Pith. sign in

REVIEW 3 major objections 5 minor 69 references

This paper argues that convergence of Transformer-based neural quantum states for the Hubbard model is set by a coupled balance among update stability, ansatz expressivity, and Monte Carlo sampling quality, not by any single hyperparameter.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 00:48 UTC pith:2QDGBVMN

load-bearing objection An honest but thin ablation of an existing ansatz: the claims are plausible, but the width/determinant trends are confounded by pretraining quality, and the evidence is mostly qualitative. the 3 major comments →

arxiv 2607.14875 v1 pith:2QDGBVMN submitted 2026-07-16 cond-mat.str-el cond-mat.dis-nnphysics.comp-ph

Optimization dynamics of Transformer backflow neural quantum states for the two-dimensional Hubbard model

Zong-Yu Liao , Jia-Qi Wang , Rong-Qiang He , Zhong-Yi Lu This is my paper
classification cond-mat.str-el cond-mat.dis-nnphysics.comp-ph
keywords neural quantum statesTransformerHubbard modelvariational Monte Carlobackflow determinantsMARCH optimizeroptimization dynamicsfermionic sign problem
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 asks what actually controls the convergence of a Transformer-based neural quantum state when it is asked to approximate the ground state of the two-dimensional Hubbard model. The authors argue that practical accuracy is not set by any single choice—network size alone, or optimizer alone—but by the coupled balance among three factors: how strongly the parameter update is damped, how expressive the variational wave function is, and how much noise the Monte Carlo sampling injects into each update. To show this, they run a controlled ablation on a doped 4x4 Hubbard cluster, varying the update-norm threshold, Transformer width, number of determinant channels, and Monte Carlo batch size, and they find a consistent ranking: a moderate update constraint, a wider network, more determinants, and larger batches each improve the final energy. If this picture is right, it gives a practical recipe for training these wave functions and explains why simply scaling up the network can fail if the optimizer and sampling are not adjusted with it.

Core claim

The paper's central claim is that the optimization performance of the Transformer backflow NQS is governed by three coupled factors: the stability of the MARCH update, the expressive power of the variational ansatz, and the statistical quality of the Monte Carlo estimates. On the 4x4 doped Hubbard benchmark, a moderate update-norm threshold, a wider Transformer, more determinant channels, and larger Monte Carlo batches each improve the late-time energy, and the same multi-stage workflow—neural-backflow initialization, supervised pre-training, MARCH—remains stable for half filling, U=4, open boundary conditions, and an 8x8 doped lattice, with the easier regimes reaching lower relative errors.

What carries the argument

The central object is the multi-determinant Transformer backflow neural quantum state: a wave function written as a sum over K Slater determinants, where each determinant is built from configuration-dependent backflow orbitals generated by a Transformer encoder. Its parameters are optimized by the Moment-Adaptive ReConfiguration Heuristic (MARCH), a stochastic-reconfiguration update combined with adaptive first- and second-moment estimators and a time-decaying norm constraint on the parameter step. A multi-stage initialization protocol—first a lightweight neural-network backflow is optimized, then the Transformer is supervised to reproduce that backflow's orbital matrices, and only then is t

Load-bearing premise

The multi-stage initialization—first optimizing a cheap neural backflow, then pre-training the Transformer to copy it—is assumed to place the network in a good variational basin; if that pre-training instead anchors the Transformer in the wrong region of parameter space, the reported hyperparameter trends would reflect the initialization quality rather than the MARCH dynamics themselves.

What would settle it

Take the baseline doped 4x4 U=8 system and run the same MARCH optimization with c0, d, K, B at their best settings but with the Transformer initialized randomly (or with corrupted pre-training labels). If the late-time energy error does not degrade substantially, or if the ranking among hyperparameters changes, then the pre-training step—not the MARCH update dynamics—is the load-bearing element in the paper's conclusions.

