Linear Resistivity from Spatially Random Interactions and the Uniqueness of Yukawa Coupling

Sang-Jin Sin; Yi-li Wang

REVIEW 4 major objections 5 minor 2 cited by

Linear Resistivity from Spatially Random Interactions and the Uniqueness of Yukawa Coupling

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Within the class of spatially random scalar couplings, only the (2+1)-dimensional Yukawa-type interaction produces linear-in-temperature resistivity.

desk verdict Systematic classification of spatially random scalar couplings, with a plausible but not fully rigorous conclusion that only 2D Yukawa gives linear-T resistivity. read the letter →

arxiv 2507.09442 v6 pith:JH2UW63J submitted 2025-07-13 hep-th cond-mat.str-el

Sang-Jin Sin , Yi-Li Wang This is my paper · ORCID

classification hep-thcond-mat.str-el

keywords strangemetalslinearresistivityspatiallyrandominteractionsYukawacouplinglarge-NtheorycriticalFermisurfaceKuboformulanon-Fermiliquid

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 asks which spatially random couplings between a Fermi surface and a critical boson can produce the linear-in-temperature resistivity that defines strange metals. Scanning the family of scalar interactions of the form $(\psi^\dagger\psi)^n\phi^m$ in $d\ge2$ spatial dimensions, it finds exactly one viable candidate: the Yukawa-type coupling with $n=m=1$ in $(2+1)$ dimensions. All other combinations give resistivity scaling $\rho\sim T^{m+2n-2}$ in two dimensions, or no solution in higher dimensions, so linearity is not a generic consequence of spatial randomness. The result matters because it turns a suggestive mechanism into a uniqueness statement, tying strange-metal linear resistivity to a specific interaction type and to two spatial dimensions.

What carries the argument

The load-bearing object is the exponent $\varsigma$ in the zero-temperature electron self-energy, $\Sigma(i\omega)\sim -i c_F g^2 \omega^{\varsigma}$; in $d=2$ it equals $2n+m-2$, and linear resistivity requires $\varsigma=1$, which forces $n=m=1$. The calculation is carried by the $G$–$\Sigma$ saddle-point equations of the replicated disorder-averaged action, together with a self-consistent power-law ansatz for the boson propagator $D(i\Omega,q)\simeq 1/(q^2+c_B|\Omega|^\eta)$. The same scaling data are inserted into the Kubo formula, and the vertex corrections are shown to vanish in the spatially random case because disorder averaging decouples internal momenta.

What would settle it

Compute the full boson self-energy $\Pi(\Omega,q)$ beyond the leading power-law ansatz for the $d=3$, $n=m=1$ model: if it develops momentum-dependent terms or non-power-law corrections, the exponent $\eta$ that drives the exclusion of $d\ge3$ is not fixed self-consistently, and the uniqueness conclusion would need revision.

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Extended reading notes

Core claim

The paper's central claim is a classification of the family of spatially random scalar couplings $(\psi^\dagger\psi)^n\phi^m$ between fermions and bosons in $d\ge2$ spatial dimensions. Using the $G$–$\Sigma$ large-$N$ saddle-point equations for the replicated, disorder-averaged action and assuming a dirty-metal fermion propagator dominated by potential disorder, the authors derive power-law self-energies with exponents controlled by $n$, $m$, and $d$. The Kubo formula converts the electron self-energy exponent $\varsigma$ into resistivity scaling $\rho\sim T^\varsigma$; requiring $\varsigma=1$ for linear-$T$ resistivity leaves $d=2$, $n=m=1$ as the only consistent solution. Spatially uniform versions of the same couplings are also considered and are excluded because the linear self-energy contribution is cancelled by a vertex correction, so the unique scalar route is the spatially random Yukawa interaction in $(2+1)$ dimensions.

Load-bearing premise

The classification stands on two assumptions: the boson propagator keeps the strict power-law form $D\sim 1/(q^2+c_B|\Omega|^\eta)$, and potential-disorder scattering dominates over the random interaction in the fermion propagator; if either fails, the exponent count that rules out higher dimensions can change.

