnetization and the thermal Hall effect,” Phys. Rev. Lett.107, 236601 (2011)

    2011

  18. [18]

    Nonlocal transport and the hydrodynamic shear viscosity in graphene,

    I. Torre, A. Tomadin, A. K. Geim, and M. Polini, “Nonlocal transport and the hydrodynamic shear viscosity in graphene,” Phys. Rev. B92, 165433 (2015)

    2015

  19. [19]

    Electron viscosity, current vortices and negative nonlocal resistance in graphene,

    L. Levitov and G. Falkovich, “Electron viscosity, current vortices and negative nonlocal resistance in graphene,” Nat. Phys.12, 672 (2016)

    2016

  20. [20]

    Negative magnetoresistance in viscous flow of two-dimensional electrons,

    P. S. Alekseev, “Negative magnetoresistance in viscous flow of two-dimensional electrons,” Phys. Rev. Lett.117, 166601 (2016)

    2016

  21. [21]

    Nonlocal transport and the Hall viscosity of two-dimensional hydrodynamic electron liquids,

    F. M. D. Pellegrino, I. Torre, and M. Polini, “Nonlocal transport and the Hall viscosity of two-dimensional hydrodynamic electron liquids,” Phys. Rev. B96, 195401 (2017)

    2017

  22. [22]

    Boundary conditions of viscous electron flow,

    E. I. Kiselev and J. Schmalian, “Boundary conditions of viscous electron flow,” Phys. Rev. B99, 035430 (2019)

    2019

  23. [23]

    Diffusive and hydrodynamic magnetotransport around a density perturbation in a two-dimensional electron gas,

    P. Shubham Parashar and M. M. Fogler, “Diffusive and hydrodynamic magnetotransport around a density perturbation in a two-dimensional electron gas,” Phys. Rev. B113, 155443 (2026). 16 Supplemental Material: Thermal and viscous contrast in quantum Hall scanning-probe images P. Shubham Parashar Independent Researcher, San Diego, California, USA psparash@ucs...

    2026