Pith. sign in

REVIEW 3 major objections 3 minor 4 cited by

Solving the Hubbard model with Neural Quantum States

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A transformer-based neural network solves the 2D Hubbard model up to 16x16 and finds that adding next-nearest-neighbor hopping puts a half-filled stripe with period 4 at the ground state at 1/8 doping.

desk verdict Transformer NQS at 16x16 with real benchmarks, but the t'=-0.2 half-filled stripe claim needs an unbiased search and error bars before it can be called established. read the letter →

arxiv 2507.02644 v2 pith:AIF2XRJL submitted 2025-07-03 cond-mat.str-el cs.AIquant-ph

Yuntian Gu , Wenrui Li , Heng Lin , Bo Zhan , Ruichen Li , Yifei Huang , Di He , Yantao Wu
show 4 more authors
Tao Xiang Mingpu Qin Liwei Wang Dingshun Lv
This is my paper · ORCID
classification cond-mat.str-elcs.AIquant-ph PACS 71.10.Fd71.27.+a
keywords HubbardmodelneuralquantumstatestransformerstripeordercupratesvariationalMonteCarlobackfloworbitalsMARCHoptimizer
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 claims that modern transformer neural networks, trained with a new optimizer called MARCH, can produce variational ground-state energies for the 2D Hubbard model that are accurate enough to settle the nature of the ground state at light doping. Using these methods under periodic boundary conditions on lattices up to 16×16, the authors find that adding next-nearest-neighbor hopping $t'=-0.2$ — the value relevant to cuprates — makes the ground state at hole doping $\delta=1/8$ a half-filled stripe with wavelength $\lambda=4$, matching experimental observations in cuprates. They also confirm that the pure Hubbard model hosts a filled stripe with wavelength $\lambda=8$ under periodic boundary conditions, and that the stripe direction prefers the longer side of the lattice. If correct, the result tightens the connection between the Hubbard model and high-temperature superconductor physics, and demonstrates that neural quantum states are competitive with the best tensor-network methods on genuinely two-dimensional fermion problems.

What carries the argument

The central object is the transformer-based neural quantum state, a wavefunction ansatz in which each lattice site is embedded as a token and processed by multi-head self-attention layers that can aggregate information across arbitrary distances; the output is mapped to backflow orbitals, and the wavefunction amplitude is the sum of Slater determinants of those orbitals. The transformer's built-in long-range connectivity — absent in RBM and MLP ansatze — is what makes large periodic-boundary systems tractable. The second piece of machinery is the MARCH optimizer (Moment-Adaptive ReConfiguration Heuristic), which augments stochastic reconfiguration with first- and second-moment estimates of the gradient, adapting the step size per parameter; this is what makes convergence fast enough for 100,000-step runs on 16×16 lattices. A temporary staggered pinning field is used to prepare candidate stripe states, which are then relaxed without the field, and this preparation is what allows the energy comparison between horizontal and vertical stripe states.

What would settle it

Run an independent high-accuracy calculation (e.g., auxiliary-field quantum Monte Carlo at 1/8 doping, or large-scale DMRG on 8- or 12-leg cylinders with the same $t'$ and periodic boundaries in the long direction) and compare to the NQS variational energy of the $\lambda=4$ horizontal stripe. If either method finds a uniform d-wave or a vertical-stripe state with lower energy per site than the horizontal half-filled stripe on the same lattice (beyond the ~0.001-per-site gaps reported), the central claim fails. A simpler version: relax the $t'=-0.2$, 32x8 system without any pinning field and check whether a $\lambda=4$ horizontal stripe emerges spontaneously from an unbiased initialization; if the optimizer settles into a non-stripe state with equal or lower energy, the pinning-field protocol biased the search.

