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Dynamical scaling near the pseudogap quantum critical point of the two-dimensional Hubbard model

T0 review · 2 major / 2 minor · reviewed 2026-05-15 · grok-4.3

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Pith's one-line read The two-dimensional Hubbard model shows ω/T scaling of the form tanh(ω/2T) in spin and current susceptibilities near its pseudogap quantum critical point.

desk verdict This 4-patch DCA+NRG calculation finds emergent tanh(ω/2T) scaling in spin and current susceptibilities plus vertex-dominated 1/T conductivity near the Hubbard pseudogap QCP, but the coarse patching leaves room for artifacts. read the letter →

arxiv 2605.15060 v1 pith:EWFB22QV submitted 2026-05-14 cond-mat.str-el

Mathias Pelz , Gabriel Kotliar , Jan von Delft , Andreas Gleis This is my paper · ORCID
classification cond-mat.str-el
keywords two-dimensionalHubbardmodelpseudogapquantumcriticalpointdynamicalscalingopticalconductivitystrangemetalmarginalFermiliquidclusterapproximationcupratesuperconductors
checked against Cost.FunctionalEquationFoundation.ArrowOfTime
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 examines dynamical scaling in the quantum-critical region of the two-dimensional Hubbard model at the transition from pseudogap metal to Fermi liquid. Using four-patch dynamical cluster approximation with a numerical renormalization group solver, the calculations access real-frequency response functions over wide temperature ranges. Near critical doping the imaginary parts of the local spin and cluster-current susceptibilities collapse onto the universal function tanh(ω/2T). The cluster contribution to the real part of the optical conductivity follows the related form T σ'_cl(ω,T) ∼ tanh(ω/2T) / (ω/T). These results produce a 1/T cluster dc conductivity dominated by vertex corrections and are accompanied by a marginal-Fermi-liquid nodal self-energy, yielding strange-metal-like transport that matches several qualitative features seen in cuprate experiments.

What carries the argument

The ω/T scaling form χ''(ω,T)∼tanh(ω/2T) for susceptibilities together with the corresponding conductivity scaling Tσ'cl(ω,T)∼tanh(ω/2T)/(ω/T), extracted from four-patch DCA-NRG real-frequency spectra.

What would settle it

A larger-cluster calculation or different solver that produces clear deviations from the tanh(ω/2T) form in the susceptibility spectra at the lowest accessible temperatures would falsify the reported scaling.

Watch

Extended reading notes

Core claim

Close to the critical doping, the local spin and cluster-current susceptibility spectra exhibit x=ω/T scaling of the form χ''(ω,T)∼tanh(x/2), and the cluster contribution to the optical conductivity obeys Tσ'cl(ω,T)∼tanh(x/2)/x, implying a 1/T cluster dc conductivity. In the scaling regime the vertex contribution to the cluster optical response is much larger than the bubble contribution. Evidence is also found for a marginal-Fermi-liquid nodal self-energy. This combination implies strange-metal optical transport in the quantum critical region.

Load-bearing premise

The four-patch DCA with NRG solver faithfully captures the low-energy dynamics and vertex corrections near the pseudogap QCP without significant finite-size or approximation artifacts.

Editorial extensions

If this is right

  • Vertex contributions dominate the optical response over bubble terms inside the scaling regime.
  • The cluster dc conductivity falls as 1/T.
  • A marginal-Fermi-liquid self-energy appears at the nodes.
  • The scaling produces strange-metal optical transport throughout the quantum-critical fan.
  • The forms match several qualitative experimental features observed in cuprates.

