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REVIEW 4 major objections 6 minor 42 references

Scale-covariant liquid in nonlocal high Tc strange metals

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Strange-metal scaling laws can come from nonlocal repulsions, not from a quantum critical point.

desk verdict Genuinely new relation gamma = 2 - 1/alpha with an honest limitations section, but the model cannot reach the observed gamma > 1.5 without an extra fitted channel, and alpha(p) is read off from the same data it explains. read the letter →

arxiv 2608.03013 v1 pith:DPIAB6AD submitted 2026-08-04 cond-mat.str-el

Jian Xian Sim This is my paper · ORCID
classification cond-mat.str-el
keywords strangemetalhigh-Tcsuperconductivitynonlocalinteractionshydrodynamicscreeningscalecovarianceself-energyexponentARPESmarginalFermiliquid
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

High-temperature superconducting cuprates show a nodal ARPES self-energy that obeys a power law in frequency and temperature with an exponent that drifts continuously with doping. This paper proposes that the exponent is a direct measure of spatial nonlocality: poorly screened repulsive interactions of the form $V_\alpha(r) = g_\alpha/|r|^\alpha$, with a doping-dependent $\alpha$ between 1 and 3, interpolate between Mott-insulating and Fermi-liquid limits. Combining that interaction with hydrodynamic screening of charge fluctuations yields a scale-covariant inelastic decay rate $\Gamma(\omega,T)\propto T^\gamma\,\Phi(\omega/T)$ with $\gamma = 2 - 1/\alpha$ for $1<\alpha<2$. If the proposal is right, the exponents fitted by spectroscopy are measurements of the effective interaction range, and scale-covariant strange-metal behavior does not require a quantum critical point. The same mechanism also distinguishes the underdoped regime as qualitatively different, outside the derivation.

What carries the argument

The load-bearing object is the dynamically screened interaction $W^R(q,\omega)$ (Eqn. 2), obtained by combining the bare power-law interaction $V_\alpha(q)\sim 1/|q|^{2-\alpha}$ with a diffusive irreducible polarization $\Pi_{\mathrm{irr}}(q,\omega)=\chi D q^2/(D q^2-i\omega)$. For $\alpha<2$ its dissipative part peaks at the superdiffusive hydrodynamic pole $\omega=-iD_\alpha|q|^\alpha$, and the self-energy is evaluated in a $G_0W$ approximation with two simplifying approximations: small-angle scattering ($q\ll k_F$, linearized fermion dispersion) and tangential-momentum dominance ($|q|\approx q_\parallel$). This machinery converts the interaction's spatial power law into the scaling exponent $\gamma=2-1/\alpha$ of the decay rate.

What would settle it

Measure the frequency- and momentum-resolved charge response in the strange-metal regime and look for the predicted superdiffusive pole $\omega=-iD_\alpha|q|^\alpha$ at intermediate $q$ and $\omega$; if the dissipative spectrum does not peak along such a pole, the mechanism fails. Alternatively, independently determine $\alpha(p)$ from the real-space interaction (e.g. via the momentum dependence of the static susceptibility) and check whether the ARPES exponent satisfies $\gamma=2-1/\alpha$ across doping.

Watch

Extended reading notes

Core claim

The paper's central claim is that the continuously doping-dependent power-law self-energy observed at the node of high-$T_c$ cuprates is not a quantum-critical signature but a hydrodynamic consequence of poorly screened, spatially nonlocal repulsions. Its central result, Eqn. 6, gives $\Gamma_{\mathrm{inel}}(k_F,\omega,T)\approx A_\gamma T^\gamma \Phi_\gamma(\omega/T)$ with $\gamma=2-1/\alpha$, obtained from the dynamically screened interaction $W(q,\omega)$ whose superdiffusive pole follows from the $1/r^\alpha$ interaction. The derivation uses a diffusive irreducible polarization in the RPA combination $W^{-1}=V^{-1}+\Pi_{\mathrm{irr}}$; for $\alpha<2$ the characteristic momentum at energy transfer $\Omega$ scales as $q_\Omega\sim(\Omega/D_\alpha)^{1/\alpha}$, making tangential scattering dominate and producing scale covariance. The paper identifies the optimal-doping marginal Fermi liquid with $\alpha\to 1^+$ ($\gamma\to 1^+$), and increasing doping with larger $\alpha$ and larger fitted $\gamma$, up to the diffusive value $\gamma=1.5$ for $\alpha>2$.