Watch

Extended reading notes

Core claim

The paper's central claim is physical and computational. Computationally, it claims that a transformer-based neural quantum state — a wavefunction ansatz in which each lattice site is embedded as a token and processed by multi-head self-attention layers that aggregate information across arbitrary distances, with the amplitude given by a sum of Slater determinants of backflow orbitals — combined with the new MARCH optimizer reaches variational ground-state energies for the 2D Hubbard model that are lower than the best PEPS results (bond dimension $D \ge 20$) on every system tested, and lower than DMRG once the lattice width exceeds 6. Physically, for the Hubbard model with next-nearest-neighbor hopping $t' = -0.2$ at hole doping $\delta = 1/8$ and $U = 8$ under periodic boundary conditions, the ground state is claimed to be a half-filled stripe with wavelength $\lambda = 4$, and on rectangular lattices this stripe runs horizontally, along the longer side, because the kinetic energy favors a single long 'river' of holes. For the pure Hubbard model ($t' = 0$), the ground state is claimed to be a filled stripe with wavelength $\lambda = 8$ under periodic boundary conditions, with the stripe-direction crossover seen in open-boundary systems identified as a finite-size boundary effect. The paper also claims that after optimization the different attention heads specialize: one encodes short-range correlations, one nearest-neighbor hopping, one the antiferromagnetic checkerboard pattern, and one delocalized phase information.

Load-bearing premise

The results assume the reported variational energies — given without error bars — are accurate enough to tell apart ground states that differ by only about 0.001 in energy per site, and that for the $t'=-0.2$ case the true ground state is among the stripe states deliberately prepared with the pinning field rather than a state of some other order.

Editorial extensions

If this is right

  • The ground state of the $t'=-0.2$ Hubbard model at 1/8 doping is a half-filled stripe with wavelength $\lambda=4$ under periodic boundary conditions, matching the period of stripe order seen in cuprate superconductors.
  • Previously reported stripe wavelengths fluctuating around 4 in cylinder DMRG studies are finite-size and boundary effects; periodic-boundary results pin the wavelength to exactly 4.
  • For rectangular lattices, the $t'=-0.2$ stripe runs along the longer direction, with the kinetic energy favoring a single long 'river' of holes over fragmented short stripes.
  • The pure Hubbard model ($t'=0$) has a filled stripe with wavelength $\lambda=8$ under periodic boundary conditions, and the stripe-direction crossover with system width seen under open boundaries disappears under PBC.
  • Transformer-based NQS with the MARCH optimizer produce lower variational energies than PEPS with bond dimension at least 20 on all tested 2D systems and outperform DMRG for widths above 6, making periodic-boundary simulations up to 16×16 practical.

Reading between the lines

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

  • The pinning-field search means the half-filled-stripe conclusion is conditional on stripe order being the right sector; a definitive test would be a large-scale auxiliary-field quantum Monte Carlo or DMRG run at 1/8 doping comparing the stripe energy against a uniform d-wave state, which the paper did not do.
  • If the $\lambda=4$ stripe is confirmed, the next testable step for the $t'$ Hubbard model is to compute pairing correlations in the same variational states; the paper establishes stripes but does not address whether d-wave superconductivity coexists with them in this parameter regime.
  • The attention-head analysis suggests a practical diagnostic: attention maps of optimized neural quantum states could be used to detect emergent order (antiferromagnetic, stripe, or other patterns) in other frustrated or doped fermion models where the order is not known in advance.
  • Extrapolating the reported $E_{\mathrm{vert}} - E_{\mathrm{hori}}$ data as a function of $L_x$ suggests stripe orientation may remain horizontal in the thermodynamic limit for $t'=-0.2$, but the paper's largest symmetric test lattice (32×16) is beyond current capacity; a targeted calculation on that size would settle the orientation question.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The manuscript presents a transformer-based neural quantum state (NQS) ansatz combined with a new optimizer, MARCH, and applies it to the two-dimensional Hubbard model at U=8 and hole doping delta=1/8 for both t'=0 and t'=-0.2, using open and periodic boundary conditions on lattices up to 16x16. The authors report variational energies that are competitive with or lower than existing DMRG and PEPS results for several OBC systems, an AFQMC consistency check at half-filling, an analysis of attention-head specialization, and two central physics claims: a filled stripe with wavelength lambda=8 in the pure Hubbard model and a half-filled stripe with wavelength lambda=4, oriented horizontally, in the t'=-0.2 model. The half-filled stripe and its orientation are the headline results of the paper.

