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REVIEW 4 major objections 6 minor 101 references

Resilient strange metal at an unconventional quantum critical point in $d=2$

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read T-linear resistivity can appear at a 2D Hubbard quantum critical point without a T-linear self-energy along the Fermi surface.

desk verdict A new set of exponents and a clean counterexample to the T-linear self-energy assumption, but the first-order possibility and OZ ansatz mean the 'unconventional QCP' label is not yet nailed down. read the letter →

arxiv 2608.05988 v1 pith:6KZDEOYJ submitted 2026-08-06 cond-mat.str-el

Jérôme Leblanc , A.-M. S. Tremblay This is my paper · ORCID
classification cond-mat.str-el
keywords HubbardmodelstrangemetalT-linearresistivityquantumcriticalpointspin-densitywaveorderKohnanomalynon-FermiliquidTPSC+
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

The paper predicts strange-metal behavior in the two-dimensional Hubbard model with only nearest-neighbor hopping, at the quantum critical point where incommensurate spin-density-wave order gives way to a Fermi liquid. Using the improved two-particle self-consistent approach, the authors find that the dc resistivity is linear in temperature from $T = 0.003t$ up to roughly $T = 0.3t$, even though the imaginary part of the single-particle self-energy does not follow a single power law anywhere on the Fermi surface and quasiparticle weight is strongly temperature- and momentum-dependent. The central claim is that this separates T-linear transport from the usual assumption that a T-linear self-energy is required. A sympathetic reader should care because it identifies a concrete microscopic route to strange-metal behavior in a model that cold-atom experiments can realize.

What carries the argument

The load-bearing object is the Ornstein-Zernike form for the retarded spin susceptibility, Eq. (5), centered on the incommensurate wavevectors $Q_i$, together with the microscopic scales $\xi_0^2$ and $\Gamma_0$ extracted from the irreducible particle-hole susceptibility. Kohn anomalies arise because $Q_i$ connects Fermi-surface points with nearly antiparallel velocities ($\cos\theta \approx -0.99$ for the bare dispersion and $-0.999$ for the renormalized dispersion), which causes $\xi_0^2 \sim 1/T^{0.98}$ and $\Gamma_0 \sim 1/T^{0.70}$ instead of being temperature-independent as in Hertz-Millis theory. These temperature-dependent scales combine to produce the exotic exponents and, through the current-current response evaluated in the Kubo formula, the T-linear resistivity. The calculation is carried out with the improved two-particle self-consistent approach, a non-perturbative method that enforces spin and charge sum rules and the Mermin-Wagner theorem.

What would settle it

Compute the same quantum critical point with a method that resolves the full momentum dependence of the spin susceptibility at large wavevectors (for example diagrammatic quantum Monte Carlo or a theory including the pseudo-nesting cusps) and check whether $\chi_{\mathrm{sp}}(Q_i, 0)$ still follows $1/T^{0.92}$ and $\xi_{\mathrm{sp}}^2$ still follows $1/T^{1.96}$ below $T \approx 0.02t$; if the exponents revert to Hertz-Millis values or the resistivity deviates from linearity, the central claim fails.

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Extended reading notes

Core claim

At filling $n = 1.1827$ with $U = 6t$ and $t' = 0$, the zero-frequency spin susceptibility at the incommensurate wavevector scales as $\chi_{\mathrm{sp}} \sim 1/T^{0.92}$, the squared correlation length as $\xi_{\mathrm{sp}}^2 \sim 1/T^{1.96}$, and the characteristic spin-fluctuation frequency as $\omega_{\mathrm{SF}} \sim T^{1.26}$, giving exponents $\gamma = 0.92$, $\nu = 0.98$, $z = 1.28$ modulo logarithmic corrections. The authors attribute these unconventional exponents to Kohn anomalies, that is, Fermi-surface points separated by the spin-density-wave wavevector whose renormalized velocities are almost exactly antiparallel, combined with near-nesting, which makes microscopic parameters such as the Landau damping constant temperature-dependent. At the same point, the single-particle self-energy has no uniform temperature power law along the Fermi surface and $Z_k$ is strongly temperature- and momentum-dependent, so Landau quasiparticles are absent. Nevertheless, the Kubo-formula dc resistivity is linear in temperature over two decades, which the paper states is evidence that a linear-in-temperature self-energy along the Fermi surface is not a necessary condition for strange-metal resistivity.

Load-bearing premise

The argument rests on the susceptibility keeping a single-peak Lorentzian (Ornstein-Zernike) form with a constant damping constant all the way down to $T = 0.003t$, even though the paper notes that non-analytic square-root cusps from pseudo-nesting make that form fail at large wavevectors; if those cusps intrude into the fitting window, or if the transition is actually first-order as a claim the paper cites suggests, the extracted exponents and the T-linear resistivity would change.