Load-bearing premise

The argument collapses if, over the relevant mesoscopic range, the true effective interaction between charge carriers is not a clean power law $V(r)\sim 1/r^\alpha$ (or if the irreducible polarization is not diffusive), because then the screened interaction lacks the superdiffusive pole from which $\gamma=2-1/\alpha$ is derived.

Editorial extensions

If this is right

  • Fitted nodal self-energy exponents become direct measurements of the effective interaction exponent $\alpha$ via $\gamma=2-1/\alpha$, turning ARPES into a probe of spatial nonlocality.
  • Scale-covariant strange-metal behavior can arise without a quantum critical point; spatial nonlocality is a sufficient organizing principle.
  • Around optimal doping the theory reduces to a marginal Fermi liquid ($\gamma=1$), and the quasiparticle residue vanishes logarithmically as $\alpha\to 1^+$.
  • If transport lifetime is proportional to inelastic lifetime, in-plane resistivity scales as $T^\gamma$ with doping-dependent $\gamma$, matching the reported resistivity exponents within the superconducting dome.
  • The superconducting state restores local screening, so the abrupt nonlocal-to-local transition in the normal state has no counterpart deep in the superconducting phase.

Reading between the lines

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

  • A natural next step the paper leaves implicit is to extract $\alpha(p)$ from independent measurements, such as momentum-resolved electron energy loss or tunneling spectra, and test the predicted $\gamma(p)$ relation.
  • The predicted superdiffusive pole $\omega=-iD_\alpha|q|^\alpha$ could be tested directly in quantum simulators with tunable long-range interactions, where the spatial exponent can be varied continuously.
  • The sharp prediction of a fixed $\gamma=1.5$ for $\alpha>2$ gives a threshold: if overdoped samples show fitted exponents passing through 1.5 in a way that tracks screening, that would cleanly distinguish nonlocal hydrodynamic scattering from competing Fermi-liquid-like contributions.
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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

4 major / 6 minor

Summary. This paper proposes that the continuously doping-dependent scale-covariant self-energy observed in nodal ARPES measurements of high-Tc cuprates can be explained by poorly screened, long-ranged effective repulsions V_alpha(r)=g_alpha/|r|^alpha. The author constructs a hydrodynamic screening model in which a diffusive irreducible polarization is combined with the nonlocal interaction via RPA, obtaining a superdiffusive pole omega=-i D_alpha |q|^alpha for alpha<2. Within G0W and after small-angle and tangential-momentum approximations, the inelastic quasiparticle decay rate takes the scale-covariant form Gamma_inel ~ A_gamma T^gamma Phi_gamma(omega/T) with gamma=2-1/alpha for 1<alpha<2, recovering a Marginal Fermi liquid as alpha->1 and diffusive gamma=1.5 for alpha>2. The paper interprets ARPES exponents as direct measures of the effective spatial nonlocality and argues that quantum criticality is not necessary. The underdoped regime is explicitly left beyond the calculation.

