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REVIEW 3 major objections 5 minor 179 references

Maximum of the compressibility in the 2D Hubbard model marks the crossover from a pseudogapped electronic liquid to a correlated Fermi liquid, and practically coincides with the doping where the antinodal spin-density-wave precursor crosses

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2026-08-03 03:55 UTC pith:NB6YUUF6

load-bearing objection A solid TPSC+ study that ties the cold-atom compressibility maximum to the antinodal SDW-precursor crossing and predicts a Knight-shift maximum, but the headline coincidence at U=7 rests on an approximation that underestimates spin correlations by ~30%. the 3 major comments →

arxiv 2602.06298 v3 pith:NB6YUUF6 submitted 2026-02-06 cond-mat.str-el

Pseudogap, Fermi liquid, Van Hove singularity and maxima of the compressibility and of the Knight shift as a function of doping in the two-dimensional Hubbard model

Y.M. Vilk , A.-M.S. Tremblay This is my paper
classification cond-mat.str-el
keywords pseudogapcompressibility maximumKnight shiftspin-density-wave fluctuationstwo-dimensional Hubbard modelTPSC+Van Hove singularityincommensurate spin fluctuations
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 argues that the doping at which the compressibility of the two-dimensional Hubbard model reaches a maximum is the thermodynamic fingerprint of the crossover from a pseudogapped electronic liquid to a correlated Fermi liquid. Using a non-perturbative approximation called TPSC+, it shows that this maximum nearly coincides with the doping at which the precursor of the lower spin-density-wave band at the antinodal point (π,0) crosses zero frequency, shifting the Van Hove singularity in the density of states from occupied to unoccupied energies. The paper also predicts that the uniform spin susceptibility (the Knight shift) has a maximum at the same doping at low temperature, and that both maxima appear for both weak and intermediate interaction strengths, driven by thermally excited critical spin fluctuations with a correlation length of only a few lattice spacings. A sympathetic reader would care because this explains recent cold-atom observations of a compressibility peak and connects a thermodynamic anomaly to a specific single-particle spectral event.

Core claim

On the paper's own terms: for the nearest-neighbor square-lattice Hubbard model, TPSC+ predicts that at low temperature the doping-dependent compressibility κ(δ) has a maximum at δ_max, and this maximum is located at essentially the same doping where the lower (π,π) spin-density-wave precursor band at the antinodal point crosses ω=0. For U=7 and T=0.0714 the numbers are δ_max=0.15 and crossing at δ=0.16; for U=3.69 they are 0.1 and 0.11. The same spectral crossing produces a maximum in the uniform spin susceptibility χ_sp(0,0)(δ), which survives to higher temperatures than the compressibility maximum because it does not require the Fermi function to resolve the peak. In both interaction regi

What carries the argument

The central object is the precursor of the lower (π,π) spin-density-wave band at the antinodal point k=(π,0), a peak in the spectral function below the Fermi level whose position relative to ω=0 changes with doping. The identity that carries the argument is δ_max ≈ δ_{ω=0}: the doping of the compressibility and spin-susceptibility maxima is (almost) the doping at which this precursor crosses zero. The machinery producing the precursor is the TPSC+ self-energy built from static classical spin fluctuations, with two analytical criteria: at regular Fermi-surface points a pseudogap opens when the correlation length ξ exceeds v_F/(πT), while at the antinodal Van Hove point the easier condition ξ>

Load-bearing premise

The load-bearing premise is that TPSC+ is quantitatively trustworthy at U≈7 and T≈0.0714, the regime where the paper itself reports that it underestimates spin correlations by 27–33% and underestimates pseudogap effects, so that the predicted value of δ_max and its coincidence with the spectral crossing are not shifted by the approximation.

What would settle it

A determinant or diagrammatic quantum Monte Carlo calculation at U=7 and T=0.0714 in the thermodynamic limit, locating both the maximum of κ(δ) and the doping at which the antinodal SDW precursor crosses ω=0; if these two dopings differ beyond numerical uncertainty, or if χ_sp(0,0)(δ) has no maximum at T=0.1, the central claim is refuted.

