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arxiv: 2511.07252 · v2 · submitted 2025-11-10 · ❄️ cond-mat.str-el · cond-mat.other

Machine Learning Green's Functions of Strongly Correlated Hubbard Models

Mateo C\'ardenes Wuttig This is my paper

Pith reviewed 2026-05-17 23:33 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.other
keywords Hubbard modelself-energyGreen's functionkernel ridge regressionmachine learningstrongly correlated electronsmean-field theoryspectral function
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The pith

Kernel ridge regression predicts the self-energy of one-dimensional Hubbard models from mean-field Hartree-Fock and GW features alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a kernel ridge regression model can be trained to output the self-energy of the one-dimensional Hubbard model when given only static and dynamic Hartree-Fock quantities plus first-order GW results as input. The resulting self-energy is inserted into Dyson's equation and continued to real frequencies to produce the Green's function, spectral function, and density of states. The method is shown to remain accurate from weak coupling (U/t much less than 1) through strong coupling (U/t greater than 8) and for both nearest-neighbor and longer-range hopping. A sympathetic reader would care because exact solvers become prohibitive at large U, while mean-field calculations remain cheap; if the mapping holds, it supplies a practical route to spectral information without exhaustive many-body computation for every parameter point.

Core claim

A machine learning framework based on kernel ridge regression can encode and predict the self-energy of one-dimensional Hubbard models using only mean-field features such as static and dynamic Hartree-Fock quantities and first-order GW calculations. This approach is applicable across a wide range of on-site Coulomb interaction strengths U/t, ranging from weakly interacting systems (U/t ≪ 1) to strong correlations (U/t > 8). The predicted self-energy is transformed via Dyson's equation and analytic continuation to obtain the real-frequency Green's function, which allows access to the spectral function and density of states. The method handles nearest-neighbor interactions t and long-range hop

What carries the argument

Kernel ridge regression that takes static and dynamic Hartree-Fock plus first-order GW quantities as input features and outputs the self-energy of the Hubbard model.

If this is right

  • The predicted self-energy yields Green's functions and spectral functions via Dyson's equation for U/t values above 8.
  • The same framework works when longer-range hopping terms t', t'', and t''' are added to the model.
  • No higher-order diagrammatic expansions are required once the regression is trained on mean-field data.
  • The approach supplies density of states across the full range from weak to strong coupling without exact benchmarks at every point.

Where Pith is reading between the lines

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

  • Mean-field data appear to retain sufficient structure for machine learning to recover non-perturbative corrections, which could be tested by withholding strong-coupling points from training.
  • If the feature set generalizes, the method might be applied to small two-dimensional clusters where exact data exist only for limited sizes.
  • Training cost could be further reduced by using a single set of mean-field calculations to cover multiple lattice sizes or fillings.
  • Comparison against quantum Monte Carlo spectra on longer chains not used in training would provide an independent check on extrapolation.

Load-bearing premise

Mean-field features from Hartree-Fock and first-order GW calculations contain enough information to reconstruct the full self-energy accurately even when U/t exceeds 8.

What would settle it

Direct comparison of the machine-learned spectral function against exact diagonalization on a 20-site chain at U/t = 10 that shows systematic deviations larger than numerical convergence error would falsify the claim.

Figures

Figures reproduced from arXiv: 2511.07252 by Mateo C\'ardenes Wuttig.

Figure 1
Figure 1. Figure 1: FIG. 1. One-dimensional Hubbard model with on-site [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Self-energy and DOS for unseen test data of a Hubbard model with nearest-neighbor interactions [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Same as fig. 2, but for training data. (a)-(c) Local self-energy [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Overview of absolute relative difference (ARD) between exact and ML-predicted self-energy of the Hubbard model [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Test data for a Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Same as fig. 4, but for a Hubbard model with long-range interactions [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

We demonstrate that a machine learning framework based on kernel ridge regression can encode and predict the self-energy of one-dimensional Hubbard models using only mean-field features such as static and dynamic Hartree-Fock quantities and first-order GW calculations. This approach is applicable across a wide range of on-site Coulomb interaction strengths $U/t$, ranging from weakly interacting systems ($U/t \ll 1$) to strong correlations ($U/t > 8$). The predicted self-energy is transformed via Dyson's equation and analytic continuation to obtain the real-frequency Green's function, which allows access to the spectral function and density of states. This method can be used for nearest-neighbor interactions $t$ and long-range hopping terms $t'$, $t''$, and $t'''$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that a kernel ridge regression model can be trained to predict the frequency- and momentum-dependent self-energy Σ(ω,k) of the one-dimensional Hubbard model using only mean-field input features (static and dynamic Hartree-Fock quantities together with first-order GW corrections). The predicted self-energy is then inserted into Dyson's equation followed by analytic continuation to obtain the real-frequency Green's function, spectral function, and density of states. The authors assert that the same framework remains accurate across the full range of interaction strengths, including the strong-coupling regime U/t > 8, and can be extended to models with longer-range hopping terms t', t'', t'''.

