Derivation and application of a general scaling relation between the dc and asymptotic high-frequency optical Hall responses

E. Abelev; N. P. Armitage; R. Romero III

REVIEW 3 major objections 4 minor 1 cited by

Derivation and application of a general scaling relation between the dc and asymptotic high-frequency optical Hall responses

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A general scaling relation links low-frequency and high-frequency optical Hall responses in any time-reversal-broken metal, and predicts a measurable dc anomalous Hall effect in the cuprate pseudogap.

desk verdict Useful KK scaling relation between dc and high-frequency Hall response; the general-law claim is ahead of the evidence, but the cuprate AHE prediction makes it worth refereeing. read the letter →

arxiv 2607.15043 v1 pith:URIH2WCF submitted 2026-07-16 cond-mat.str-el

E. Abelev , R. Romero III , N. P. Armitage This is my paper

classification cond-mat.str-el

keywords HallconductivityKramers-Kronigrelationmagneto-opticalKerreffecttime-reversalsymmetrybreakinganomalouscupratesuperconductivityopticalcyclotronresonance

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

This paper establishes a general, Kramers-Kronig-based scaling law: in any time-reversal-symmetry-broken conductor, the high-frequency optical Hall conductivity should fall off as minus the dc Hall conductivity times the square of the characteristic gap scale divided by frequency. The authors show the law is exact for a single cyclotron resonance, holds approximately for ferromagnetic and superconducting models, and agrees reasonably with measured ferromagnet data. Its practical payoff is a prediction: the ~0.8 microradian polar Kerr rotation seen near the pseudogap temperature in the cuprates implies a dc anomalous Hall effect of order 0.1 Ω⁻¹cm⁻¹, small but within reach of existing experiments. A sympathetic reader would care because the relation turns a single optical measurement into a low-frequency transport prediction, and vice versa, without detailed microscopic modeling.

What carries the argument

The central object is Eq. (8), a two-line Kramers-Kronig identity: σ_xy(ω ≫ ω_g) = −σ_xy(0)(δ/ω)², where δ is the characteristic absorption peak (2Δ for superconductors, E_g for ferromagnets, ω_c for cyclotron resonance). Its derivation rests on the 'narrow-band assumption': the circular-dichroic absorption σ_1r − σ_1l is nonzero only in a narrow, roughly symmetric interval of width δ(2Δ), so the running frequency ω′ inside the Kramers-Kronig integral can be replaced by its mean value, 2Δ, and pulled out. The remaining integral is the difference of spectral weights between right and left circular channels; the ratio of the low- and high-frequency limits then cancels all microscopic details,

What would settle it

Measure the dc anomalous Hall conductivity of a clean YBa₂Cu₃O₆₊ₓ crystal in the pseudogap state and compare with the value −σ_xy(0.8 eV)(0.8 eV/δ)² predicted from the measured Kerr rotation, using δ = 3.5kT*; a null result below ~0.01 Ω⁻¹cm⁻¹ or with the wrong sign would falsify Eq. 8's applicability (or the assumed gap scale) in cuprates. Alternatively, in any ferromagnet with a broad asymmetric Hall absorption tail, verify whether the high-frequency real part of σ_xy actually changes sign and follows −σ_xy(0)(δ/ω)²; the untruncated Dirac model is an explicit counterexample where neither hap

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Extended reading notes

Core claim

Starting from the Kramers-Kronig relation for the Hall conductivity, the paper shows that when the dissipative Hall difference (absorption of right versus left circular polarization) is confined to a narrow frequency band of width δ(2Δ) centered at 2Δ, the low-frequency Hall value σ_xy(0) and the high-frequency asymptote are locked together by σ_xy(ω ≫ 2Δ) = −σ_xy(0)(2Δ/ω)². The same form holds with 2Δ replaced by an interband energy E_g for ferromagnets and TRSB superconductors. The minus sign matters: the high-frequency Kerr or Faraday signal must have the opposite sign from the dc Hall effect when the material is in the asymptotic regime. The authors verify the relation on a sharp cyclotr

Load-bearing premise

The dissipative right-left circular absorption difference must be confined to a narrow, roughly symmetric band centered at the gap scale 2Δ (or E_g), so that the frequency inside the Kramers-Kronig integral can be treated as a constant; where that fails — as in the untruncated Dirac continuum — the scaling degrades or breaks.