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

If this is right

  • If the central claim is correct, the cold-atom compressibility maximum is a pseudogap-to-Fermi-liquid crossover, not a signature of Mott criticality, and it should appear even at weak interaction strengths.
  • The predicted maximum in the uniform spin susceptibility χ_sp(0,0)(δ), measurable as a Knight shift, should appear at somewhat higher temperatures than the compressibility maximum and could be easier to observe experimentally.
  • As temperature decreases, δ_max(T) moves away from half-filling toward the quantum critical point of the spin-density-wave transition.
  • At sufficiently low temperature, incommensurate spin fluctuations should produce more than two SDW precursor peaks in the spectral function and density of states, a signature accessible to photoemission-type measurements.
  • The large charge compressibility deep in the pseudogap regime, combined with an effective charge vertex that can turn negative, raises the possibility of SDW-driven charge-density-wave or stripe precursors at very low temperature.

Where Pith is reading between the lines

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

  • If the spectral-crossing mechanism is right, the compressibility maximum is a thermodynamic proxy for the antinodal pseudogap edge, so the crossover line δ_max(T) could be mapped with equation-of-state measurements alone, without resolving the spectrum.
  • The same crossing logic suggests that other thermodynamic quantities sensitive to the Van Hove singularity, such as the doping derivative of entropy or the specific heat coefficient, should show extrema at nearby dopings; this is a testable extension not made in the paper.
  • Because TPSC+ underestimates spin correlations by 27–33% at U=7, exact numerical methods may place δ_max and the spectral crossing at slightly different dopings; if the two features remain tied, the mechanism survives, but if they separate, the quantitative coincidence is an artifact of the approximation.
  • The predicted negative effective charge vertex at low temperature implies that density correlations should develop incommensurate structure; a low-temperature measurement of the density structure factor near δ_max could test the precursor-to-stripe scenario.

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

3 major / 5 minor

Summary. The paper uses the TPSC+ approach for the nearest-neighbor square-lattice Hubbard model to study the doping-driven crossover from a pseudogapped electronic liquid to a correlated Fermi liquid. It reports that the isothermal compressibility κ(δ) has a maximum that, for U=7 and T=1/14, occurs at δ_max≈0.15, practically coinciding with the doping δ≈0.16 at which the lower antinodal SDW precursor crosses ω=0 (Sec. V A, Figs. 6–7). The same mechanism is claimed to produce a maximum in the uniform spin susceptibility χ_sp(0,0)(δ) at comparable doping, and both maxima are predicted for weak and intermediate U. The paper also discusses finite-size effects, incommensurate spin fluctuations at low T, and a breakdown of the equivalence between ∂n/∂μ and χ_ch(0,0) within TPSC+ (Sec. VI).

Significance. If the central coincidence is robust, the paper provides a concrete microscopic mechanism connecting a thermodynamic anomaly (the compressibility maximum) to the reconstruction of the single-particle spectrum, and it makes a falsifiable prediction for the Knight shift maximum versus doping. The calculations are internally consistent, use sum-rule-fixed vertices rather than fitted parameters, and reproduce the cold-atom compressibility quantitatively at U=3.69. The finite-size analysis and the explicit discussion of the method's limitations are also valuable. However, the quantitative coincidence at U≈7 rests entirely on an approximation that the paper itself reports underestimates spin correlations by 27–33% at that interaction, so the robustness of the central claim is not yet established.

