Strange diffusivity of incoherent metal in half-filled two-dimensional Hubbard model

Aaram J. Kim; Boris Svistunov; Evgeny Kozik; Igor S. Tupitsyn; Nikolay V. Prokof'ev; Youngmin Eom

REVIEW 3 major objections 4 minor 52 references

Strange diffusivity of incoherent metal in half-filled two-dimensional Hubbard model

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The half-filled 2D Hubbard model's incoherent metal has a charge diffusion constant D ~ 1/√T across all anomalous-resistivity regimes down to the metal-insulator crossover, plus a Pseudogap Metal state where insulating compressibility coexi

desk verdict D~T^-1/2 diffusion in the half-filled Hubbard model is the new result; the analytic continuation is the main caveat, but the paper deserves serious refereeing. read the letter →

arxiv 2509.00281 v1 pith:NQAANOZB submitted 2025-08-29 cond-mat.str-el cond-mat.quant-gas

Youngmin Eom , Igor S. Tupitsyn , Nikolay V. Prokof'ev , Boris Svistunov , Evgeny Kozik , Aaram J. Kim This is my paper · ORCID

classification cond-mat.str-elcond-mat.quant-gas

keywords HubbardmodelstrangemetalchargediffusionNernst-EinsteinrelationdiagrammaticMonteCarloanalyticcontinuationpseudogapopticalconductivity

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 studies charge transport in the half-filled two-dimensional Hubbard model using numerically exact diagrammatic Monte Carlo combined with controlled analytic continuation. It claims that over a broad temperature range where the DC resistivity is anomalously temperature-dependent (~T^α with 0<α≲1), the charge diffusion constant D extracted from the Nernst-Einstein relation follows a robust ~1/√T law. It also identifies a Pseudogap Metal regime in which the charge compressibility is insulator-like while transport remains metallic. If correct, this establishes a simple, universal-looking diffusive signature of incoherent charge transport and sharpens the picture of how the system enters the insulating state.

What carries the argument

The central computational object is the imaginary-time current-current correlator Λ(iωn), continued via its spectral representation to the optical conductivity σ(ω) using the SOCC/MCC method with a multiscale high-frequency tail from Bold4. The central physical identity is the Nernst-Einstein relation D = σDC/κ, which particle-hole symmetry at half-filling makes exact because thermoelectric response vanishes. Its diagnostic power lies in decomposing anomalous resistivity into a universal diffusion law (1/√T) and a non-universal compressibility; the diagrammatic spin-resolved decomposition separates bubble, same-spin, and opposite-spin vertex contributions that explain the Drude-to-continuum

What would settle it

A measurement of the optical conductivity in the half-filled 2D Hubbard model that does not rest on maximum-smoothness analytic continuation—for example a real-time dynamical quantum Monte Carlo calculation or a cold-atom density-response measurement at half-filling—should find σDC at U=4,T=0.4-4 within the stretch-test window; if σDC deviates significantly, then D=σ/κ will not follow the reported 1/√T law.

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Extended reading notes

Core claim

For the half-filled 2D Hubbard model, the authors compute the current-current correlation function directly in the thermodynamic limit with diagrammatic Monte Carlo, splice the high-frequency tail with the self-consistent Bold4 diagrammatic theory, and continue to real frequencies with a stretch-test-controlled stochastic method. They find that the DC resistivity's anomalous scaling between high and low temperatures is the product of a near-universal diffusivity D∼1/√T multiplied by a compressibility that is non-universal; at lower temperatures the compressibility turns over and the system enters a Pseudogap Metal state with ∂κ/∂T>0 but metallic ρDC(T), until the metal-insulator crossover. D

Load-bearing premise

The numerical analytic continuation must recover the true optical conductivity near zero frequency; the smoothness-constrained spectrum is validated by a stretch test but is not uniquely determined by the Matsubara data.

Editorial extensions

If this is right

The 1/√T diffusivity extends through Diffusive Metal I, Diffusive Metal II, and the Pseudogap Metal, so in that window the anomalous resistivity exponent α is controlled mostly by the compressibility, not by the diffusion constant.
Because particle-hole symmetry makes the Nernst-Einstein extraction exact at half-filling, the reported D is a genuine charge diffusion constant rather than a thermoelectric mixture.
The Pseudogap Metal state is defined by dκ/dT>0 together with metallic transport, showing that insulating charge response and metallic conduction can coexist without an actual gap in the conductivity.
The opposite-spin vertex corrections deplete the low-frequency Drude peak and feed a high-frequency continuum, explaining why the central optical peak is suppressed before the system becomes insulating.