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

If this is right

  • For doped Hubbard systems, increasing the number of determinant channels K is a direct and reliable way to lower the variational energy, because multi-determinant structure is the main carrier of fermionic expressive power.
  • A moderate update-norm threshold should be treated as a regularizer that trades off speed against stability; setting it too small stalls convergence even for an expressive ansatz.
  • Monte Carlo batch size is not just a cost parameter: it controls the noise in the MARCH direction, so larger batches are required to unlock the benefit of wider networks or more determinants.
  • The NNB-initialization, supervised pre-training, MARCH workflow transfers across filling, interaction strength, boundary conditions, and lattice size up to 8x8, suggesting it is a general protocol for Transformer backflow NQS calculations.
  • The three-factor coupling implies that capacity increases (d or K) should be accompanied by matched improvements in update stability and sampling quality; otherwise the added parameters are not effectively used.

Where Pith is reading between the lines

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

  • The ablation isolates each hyperparameter one at a time, so the paper does not test interactions between, say, K and B; a natural extension is to scan the joint (K, B) and (d, c0) planes to see whether the reported trends cross or trade off.
  • Because the pre-training target is the single-determinant NNB orbital matrix, the Transformer is initialized in a basin anchored to a single-determinant picture; one could test whether pre-training on a multi-determinant target changes the optimal K.
  • The paper notes the O(N^3) scaling of the backflow Transformer; if that holds, the workflow may become a practical tool for 16x16 or larger doped lattices, where the three-factor balance would need to be re-checked against an approximate reference.
  • A cheap way to test the three-factor coupling in practice: for a fixed wall-clock budget, compare a strategy that spends all compute on a large batch with one that splits it between more training steps and moderate batch sizes; the paper's logic predicts the latter will often win because the MARCH direction benefits from more reliable estimates per step.

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 studies the optimization dynamics of a multi-determinant Transformer backflow neural quantum state for the two-dimensional Hubbard model. The workflow has three stages: optimize a lightweight neural-network backflow (NNB), pretrain the Transformer to reproduce the NNB orbital matrices, and then run main energy optimization with the MARCH optimizer in variational Monte Carlo. Using the doped 4×4 PBC Hubbard model at U=8 as a baseline, the authors ablate the update-norm threshold c0, Transformer width d, number of determinant channels K, and Monte Carlo batch size B. They report that a moderate c0 improves convergence efficiency, larger d and K improve accuracy, and larger B reduces sampling noise, and they interpret these as evidence that optimization performance is governed by coupled factors: MARCH update stability, ansatz expressivity, and Monte Carlo sampling quality. The same workflow is also tested at half filling, U=4, OBC, and on an 8×8 doped lattice.

Significance. If the central claim is supported, the paper would provide practically useful guidance for training Transformer-based fermionic NQSs, a topic of current interest. The use of external exact-diagonalization and DMRG reference energies is a strength: the reported energy errors are not circularly defined. The multi-stage workflow is a concrete, transferable protocol, and the extension to several physical settings is valuable. However, the evidence for the central claim is currently limited by confounded ablations and insufficient statistical reporting.

major comments (3)
  1. [§II.C, §III.C–D, Fig. 2(b)–(c)] The width and determinant ablations do not isolate post-pretraining expressivity, because the supervised pretraining target is the same NNB matrix for all models while model capacity varies. In Eq. (23), L_pre sums the squared Frobenius error between each Transformer channel M_θ^(k) and M_NNB; a d=64/K=8 network can fit this target more closely than d=32/K=2 under a fixed pretraining schedule. The paper does not report pretraining loss, initial main-training energy, or distance to the NNB at hand-off. The late-time ordering in Fig. 2(b,c) could therefore be inherited from better initial states rather than from MARCH update stability or post-pretraining expressivity. Since two of the three claimed coupled factors rest on these ablations, the central claim is not yet established. Please report these diagnostics or otherwise match pretraining quality across models.
  2. [§III.B, Fig. 2(a)] Only two initial norm thresholds, c0=0.1 and c0=0.5, are compared. The conclusion that c0=0.5 is 'moderate' and provides a better balance is stronger than the evidence: with two values one can only say that 0.5 beats 0.1, not that an intermediate optimum exists. A scan over at least three values, or a quantitative argument about the constraint schedule, is needed to support the 'moderate threshold' claim.
  3. [§III.A, Fig. 2] No error bars, seed counts, or aggregate statistics are reported. The text states that shades denote independent random seeds, but the number of seeds is not given and the trajectories are only qualitatively described. The late-time differences in Fig. 2 are on the 10^-3 scale, where run-to-run fluctuations may be comparable; for example, the d=32 and d=64 curves in Fig. 2(b) appear to overlap at several steps. A quantitative summary (mean ± standard error, number of seeds) is required to establish the reported trends.
minor comments (5)
  1. [Eq. (4)] The symbol H_h is used both for the output of attention head h and as the head index in the surrounding text; this is confusing and should be relabeled (e.g., O_h).
  2. [Fig. 4] The y-axis label says E while the caption and text define the quantity as energy per site. Use a consistent label such as E/N to avoid ambiguity.
  3. [Eq. (35)] Clipping only the real part of the local energy while leaving the imaginary part unchanged is an asymmetric operation. The text justifies it by reference to FermiNet, but a brief comment on why this does not bias the complex-valued optimization would be useful.
  4. [§III.A] The smoothing window (200 steps, 800 steps for U=4) is stated, but the raw data are not shown. Showing at least one unsmoothed trajectory would help the reader assess the noise level.
  5. [General] The reference list includes several very recent preprints; please verify that all citations are available and accurately assigned, especially Refs. [47], [54], and [55].