Editorial extensions

If this is right

A measured dc anomalous Hall effect directly fixes the magnitude of the high-frequency optical Hall response (and Kerr/Faraday rotation) once the dominant absorption peak δ is known.
In any material where the low-frequency and high-frequency Hall signals have the same sign, the data are not yet in the asymptotic regime of the relation; the sign change is a necessary diagnostic.
The relation converts the existing 0.8 μrad polar Kerr signal in the cuprate pseudogap into a concrete, testable dc Hall prediction of order 0.1 Ω⁻¹cm⁻¹, with Hall angle ~10⁻⁵.
Because the predicted dc Hall value scales as δ², a doubling of the pseudogap gap scale changes the prediction by a factor of four; the testability is robust to that uncertainty.
The same scaling applies to conductors in a magnetic field, with the cyclotron frequency as the scale — already exact in the Drude model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The relation can be read as a sum rule: the frequency-integrated circular dichroism is fixed by the dc Hall conductivity. That suggests a direct experimental test using broadband circular-polarization absorption measurements in ferromagnets, without needing a Kerr measurement.
If the cuprate Hall effect is not found at the predicted level, the failure would most naturally indict either the assumed gap scale δ = 3.5kT* or the interpretation of the Kerr signal as a bulk optical Hall response — a useful diagnostic for the pseudogap order debate.
The authors' treatment suggests that reporting the optical Hall conductivity at two well-separated frequencies (one dc/low-frequency, one ≫δ) is enough to extract the gap scale δ from any TRSB material; existing datasets for ferromagnets could be re-analyzed that way.
Since the relation is parameter-free except for δ, it may provide a 'quantum-weight'-style bound on the dc Hall response from optical measurements, complementing the quantum-geometric bounds the authors mention, even though their derivation is purely causal.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The useful nugget here is Eq. 8: via Kramers-Kronig, for a dissipative Hall conductivity concentrated in a narrow symmetric band around a scale δ, the high-frequency tail of σxy behaves as -σxy(0)(δ/ω)^2. The derivation is simple, transparent, and self-contained. I don't think this exact scaling for arbitrary TRSB states appears in the cited literature; the cyclotron version is textbook, and the generalization to superconductors and to the cuprate prediction is new.

The paper does several things well. It states its key assumption explicitly — that σ1r−σ1l is nonzero only in a narrow range — and it flags deviations. It tests the relation on a cyclotron model (exact), on Dirac models (with a factor-of-4 underestimate in the truncated case), and on digitized theory and experiment. The parametric plots are an honest way to show the asymptotic approach. The final prediction — that cuprates should have a dc AHE of order 0.1 Ω−1cm−1, inferred from the 0.8 μrad Kerr signal and a BCS-like 2ΔPG — is concrete, falsifiable, and within experimental reach.

The soft spots are the ones the stress-test note points out. The 'general' claim is not established. The narrow-band assumption is strong: the effective scale δ_eff = sqrt(M1/M_{-1}) can differ by large factors from the peak δ when spectral weight is broad or asymmetric, and the paper never computes these moments. Instead, δ is read from the same spectra that are being tested, which can improve the apparent collapse. The sign-selection filter — keeping only data where the dc and high-frequency Hall signs are opposite — is a reasonable minimal asymptotics check but also discards cases that might disprove the relation. And 'small factor of about 4' is a strange description of a factor-of-4 error; it shows the relation is a heuristic, not a law. The paper itself acknowledges the offset in theory comparisons and the strong sensitivity of the cuprate number to the assumed 2ΔPG.

None of this makes the paper unserious. It is honestly written, the math is not circular, and the prediction is worth testing. My net: treat Eq. 8 as a useful rough bridge for single-mode-like systems, not a universal sum rule. I would send it to peer review and ask that the authors compute the moment ratio for their examples, report the validation without the sign filter, and soften 'general' to 'approximate.'