Significance. If the half-filled stripe claim were fully established, the paper would be an important step for both NQS methodology and Hubbard-model physics: it would support the t' Hubbard model as a minimal model for cuprate stripe physics and would demonstrate that NQS can resolve competing orders at 1/8 doping on two-dimensional PBC lattices beyond cylinder DMRG. The paper's concrete strengths are its extensive benchmarks (Table S1 against AFQMC; Table S2 and Figure 2C against PEPS and DMRG), the detailed architecture and hyperparameter documentation in the Supplementary Materials, and the use of PBC to remove open-boundary stripe-orientation biases. However, the central t'=-0.2 stripe identification is not yet supported to variational accuracy: the searches are seeded by a pinning field, no non-stripe variational competitor is included, and the orientation decision rests on energy differences without reported error bars. The significance is therefore conditional pending a stronger unbiased comparison or an explicit energy-resolution estimate.

major comments (3)
  1. [SM Section 3, Eq. (S29); main text 'Half-filled stripe order in t'-Hubbard model'] The half-filled stripe conclusion for t'=-0.2 is obtained exclusively from pinning-field-seeded optimizations. SM Section 3 states that psi1 is optimized under an AFM pinning field of the form of Eq. (S29), that psi2 is initialized by maximizing fidelity to psi1, and that only then is psi2 optimized with the field removed; the section further says this procedure is used 'to stabilize a vertical or horizontal stripe.' No unbiased optimization is reported, and no non-stripe variational candidate such as a uniform d-wave state, a paired state, or another charge-ordered state is included as a competitor. Without such a comparison, the main-text assertion that 'the ground state for the t'=-0.2 Hubbard model is clearly the half-filled stripe' is not tested by the evidence. Please add, for at least one representative system (e.g., 16x16 or 32x8 with t'=-0.2), either an unbiased optimization that converges to the stripe without a pinning seed, or an energy comparison against one or more non-stripe variational ansatze.
  2. [Figure 5, Figure S32] The stripe-orientation claim rests on energy differences between vertical and horizontal stripes that are on the order of 0.001-0.002 per site, but no statistical or systematic uncertainties are reported for these variational energies. Figure S32 shows variation of both vertical and horizontal energies as a function of the number of attention heads and determinants, and the spread of those values is comparable to the orientation gap. The conclusion that 'the horizontal stripe has lower energy for all the systems' (Figure 5 caption) is therefore not established at the reported resolution. Please provide energy uncertainties from independent optimization runs or MCMC re-sampling, and show that the vertical-horizontal gap exceeds this resolution for at least the largest system studied.
  3. [Table S1; Figure 2C; SM Table S2] The AFQMC consistency check is performed only at half-filling, whereas the half-filled stripe claim is made at delta=1/8 doping. The OBC comparisons against DMRG and PEPS in Figure 2C and Table S2 are for the pure Hubbard model and, while encouraging, do not validate the energy differences that select stripe filling, stripe wavelength, or stripe orientation in the t'=-0.2 doped case. Please either provide a doped benchmark on a system where a more accurate reference is available (for example AFQMC on a small PBC lattice where the sign problem is manageable, or DMRG on a wider cylinder for t'=-0.2), or explicitly qualify the stripe conclusion as a variational result whose accuracy is not independently benchmarked at delta=1/8.
minor comments (3)
  1. [SM Figures S11-S30] The captions of Figures S11-S30 omit the minus sign on the energies, e.g., 'E = 0.6995' should read 'E = -0.6995'; this makes the density figures inconsistent with the main-text energies, which are negative.
  2. [SM Section 1.1; Table S4 caption] There are minor wording and typographical issues: 'transformered' should be 'transformed' in SM Section 1.1, and 'Dopping' should be 'Doping' in the Table S4 caption.
  3. [SM Tables S5-S7] The hyperparameter tables list a single configuration, but the main text notes that hyperparameters may differ between systems; please list the actual hyperparameters used for each reported system size, or state explicitly which entries are common across all calculations, to make the results reproducible.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the physical conclusions are produced by variational energy minimization against external benchmarks, not by construction from fitted inputs.