Circularity Check

0 steps flagged

No circularity: reported energy errors are benchmarked against external ED/DMRG references and the optimization trends are empirical hyperparameter comparisons.

full rationale

The paper's quantitative target is the relative energy error ε_E = |E(r) - E_ref| / |E_ref|, with E_ref taken from exact diagonalization for the 4×4 doped U=8 baseline (−11.86884) and from independent benchmark/DMRG results [62] for other settings; the quantity being minimized is therefore not defined in terms of the fitted model. The 'three coupled factors' statement in the Discussion is a qualitative summary of controlled ablations over c0, d, K, and B, not a first-principles derivation, and none of the reported quantities is produced by fitting to the same data that defines the result. The NNB initialization and supervised pre-training are explicitly fixed initialization procedures within each ablation group and are not presented as predictions; the paper states they are 'not treated as independent ablation variables.' The only self-citations (Refs. [45,46]) are introductory examples of NQS architectures and are not load-bearing; the Transformer backflow ansatz and MARCH workflow are attributed to external references [47]. No uniqueness theorem, self-citation chain, or definitional identification of a fit with a predicted quantity is used. The possibility that NNB pre-training quality confounds the width/determinant ablations is a validity concern, not circularity, because the energy comparisons remain anchored to external references.

Axiom & Free-Parameter Ledger

10 free parameters · 5 axioms · 0 invented entities

The paper makes empirical claims about hand-chosen hyperparameters; no new particles, forces, or conserved quantities are introduced. The load-bearing assumptions are standard VMC sampling/local-energy assumptions plus the multi-stage initialization acting as a benign prior. The only external benchmark is the reference energy, so verification depends on ED/DMRG values being correct.