Referee Report

3 major / 4 minor

Summary. Based on Kramers-Kronig relations, the paper derives a relation between the quasi-dc Hall conductivity and the asymptotic high-frequency Hall conductivity in time-reversal-symmetry-broken systems: σ1,xy(ω≫δ) = −σ1,xy(0)(δ/ω)^2. The derivation assumes the dissipative Hall absorption σ1r−σ1l is confined to a narrow, roughly symmetric band of width δ(2Δ) centred at 2Δ (or Eg). The relation is exact for a single cyclotron resonance, approximately obeyed in several theoretical models of TRSB superconductors and ferromagnets, and used as a heuristic against experimental ferromagnet data. The paper then uses the measured Kerr rotation in cuprates at 0.8 eV to predict a dc anomalous Hall effect of about 0.1 Ω^-1 cm^-1, assuming δ=3.5kT*. The authors acknowledge that the relation is approximate and that errors are introduced when the Hall absorption is not narrow.

Significance. If established, a simple two-scale scaling relation of this type would be valuable: it connects dc transport measurements to high-frequency optical/Kerr measurements and enables order-of-magnitude estimates when only one response is accessible. The derivation is transparent and self-contained, the single-mode cyclotron limit is handled correctly, and the authors are honest about the narrow-absorption assumption and provide a falsifiable prediction. However, as written the 'general' claim is not supported: the key approximation is uncontrolled for broad or asymmetric Hall absorptions, the validation protocol is partly circular, and the cuprate prediction depends sensitively on an assumed energy scale. The paper is best viewed as proposing a useful heuristic, not a theorem.

major comments (3)

[§II, Eqs. (3)–(8); Fig. 3c] The central step is the replacement of ω′ by δ=2Δ in both the first-moment integral (Eq. 3) and the inverse-moment integral (Eq. 6). For a general absorptive Hall spectrum Δσ(ω)=σ1r−σ1l, the exact low- and high-frequency limits are σ1,xy(0)=(1/π)∫Δσ/ω′ dω′ and σ1,xy(ω≫...)=−(1/π)(1/ω^2)∫ω′Δσ dω′. Equation (8) is therefore exact only if the chosen δ satisfies δ^2=[∫ω′Δσ dω′]/[∫Δσ/ω′ dω′]. The paper does not verify this moment condition for any of the digitized spectra; it instead sets δ to the peak of Im σxy. The truncated Dirac example of §III (red curve, Fig. 3c) is a direct counterexample: the paper reports a factor-of-≈4 underestimation, and the text attributes this to spectral weight asymmetrically extended to higher frequencies. That is not an 'order-unity' correction, and it shows the approximation is uncontrolled. The authors should either restrict the claim to single-mode-like sp
[§III and §IV, Figs. 4 and 5] The experimental/theoretical validation is weakened by two selection choices. (i) In §III the authors write 'We only included calculations where the high frequency optical Hall conductivity have an opposite sign than the low frequency Hall effect,' and the same requirement is applied to experiments in §IV. Since Eq. (8) itself predicts opposite signs for a single-mode spectrum, excluding same-sign datasets removes the most direct failures (e.g. the untruncated Dirac model, whose Im σxy does not change sign); it cannot be used as evidence of generality. (ii) The scale δ is read from the same spectrum being tested ('we identify the most prominent peak in the dissipative response with the scale δ'; in §IV δ is set where Im|σxy| reaches a maximum). This makes the agreement in parametric plots partly built in: any spectrum with a dominant absorption peak will have a crossover around ω≈δ. A co
[§V, cuprate prediction] The headline prediction of a dc anomalous Hall conductivity ≈0.1 Ω^-1 cm^-1 depends on two inputs not established by the paper's own analysis: the Hall conductivity at 0.8 eV inferred from Kerr data, and the assumption δ=2Δ_PG=3.5kT*. The latter is particularly consequential because the prediction scales as δ^-2; a factor of two in δ changes the answer by four, as the authors acknowledge. More importantly, the relation also predicts a sign reversal between the dc and high-frequency Hall responses, but the sign of the cuprate dc AHE is not given. With an assumed δ and no independent constraint on the width/position of the Hall absorption, the prediction is an order-of-magnitude estimate rather than a definite result. The paper should present a range over plausible T* values and should state the sign and falsifiability conditions.