full rationale

The paper's central results (the half-filled stripe for t'=-0.2 at delta=1/8 and its horizontal orientation) are outputs of an unbiased variational minimization of a general neural-network ansatz: the NQS wavefunction is a sum of determinants of backflow orbitals (Eq. S8), and the optimization target is the unconstrained energy (Eq. S3). Although Supplementary Section 3 initializes stripe-ordered states with a temporary pinning field (Eq. S29), the field is removed before the final optimization, and the remaining bias is only in the initialization; no equation defines the stripe order into the ansatz or sets the stripe energy equal to anything fitted from the data. The energy comparison between vertical and horizontal stripes (Figure S32 and Figure 5) is a genuine variational comparison, not a fitted parameter renamed as a prediction. Benchmarks against AFQMC (Table S1), DMRG, and PEPS (Table S2, Fig. 2C) are external, and the fact that some cited benchmark papers (e.g., refs 47 and 49) share co-authors does not make those comparisons load-bearing self-citations, since they do not assume the present stripe conclusion. The lack of an unpinned non-stripe competitor and the absence of reported error bars on variational energies are correctness and robustness concerns, not circularity. Overall, the derivation chain is self-contained and no circular step can be exhibited from the paper's equations.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The central claims rest on the expressivity of the transformer ansatz, the convergence of the MARCH optimizer, the reliability of the pinning-field relaxation, and the standard variational Monte Carlo framework. The free parameters are architectural and optimization hyperparameters chosen by hand, which influence the accuracy of the variational energies and hence the physical conclusions.

free parameters (6)
  • Transformer hidden dimension = 256 (384 for 'large' runs)
    Architectural hyperparameter; larger values increase expressivity at higher computational cost, affecting the variational accuracy.
  • Number of transformer layers = 4
    Depth of the network; chosen for expressivity/optimizability tradeoff.
  • Number of determinants in ansatz = 4 (up to 6 in stripe-orientation tests)
    Number of Slater determinants summed in equation S8; directly controls the representational power of the wavefunction.
  • MARCH damping lambda = 0.001
    Regularization parameter in the stochastic reconfiguration update (Table S7); influences convergence and final energy.
  • Pinning field strength h_m = 0.2
    Temporary antiferromagnetic field strength used to stabilize stripe patterns during optimization (SM Section 3).
  • Local energy clipping threshold = 5.0
    Clipping of local energies in the VMC loss to control variance; affects the effective Hamiltonian used in optimization.
assumptions (5)
  • domain assumption The Hubbard model with U=8 and t'=-0.2 is an appropriate minimal model for cuprate high-Tc superconductors.
    The physical relevance of the stripe order claim depends on this connection, which is widely discussed in the literature but not established from first principles.
  • standard math The variational principle and Monte Carlo sampling give unbiased estimates of ground-state properties when the ansatz is sufficiently expressive.
    Standard VMC assumption; the paper provides no error bars, so the unbiasedness of the reported energies is unverified.
  • domain assumption The transformer-based sum of Slater determinants can approximate the true ground state closely for the system sizes and dopings studied.
    Benchmarks on half-filling and small doped systems support this, but the accuracy at 16x16 with delta=1/8 and t'=-0.2 is not directly benchmarked against an exact method.
  • ad hoc to paper The pinning-field procedure, followed by relaxation, converges to the true ground state in the symmetry sector of the pinned stripe.
    The temporary pinning field (SM Section 3) guides the wavefunction into a stripe pattern; the subsequent relaxation is assumed to remove the pinning bias, but this is not proven.
  • ad hoc to paper The attention head specialization after optimization reflects physically meaningful correlation scales.
    The claim that heads encode local, hopping, AFM, and delocalized correlations is based on visual inspection of attention maps (Figure 3), without a quantitative measure.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Solving the Hubbard model with Neural Quantum States." pith.science (2026). https://pith.science/paper/AIF2XRJL

@misc{pith2026250702644,
  author       = {Pith},
  title        = {Pith review of: Solving the Hubbard model with Neural Quantum States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIF2XRJL}},
  note         = {Machine review of arXiv:2507.02644}
}
read the original abstract

The rapid development of neural quantum states (NQS) has established it as a promising framework for studying quantum many-body systems. In this work, by leveraging the cutting-edge transformer-based architectures and developing highly efficient optimization algorithms, we achieve the state-of-the-art results for the doped two-dimensional (2D) Hubbard model, arguably the minimum model for high-Tc superconductivity. Interestingly, we find different attention heads in the NQS ansatz can directly encode correlations at different scales, making it capable of capturing long-range correlations and entanglements in strongly correlated systems. With these advances, we establish the half-filled stripe in the ground state of 2D Hubbard model with the next nearest neighboring hoppings, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Neural Network-Augmented Pfaffian Wave-functions for Scalable Simulations of Interacting Fermions

    cond-mat.str-el 2025-07 conditional novelty 7.0 of 10

    A neural-network Pfaffian wavefunction with hidden fermions achieves state-of-the-art variational accuracy on the 2D Hubbard model and scales via low-rank updates and sublattice symmetry.