free parameters (10)
  • c0 (initial update-norm threshold) = 0.5 baseline; 0.1 variant
    Central comparison variable; chosen by hand, not derived. Only two values are tested, so the 'moderate' conclusion is not established.
  • d (Transformer hidden width) = 64 baseline; 32 and 128 variants
    Central expressivity knob; chosen by hand.
  • K (number of determinant channels) = 4 baseline; 2 and 8 variants
    Central expressivity knob; chosen by hand.
  • B (Monte Carlo batch size) = 4096 baseline; 1024 variant
    Central sampling-noise knob; chosen by hand.
  • lambda (MARCH regularization) = 0.001
    Optimizer hyperparameter chosen by hand and not ablated.
  • k0 (norm-threshold decay scale) = 8000
    Schedule parameter chosen by hand and not ablated.
  • kappa_clip (local-energy clipping scale) = 5
    Clipping scale chosen by hand following the FermiNet prescription.
  • d_hid (NNB hidden dimension) = 256 baseline; 64 for U=4, OBC, 8x8
    NNB capacity chosen by hand.
  • L (number of Transformer blocks) = unspecified
    Architectural parameter not given in the paper; the central results depend on it.
  • H (number of attention heads) = unspecified
    Architectural parameter not given in the paper; the central results depend on it.
axioms (5)
  • domain assumption Monte Carlo sampling from |psi(n)|^2 is ergodic and the clipped local-energy estimator does not bias the final converged energy.
    Used implicitly in Section II.C.3 for all VMC energy estimates. Clipping only the real part of the local energy may introduce bias in the optimization direction.
  • domain assumption The reference energies from exact diagonalization and DMRG are the true ground-state energies.
    The 4x4 doped reference -11.86884 is quoted without a citation; if this value is wrong, the central error metric epsilon_E loses meaning.
  • domain assumption The NNB plus supervised pretraining initialization does not trap the Transformer in the NNB's variational basin.
    Load-bearing for the transferability of the reported trends; described in Section II.C. If false, the ablations measure initialization quality rather than optimizer dynamics.
  • domain assumption A Transformer with one-hot local states and positional encoding can represent the needed 2D correlations on 4x4 and 8x8 lattices.
    Relies on the demonstration in Ref. [47]; no convergence guarantee is provided in this paper.
  • standard math Slater determinants and their sum enforce fermionic antisymmetry.
    Standard linear algebra used in Eqs. (9)-(10) and (14).

pith-pipeline@v1.3.0-alltime-deepseek · 14997 in / 12027 out tokens · 106444 ms · 2026-08-02T00:48:03.831850+00:00 · methodology

0 comments
read the original abstract

Building on the multi-determinant Transformer backflow neural quantum state (NQS) ansatz and the associated multi-stage training workflow for the doped two-dimensional Hubbard model, we investigate how the optimization dynamics of the NQS depend on several key optimization and architectural hyperparameters. The workflow consists of neural-network backflow (NNB) initialization, supervised Transformer pre-training, and main energy optimization using the Moment-Adaptive ReConfiguration Heuristic (MARCH) within variational Monte Carlo. Using the doped $4\times4$ periodic Hubbard model at $U=8$ as a baseline, we examine how the update-norm threshold, Transformer width, number of determinant channels, and Monte Carlo batch size affect convergence. We find that a moderate update constraint improves the efficiency of MARCH optimization, larger Transformer width and more determinant channels improve the expressive capacity of the ansatz, and larger Monte Carlo batches reduce sampling noise in the update direction. We further test the same workflow at half filling, weaker interaction strength, open boundary conditions, and on a larger $8\times8$ doped lattice. These results identify practical optimization trends for Transformer backflow NQSs and highlight the balance between ansatz expressivity, MARCH update stability, and Monte Carlo sampling quality.

Figures

Figures reproduced from arXiv: 2607.14875 by Jia-Qi Wang, Rong-Qiang He, Zhong-Yi Lu, Zong-Yu Liao.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic architecture of the main-training Transformer backflow NQS. The input configuration is mapped to local one-hot tokens [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Baseline ablation results for the doped 4 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Optimization behavior across different 4 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Optimization on the larger doped 8 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

69 extracted references · 1 canonical work pages

  1. [1]

    This NNB serves as the first stage of our training protocol

    Neural Network Backflow Architecture To provide a computationally efficient initialization, we construct a NNB architecture. This NNB serves as the first stage of our training protocol. It maps the real-space elec- tronic configuration to a set of spin-dependent backflow or- bitals, which are subsequently used to evaluate the wave func- tion amplitude via...

  2. [2]

    The purpose of this stage is to transfer the coarse physical structure learned by the lightweight NNB to the more expressive Transformer representation

    Pre-training After the NNB wave function has been optimized, we use it to initialize the Transformer ansatz through a supervised pre- training stage. The purpose of this stage is to transfer the coarse physical structure learned by the lightweight NNB to the more expressive Transformer representation. In this stage, the optimized NNB is used as the sampli...

  3. [3]

    LetNp be the number of real variational parameters, and letθ∈R Np denote the flattened parameter vector

    Main training The main optimization is performed using MARCH. LetNp be the number of real variational parameters, and letθ∈R Np denote the flattened parameter vector. For each configuration n, we define the logarithmic derivative Oα (n) =∂logψ θ (n) ∂ θα ,α=1, . . . ,N p.(24) For a Monte Carlo batch{n i}B i=1, the centered logarithmic derivatives are coll...