Significance. If the central mapping holds, the paper would provide a simple, quantum-criticality-free route to strange-metal scale covariance and would turn ARPES self-energy exponents into a measure of the spatial range of effective interactions. The derivation from Eq. (5) to Eq. (6) is internally coherent, and the End Matter supplies a complete Fermi golden rule account of the four emission/absorption processes and an exact evaluation of the tangential momentum integral. The marginal-Fermi-liquid limit is a useful check. However, the model's predictive content is weakened by the fact that alpha(p) is a free parameter inferred from the same ARPES data, and by the need for an additional, undetermined inelastic channel to reach the experimentally observed gamma~1.68. These issues limit the strength of the claims as currently stated.

major comments (4)
  1. [Main Text, Eq. (6) and item (vi)] For 1<alpha<2, Eq. (6) predicts gamma in (1,1.5); for alpha>2 the text fixes gamma=1.5. No admissible alpha produces gamma>1.5. The ARPES data of Ref. [17] that the paper claims to capture include gamma=2 beta ~1.68 at p~0.29. The response in item (vi) introduces an extra Fermi-liquid-like inelastic channel with undetermined coefficients and notes that the two forms are hard to distinguish. This is an external correction, not a prediction of the model, and it breaks the claimed one-to-one relation between the measured self-energy exponent and the nonlocality parameter alpha exactly in the overdoped regime emphasized in the abstract. The abstract's statement that the model 'naturally captures the optimally doped to overdoped regimes' is therefore not supported for the full stated doping range.
  2. [Setup] The paper explicitly states that it will not deduce from microscopics how alpha varies with doping, but will constrain alpha(p) from ARPES data. Since Eq. (6) is then inverted to read alpha from the same exponents that the model is said to 'capture', the comparison is partly a restatement of the input rather than an independent test. The mapping gamma=2-1/alpha is derived, not fitted, but the predictive content of the paper would be substantially strengthened by an independent constraint on alpha(p), for example from transport or from a microscopic calculation of the screened interaction.
  3. [Abstract and Fig. 1] The abstract presents 1<=alpha<=3 as a continuous interpolation between Mott insulating and Fermi liquid limits, but the calculated observable does not realize this interpolation: the formula gamma=2-1/alpha is derived only for 1<alpha<2, alpha>2 gives a fixed gamma=1.5, and alpha<1 is explicitly left untreated. The paper should state precisely which observable interpolates with alpha and should reconcile the abstract and Fig. 1 (whose caption restricts the captured regime to 1<=alpha<=2) with the body.
  4. [Screening, nonlocality and hydrodynamics] The entire scaling result rests on the postulate that the effective interaction is an exact power law V_alpha(r)=g_alpha/|r|^alpha over the mesoscopic range relevant to hydrodynamics. This is acknowledged as a postulate, but no microscopic estimate or experimental diagnostic for the power-law form is provided. Because the superdiffusive pole and therefore Eq. (6) fail if the interaction is not a clean 1/r^alpha over the relevant range, this assumption is load-bearing; the manuscript should either justify it from a microscopic model or identify an observable that can falsify it independently of the self-energy exponent.
minor comments (6)
  1. [References] Reference [23] contains a malformed DOI and reference [24] is a private communication; please update or replace them with published sources.
  2. [Fig. 1] Figure 1 caption states 'locality exponent 1≤alpha≤2' while the abstract and text allow 1≤alpha≤3; please reconcile the ranges.
  3. [Eq. (6)] After Eq. (6), the phrase 'for γ, α > 1' should read 'for α > 1' because gamma=2-1/alpha is determined by alpha.
  4. [Eq. (1)] In Eq. (1), the parameter C comparing the relative strengths of omega and T is not related to the scaling function Phi_gamma in Eq. (6); please define the connection.
  5. [Item (vi)] In item (vi), the proposed form 'Γ∼µ(ω^{3/4}+T^{3/4})^2 + λ(ω^2+T^2)' uses ambiguous notation (µ vs. μ, and the exponent structure of the first term); please clarify.
  6. [End Matter] The End Matter would benefit from a short derivation of F_alpha(0)=π/(alpha sin(π/2alpha)) rather than quoting it.