  2. Thermalization Dynamics in the Two-Dimensional Hubbard Model with Neural-Network Quantum States

    cond-mat.str-el 2026-06 unverdicted novelty 6.0 of 10

    Real-time dynamics in the 2D Hubbard model show thermalization of double occupancy below a critical U_c but clear breakdown of thermalization above it.

  3. Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model

    cond-mat.str-el 2025-10 conditional novelty 6.0 of 10

    For the doped square-lattice Hubbard model, hidden-fermion and backflow neural quantum states with a Vision Transformer backbone reach nearly equal variational energies; translation-equivariant attention is outperform...

  4. A Sketch-and-Project Analysis of Subsampled Natural Gradient Algorithms

    cs.LG 2025-08 conditional novelty 6.0 of 10

    For linear least squares, SNGD and SPRING are proved equivalent to accelerated regularized Kaczmarz methods, yielding the first fast rates and first SPRING guarantee; the general quadratic analysis holds under strong ...

Reference graph

Works this paper leans on

59 extracted references · 38 canonical work pages · cited by 4 Pith papers

  1. [1]

    Hubbard, Electron Correlations in Narrow Energy Bands

    J. Hubbard, Electron Correlations in Narrow Energy Bands. Proc. R. Soc. Lond. A 276 (1365), 238–257 (1963), doi:10.1098/rspa.1963.0204, http://www.jstor.org/stable/2414761

    arXiv 1963

  2. [2]

    M. Qin, T. Sch ¨afer, S. Andergassen, P. Corboz, E. Gull, The Hubbard model: A computational perspective. Annual Review of Condensed Matter Physics 13 (1), 275–302 (2022)

    work page 2022

  3. [3]

    D. P. Arovas, E. Berg, S. A. Kivelson, S. Raghu, The hubbard model. Annual review of condensed matter physics 13 (1), 239–274 (2022)

    work page 2022

  4. [4]

    P. W. Anderson, The Resonating Valence Bond State in La 2CuO4 and Superconductivity. Science 235 (4793), 1196–1198 (1987), doi:10.1126/science.235.4793.1196, https://www. science.org/doi/abs/10.1126/science.235.4793.1196

    arXiv 1987

  5. [5]

    F. C. Zhang, T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides.Phys. Rev. B 37, 3759–3761 (1988), doi:10.1103/PhysRevB.37.3759, https://link.aps.org/doi/10. 1103/PhysRevB.37.3759

    doi:10.1103/physrevb.37.3759 1988

  6. [6]

    S. R. White, Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992), doi:10.1103/PhysRevLett.69.2863, https://link.aps.org/doi/ 10.1103/PhysRevLett.69.2863

    doi:10.1103/physrevlett.69.2863 1992

  7. [7]

    J. I. Cirac, D. P´erez-Garc´ıa, N. Schuch, F. Verstraete, Matrix product states and projected entan- gled pair states: Concepts, symmetries, theorems. Rev. Mod. Phys. 93, 045003 (2021), doi:10. 1103/RevModPhys.93.045003, https://link.aps.org/doi/10.1103/RevModPhys.93. 045003

    doi:10.1103/revmodphys.93 2021

  8. [8]

    T. Xiang, Density Matrix and Tensor Network Renormalization (Cambridge University Press) (2023), https://books.google.com/books?hl=en&lr=&id=E5fxEAAAQBAJ&oi= fnd&pg=PP1&ots=Hqdq-ApAx0&sig=xi-IvwDkPLk7CTWPcB_d5zRtFZk

    work page 2023

Show all 59 references
  1. [9]

    Verstraete, J

    F. Verstraete, J. I. Cirac, Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions (2004), https://arxiv.org/abs/cond-mat/0407066. 16

    2004 arXiv

  2. [10]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996), doi:10.1103/RevModPhys.68.13, https://link.aps.org/doi/10.1103/ RevModPhys.68.13

    1996 doi

  3. [11]