Editorial extensions

If this is right

In $(2+1)$ dimensions, every scalar coupling of this family obeys $\rho\sim T^{m+2n-2}$, so adding extra fermion or boson fields pushes resistivity farther from linearity.
No choice of $n,m$ in $d\ge3$ solves the consistency conditions, so for scalar couplings the linear-resistivity mechanism is intrinsically two-dimensional.
Spatially uniform versions of the same couplings cannot produce linear resistivity because a vertex correction cancels the linear self-energy contribution, leaving only the spatially random case.
Taken with the paper's companion vector-coupling result, the classification singles out two minimal building blocks for linear-$T$ resistivity in two dimensions: a scalar Yukawa coupling and a vector ('QED-like') coupling.

Reading between the lines

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

If the assumption $|\Sigma_v|\gg|\Sigma_g|$ is relaxed, the fermion propagator would itself be dressed by the random interaction; in that strong-coupling regime other $(n,m)$ combinations might reach linear scaling, so the uniqueness claim is tied to the weak-coupling dirty-metal limit.
Introducing a finite correlation length into the disorder variance would add a new energy scale; the classification's dependence on delta-correlated randomness suggests linearity could be restored or destroyed in a dimension-dependent way, which is a directly testable extension.
The appendix's demonstration that the golden-rule scattering estimate fails away from $n=m=1$ in $d=2$ implies that transport in these models cannot be inferred from single-particle scattering rates, making the Kubo calculation the necessary arbiter for any analogous system.

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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 genuinely new thing here is the systematic sweep over (ψ†ψ)^n φ^m couplings in arbitrary dimension d. Previous work had the specific n=m=1 Yukawa and vector cases; this paper asks the classification question and answers it with a scaling analysis. That is a legitimate and useful extension, and the conclusion—only n=m=1 in d=2 yields linear resistivity—is consistent with what was known. I also give the authors credit for being upfront about where the calculation is shaky: they admit the coefficient integral in eq. (2.36) is generally divergent, and they postpone regularization until they find a linear candidate. That is honest, and it shapes how I read the rest.

The soft spots are real but not disqualifying. The central exponents in eqs. (2.41)–(2.50) come from integrals I_{α,β} that are never evaluated beyond a scaling statement; the Θ(β) factors in eq. (2.36) look under-derived, and the identity (A.2) is asserted, not proved, and is used in log-sensitive regimes where pure power behavior is doubtful. The paper also leans on the weak-coupling assumption |Σ_v| >> |Σ_g| without checking whether it holds in the strong-coupling large-N regimes that motivated the SYK-rised program. And the 'uniqueness' language is too broad: their earlier vector coupling also gives linear resistivity, so the uniqueness is only within the scalar class—they do acknowledge this in the conclusion, but the abstract overstates it.

That said, I don't think the stress-test concern kills the paper. The divergence is in the coefficient, not in the frequency exponent, and for the one case that matters (n=m=1, d=2) they handle the logs and the divergence explicitly. The scaling exponents for the excluded cases could in principle be modified by a cutoff, but the authors' self-consistency check for η is at least plausible. This is not a rigorous proof, but it is a clear, testable classification that a specialist would want to see tightened rather than dismissed.

Who benefits: people working on SYK-rised models and strange-metal transport, especially those deciding which couplings to study. It deserves a serious referee—someone who can check the scaling integrals and the Θ(β) step. I would send it out, with the expectation of major revision rather than acceptance as is.

Referee Report

4 major / 5 minor

Summary. The paper classifies spatially random scalar couplings of the form (ψ†ψ)^n φ^m in d≥2 spatial dimensions, working in a large-N G−Σ formalism with quenched disorder. It derives scaling forms for the fermion and boson self-energies, computes the Kubo conductivity, and concludes that only the Yukawa-type case n=m=1 in d=2 yields linear-in-temperature resistivity. It also argues that spatially uniform versions of these couplings cannot produce linear resistivity. The central claim is that the 2D spatially random Yukawa coupling is the unique scalar SYK-rised interaction giving strange-metal transport.