Reading between the lines

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

  • Similar ω/T scaling may appear in other response functions or on larger clusters once computational limits are overcome.
  • The results suggest vertex corrections must be retained to obtain correct transport in quantum-critical regimes of strongly correlated models.
  • The 1/T conductivity and marginal self-energy together point to a possible route toward understanding linear-in-T resistivity in the strange-metal phase.
  • Extending the same scaling analysis to the charge susceptibility could test whether the reported forms are universal across channels.
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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

2 major / 2 minor

Summary. The manuscript presents a numerical study of dynamical scaling near the pseudogap quantum critical point in the two-dimensional Hubbard model. Employing a four-patch dynamical cluster approximation (DCA) solved with the numerical renormalization group (NRG), the authors compute real-frequency response functions over wide temperature and frequency ranges. They report that near the critical doping, the imaginary part of the local spin and cluster-current susceptibilities follow ω/T scaling of the form χ''(ω, T) ∼ tanh(ω/(2T)), while the cluster optical conductivity satisfies T σ'_cl(ω, T) ∼ tanh(ω/(2T)) / (ω/T), leading to a 1/T dc conductivity. Vertex corrections dominate the conductivity, and the nodal self-energy shows marginal Fermi-liquid behavior, suggesting strange-metal transport in the quantum critical fan. These findings are claimed to qualitatively match several experimental observations in cuprate superconductors.

Significance. If the reported scaling forms are robust, this work supplies direct numerical evidence for emergent ω/T scaling and vertex-dominated strange-metal transport arising from the Hubbard model at the pseudogap QCP without fitted parameters. The NRG-enabled access to real-frequency data over several decades is a technical strength that allows clean extraction of the tanh forms. The results offer a microscopic route to several cuprate anomalies, though their quantitative reliability hinges on controlling cluster-size effects.

major comments (2)
  1. [Methods (DCA implementation)] The four-patch DCA is the central methodological choice, yet the manuscript provides no convergence tests with larger clusters (e.g., 8- or 16-patch) or alternative solvers. Because long-wavelength fluctuations control the pseudogap QCP and the Brillouin-zone sampling is coarse near the nodal/antinodal points, the precise tanh(ω/2T) shape and the reported dominance of vertex over bubble contributions could be artifacts of the restricted momentum resolution rather than intrinsic properties of the 2D Hubbard model.
  2. [Results (optical conductivity)] The claim that the vertex contribution to the cluster optical conductivity greatly exceeds the bubble contribution is load-bearing for the strange-metal interpretation. Without quantitative ratios, error estimates, or explicit plots of vertex/bubble decomposition across the scaling regime (e.g., in the figures showing Tσ'_cl), it is impossible to judge how large the dominance is or whether it persists down to the lowest temperatures accessed.
minor comments (2)
  1. [Abstract] The scaling variable x = ω/T is used throughout but should be defined explicitly on first appearance in the abstract and main text.
  2. [Figure captions] Scaling-collapse figures should indicate the temperature range, number of independent NRG runs, and any broadening parameters used to ensure the tanh form is not sensitive to numerical details.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the positive overall assessment and for the detailed, constructive comments. We address each major point below. Where feasible we have revised the manuscript to incorporate additional discussion and data; we also note one computational limitation that prevents a direct response.

read point-by-point responses

  1. Referee: [Methods (DCA implementation)] The four-patch DCA is the central methodological choice, yet the manuscript provides no convergence tests with larger clusters (e.g., 8- or 16-patch) or alternative solvers. Because long-wavelength fluctuations control the pseudogap QCP and the Brillouin-zone sampling is coarse near the nodal/antinodal points, the precise tanh(ω/2T) shape and the reported dominance of vertex over bubble contributions could be artifacts of the restricted momentum resolution rather than intrinsic properties of the 2D Hubbard model.

    Authors: We agree that cluster-size convergence is important. The four-patch DCA was deliberately chosen to resolve the nodal-antinodal differentiation that underlies the pseudogap, and prior benchmark studies have shown it reproduces the essential physics of the 2D Hubbard model for both single-particle and two-particle quantities. Performing NRG calculations on 8- or 16-patch clusters is currently prohibitive because of the exponential growth of the impurity Hilbert space. In the revised manuscript we have added a dedicated paragraph in Sec. II that discusses this limitation, cites the relevant benchmark literature, and explains why the observed ω/T scaling is unlikely to be an artifact: the same tanh form appears consistently in the local spin susceptibility, the cluster current susceptibility, and the self-energy, all of which are less sensitive to long-wavelength sampling than the conductivity. We therefore maintain that the reported scaling reflects intrinsic behavior, while acknowledging that larger-cluster studies would be desirable. revision: partial

  2. Referee: [Results (optical conductivity)] The claim that the vertex contribution to the cluster optical conductivity greatly exceeds the bubble contribution is load-bearing for the strange-metal interpretation. Without quantitative ratios, error estimates, or explicit plots of vertex/bubble decomposition across the scaling regime (e.g., in the figures showing Tσ'_cl), it is impossible to judge how large the dominance is or whether it persists down to the lowest temperatures accessed.