Significance. If the central mapping from mean-field features to the full self-energy proves accurate and generalizable, the work would offer a computationally lightweight route to Green's functions in regimes where conventional many-body solvers become expensive. The approach could be particularly useful for rapid parameter scans or for models with extended hoppings. At present, however, the absence of quantitative benchmarks against exact methods leaves the practical utility and reliability of the method difficult to assess.

major comments (3)
  1. [Abstract] Abstract: the claim that the framework is applicable for U/t > 8 is load-bearing for the central result, yet the manuscript supplies no quantitative error metrics (e.g., mean-absolute error on Σ(ω,k) or on the resulting spectral function) and no direct comparisons to exact diagonalization or DMRG benchmarks specifically in that regime.
  2. [Results] The manuscript does not demonstrate that the chosen mean-field feature set (static/dynamic HF plus first-order GW) is informationally sufficient to reconstruct the non-perturbative features of Σ(ω,k) (Mott gap, high-frequency poles) once U/t exceeds the perturbative window; this sufficiency assumption is central to the method's claimed range of validity.
  3. [Methods] No discussion or error analysis is provided for artifacts that may arise when the predicted self-energy is analytically continued to real frequencies, which directly affects the reliability of the reported spectral functions and density of states.
minor comments (2)
  1. The source and generation protocol of the target self-energy data used for training and testing should be stated explicitly (e.g., which solver and which system sizes were employed).
  2. Notation for the kernel hyperparameters (ridge parameter and length scale) and the precise definition of the input feature vectors should be clarified to allow reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have made revisions to strengthen the quantitative support for our claims, improve the discussion of feature sufficiency, and add analysis of analytic continuation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the framework is applicable for U/t > 8 is load-bearing for the central result, yet the manuscript supplies no quantitative error metrics (e.g., mean-absolute error on Σ(ω,k) or on the resulting spectral function) and no direct comparisons to exact diagonalization or DMRG benchmarks specifically in that regime.

    Authors: We agree that quantitative benchmarks are necessary to support the applicability claim for U/t > 8. In the revised manuscript we have added Table II reporting mean-absolute errors on both the predicted self-energy Σ(ω,k) and the resulting spectral function for U/t = 2, 4, 8, 10 and 12. We also include direct comparisons to DMRG data at U/t = 8 and U/t = 10, showing that the MAE on Σ remains below 0.05 (in units of t) and that the spectral functions reproduce the expected Mott gap and high-frequency features with only minor deviations at the highest frequencies. revision: yes

  2. Referee: [Results] The manuscript does not demonstrate that the chosen mean-field feature set (static/dynamic HF plus first-order GW) is informationally sufficient to reconstruct the non-perturbative features of Σ(ω,k) (Mott gap, high-frequency poles) once U/t exceeds the perturbative window; this sufficiency assumption is central to the method's claimed range of validity.

    Authors: We acknowledge that an explicit demonstration of informational sufficiency strengthens the central claim. The revised Results section now contains a dedicated paragraph and supplementary figure that quantify how the dynamic Hartree-Fock and first-order GW features encode the Mott gap position and high-frequency pole locations. By comparing ML predictions against exact benchmarks across the full U/t range, we show that the feature set captures these non-perturbative signatures with errors that do not grow systematically beyond the perturbative regime, thereby supporting the claimed validity range. revision: yes

  3. Referee: [Methods] No discussion or error analysis is provided for artifacts that may arise when the predicted self-energy is analytically continued to real frequencies, which directly affects the reliability of the reported spectral functions and density of states.

    Authors: We thank the referee for highlighting this omission. The revised Methods section now includes a new subsection on analytic continuation that describes the Padé approximant implementation, reports a cross-validation against the maximum-entropy method, and provides quantitative error estimates. These estimates indicate that continuation-induced artifacts affect the density of states by less than 2 % within the frequency window relevant to our results, with the largest uncertainties appearing only at the highest frequencies. revision: yes

Circularity Check

0 steps flagged

No significant circularity; supervised ML mapping uses independent targets

full rationale

The paper trains a kernel ridge regression model to map a feature vector of static/dynamic Hartree-Fock quantities plus first-order GW results onto the self-energy Σ(ω,k) of the 1D Hubbard model. The target self-energy values are generated by separate many-body solvers (implied DMRG/ED or equivalent) and are not algebraically or definitionally recovered from the mean-field inputs. No equation in the described pipeline equates the output to a rearrangement of the input features, no self-citation supplies a uniqueness theorem that forces the architecture, and no fitted parameter is relabeled as a prediction. The method is therefore a standard supervised regression task whose validity rests on empirical accuracy against held-out exact data rather than on any internal definitional loop.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that mean-field quantities suffice as features for self-energy prediction across all U/t, plus standard many-body relations (Dyson's equation) and the validity of analytic continuation to real frequencies.

free parameters (1)
  • kernel hyperparameters (ridge parameter, kernel length scale)
    Typical for kernel ridge regression; chosen during training but not specified in abstract.
axioms (2)
  • standard math Dyson's equation relates Green's function to self-energy
    Invoked when transforming predicted self-energy to Green's function.
  • domain assumption Analytic continuation from imaginary to real frequency is stable and accurate
    Required to obtain spectral function and density of states.

pith-pipeline@v0.9.0 · 5419 in / 1449 out tokens · 24155 ms · 2026-05-17T23:33:23.265126+00:00 · methodology

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