major comments (3)
  1. [Sec. V A, Figs. 6–7] The central claim, δ_max≈0.15 with the antinodal precursor crossing at δ=0.16, is a TPSC+ result at U=7, T=0.0714. At the same U, Sec. III C reports that TPSC+ underestimates equal-time spin correlations by 27–33%. Since the proposed mechanism is SDW fluctuations, a doping-dependent error of this size could shift both δ_max and the crossing doping. The near coincidence could survive if the error is uniform in doping, but that is not demonstrated. A robustness test—for example, a comparison with DiagMC or DQMC near T≈0.07, or a controlled sensitivity study in U_sp/χ_sp—is needed before the coincidence can be stated as more than an internally consistent TPSC+ result.
  2. [Sec. VI, Fig. 14] The paper shows that TPSC+ does not satisfy the exact identity κ=∂n/∂μ=χ_ch(0,0). At T=0.025, the first-level χ_ch(0,0)(δ) is monotonic and shows no maximum, while κ(δ) does; at T=0.15 neither shows a well-defined maximum. No comparison is shown at the T=0.0714 used for the central claim. This leaves open the possibility that the maximum in κ(δ) and its coincidence with the spectral crossing are artifacts of the second-level self-energy rather than a robust property. The manuscript should either provide a consistency check at T≈0.0714 or explicitly justify why the second-level result is the reliable one in that regime.
  3. [Abstract] The abstract states that TPSC+ correctly predicts a maximum in the temperature dependence of the Knight shift, χ_sp(0,0)(T), consistent with cold-atom experiments (Ref. [45]). The body of the paper contains no such calculation or quantitative comparison; the only χ_sp results are versus doping. This claim is currently unsupported and should either be backed by a dedicated figure and comparison or removed from the abstract and conclusions.
minor comments (5)
  1. [Sec. V A] The text near Fig. 7 says the U=3.69 case has "a smaller δ_max = 0.15", but earlier in the same section and in Figs. 4–5 δ_max=0.1 for U=3.69. This appears to be a typo and should be corrected.
  2. [Fig. 9 caption] The caption lists k=(3π/4, π/4) while the main text refers to k=(3π/8, π/8) for the same panel. Please make the notation consistent.
  3. [Sec. IX] There are several typos in the conclusion, e.g., "χ_sp(,0,0)" on two occasions. These should be cleaned up.
  4. [Fig. 4 caption] The caption says "The doping-dependent isothermal compressibility κ(δ) shows a maximum for T=0.1", but the figure also includes T=0.15, where no maximum is visible. Clarify which curves show maxima.
  5. [Eq. (14)] The factor \tilde g_{fb} is used in Eq. (14) before its values are introduced later in Sec. II B 2. A brief definition at first use would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the TPSC+ predictions are computed, not fitted to the claimed maxima; the δ_max/crossing coincidence is an internal consistency check, not a construction.

full rationale

The central claims are generated by solving the TPSC+ equations self-consistently. The compressibility maximum is obtained by numerical differentiation of n(μ) and is not used to set any parameter; no parameter is fitted to the experimental κ(δ) maximum. The claimed coincidence between δ_max≈0.15 and the antinodal precursor crossing δ≈0.16 is a diagnostic read off the same calculated Green's function (Eqs. 15–16), not a definition: the two dopings differ, so they are not identical by construction. Eq. (16) is an exact relation, and interpreting the maximum through the motion of a DOS peak is an elucidation, not a circular derivation. The g-tilde weights in the self-energy were fixed previously by Monte Carlo benchmarking [56,68], and the predicted observables are not the fit targets. Self-citations to [55,56,68] validate a method, not the specific result, and no uniqueness theorem is imported. The acknowledged limitations (TPSC+ underestimates spin correlations by 27–33% at U=7, underestimates pseudogap effects, and ∂n/∂μ disagrees with χ_ch(0,0) at low T) are accuracy concerns that affect any TPSC+ prediction; they do not make the derivation circular. The paper is benchmarked against external cold-atom data and DiagMC, so it is not relying on an unverified self-citation chain. The abstract's assertion about a maximum in χ_sp(0,0)(T) is not explicitly demonstrated in the body, but that is a support gap, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central claims rest on the TPSC+ approximation scheme, whose vertex closures are treated as axioms, and on Padé analytic continuation. The only hand-tuned numerical factor is the g-tilde combination, fitted to earlier Monte Carlo benchmarks. No new particles, forces, or conserved quantities are introduced.