  4. [4]

    As shown in Fig

    Half-filled4×4PBC benchmark We first consider the half-filled system as a reference case away from the doped baseline. As shown in Fig. 3(a), the optimization remains stable under the same training protocol, indicating that the method is not restricted to doped systems. The comparison with the doped baseline also shows that the half-filled case is optimiz...

  5. [5]

    Weaker-interaction doped4×4PBC system We next reduce the interaction strength while keeping the same doped 4×4 PBC geometry, as shown in Fig. 3(b). Com- pared with the strongly correlatedU=8 doped baseline, the U=4 calculation reaches higher accuracy under the same overall workflow. This behavior is consistent with the re- duced correlation strength: the ...

  6. [6]

    This changes the boundary geometry while leaving the optimiza- tion protocol unchanged

    Open-boundary4×4systems We then test the same training strategy under OBC. This changes the boundary geometry while leaving the optimiza- tion protocol unchanged. The results in Fig. 3(c, d) show that the workflow remains stable for both half-filled and doped OBC systems, suggesting that the method is not tied to PBC

  7. [7]

    This test probes whether the approach can be used beyond the 4×4 benchmark systems

    Larger8×8doped PBC lattice Finally, we apply the same overall workflow to a larger doped lattice. This test probes whether the approach can be used beyond the 4×4 benchmark systems. As shown in Fig. 4, the optimization remains stable on the larger lattice, although the problem is visibly more challenging than the smaller sys- tems. 9 0 500 1k 1.5k 2k 2.5k...

  8. [8]

    M. Qin, T. Sch¨afer, S. Andergassen, P. Corboz, and E. Gull, The hubbard model: A computational perspective, Annual Review of Condensed Matter Physics13, 275 (2022). 10

    2022

  9. [9]

    D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The hub- bard model, Annual Review of Condensed Matter Physics13, 239 (2022)

    2022

  10. [10]

    Imada, A

    M. Imada, A. Fujimori, and Y . Tokura, Metal-insulator transi- tions, Rev. Mod. Phys.70, 1039 (1998)

    1998

  11. [11]

    Kohno, Mott transition in the two-dimensional hubbard model, Phys

    M. Kohno, Mott transition in the two-dimensional hubbard model, Phys. Rev. Lett.108, 076401 (2012)

    2012

  12. [12]

    Fratino, P

    L. Fratino, P. S ´emon, M. Charlebois, G. Sordi, and A.-M. S. Tremblay, Signatures of the mott transition in the antiferromag- netic state of the two-dimensional hubbard model, Phys. Rev. B 95, 235109 (2017)

    2017

  13. [13]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dy- namical mean-field theory of strongly correlated fermion sys- tems and the limit of infinite dimensions, Rev. Mod. Phys.68, 13 (1996)

    1996

  14. [14]

    Wietek, R

    A. Wietek, R. Rossi, F. ˇSimkovic, M. Klett, P. Hansmann, M. Ferrero, E. M. Stoudenmire, T. Sch ¨afer, and A. Georges, Mott insulating states with competing orders in the triangular lattice hubbard model, Phys. Rev. X11, 041013 (2021)

    2021

  15. [15]

    Mazurenko, C

    A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kan ´asz- Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom fermi-hubbard antiferromagnet, Na- ture545, 462 (2017)

    2017

  16. [16]

    L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol, and M. W. Zwierlein, Observation of spatial charge and spin cor- relations in the 2d fermi-hubbard model, Science353, 1260 (2016)

    2016

  17. [17]

    M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, Site-resolved measurement of the spin-correlation function in the fermi-hubbard model, Science353, 1253 (2016)

    2016

  18. [18]

    C. N. Varney, C.-R. Lee, Z. J. Bai, S. Chiesa, M. Jarrell, and R. T. Scalettar, Quantum monte carlo study of the two- dimensional fermion hubbard model, Phys. Rev. B80, 075116 (2009)

    2009

  19. [19]