Significance. If the classification is correct, it is a meaningful result: it sharpens the recent Patel et al. mechanism for linear resistivity by showing that, within a broad class of spatially random scalar couplings, linear-T resistivity is not generic but requires both a specific interaction form and a specific dimensionality. The paper is self-contained, does not fit free parameters, and reproduces the known n=m=1 result as a special case. The extension to all (n,m,d) is a natural and potentially useful contribution. However, the proof of the classification is carried by a scaling analysis of divergent integrals and an unproved integral identity; because every exclusion of a non-Yukawa coupling depends on those exponents, the central claim is not yet established at the level of rigor required for a journal publication.

major comments (4)

[Sec. 2.2, Eqs. (2.30)-(2.36)] The evaluation of I_{α,β} is the load-bearing step, but for β=0 the formula in Eq. (2.30) is not well-defined: there is no q_β integration or final boson propagator, and the subsequent Θ(β) prescription in Eq. (2.36) is introduced ad hoc. Since β=0 corresponds to the boson self-energy for m=1, and since m=1 is exactly the case that survives the classification, this step needs a separate, explicit derivation rather than a theta-function shortcut.
[Sec. 2.2, end of Sec. 2.2] The text acknowledges that the coefficient integral ∫du is generally divergent and defers regularization, yet the exponents in Eqs. (2.41)-(2.50) and hence all the 'no solution' conclusions in Sec. 3 are obtained from this unregularized integral. If a UV cutoff is needed, it can introduce power-law or logarithmic corrections to these exponents; without a controlled regularization for the excluded cases, the statement that no other (n,m,d) can yield linear resistivity is not established.
[Appendix A, Eq. (A.2)] Identity (A.2) is used to replace the fermion frequency integrals by |B|^{n+λ}, but it is not proved and, as stated, it is not valid for the unrestricted integrals appearing in Eq. (2.33): for n=1 and λ=1, the integral ∫ dx sgn(x)|B-x| diverges, and the finite result |B|^{n+λ} holds only with a restricted domain or a regulator. The d=2 case is log-sensitive, so this replacement needs a rigorous justification or a regularized derivation.
[Sec. 3.1, Eqs. (3.9)-(3.16)] The classification solves only necessary conditions for ς=1, but it does not verify self-consistency of the assumed propagator ansatz (2.26)-(2.29) for the models it excludes. One should check that a solution for η actually exists in each excluded case and that the conductivity formula (3.8), which assumes a small power-law correction to the Drude-like term, remains valid when ς<0 or when divergences are present.

minor comments (5)

[Abstract] The sentence 'strange-metal property replies on both dimensions and interaction type' contains a typo ('replies' should be 'relies') and the final sentence is grammatically awkward.
[Sec. 1, last paragraph] The sentence ending with 'across var(2 + 1)-dimensional Yukawa-type couplings are the only class that produces strange-metal behaviour' is garbled and incomplete; it should be rewritten.
[Sec. 2.1, after Eq. (2.16)] An editorial note beginning with 'comments' is inserted in the main text; this should be removed or integrated into the prose.
[Sec. 2.2, paragraph before Eq. (2.26)] The sentence 'Since bosonic self-energy and bosonic self-energy are structurally similar' repeats 'bosonic self-energy'; the second occurrence should be 'fermionic self-energy'.
[Sec. 4 and Appendix B] There are typos: 'sptatially' in Sec. 4 and 'depite' in Appendix B; these should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the uniqueness claim is derived from an explicit large-N scaling analysis, not from a fitted parameter or a self-citation chain.