Reading between the lines

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

One implication beyond the paper's claims is that D∼1/√T may be universal across doping, with all material-specific and model-specific variation residing in the compressibility; the cold-atom data at substantial doping already point in this direction.
A testable extension would be a real-time measurement of charge diffusion in an ultracold-atom realization at half-filling: if the Pseudogap Metal picture is right, density-response (compressibility) and cloud-expansion (transport) probes should show opposite temperature trends.
The spin-resolved vertex mechanism suggests an optical sum-rule-style diagnostic: the same-spin vertex adds spectral weight in a window set by temperature, while the opposite-spin vertex removes it from the Drude peak; this separation could be probed by spin-resolved or polarized light experiments in analog systems.
If the 1/√T law is the universal incoherent-metal signature, then future analytic-continuation studies should report D rather than only ρDC, since D is the quantity that exposes the regularity behind the non-universal resistivity exponents.

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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

Colleague,

The paper you'd want to know about: it reports a clean D ~ T^{-1/2} scaling of the charge diffusion constant in the half-filled 2D Hubbard model over a wide temperature window, obtained by combining connected determinant Monte Carlo (CDet) with a self-consistent Bold4 scheme for the high-frequency tail and then doing numerical analytic continuation (NAC). That scaling hasn't been seen at half-filling before, and the paper connects it to the cold-atom data on the doped side, suggesting a possible universal diffusive law for incoherent metals. The other genuinely new piece is the identification of a 'Pseudogap Metal' regime where the compressibility is insulating-like while the DC conductivity is still metallic.

The methodology is careful: the diagrammatic series converge to order 8–10 with Padé extrapolation and controlled errors; the high-frequency tail from Bold4 is substituted only where the CDet error bars make it beneficial; and the stretch test is used to define error bars on σ(0). They also cross-check one point with SOM. No fitted parameter is dressed as a prediction—D is read off the computed σ and κ. The spin-resolved decomposition of the current correlator, with an exact selection rule based on Furry's theorem, is a nice formal contribution.

The main soft spot is exactly the NAC step. The stretch test quantifies how far one can shift σ(0) before the spectrum develops wiggles under a smoothness constraint; it does not certify that the true spectrum is smooth. A less smooth spectrum—say a narrower Drude peak—could give a different σ(0), and if the bias varies with T it could distort the scaling exponent. The ad hoc O(1/ω³) error on the Bold4 tail is honest but doesn't close that gap. The absence of released data and code makes it harder to pressure-test this; a reader can't re-run the inversion or examine the stretch test protocol in detail.

Still, the scaling is observed across a decade of temperature for three U values, and the compressibility is computed directly, not continued, so the D(T) behavior is not simply an artifact of the continuum inversion. The NAC concern is a real caveat, but it's a caveat, not a refutation. This paper deserves a serious referee—ideally someone deep in the NAC literature—and the referee should focus on the continuation error bars and ask for reproducibility materials.

Referee Report

3 major / 4 minor

Summary. The paper studies charge transport across the metal-insulator crossover in the half-filled two-dimensional Hubbard model. The authors compute the current-current correlation function directly in the thermodynamic limit with connected determinant diagrammatic Monte Carlo (CDet) up to order 8-10, combine it with a self-consistent Bold4 high-frequency tail, and use numerical analytic continuation (SOCC/MCC with a 'stretch test') to obtain the optical conductivity. Combining the DC conductivity with the compressibility computed earlier by the same group, they extract the charge diffusion constant D = sigma_DC / kappa. The central claim is that, across a broad temperature range where the DC resistivity has anomalous scaling rho_DC ~ T^alpha with 0<alpha<1, the diffusion constant displays a robust ~1/sqrt(T) 'strange metal' behavior. They also identify a 'Pseudogap Metal' regime characterized by insulating charge compressibility coexisting with metallic transport, and analyze the diagrammatic origin of this behavior, finding that opposite-spin vertex corrections transfer Drude weight to a high-frequency continuum.