    ˇSimkovic, J

    F. ˇSimkovic, J. P. F. LeBlanc, A. J. Kim, Y . Deng, N. V . Prokof’ev, B. V . Svistunov, and E. Kozik, Extended crossover from a fermi liquid to a quasiantiferromagnet in the half-filled 2d hubbard model, Phys. Rev. Lett.124, 017003 (2020)

    2020

  20. [20]

    Borejsza and N

    K. Borejsza and N. Dupuis, Antiferromagnetism and single- particle properties in the two-dimensional half-filled hubbard model: A nonlinear sigma model approach, Phys. Rev. B69, 085119 (2004)

    2004

  21. [21]

    Zheng, C.-M

    B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional hubbard model, Science358, 1155 (2017)

    2017

  22. [22]

    H. Xu, H. Shi, E. Vitali, M. Qin, and S. Zhang, Stripes and spin- density waves in the doped two-dimensional hubbard model: Ground state phase diagram, Phys. Rev. Res.4, 013239 (2022)

    2022

  23. [23]

    Xiao, Y .-Y

    B. Xiao, Y .-Y . He, A. Georges, and S. Zhang, Tempera- ture dependence of spin and charge orders in the doped two- dimensional hubbard model, Phys. Rev. X13, 011007 (2023)

    2023

  24. [24]

    Chang and S

    C.-C. Chang and S. Zhang, Spin and charge order in the doped hubbard model: Long-wavelength collective modes, Phys. Rev. Lett.104, 116402 (2010)

    2010

  25. [25]

    P. Mai, S. Karakuzu, G. Balduzzi, S. Johnston, and T. A. Maier, Intertwined spin, charge, and pair correlations in the two-dimensional hubbard model in the thermodynamic limit, Proceedings of the National Academy of Sciences119, e2112806119 (2022)

    2022

  26. [26]

    E. W. Huang, C. B. Mendl, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Stripe order from the perspective of the hubbard model, npj Quantum Materials3, 22 (2018)

    2018

  27. [27]

    Jiang and T

    H.-C. Jiang and T. P. Devereaux, Superconductivity in the doped hubbard model and its interplay with next-nearest hop- pingt ′, Science365, 1424 (2019)

    2019

  28. [28]

    Qin, C.-M

    M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollw¨ock, S. R. White, and S. Zhang (Simons Collabora- tion on the Many-Electron Problem), Absence of superconduc- tivity in the pure two-dimensional hubbard model, Phys. Rev. X10, 031016 (2020)

    2020

  29. [29]

    Jiang, T

    Y .-F. Jiang, T. P. Devereaux, and H.-C. Jiang, Ground-state phase diagram and superconductivity of the doped hubbard model on six-leg square cylinders, Phys. Rev. B109, 085121 (2024)

    2024

  30. [30]

    D. J. Scalapino, Numerical studies of the 2d hubbard model (2006), arXiv:cond-mat/0610710 [cond-mat.str-el]

    Pith/arXiv arXiv 2006

  31. [31]

    X. Dong, L. Del Re, A. Toschi, and E. Gull, Mechanism of superconductivity in the hubbard model at intermediate interac- tion strength, Proceedings of the National Academy of Sciences 119, e2205048119 (2022)

    2022

  32. [32]

    Macridin, M

    A. Macridin, M. Jarrell, T. Maier, and G. A. Sawatzky, Physics of cuprates with the two-band hubbard model: The validity of the one-band hubbard model, Phys. Rev. B71, 134527 (2005)

    2005

  33. [33]

    Maier, M

    T. Maier, M. Jarrell, T. Pruschke, and J. Keller,d-wave super- conductivity in the hubbard model, Phys. Rev. Lett.85, 1524 (2000)

    2000

  34. [34]

    E. Gull, O. Parcollet, and A. J. Millis, Superconductivity and the pseudogap in the two-dimensional hubbard model, Phys. Rev. Lett.110, 216405 (2013)

    2013

  35. [35]

    Zhang, J

    S. Zhang, J. Carlson, and J. E. Gubernatis, Constrained path monte carlo method for fermion ground states, Phys. Rev. B55, 7464 (1997)

    1997

  36. [36]