full rationale

The paper's central claim is a classification of spatially random scalar couplings derived from the saddle-point equations of the G-Σ action and the Kubo formula. The exponents in Eqs. (2.41)-(2.50) are obtained by evaluating the integral I_{α,β} under an explicitly stated scaling ansatz for the boson propagator, Eq. (2.29), and the self-consistency condition fixes η from the same equations rather than importing the target result. The condition ς=1 is then solved algebraically for n,m,d, so the conclusion that only n=m=1 in d=2 works is a consequence of the calculation, not an input. The n=m=1 case agrees with the independent result of Patel et al. [8], but it is reproduced here from the same formalism rather than assumed. Self-citations [9], [11], and [12] appear in contextual or concluding remarks and are not load-bearing for the scalar uniqueness claim; the cancellation of vertex corrections and MT/AL diagrams is cited to the external references [8,19]. The unregularized integrals and the unproved identity (A.2) are potential correctness concerns, but they are not circular: they do not make the derivation equivalent to its inputs. The paper is therefore not significantly circular, with only a nominal score reflecting the presence of self-citations in otherwise non-essential framing.

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

The central claim rests on the large-N replica-symmetric saddle point, the criticality tuning of the boson mass, the dirty-propagator approximation, and a power-law ansatz for the boson self-energy. No parameters are fitted to data; the unspecified coefficients in the scaling formulas do not affect the exponents. No new entities are introduced.

assumptions (6)

domain assumption Large-N limit with replica symmetry: replica off-diagonal components of G and D are neglected.
Invoked in Section 2.1 following Ref. [16]; the disorder average is performed using replicas and replica symmetry is assumed from the outset.
domain assumption Impurity (potential) disorder dominates over the fermion-boson interaction, |Σ_v| >> |Σ_g|.
Stated before Eq. (2.24); it justifies replacing the electron propagator by the dirty-metal form G ≈ 1/(i sgn(ω)Γ/2 - ξ).
domain assumption The boson mass is tuned to criticality, m_b^2 - Π(0,0) = 0.
Used after Eq. (2.24) to set the bosonic propagator to the massless form (2.25).
domain assumption The bosonic self-energy obeys a pure power-law scaling Π(iΩ)-Π(0) ∼ -c_B |Ω|^{η'}, and the propagator is taken as D ≈ 1/(q^2 + c_B |Ω|^η) with η ≤ 2.
Eqs. (2.26)-(2.29); the self-consistency of η is only partially solved in the text.
domain assumption Vertex corrections (Maki-Thompson and Aslamazov-Larkin diagrams) vanish in the spatially random models because disorder averaging decouples spatial momenta.
Section 3, around Eq. (3.5); the vanishing argument is a symmetry statement and is not demonstrated for all diagrams.
standard math Standard frequency and momentum integrals, including the sign-function integral in Appendix A (Eq. A.2).
Used to evaluate the scaling of I_{α,β}; the result |B|^{n+λ} is a standard convolution.

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Cite this review

Pith. "Pith review of Linear Resistivity from Spatially Random Interactions and the Uniqueness of Yukawa Coupling." pith.science (2026). https://pith.science/paper/JH2UW63J

@misc{pith2026250709442,
  author       = {Pith},
  title        = {Pith review of: Linear Resistivity from Spatially Random Interactions and the Uniqueness of Yukawa Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JH2UW63J}},
  note         = {Machine review of arXiv:2507.09442}
}

abstract

Recent studies have shown that a spatially random Yukawa-type interaction between a Fermi surface and critical bosons can produce linear-in-temperature resistivity, the defining signature of strange metals. In this article, we systematically classify all scalar couplings of the form $(\psi^{\dagger}\psi)^n\phi^m$ in arbitrary dimensions to identify possible candidates for strange-metal behaviour within this disordered framework. We find that only spatially random Yukawa-type interaction in $(2+1)$ dimensions can yield linear resistivity. This indicates that linear resistivity is not a universal property of all spatially random scalar coupling, and strange-metal property replies on both dimensions and interaction type.

Discussion (0). Continue with ORCID to comment.

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

P.A. Lee, N. Nagaosa and X.-G. Wen,Doping a mott insulator: Physics of high-temperature superconductivity,Rev. Mod. Phys.78(2006) 17

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work page 2017

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work page Pith review arXiv 2018

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doi:10.1146/annurev-conmatphys-031119-050558 2020

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P.W. Phillips, N.E. Hussey and P. Abbamonte,Stranger than metals,Science377(2022) abh4273 [2205.12979]

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arXiv 2023