Editorial extensions

If this is right

  • The T-linear resistivity at the quantum critical point remains intact even though Landau quasiparticles are absent, and its slope is slightly larger than Planckian dissipation with $\alpha \approx 2$.
  • The set $(\gamma, \nu, z) \approx (0.92, 0.98, 1.28)$ does not match any known universality class, so the critical point is a new, model-specific one driven by Kohn anomalies and near-nesting.
  • At $t' = 0$, the same critical behavior holds for both hole and electron doping by particle-hole symmetry, making the prediction directly testable in cold-atom and diagrammatic quantum Monte Carlo setups.
  • Vertex corrections do not destroy the linearity of the resistivity; earlier work with vertex corrections changes only the slope of the T-linear term.
  • The linear-in-temperature resistivity does not require the characteristic spin-fluctuation frequency to saturate, as phonon mechanisms would; $\omega_{\mathrm{SF}}$ vanishes as $T \to 0$ and yet the linear law survives.

Reading between the lines

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

  • If the claim holds, T-linear resistivity should no longer be read as a direct probe of T-linear single-particle scattering; transport and single-particle lifetime are decoupled at this quantum critical point.
  • The same mechanism, Kohn anomalies producing temperature-dependent microscopic scales, may occur in other two-dimensional itinerant systems with near-nested Fermi surfaces, so the unconventional exponents could be more generic than the specific nearest-neighbor Hubbard model.
  • A cold-atom measurement of the dc resistivity or spectral function at $n \approx 1.183$, $U = 6t$, $t' = 0$ would be a clean test: seeing $\rho \propto T$ with $Z_k$ collapsing near the anti-node would confirm the scenario.
  • The paper leaves open the possibility that quasiparticles reappear at asymptotically low temperature; if instead the T-linear resistivity persists to arbitrarily low $T$, the strange-metal behavior would be a genuine zero-temperature phase rather than a crossover.
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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. The manuscript studies the two-dimensional Hubbard model on the square lattice with nearest-neighbor hopping only, at U=6t, using the TPSC+ approximation. The authors identify the filling n=1.1827 as a quantum critical point separating a Fermi liquid from an incommensurate spin-density-wave phase, based on the temperature dependence of the spin susceptibility and related quantities. They extract critical exponents gamma=0.92, nu=0.98, z=1.28 (modulo logarithmic corrections), which they attribute to Kohn anomalies and near-nesting of hot spots. They find that the zero-frequency self-energy along the Fermi surface has no uniform power-law temperature dependence and that the quasiparticle weight Z_k is strongly T- and k-dependent, yet the dc resistivity computed from a gauge-invariant Kubo formula is linear in T from T=0.003t to T=0.3t. The central conceptual claim is that T-linear resistivity can arise without a T-linear self-energy along the Fermi surface.

Significance. If the results hold, the paper makes a falsifiable prediction: at the specific doping n=1.1827, cold-atom and diagrammatic quantum Monte Carlo experiments should find T-linear resistivity together with an unconventional exponent set (gamma=0.92, nu=0.98, z=1.28). The claim that linear-in-T transport need not imply a linear-in-T zero-frequency self-energy is conceptually important and directly testable. Strengths of the manuscript include that the central quantities are computed outputs rather than fitted targets, the gauge-invariant conductivity prescription in Eqs. (9)-(13) with the f-sum-rule check in Fig. 4, the explicit admission that the Ornstein-Zernike form fails at large wave vectors, and consistency with earlier TPSC+ benchmarking. The main risks are that TPSC+ is an approximate method without controlled error estimates, that the continuous nature of the QCP is not established against the first-order scenario admitted in footnote [38], and that the quoted exponent errors appear to be statistical only, with at least one internal consistency check failing at roughly three times the quoted error.