    M. Qin, H. Shi, and S. Zhang, Benchmark study of the two-dimensional hubbard model with auxiliary-field quantum monte carlo method, Phys. Rev. B94, 085103 (2016)

    2016

  37. [37]

    M. Qin, H. Shi, and S. Zhang, Numerical results on the short- range spin correlation functions in the ground state of the two- dimensional hubbard model, Phys. Rev. B96, 075156 (2017)

    2017

  38. [38]

    Qin, Self-consistent optimization of the trial wave func- tion within the constrained path auxiliary field quantum monte carlo method using mixed estimators, Phys

    M. Qin, Self-consistent optimization of the trial wave func- tion within the constrained path auxiliary field quantum monte carlo method using mixed estimators, Phys. Rev. B107, 235124 (2023)

    2023

  39. [39]

    S. R. White and D. J. Scalapino, Stripes on a 6-leg hubbard ladder, Phys. Rev. Lett.91, 136403 (2003)

    2003

  40. [40]

    Hager, G

    G. Hager, G. Wellein, E. Jeckelmann, and H. Fehske, Stripe formation in doped hubbard ladders, Phys. Rev. B71, 075108 (2005)

    2005

  41. [41]

    Ehlers, S

    G. Ehlers, S. R. White, and R. M. Noack, Hybrid-space den- sity matrix renormalization group study of the doped two- dimensional hubbard model, Phys. Rev. B95, 125125 (2017)

    2017

  42. [42]

    Corboz, Improved energy extrapolation with infinite pro- jected entangled-pair states applied to the two-dimensional hub- bard model, Phys

    P. Corboz, Improved energy extrapolation with infinite pro- jected entangled-pair states applied to the two-dimensional hub- bard model, Phys. Rev. B93, 045116 (2016)

    2016

  43. [43]

    Scheb and R

    M. Scheb and R. M. Noack, Finite projected entangled pair states for the hubbard model, Phys. Rev. B107, 165112 (2023)

    2023

  44. [44]

    Ponsioen, S

    B. Ponsioen, S. S. Chung, and P. Corboz, Period 4 stripe in the extended two-dimensional hubbard model, Phys. Rev. B100, 195141 (2019)

    2019

  45. [45]

    Zhang, J.-W

    C. Zhang, J.-W. Li, D. Nikolaidou, and J. von Delft, Frustration- induced superconductivity in thet−t ′ hubbard model, Phys. Rev. Lett.134, 116502 (2025)

    2025

  46. [46]

    Knizia and G

    G. Knizia and G. K.-L. Chan, Density matrix embedding: A simple alternative to dynamical mean-field theory, Phys. Rev. Lett.109, 186404 (2012)

    2012

  47. [47]

    Zheng and G

    B.-X. Zheng and G. K.-L. Chan, Ground-state phase diagram of the square lattice hubbard model from density matrix em- 11 bedding theory, Phys. Rev. B93, 035126 (2016)

    2016

  48. [48]

    Wu, Z.-H

    X. Wu, Z.-H. Cui, Y . Tong, M. Lindsey, G. K.-L. Chan, and L. Lin, Projected density matrix embedding theory with ap- plications to the two-dimensional hubbard model, Journal of Chemical Physics151, 064108 (2019)

    2019

  49. [49]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science355, 602 (2017)

    2017

  50. [50]

    Nomura, A

    Y . Nomura, A. S. Darmawan, Y . Yamaji, and M. Imada, Re- stricted boltzmann machine learning for solving strongly corre- lated quantum systems, Phys. Rev. B96, 205152 (2017)

    2017

  51. [51]

    Glasser, N

    I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, and J. I. Cirac, Neural-network quantum states, string-bond states, and chiral topological states, Phys. Rev. X8, 011006 (2018)

    2018

  52. [52]

    Wang, H.-Q

    J.-Q. Wang, H.-Q. Wu, R.-Q. He, and Z.-Y . Lu, Variational op- timization of the amplitude of neural-network quantum many- body ground states, Phys. Rev. B109, 245120 (2024)

    2024

  53. [53]