minor comments (4)

[Abstract and affiliations] Typos: 'Kramers-Krong' should be 'Kramers-Kronig'; 'that that' is repeated; the affiliation reads 'Department of Department of Physics and Astronomy.'
[§II, notation] The symbol δ is used both as the width of the absorption window, δ(2Δ)=2Δr−2Δl, and later as the full energy scale appearing in Eq. (8). This dual use should be clarified with different notation, e.g. Γ for the width and δ for the centre scale.
[§IV, Figs. 4 and 5] The parametric plots do not show error bars or a quantitative deviation metric; 'reasonable agreement' is supported only by inspection. A plot of |σxy(ω)|/|σxy(0)| versus (δ/ω)^2, or of the ratio as a function of ω/δ, would be more informative and would make the claimed factor-of-order-unity accuracy testable.
[Eq. (14)] The conversion from Kerr rotation to σxy should be derived or cited with the relevant sign convention and thin-film/reflection assumptions. Currently the expression is introduced without enough context for the reader to assess the reliability of the extracted Hall conductivities.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: Eq. (8) is a genuine KK consequence under the paper's explicitly stated narrow-band assumption; the parametric validation is self-referential in its choice of δ and sign-preselection, but disclosed and not by construction.

other [Section III (Dirac-model parametric analysis, Fig. 3c) and Section IV (experimental parametric plots, Figs. 4-5)]
"For the analysis we identify the most prominent peak in the dissipative response with the scale δ."

The empirical tests of Eq. 8 select the scale δ from the same spectra they are verifying (peak of the dissipative Hall response), and datasets are admitted only when high- and low-frequency responses already have opposite signs, so the sign content of Eq. 8 is enforced by the inclusion criterion rather than tested. The magnitude test then checks only whether the first-moment ratio M1/(M0·δ_peak) ≈ 1; for near-symmetric single-peak spectra this nearly reproduces the moment relation δ_eff² = M1/M_{-1} that the derivation itself builds in, so such data cannot strongly falsify Eq. 8. However, the paper openly discloses the scheme and reports a factor ~4 deviation for the truncated Dirac model, showing the test can fail; this is a validation-leniency caveat, not a reduction of the central deriv

full rationale

The central result Eq. (8) is derived, not assumed: Eqs. (3)-(7) are two KK integrals (the high-frequency asymptote and the dc limit) over the same dissipative difference σ1r−σ1l, and Eq. (8) follows by eliminating the common spectral weight γσN δ(2Δ). Neither side is defined in terms of the other: σ1,xy(0) is the dc Hall conductivity and σ1,xy(ω≫2Δ) is the measured high-frequency optical Hall response; the nontrivial content is that their ratio is fixed by (2Δ/ω)² independently of the oscillator strength. The derivation is self-contained once one grants the paper's own stated key assumption of a narrow, roughly symmetric Hall absorption, and the paper flags this assumption explicitly ('This is the key assumption of this analysis') and demonstrates its failure in the untruncated Dirac continuum model (no sign change) and its partial failure in the truncated model (factor ~4 error). Thus Eq. 8 is not protected from falsification by construction. The validation is lenient—δ is taken from the peak of the same spectra, and only opposite-sign datasets are included—but that leniency is disclosed and the reported order-unity deviations show the test retains discriminating power. The cuprate prediction uses an assumed scale δ = 3.5kT* whose sensitivity is acknowledged ('A factor of two uncertainty in this scale changes the predicted Hall conductivity by a factor of four'), so it is a genuine extrapolation rather than a restatement of fitted inputs. The only self-citation used in the narrative (Ref. [2], Onsager-reciprocity constraint on Kerr response) is corroborated by an external reference (Ref. [36]) and by standard physics, and it is not load-bearing for the derivation. Overall the derivation chain is independent and non-circular, with only a mild, self-disclosed validation caveat; hence score 1.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central relation contributes a clean KK derivation, but its applicability and the cuprate prediction depend on assumptions about spectral narrowness and an assumed gap scale. No new entities are introduced.