free parameters (1)
  • g-tilde and g-tilde_fb vertex weighting factors = g-tilde_fb=3/8, g-tilde=1/4 for U<3; g-tilde_fb=1/4, g-tilde=3/8 for U≥3
    These factors enter the feedback self-energy (Eq. 14) and the full self-energy (Eq. 15). The paper states they were chosen by comparing to Monte Carlo results for U<3 versus U≥3 in previous work [56,68]. They affect spectral functions and hence the location and sharpness of the computed maxima.
axioms (5)
  • domain assumption TPSC+ closure for the spin vertex: U_sp = U <n_up n_down> / (<n_up><n_down>) (Eq. 12)
    This ansatz is load-bearing for the renormalized spin susceptibility and the self-energy feedback. It is benchmarked but not derived from first principles.
  • domain assumption Momentum- and frequency-independent charge vertex U_ch determined from the charge sum rule
    Section VI shows that this approximation fails deep in the pseudogap regime, where ∂n/∂μ and χ_ch(0,0) differ qualitatively. The charge-response discussion and the stripe precursor speculation depend on this assumption.
  • domain assumption Padé analytic continuation reliably resolves the spectral function, including closely spaced incommensurate quasi-singularities
    All real-frequency results are obtained from Padé approximants of Matsubara data. The multi-peak spectral predictions in Sec. V B and Figs. 9 and 13 depend on the stability of this continuation.
  • domain assumption The single-band nearest-neighbor Hubbard model is a sufficient description of the cold-atom simulator and of the cuprate-relevant physics discussed
    The entire analysis is carried out on the nearest-neighbor one-band Hubbard model with t=1. The relevance to cuprates is argued qualitatively in Sec. VIII D, not derived.
  • standard math Mermin-Wagner theorem applies, so the antiferromagnetic transition is at T=0 and the pseudogap is driven by critical thermal fluctuations
    This is a rigorous theorem for the 2D Hubbard model with short-range interactions, and it underpins the interpretation of the crossover in terms of thermal SDW fluctuations.

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read the original abstract

Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a Hubbard model simulator with nearest-neighbor hoppings \cite{kendrick2025pseudogap} showed that the isothermal compressibility $\kappa(\delta)$ has a maximum as a function of doping $\delta$. Here we use the two-particle self-consistent plus (TPSC+) approach to explain these experiments and connect the maximum in $\kappa(\delta)$ to the single-particle spectrum transformation from the pseudogapped to the metallic regime, elucidating the nature of the pseudogap (PG). The maximum in $\kappa(\delta)$ practically coincides with the doping where the precursor of the lower $(\pi,\pi)$ spin density wave (SDW) band at the antinodal point crosses zero frequency $\omega=0$. The Knight shift, $\chi_{sp}(0,0)(\delta)$, should also exhibit a maximum versus doping. Additionally, TPSC+ correctly predicts a maximum in the temperature dependence of the Knight shift, $\chi_{sp}(0,0)(T)$, consistent with recent ultracold atom experiments \cite{chalopin2026observation}.These maxima should exist at low temperatures ($T$) in both the intermediate $U \approx U_{Mott}$ and weak $U < U_{Mott}$ interaction limits due to critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at $\delta_{max}(T)$ can be small, controlled by dynamic rather than static critical thermal fluctuations. The SDW fluctuations are incommensurate at $\delta=\delta_{max}$. At low $T$, multiple peaks in the incommensurate spin susceptibility lead to more than two SDW precursor peaks in the spectral function and density of states. By accessing parameter regimes relevant to cuprates, including further-neighbor hopping ($t', t''$) and low temperatures, our work provides a high-impact tool for further studies.

Figures

Figures reproduced from arXiv: 2602.06298 by A.-M.S. Tremblay, Y.M. Vilk.

Figure 1
Figure 1. Figure 1: Comparison of the TPSC+ doping dependence of the com [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the experimental equal-time spin cor [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of the finite system size on the doping dependence [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dependence of the compressibility κ on doping for four different temperatures, T = (0.15, 0.1, 0.0714, 0.025), and two dif￾ferent interaction strengths, U = (3.69, 7). The smaller interaction strength represents a weak-interaction case, U = 3.69 < UMott, while the larger interaction strength, U = 7 ≈ UMott, corresponds to an intermediate-interaction case. The results are obtained in the thermodynamic limit… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the density of states with doping for [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dependence of the uniform spin susceptibility [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the spectral function A(k, ω) with doping for U = 7 and T = 0.0714. For these parameters, a well-defined pseudogap is present at half-filling (δ = 0) for both the nodal and antinodal momentum points. As doping increases, the peak corre￾sponding to the precursor of the lower AFM band moves closer to ω = 0 for both nodal and antinodal points. At doping δ = 0.13, the peak at the nodal point has a… view at source ↗
Figure 9
Figure 9. Figure 9: Effect of the incommensurate spin fluctuations on the [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Part of the Brillouin zone and two of the directions used [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the spectral function Ak(ω) as momentum crosses the Fermi surface (EDC curves) along two directions defined in [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison between the doping dependence of [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Semi-logarithmic plot for the dependence of the corre [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗

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