Circularity Check

2 steps flagged · score 6.0 of 10

Central 'direct probe of nonlocality' claim reduces to re-labeling the fitted ARPES exponent; the derived gamma = 2 - 1/alpha relation itself is non-circular.

  1. self definitional [Setup (after Eqn 1); Abstract]
    "We will not deduce from microscopics how α varies with doping p. Rather, we make several microscopically-agnostic calculations and constrain the evolution of α(p) from ARPES data. ... In our theory, spectroscopy-fitted exponents directly probe the charged fluid's effective spatial nonlocality."

    The only bridge between the observable gamma and the model parameter alpha is Eqn 6, gamma = 2 - 1/alpha. Because alpha(p) is explicitly fixed from the ARPES gamma(p) data, the statement that ARPES exponents 'directly probe' alpha is a restatement of the fitting condition: given gamma, alpha is defined as 2 - 1/gamma. The doping dependence of alpha is inherited from the fitted beta(p), not independently predicted. The derivation of Eqn 6 is non-circular, but the paper's load-bearing interpretive claim reduces to a dictionary between two parametrizations of the same fitted exponent.

  2. renaming known result [After Eqn 6]
    "Comparing to Eqn 1 proposed by [18], we identify γ= 2β, up to the dimensionless function."

    Eqn 1 is an empirical power-law form with beta(p) already fitted to ARPES data. Since alpha is not independently constrained, identifying gamma = 2 beta and then using gamma = 2 - 1/alpha simply defines alpha = 2 - 1/(2 beta). The continuously doping-dependent nonlocality exponent alpha is therefore the known 'power law liquid' exponent beta re-expressed in new coordinates, and the claim that spectroscopy 'directly probes' nonlocality is a renaming of the empirical pattern rather than an independent prediction.

full rationale

The analytic derivation leading to Eqn 6 is self-contained: given the postulated V_alpha(r) ~ 1/r^alpha and a diffusive irreducible polarization, the superdiffusive pole and the resulting gamma = 2 - 1/alpha follow from a well-defined integral, with alpha entering as a model parameter rather than as a fitted quantity inside the derivation. No load-bearing self-citation chain or imported uniqueness theorem is present; the references to Nishikawa/Saito and Kiselev are external and used as supporting evidence for superdiffusion. However, the paper's central interpretive statement that ARPES exponents 'directly probe' the effective spatial nonlocality is circular in the specific sense that alpha(p) is constrained from those same ARPES data through Eqn 6. Thus the optimal-to-overdoped 'capture' is partly a consistency check and partly a reparametrization of the empirical beta(p), not a prediction of gamma(p). The acknowledged need for an additional Fermi-liquid-like channel to reach gamma ~ 1.68 also limits the full-range claim, but that is an incompleteness/correctness issue rather than a circularity. Overall, one or more of the headline claims reduce by construction, warranting a partial-circularity score of 6.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model has one principal free parameter, alpha(p), plus scale-setting constants. It relies on a chain of plausible but unverified assumptions: the power-law form of the interaction, diffusive irreducible polarization with RPA, the superdiffusive hydrodynamic pole, and the G0W small-angle approximations. No fundamentally new entity is introduced; the nonlocality is encoded in the interaction exponent alpha.