    Knizia, G

    G. Knizia, G. K.-L. Chan, Density Matrix Embedding: A Simple Alternative to Dynami- cal Mean-Field Theory. Phys. Rev. Lett. 109, 186404 (2012), doi:10.1103/PhysRevLett.109. 186404, https://link.aps.org/doi/10.1103/PhysRevLett.109.186404

    2012 doi

  4. [12]

    Gubernatis, N

    J. Gubernatis, N. Kawashima, P. Werner, Quantum Monte Carlo Methods: Algorithms for Lattice Models (Cambridge University Press) (2016)

    2016

  5. [13]

    Becca, S

    F. Becca, S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press) (2017)

    2017

  6. [14]

    Zhang, Auxiliary-Field Quantum Monte Carlo for Correlated Electron Systems , Vol

    S. Zhang, Auxiliary-Field Quantum Monte Carlo for Correlated Electron Systems , Vol. 3 of Emergent Phenomena in Correlated Matter: Modeling and Simulation , Ed. E. Pavarini, E. Koch, and U. Schollw¨ock (Verlag des Forschungszentrum J¨ ulich, 2013)

    2013

  7. [15]

    J. P. F. LeBlanc, et al., Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms. Phys. Rev. X 5, 041041 (2015), doi:10. 1103/PhysRevX.5.041041, https://link.aps.org/doi/10.1103/PhysRevX.5.041041

    2015 doi

  8. [16]

    Zheng, et al., Stripe order in the underdoped region of the two-dimensional Hubbard model

    B.-X. Zheng, et al., Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358 (6367), 1155–1160 (2017), doi:10.1126/science.aam7127,https://www. science.org/doi/abs/10.1126/science.aam7127

    2017 doi

  9. [17]

    Xu, et al

    H. Xu, et al. , Coexistence of superconductivity with partially filled stripes in the Hubbard model. Science 384 (6696), eadh7691 (2024)

    2024

  10. [18]

    Nomura, A

    Y. Nomura, A. S. Darmawan, Y. Yamaji, M. Imada, Restricted Boltzmann machine learning for solving strongly correlated quantum systems. Physical Review B 96 (20), 205152 (2017)

    2017

  11. [19]

    K. Inui, Y. Kato, Y. Motome, Determinant-free fermionic wave function using feed-forward neural networks. Physical Review Research 3 (4), 043126 (2021). 17

    2021

  12. [20]

    D. Luo, B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions. Physical review letters 122 (22), 226401 (2019)

    2019

  13. [21]

    Robledo Moreno, G

    J. Robledo Moreno, G. Carleo, A. Georges, J. Stokes, Fermionic wave functions from neural- network constrained hidden states.Proceedings of the National Academy of Sciences 119 (32), e2122059119 (2022)

    2022

  14. [22]

    Zhou, Z.-W

    Y.-T. Zhou, Z.-W. Zhou, X. Liang, Solving Fermi-Hubbard-type models by tensor representa- tions of backflow corrections. Physical Review B 109 (24), 245107 (2024)

    2024

  15. [23]

    A. Chen, M. Heyl, Empowering deep neural quantum states through efficient optimization. Nature Physics 20 (9), 1476–1481 (2024)

    2024

  16. [24]

    Rende, L

    R. Rende, L. L. Viteritti, L. Bardone, F. Becca, S. Goldt, A simple linear algebra identity to optimize large-scale neural network quantum states. Communications Physics 7 (1), 260 (2024)

    2024

  17. [25]

    Goldshlager, N

    G. Goldshlager, N. Abrahamsen, L. Lin, A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions.Journal of Computational Physics 516, 113351 (2024)

    2024

  18. [26]

    Carleo, M

    G. Carleo, M. Troyer, Solving the quantum many-body problem with artificial neural networks. Science 355 (6325), 602–606 (2017)

    2017

  19. [27]

    D. Pfau, J. S. Spencer, A. G. Matthews, W. M. C. Foulkes, Ab initio solution of the many- electron Schr ¨odinger equation with deep neural networks. Physical review research 2 (3), 033429 (2020)

    2020

  20. [28]

    W. Ren, W. Fu, X. Wu, J. Chen, Towards the ground state of molecules via diffusion Monte Carlo on neural networks. Nature Communications 14 (1), 1860 (2023)