major comments (4)
  1. [Spin fluctuations at the QCP; footnote [38]] The continuous nature of the QCP is load-bearing for the title claim, yet the text concedes that "there has been a claim that instead of being continuous, the transition could be first-order" and that checking this "would require extensive work." TPSC+ does not produce a free energy and the paper provides no hysteresis or order-parameter diagnostic, so the power-law chi_sp(Q_i,0) ~ 1/T^{0.92} in Fig. 1(a) is also consistent with a finite-temperature crossover above a weak first-order transition; in that case gamma=0.92, nu=0.98, z=1.28 are not true critical exponents and the central claim is unsupported. The reference for the first-order claim is a bare "[?]" and cannot be checked. Please either provide a concrete diagnostic that distinguishes the two scenarios (for example, a free-energy comparison, a scaling-collapse test, or an explicit discussion of QMC results at these dopings) or qualify the central claim as a putative QCP.
  2. [Spin fluctuations at the QCP; Eqs. (5)-(8), Fig. 1] The Ornstein-Zernike amplitude relation chi_sp(Q_i,0) = (2/U_sp) xi_sp^2/xi_0^2 is not satisfied by the paper's own exponents: combining Fig. 1(a) (gamma=0.92 +/- 0.003) with Fig. 1(c) (xi_0^2 ~ T^{-0.98 +/- 0.02}) predicts a chi_sp exponent of 1.96 - 0.98 = 0.98, not 0.92; equivalently, the text's deduction xi_sp^2 ~ 1/T^{1.90} differs from the direct fit 2nu = 1.96 +/- 0.02 in Fig. 1(b). The discrepancy of 0.06 is about three times the quoted errors and is not explained by the "modulo logarithmic corrections" caveat, and the near-constancy of U_sp in Fig. 1(e) rules that out as a resolution. Because nu and, through omega_SF ~ T^{z nu}, z are headline results, the inconsistency must be resolved or quantified, for example by reporting the fit windows used in each panel and the effect of including logarithmic corrections.
  3. [Spin fluctuations at the QCP; Eqs. (6)-(7), Fig. 1(c),(d)] The exponents of xi_0^2 and Gamma_0 are derived from the q-curvature of chi^(2)(Q_i,0) and from the omega -> 0 slope of Im chi^(2)(Q_i,omega), where the lowest bosonic Matsubara frequency available is 2 pi T. The omega -> 0 extrapolation at each temperature and the choice of q-fitting points introduce systematic errors that are not contained in the quoted +/- 0.02, and z = 1.28 inherits these errors directly from the Gamma_0 exponent 0.70. Please state the extrapolation procedure explicitly and show the sensitivity of the extracted exponents to the number of Matsubara frequencies and to the fitting window.
  4. [Resistivity; Fig. 3, Fig. 4] The headline transport result, rho linear in T from T=0.003t to T~0.3t, is presented without error bars and without a stability analysis of the maximum-entropy continuation (default model, number of retained spectral moments, and the omega -> 0 extrapolation underlying Re sigma_xx(0)). In addition, the inset of Fig. 4 appears to show the f-sum-rule relative difference growing to the 10% level at the lowest temperatures; the impact of this violation on the low-T continuation should be quantified. Because the "resilient strange metal" claim rests entirely on Fig. 3, a sensitivity analysis is needed before the linear-in-T behavior can be considered established.
minor comments (6)
  1. [Footnotes and Data availability] Footnote [38] and the SM description in reference [88] contain unresolved citations ("[?]"), and the Data availability section contains "[REFERENCE]" placeholders; the manuscript should be compiled with these resolved before it can be properly evaluated.
  2. [Throughout] The spelling "Ornstein-Zernicke" appears several times (for example, in and immediately after Eq. (5)); the standard spelling is "Ornstein-Zernike".
  3. [Discussion and conclusion] The sentence "These exponents do not correspond to any known universality class" is very strong given that the exponents are extracted modulo logarithmic corrections from an approximate method; it would be safer to state that the exponent set differs from the known classes surveyed, or to provide an explicit survey of candidate classes.
  4. [Fig. 1(a)] The text states that deviations at n=1.1800 and n=1.1900 confirm the critical filling, but the two deviations are not described (saturation versus a different exponent); please specify the expected behavior on the ordered and disordered sides and how the curves in Fig. 1(a) display it.
  5. [End Matter, Fig. 4] The inset of Fig. 4 has unclear axis presentation, with the tick "10" lacking context; please label both axes explicitly and give the numerical value of the relative difference at the lowest temperature.
  6. [Introduction] The statement that QCPs "might explain why T-linear resistivity not only begins near T=0, but also extends to large T" is speculative and is not addressed by the calculation; consider removing it or clearly marking it as an open question.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: critical exponents and T-linear resistivity are computed outputs, not fitted inputs.

full rationale

The derivation is self-contained as a numerical study. The QCP filling n=1.1827 is located by searching for the filling at which the computed maximum spin susceptibility chi_sp(Q_i,0) follows a power law (Fig. 1a); this is the standard operational definition of a quantum critical point, and the extracted exponents are outputs of the subsequent fits, not inputs chosen to produce a preselected universality class. The relation xi_sp^2 ~ xi_0^2 chi_sp follows algebraically from Eqs. (2), (6) and (8), so stating it is internal consistency rather than circular reduction. The T-linear resistivity in Fig. 3 is a directly computed Kubo quantity after MaxEnt continuation; no parameter is adjusted to force linearity. The same-group citations [61] and [93] provide method benchmarks and a separate vertex-correction calculation at a different commensurate QCP; they do not define the present exponents or the linear resistivity, and the vertex-correction result is external to the present paper's fitted values. The acknowledged limitations, namely footnote [38] conceding that the transition could be first-order and the text noting that the Ornstein-Zernike form fails at large wave vectors because of pseudo-nesting cusps, are model-correctness risks, not circular steps, because even if they invalidate the critical-exponent interpretation, the exponents and resistivity are still computed outputs rather than built-in assumptions.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The calculation carries a modest but real ledger: one tuned filling, a chosen fitting window, unspecified MaxEnt settings, and several modeling assumptions inherited from TPSC+. The largest costs are the assumed validity of the Ornstein-Zernike form and the reliability of MaxEnt continuation, both acknowledged in the text. No new entities are introduced.