free parameters (4)
  • alpha(p), effective interaction exponent = 1 <= alpha <= 3, inferred from ARPES exponents via gamma = 2 - 1/alpha, not independently measured
    The central model input. The paper postulates V(r) ~ 1/r^alpha with alpha monotonically increasing with doping and says it will not derive alpha from microscopics; instead it constrains alpha from the same ARPES data whose exponents it explains.
  • g_alpha, coupling prefactor = not determined
    Sets the overall scale of V_alpha and appears in A_gamma; absorbed into the normalization and not compared quantitatively to data.
  • D_alpha (or D), hydrodynamic transport coefficient = not determined
    Rescales momenta and frequency in W; enters through the combination D_alpha = D chi g_alpha and does not affect the exponent gamma.
  • omega_UV, ultraviolet cutoff = inserted by hand
    Required in the marginal Fermi liquid limit (gamma = alpha = 1) to define the logarithmic self-energy; not a physically derived scale.
assumptions (6)
  • ad hoc to paper Effective repulsion between charges has the exact power-law form V_alpha(r) = g_alpha/|r|^alpha for 1 <= alpha <= 3 over the mesoscopic scale relevant to hydrodynamics.
    Postulated rather than derived from a microscopic Hamiltonian; the doping dependence of alpha is also postulated in the Screening section.
  • domain assumption The irreducible polarization remains diffusive, Pi_irr(q,omega) = chi D q^2/(D q^2 - i omega), even when the interaction is nonlocal, and W is obtained by the RPA combination W^{-1} = V^{-1} + Pi_irr.
    Invoked in the Screening section before Eq. 2; the paper notes W can also be taken as the starting postulate, so the microscopic basis is not established.
  • domain assumption Hydrodynamic charge relaxation in 2D is superdiffusive for alpha < 2, with a pole omega = -i D_alpha |q|^alpha.
    Obtained from continuity and j = -sigma grad mu, called a hand-waving argument in footnote [28]; cited spin-chain results support the general phenomenon but not this charged 2D fluid with alpha-dependent screening.
  • domain assumption The self-energy is computed at leading order in G0W with no vertex corrections, and only the exchange of a single hydrodynamic density mode contributes.
    Standard many-body approximation, but its accuracy for a strongly nonlocal interaction is not tested.
  • domain assumption Scattering is dominated by small-angle processes with q << k_F and tangential momentum q_parallel, so the fermion dispersion can be linearized and |q| approximated by |q_parallel|.
    Approximations (A) and (B) stated in the Main Text; justified by the characteristic momentum q_Omega ~ (Omega/D_alpha)^(1/alpha) but not controlled for all phases.
  • domain assumption The measured ARPES exponent beta maps to the model scaling dimension through gamma = 2 beta.
    The paper equates Eqn 1 and Eqn 6 by identifying gamma = 2 beta, which assumes the inelastic term dominates the self-energy and that the k-dependence measured by Smit et al. does not alter the nodal exponent comparison.

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Cite this review

Pith. "Pith review of Scale-covariant liquid in nonlocal high Tc strange metals." pith.science (2026). https://pith.science/paper/DPIAB6AD

@misc{pith2026260803013,
  author       = {Pith},
  title        = {Pith review of: Scale-covariant liquid in nonlocal high Tc strange metals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPIAB6AD}},
  note         = {Machine review of arXiv:2608.03013}
}
abstract

Experiments in recent years on high $T_c$ superconductors find a puzzling nodal scale-covariant self-energy with an exponent varying continuously with doping. We propose a mechanism: nonlocality induced by poorly screened effective repulsions $V_{\alpha}(r) \sim 1/r^\alpha$, where a continuously doping-dependent exponent $1 \le \alpha \le 3$ naturally interpolates between the Mott insulating and Fermi liquid limits. We develop a phenomenology of hydrodynamic screening, finding a scale-covariant quasiparticle decay rate $\Gamma(\omega,T) \propto T^{\gamma} \Phi(\omega/T)$ in energy $\omega$ and temperature $T$, with $\gamma = 2-\frac{1}{\alpha}$ for nonlocal $ 1 < \alpha < 2$. Our results naturally capture the optimally doped to overdoped regimes, whereas the underdoped regime is qualitatively distinct. In our theory, spectroscopy-fitted exponents directly probe the charged fluid's effective spatial nonlocality. Nonlocality shows that quantum criticality is not necessary to explain scale-covariant phenomena.

Figures

Figures reproduced from arXiv: 2608.03013 by the authors.

Figure 1
Figure 1. FIG. 1. The proposed phase diagram from nonlocality. The op [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Locations in the Brillouin zone [ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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