    2023

  21. [29]

    Li, et al

    R. Li, et al. , A computational framework for neural network-based variational Monte Carlo with Forward Laplacian. Nature Machine Intelligence 6 (2), 209–219 (2024). 18

    2024

  22. [30]

    Nomura, M

    Y. Nomura, M. Imada, Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy.Physical Review X 11 (3), 031034 (2021)

    2021

  23. [31]

    C. Roth, A. Szab ´o, A. H. MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks. Physical Review B 108 (5), 054410 (2023)

    2023

  24. [32]

    Yu, Z.-J

    Q.-H. Yu, Z.-J. Lin, Solving quantum many-particle models with graph attention network. Chinese Physics Letters 41 (3), 030202 (2024)

    2024

  25. [33]

    Sorella, Generalized Lanczos algorithm for variational quantum Monte Carlo

    S. Sorella, Generalized Lanczos algorithm for variational quantum Monte Carlo. Physical Review B 64 (2), 024512 (2001)

    2001

  26. [34]

    Nightingale, V

    M. Nightingale, V. Melik-Alaverdian, Optimization of ground-and excited-state wave functions and van der Waals clusters. Physical review letters 87 (4), 043401 (2001)

    2001

  27. [35]

    Sorella, M

    S. Sorella, M. Casula, D. Rocca, Weak binding between two aromatic rings: Feeling the van der Waals attraction by quantum Monte Carlo methods. The Journal of chemical physics 127 (1) (2007)

    2007

  28. [36]

    Karthik, A

    V. Karthik, A. Medhi, Convolutional restricted Boltzmann machine correlated variational wave function for the Hubbard model on a square lattice.Physical Review B110 (12), 125125 (2024)

    2024

  29. [37]

    Vaswani, et al., Attention is all you need.Advances in neural information processing systems 30 (2017)

    A. Vaswani, et al., Attention is all you need.Advances in neural information processing systems 30 (2017)

    2017

  30. [38]

    Kaplan, et al., Scaling laws for neural language models

    J. Kaplan, et al., Scaling laws for neural language models. arXiv preprint arXiv:2001.08361 (2020)

    2020 arXiv

  31. [39]

    Tranquada, B

    J. Tranquada, B. Sternlieb, J. Axe, Y. Nakamura, S.-i. Uchida, Evidence for stripe correlations of spins and holes in copper oxide superconductors. nature 375 (6532), 561–563 (1995)

    1995

  32. [40]

    Andersen, A

    O. Andersen, A. Liechtenstein, O. Jepsen, F. Paulsen, LDA Energy Bands, Low-Energy Hamil- tonians,𝑡′,𝑡′′,𝑡⊥(𝑘), and𝐽⊥. Journal of Physics and Chemistry of Solids 56 (12), 1573–1591 (1995). 19

    1995

  33. [41]

    Hirayama, Y

    M. Hirayama, Y. Yamaji, T. Misawa, M. Imada, Ab initio effective Hamiltonians for cuprate superconductors. Physical Review B 98 (13), 134501 (2018)

    2018

  34. [42]

    W.-Y. Liu, H. Zhai, R. Peng, Z.-C. Gu, G. K.-L. Chan, Accurate Simulation of the Hubbard Model with Finite Fermionic Projected Entangled Pair States. Phys. Rev. Lett. 134, 256502 (2025), doi:10.1103/r4q9-4yvj, https://link.aps.org/doi/10.1103/r4q9-4yvj

    2025 doi

  35. [43]

    Brown, et al

    T. Brown, et al. , Language models are few-shot learners. Advances in neural information processing systems 33, 1877–1901 (2020)

    2020

  36. [44]

    Cheng, Quantum geometric tensor (fubini-study metric) in simple quantum system: A pedagogical introduction

    R. Cheng, Quantum geometric tensor (fubini-study metric) in simple quantum system: A pedagogical introduction. arXiv preprint arXiv:1012.1337 (2010)

    2010 arXiv

  37. [45]

    D. P. Kingma, J. Ba, Adam: A Method for Stochastic Optimization (2017), https://arxiv. org/abs/1412.6980

    2017 arXiv

  38. [46]

    Sutskever, J

    I. Sutskever, J. Martens, G. Dahl, G. Hinton, On the importance of initialization and momentum in deep learning, in International conference on machine learning (PMLR) (2013), pp. 1139– 1147