Significance. If the reported 1/sqrt(T) law holds, it would provide a striking universal diffusive signature of incoherent transport in the half-filled 2D Hubbard model, with a direct connection to cold-atom measurements in the doped system. The numerical work is careful: the CDet series are pushed to high order and extrapolated with Pade approximants, error bars are propagated into observables, and the analytic continuation is guarded by a stretch test and cross-checked against SOM at one representative point. The diagrammatic decomposition also benefits from an exact order-by-order selection rule for spin-resolved vertex contributions. These strengths make the paper a serious candidate for publication, provided the principal scaling claim is robust against the known ill-posedness of the analytic continuation.

major comments (3)

[SM Sec. III, Eq. (4)] The stretch test defines the sigma(0) error bar as the range over which the MCC spectrum does not develop visible wiggles; this is a smoothness-prior sensitivity test, not a uniqueness bound. Because the kernel K(i omega_n, omega) has small eigenvalues, non-smooth spectra with different sigma(0) can fit Lambda(i omega_n) within the DiagMC error bars. The main claim D(T) ~ T^{-1/2} is obtained from sigma_DC(T)/kappa(T) across a range of T, so a T-dependent continuation bias could fake or distort the power law. Please cross-validate with an independent continuation method (e.g., MaxEnt, SOM, sparse modeling, or Nevanlinna continuation) at enough (U,T) points to map D(T), and/or demonstrate with synthetic spectra that non-smooth solutions consistent with Lambda(i omega_n) change D(T) by less than the quoted error.
[SM Sec. III, Fig. S2(a)] The high-frequency tail for omega_n > 15 is replaced by Bold4 values with an ad hoc O(1/omega_n^3) uncertainty. This tail enters as data in Eq. (4) and can influence the low-frequency inversion through overall normalization and constraints. The systematic difference between Bold4 and the exact CCF is acknowledged but is not propagated into the final D(T) error bars in a well-controlled way. Please test the sensitivity of sigma(0) to the substitution threshold and to the assumed tail-error model, and include the resulting uncertainty in the reported diffusion constant.
[Fig. 2(c), Table I] The central claim of a robust ~1/sqrt(T) scaling is supported visually by reference lines, but no fitting procedure, exponent uncertainty, or temperature-window definition is reported. Provide a power-law fit D(T) = A T^{-gamma} for each U with statistical errors, state the criterion for the fitting window (e.g., DM1 through PGM), and report gamma and its uncertainty. This is needed to substantiate the universality statement and to make the claimed T^{-1/2} exponent falsifiable.

minor comments (4)

[Fig. 3] The panel labels and caption are confusing: both panels appear to be labeled 'a' and 'b', and the inset showing sigma_DC * Gamma_tr and D * Gamma_tr is not fully described. Please clarify the figure layout and the definition of Gamma_tr in the caption.
[SM Sec. IV] The temperature T_II is defined through the inflection point of a 5th-order polynomial fit to d kappa / d log T. Please comment on the stability of T_II with respect to the fit order and the fitting range.
[Table I] The quoted ranges of the resistivity exponent alpha (0 <~ alpha <~ 0.5 and 0.5 <~ alpha <~ 1) are not tied to explicit fits. These ranges should be obtained from and reported with the same fitting procedure used for the diffusion constant.
[References] Reference [48] is an arXiv preprint; if a published version exists, please cite it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the D(T) scaling is read off independently computed σDC and κ; analytic-continuation systematics are a robustness concern, not a circular reduction.

full rationale

The derivation chain is: Λ(iωn) is computed by DiagMC directly in the thermodynamic limit; the high-frequency tail is taken from Bold4 with an explicitly assigned O(1/ω_n^3) uncertainty; σ(ω) is obtained by numerical analytic continuation; σDC=σ(0); and D=σDC/κ with κ computed by DiagMC (with first-order analytic complements). Each step is an independent numerical evaluation. No quantity along this chain is defined in terms of the claimed D(T)∼1/√T scaling, and no parameter is fitted to the ρDC(T) or D(T) behavior that is then presented as a prediction. The paper explicitly frames the NAC error using a stretch test around the "as-smooth-as-possible" solution, which is a sensitivity statement about smoothness-constrained spectra, not a uniqueness certificate; the possibility that a less smooth true spectrum could bias σ(0) is a correctness/robustness risk, not a circularity of the paper's own equations. The uses of the authors' prior results (Ref. [13] for compressibility and Ref. [15] for self-energy anisotropy) are citations to separate computations of different observables, not inputs that already contain the transport conclusion; they provide independent, externally falsifiable information. Thus no load-bearing step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or interactions. The Hubbard model parameters U and t are physical inputs. The central results are numerical observations; the only 'free' choices are in the analysis procedure (e.g., polynomial fits for crossover scales), which do not enter the main claims. The key assumptions are methodological: reliability of the analytic continuation, convergence of the diagrammatic series, and the validity of the Einstein relation.