free parameters (2)

δ (energy scale in Eq. 8) for each analyzed spectrum = peak position or midpoint between peaks of Im σ_xy; values vary by material
Used to construct the parametric plots; the same spectrum supplies both σxy(0) and δ, so the comparison is not independent of the fitted scale.
2Δ_PG = 3.5 k T* (pseudogap energy scale) = not fixed numerically; ~50 meV for T*≈180 K
Assumed BCS-like relation to convert the 0.8 eV Kerr-derived Hall value to dc; the prediction scales quadratically with this parameter, and the paper admits a factor-of-two uncertainty changes the answer by four.

assumptions (6)

standard math Kramers-Kronig relations hold (linearity, causality, analyticity in the upper half-plane)
Eq. (1) is the starting point; no alternative derivation is offered.
domain assumption σ1r−σ1l is nonzero only in a narrow frequency range δ(2Δ) centered at 2Δ with δ(2Δ)≪2Δ
Explicitly called 'the key assumption of this analysis' in Section II; fails for broad/asymmetric spectra such as the untruncated Dirac model.
domain assumption ω′ in the KK integrals can be replaced by its mean value 2Δ and pulled out of the integral
Used to pass from Eq. (3) to Eq. (4) and in the low-frequency limit; valid only under the narrow-band assumption.
domain assumption For clean multiband superconductors and ferromagnets the same narrowness holds around E_g, the interband transition energy gapped/shifted by TRSB
Section II, final paragraph; stated without microscopic derivation.
domain assumption A finite normal-incidence Kerr rotation in YBCO forces broken time-reversal symmetry (Onsager reciprocity in a TRS medium forbids it)
Section V; relies on Refs. [2,36] and is the motivation for applying Eq. 8.
ad hoc to paper Pseudogap scale obeys 2Δ_PG=3.5kT*
Section V; BCS-like assumption with no direct measurement cited, and the prediction is quadratically sensitive to it.

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Pith. "Pith review of Derivation and application of a general scaling relation between the dc and asymptotic high-frequency optical Hall responses." pith.science (2026). https://pith.science/paper/URIH2WCF

@misc{pith2026260715043,
  author       = {Pith},
  title        = {Pith review of: Derivation and application of a general scaling relation between the dc and asymptotic high-frequency optical Hall responses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URIH2WCF}},
  note         = {Machine review of arXiv:2607.15043}
}

abstract

Based on the Kramers-Krong relations, we derive and apply a quite general expression that that relates the low frequency quasi-dc Hall conductivity to the asymptotic high frequency optical Hall response (and its manifestations in Kerr and Faraday effects) for time-reversal symmetry breaking (TRSB) states of matter like ferromagnets and time-reversal symmetry breaking superconductors as well as metals in magnetic field. Parametric plots shows this relation is obeyed exactly for the trivial single-mode case of sharp cyclotron resonance, and approximately for theoretical models for ferromagnets and TRSB superconductors. We also apply it to the experimental data from a variety of ferromagnetic systems and show reasonable agreement there as well. We use the relation to predict, from the size of the spontaneous Kerr effect at the pseudogap temperature of the cuprate superconductors that cuprates should exhibit an anomalous Hall effect of approximately 0.1 Ohm$^{-1}\cdot$cm$^{-1}$. This is a small value, but one within experimental reach and we encourage the search for it. Although not explicitly quantum geometric, our treatment has some similarities to efforts to set bounds on physical quantities based on quantum geometric relations and limited physical information.

Figures

Figures reproduced from arXiv: 2607.15043 by the authors.

**Figure 2.** FIG. 2. Hall conductivity of cyclotron resonance in Drude [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Hall conductivity of truncated and untruncated Dirac [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Real part of Hall conductivity predicted theoretically [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Real part of Hall conductivity experimentally mea [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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