    2013

  39. [47]

    M. Qin, H. Shi, S. Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method. Physical Review B 94 (8), 085103 (2016)

    2016

  40. [48]

    L. L. Viteritti, R. Rende, A. Parola, S. Goldt, F. Becca, Transformer wave function for two dimensional frustrated magnets: Emergence of a spin-liquid phase in the Shastry-Sutherland model. Physical Review B 111 (13), 134411 (2025)

    2025

  41. [49]

    Qin, et al

    M. Qin, et al. , Absence of superconductivity in the pure two-dimensional Hubbard model. Physical Review X 10 (3), 031016 (2020)

    2020

  42. [50]

    Hirayama, Y

    M. Hirayama, Y. Yamaji, T. Misawa, M. Imada, Ab initio effective Hamiltonians for cuprate superconductors. Phys. Rev. B 98, 134501 (2018), doi:10.1103/PhysRevB.98.134501,https: //link.aps.org/doi/10.1103/PhysRevB.98.134501. 20

    2018 doi

  43. [51]

    Chung, M

    C.-M. Chung, M. Qin, S. Zhang, U. Schollw¨ock, S. R. White, Plaquette versus ordinary𝑑-wave pairing in the𝑡 ′ -Hubbard model on a width-4 cylinder. Phys. Rev. B 102, 041106 (2020), doi: 10.1103/PhysRevB.102.041106, https://link.aps.org/doi/10.1103/PhysRevB.102. 041106

    2020 doi

  44. [52]

    S. R. White, D. J. Scalapino, Density Matrix Renormalization Group Study of the Striped Phase in the 2Dt−J Model. Phys. Rev. Lett.80, 1272–1275 (1998), doi:10.1103/PhysRevLett.80.1272, https://link.aps.org/doi/10.1103/PhysRevLett.80.1272

    1998 doi

  45. [53]

    Jiang, J

    Y.-F. Jiang, J. Zaanen, T. P. Devereaux, H.-C. Jiang, Ground state phase diagram of the doped Hubbard model on the four-leg cylinder.Phys. Rev. Res. 2, 033073 (2020), doi:10.1103/ PhysRevResearch.2.033073, https://link.aps.org/doi/10.1103/PhysRevResearch. 2.033073

    2020 doi

  46. [54]

    E. W. Huang, C. B. Mendl, H.-C. Jiang, B. Moritz, T. P. Devereaux, Stripe order from the perspective of the Hubbard model. npj Quantum Materials 3 (1), 22 (2018), doi:10.1038/ s41535-018-0097-0

    2018

  47. [55]

    Jiang, T

    H.-C. Jiang, T. P. Devereaux, Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping 𝑡′. Science 365 (6460), 1424–1428 (2019), doi:10.1126/science. aal5304, https://www.science.org/doi/abs/10.1126/science.aal5304

    2019 doi

  48. [56]

    X. Lu, F. Chen, W. Zhu, D. N. Sheng, S.-S. Gong, Emergent Superconductivity and Com- peting Charge Orders in Hole-Doped Square-Lattice 𝑡−𝐽 Model. Phys. Rev. Lett. 132, 066002 (2024), doi:10.1103/PhysRevLett.132.066002, https://link.aps.org/doi/10. 1103/PhysRevLett.132.066002

    2024 doi

  49. [57]

    ˇSimkovic, R

    F. ˇSimkovic, R. Rossi, A. Georges, M. Ferrero, Origin and fate of the pseudogap in the doped Hubbard model. Science 385 (6715), eade9194 (2024), doi:10.1126/science.ade9194, https://www.science.org/doi/abs/10.1126/science.ade9194

    2024 doi

  50. [58]

    Elfwing, E

    S. Elfwing, E. Uchibe, K. Doya, Sigmoid-weighted linear units for neural network function approximation in reinforcement learning. Neural networks 107, 3–11 (2018). 21

    2018

  51. [59]

    Y. Wu, Z. Dai, Algorithms for variational Monte Carlo calculations of fermion PEPS in the swap gates formulation (2025), https://arxiv.org/abs/2506.20106. Acknowledgements We thank ByteDance Seed for inspiration and encouragement, and Hang Li for his guidance and support. We t...

    2025 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.