assumptions (5)

domain assumption Particle-hole symmetry at half-filling gives zero thermoelectric response, so the Nernst-Einstein relation D=σ/κ holds in its simple form.
Invoked in Introduction (Eq. 2) to justify inferring D from computed σ and κ. Standard symmetry of the half-filled Hubbard model on a bipartite lattice.
domain assumption The diagrammatic expansion of the current-current correlator converges and the Padé extrapolation to infinite order is reliable within the quoted error bars.
Used throughout; series convergence and extrapolation discussed in SM Sec. II, Fig. S1. This is a standard but unproven assumption of DiagMC.
ad hoc to paper The numerical analytic continuation (SOCC/MCC) with the stretch test yields the true optical conductivity σ(ω), particularly σ(0), within the stated error bars.
The entire DC transport depends on σ(0) obtained by inverting an ill-posed Fredholm equation. The paper uses a smoothness constraint and a stretch test; the correctness of this protocol is not guaranteed by mathematics, only by the paper's validation. This is the weakest assumption.
ad hoc to paper The Bold4 self-consistent diagrammatic results accurately describe the high-frequency tail of the current correlator, with the introduced O(1/ω_n^3) error bounding the uncertainty.
SM Sec. III: the CDet results are substituted by Bold4 for ω_n>15, with an ad hoc error added. Systematic deviation from exact results is not fully controlled.
domain assumption Perfect nesting and Mermin-Wagner theorem ensure the finite-temperature transport is incoherent and the phase diagram is as described.
Background, from Refs. 13 and 17. Used to justify classification of thermal and diffusive regimes.

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Pith. "Pith review of Strange diffusivity of incoherent metal in half-filled two-dimensional Hubbard model." pith.science (2026). https://pith.science/paper/NQAANOZB

@misc{pith2026250900281,
  author       = {Pith},
  title        = {Pith review of: Strange diffusivity of incoherent metal in half-filled two-dimensional Hubbard model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQAANOZB}},
  note         = {Machine review of arXiv:2509.00281}
}

abstract

We study charge transport across the metal-insulator crossover in the half-filled two-dimensional Hubbard model, with particular emphasis on precision control. The dynamic current-current correlation function is obtained directly in the thermodynamic limit, and the optical conductivity is extracted using numerical analytic continuation. To achieve this, we develop a multiscale approach: the non-perturbative low-frequency behavior is computed using the unbiased diagrammatic Monte Carlo technique, while the high-frequency physics is captured via a self-consistent (semi-)analytic diagrammatic theory. We found that across a broad temperature range where the DC resistivity displays anomalous scaling, $\sim T^\alpha$ with $0<\alpha\lesssim 1$, the Nernst-Einstein relation implies the diffusion constant with the characteristic $\sim 1/\sqrt{T}$ "strange metal" behavior. It was also revealed that the insulating regime is entered through a peculiar non-Fermi liquid state-which we call a Pseudogap Metal-characterized by insulating charge compressibility coexisting with metallic transport. Diagrammatically, the high-temperature incoherent transport is captured by the dressed polarization bubble, whereas near the metal-insulator crossover, the effective interaction vertex between opposite-spin particles is responsible for transferring the Drude weight to a high-frequency continuum.

Figures

Figures reproduced from arXiv: 2509.00281 by the authors.

**Figure 2.** Nernst-Einstein decomposition of (a) the DC resistivity [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Optical conductivity σ(ω) for various (a) temperatures and (b) interaction strengths. The inset of panel (a) presents the momentum relaxation rate for U = 1.5, 3.2 and 4 . AΓtr where A = σDC or D, is shown in the inset of panel (b), and (blue) horizontal dashed line represents the limiting value at weak coupling and high temperature [11]. Frequency dependent conductivity- The full frequency dependence of the condu… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (a) The dressed bubble contribution to ρDC and Γtr. (b) Exact second-order vertex diagrams: two oppositespin vertices, Λpp and Λph2 cancel perfectly, leaving only the same-spin vertex (Λph1). The diagrammatic decomposition (bubble, same-spin vertex, and opposite-spin …

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