    Wang, R.-Q

    J.-Q. Wang, R.-Q. He, and Z.-Y . Lu, Generalized lanczos method for systematic optimization of neural-network quantum states, Phys. Rev. B113, 085120 (2026)

    2026

  54. [54]

    Y . Gu, W. Li, H. Lin, B. Zhan, R. Li, Y . Huang, D. He, Y . Wu, T. Xiang, M. Qin, L. Wang, and D. Lv, Solving the Hub- bard model with neural quantum states, Nature Communica- tions 10.1038/s41467-026-74028-6 (2026)

    work page doi:10.1038/s41467-026-74028-6 2026

  55. [55]

    Ibarra-Garc ´ıa-Padilla, H

    E. Ibarra-Garc ´ıa-Padilla, H. Lange, R. G. Melko, R. T. Scalet- tar, J. Carrasquilla, A. Bohrdt, and E. Khatami, Autoregressive neural quantum states of fermi hubbard models, Physical Re- view Research7, 013122 (2025)

    2025

  56. [56]

    Chen and M

    A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nature Physics20, 1476 (2024)

    2024

  57. [57]

    Rende, L

    R. Rende, L. L. Viteritti, L. Bardone, F. Becca, and S. Goldt, A simple linear algebra identity to optimize large-scale neural net- work quantum states, Communications Physics7, 260 (2024)

    2024

  58. [58]

    Robledo Moreno, G

    J. Robledo Moreno, G. Carleo, A. Georges, and J. Stokes, Fermionic wave functions from neural-network constrained hidden states, Proceedings of the National Academy of Sci- ences119, e2122059119 (2022)

    2022

  59. [59]

    Luo and B

    D. Luo and B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, Physical Re- view Letters122, 226401 (2019)

    2019

  60. [60]

    Chen, Z.-Q

    A. Chen, Z.-Q. Wan, A. Sengupta, A. Georges, and C. Roth, Neural network-augmented pfaffian wave-functions for scalable simulations of interacting fermions (2025), arXiv:2507.10705 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  61. [61]

    Y . Gu, Z. Han, W. Li, Z. Xiao, T. Xiang, M. Qin, L. Wang, and D. Lv, Pareto frontier of neural quantum states: Scalable, affordable, and accurate convolutional backflow for strongly correlated lattice fermions (2026), arXiv:2604.25775 [cond- mat.str-el]

    Pith/arXiv arXiv 2026

  62. [62]

    L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Carleo, and A. Georges, Beyond variational bias: Resolving intertwined orders in the hubbard model (2026), arXiv:2604.21978 [cond- mat.str-el]

    Pith/arXiv arXiv 2026

  63. [63]

    Elfwing, E

    S. Elfwing, E. Uchibe, and K. Doya, Sigmoid-weighted linear units for neural network function approximation in reinforce- ment learning, Neural Networks107, 3 (2018)

    2018

  64. [64]

    D. P. Kingma and J. Ba, Adam: A method for stochastic opti- mization, arXiv preprint arXiv:1412.6980 (2014)

    Pith/arXiv arXiv 2014

  65. [65]

    Sorella, M

    S. Sorella, M. Casula, and D. Rocca, Weak binding between two aromatic rings: Feeling the van der waals attraction by quantum monte carlo methods, J. Chem. Phys.127, 014105 (2007)

    2007

  66. [66]

    M. P. Nightingale and V . Melik-Alaverdian, Optimization of ground- and excited-state wave functions and van der waals clusters, Phys. Rev. Lett.87, 043401 (2001)

    2001

  67. [67]

    Sorella, Generalized lanczos algorithm for variational quan- tum monte carlo, Phys

    S. Sorella, Generalized lanczos algorithm for variational quan- tum monte carlo, Phys. Rev. B64, 024512 (2001)

    2001

  68. [68]

    D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes, Ab initio solution of the many-electron schr ¨odinger equation with deep neural networks, Physical Review Research 2, 033429 (2020)

    2020

  69. [69]

    W.-Y . Liu, H. Zhai, R. Peng, Z.-C. Gu, and G. K.-L. Chan, Ac- curate simulation of the hubbard model with finite fermionic projected entangled pair states, Phys. Rev. Lett.134, 256502 (2025)

    2025