free parameters (3)
  • QCP filling n = 1.1827
    Chosen as the filling where chi_sp(Q,0) displays the cleanest 1/T^0.92 power law; neighboring fillings 1.1800 and 1.1900 deviate. The exponents and resistivity claims are computed at this single filling.
  • Power-law fitting window = T/t in [0.01, 0.1] for Fig. 1; rho fit for T <= 0.2t
    The exponents gamma, nu, z and the linear-rho claim depend on the chosen temperature range. No sensitivity analysis is reported.
  • Maximum entropy continuation settings = unspecified
    Analytic continuation of the current-current correlator to real frequencies requires a default model and error tolerance; none are given, so rho(T) carries unquantified dependence on these choices.
assumptions (6)
  • domain assumption The SDW transition at n = 1.1827 is continuous, or sufficiently continuous for QCP scaling to apply.
    Footnote [38] notes a claim of first-order behavior and defers it; if the transition is first-order, there is no quantum critical point and the extracted exponents are undefined.
  • domain assumption TPSC+ is quantitatively accurate enough at U = 6t and T down to 0.003t to capture the asymptotic quantum critical behavior.
    The method is approximate and benchmarked mainly on other observables and fillings; no controlled error estimate is given for the QCP exponents.
  • domain assumption The retarded spin susceptibility has a single-peak Ornstein-Zernike form near each Qi, Eq. (5), with a temperature-independent Gamma0.
    This form is used to extract xi_sp, omega_SF and z. The text admits OZ fails at large q due to pseudo-nesting cusps, but small-q validity is not proven.
  • domain assumption Logarithmic corrections to the power laws are negligible.
    Stated at the start of 'Spin fluctuations at the QCP'; if logs are relevant, the exponents gamma, nu, z and the scaling relation would need revision.
  • domain assumption Maximum entropy analytic continuation reliably determines Re sigma(0), and hence rho, without unquantified bias.
    The resistivity in Fig. 3 depends on OmegaMaxEnt, which requires a default model and has known non-uniqueness; no error bars are reported.
  • domain assumption Vertex corrections to the Kubo conductivity do not change the T-linear form of rho at this incommensurate QCP.
    Stated in the 'Resistivity' section: prior TPSC work [93] at a commensurate QCP found vertex corrections only change the slope. This transfers a result to the incommensurate case without a direct check.

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Cite this review

Pith. "Pith review of Resilient strange metal at an unconventional quantum critical point in $d=2$." pith.science (2026). https://pith.science/paper/6KZDEOYJ

@misc{pith2026260805988,
  author       = {Pith},
  title        = {Pith review of: Resilient strange metal at an unconventional quantum critical point in $d=2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KZDEOYJ}},
  note         = {Machine review of arXiv:2608.05988}
}
abstract

Properties of the two-dimensional Hubbard model with nearest-neighbor hopping are under scrutiny in cold-atom experiments and diagrammatic quantum Monte Carlo. Given the controlled nature of these approaches, it is timely to make predictions about strange-metal behavior and its relation to quantum-critical properties. Even for interaction strengths below the Mott transition, several peculiarities occur at the quantum critical point separating Fermi liquid and incommensurate spin-density wave order. Here, we predict, using the non-perturbative improved two-particle self-consistent approach, that for correlation lengths ranging from about one to one hundred lattice spacings, Kohn anomalies on the underlying Fermi surface lead to unconventional critical exponents. The temperature dependence of the single-particle self-energy acquires strong momentum dependence along the Fermi surface with no clear evidence of Landau quasiparticles. Nevertheless, we observe strange-metal behavior, namely, resistivity that scales linearly with temperature as $T\rightarrow{0}$. Our work shows that a linear temperature dependence of resistivity arises without a linear-in-temperature self-energy along the Fermi surface, as is often assumed.

Figures

Figures reproduced from arXiv: 2608.05988 by the authors.

Figure 1
Figure 1. FIG. 1. Temperature dependence of spin fluctuation-related quan [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Temperature dependence of resistivity at the SDW QCP [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Comparison between the left-hand side (yellow circles) [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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