    Authors: We thank the referee for highlighting this point. The original manuscript stated the dominance qualitatively; to make the claim quantitative we have added a new figure (Fig. 7 in the revised version) that decomposes the cluster optical conductivity into bubble and vertex parts for several temperatures inside the scaling regime. The figure also shows the vertex-to-bubble ratio versus ω/T together with NRG error estimates. The ratio exceeds 5 at low frequencies and remains roughly constant down to the lowest temperatures accessed, confirming that vertex corrections dominate throughout the quantum-critical fan. This addition directly addresses the request for quantitative ratios and explicit plots. revision: yes

standing simulated objections not resolved
  • Direct convergence tests with 8- or 16-patch DCA clusters using the NRG solver are computationally infeasible with present resources.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: scaling forms emerge from direct numerical solution of Hubbard model

full rationale

The paper's central results on ω/T scaling in susceptibilities and conductivity are obtained by solving the 2D Hubbard Hamiltonian numerically via four-patch DCA with NRG as the impurity solver. The reported forms χ''(ω,T)∼tanh(x/2) and Tσ'_cl(ω,T)∼tanh(x/2)/x are presented as outputs observed in the computed real-frequency spectra near critical doping, with no parameters fitted to enforce them and no reduction of the scaling to an input ansatz or self-citation by construction. The derivation chain is therefore self-contained: the Hubbard model plus the DCA+NRG approximation constitute the sole inputs, and the scaling is an emergent numerical finding rather than a tautology.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the validity of the DCA approximation and NRG solver for real-frequency dynamics; no additional free parameters are introduced beyond the standard Hubbard U and doping, which are chosen to place the system near the known critical point.

assumptions (1)
  • domain assumption The four-patch DCA with NRG accurately reproduces the low-energy real-frequency response of the 2D Hubbard model near the pseudogap QCP.
    Invoked throughout the abstract as the basis for the reported spectra and scaling.

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Cite this review

Pith. "Pith review of Dynamical scaling near the pseudogap quantum critical point of the two-dimensional Hubbard model." pith.science (2026). https://pith.science/paper/EWFB22QV

@misc{pith2026260515060,
  author       = {Pith},
  title        = {Pith review of: Dynamical scaling near the pseudogap quantum critical point of the two-dimensional Hubbard model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EWFB22QV}},
  note         = {Machine review of arXiv:2605.15060}
}
abstract

We study dynamical scaling in the quantum-critical fan of the pseudogap-metal to Fermi-liquid transition of the two-dimensional Hubbard model. Using a four-patch dynamical cluster approximation with the numerical renormalization group as a cluster impurity solver, we access real-frequency dynamics over several decades at arbitrary temperatures. Close to the critical doping, the local spin and cluster-current susceptibility spectra exhibit $x=\omega/T$ scaling of the form $\chi''(\omega,T)\sim \tanh(x/2)$, and the cluster contribution to the optical conductivity obeys $T\sigma'_{\mathrm{cl}}(\omega,T) \sim \tanh(x/2)/x$, implying a $1/T$ cluster dc conductivity. In the scaling regime, the vertex contribution to the cluster optical response is much larger than the bubble contribution. We further find evidence for a marginal-Fermi-liquid nodal self-energy. This, together with the $1/T$ vertex contribution to the conductivity, implies strange-metal optical transport in the quantum critical region. Our results describe several qualitative aspects of several experimental observations.

Figures

Figures reproduced from arXiv: 2605.15060 by the authors.

Figure 2
Figure 2. FIG. 2. Real-frequency scaling of the local spin susceptibil [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. (a) Zero-temperature phase diagram extracted from [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Cluster contribution to the real part of the opti [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The spectral function [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Spectral part of the cluster optical conductivity, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Four types of contributions to the spectral part, [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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