pith. machine review for the scientific record. sign in

arxiv: 2604.21978 · v1 · submitted 2026-04-23 · ❄️ cond-mat.str-el · cond-mat.dis-nn· cond-mat.supr-con

Recognition: unknown

Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model

Anirvan Sengupta, Antoine Georges, Christopher Roth, Giuseppe Carleo, Luciano Loris Viteritti, Riccardo Rende

Authors on Pith no claims yet

Pith reviewed 2026-05-08 14:07 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.dis-nncond-mat.supr-con
keywords Hubbard modelvariational wave functionsstripe ordersuperconductivityintertwined ordersTransformer ansatzsymmetry restorationvariance reduction
0
0 comments X

The pith

Improving three different variational wave functions for the doped Hubbard model causes them to converge on coexisting superconducting and stripe orders.

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

The two-dimensional Hubbard model at finite doping produces conflicting computational results about its ground state because of intertwined competing orders. Three Transformer backflow fermionic wave functions, built on Slater determinants, particle-hole counterparts, and Pfaffians and initialized without mean-field pretraining, reach nearly identical state-of-the-art variational energies yet initially display qualitatively different spin, charge, and pairing correlations. Once symmetry restoration and variance reduction are applied to raise accuracy, all three wave functions converge to the same physical state containing both superconducting and stripe orders. The work shows that variational energy alone cannot identify the ground state when phases compete and that correlation functions must be followed as the wave function is systematically refined.

Core claim

Three distinct Transformer backflow ansatzes achieve nearly degenerate, state-of-the-art variational energies but initially converge to states with qualitatively different spin, charge, and pairing correlations. After symmetry restoration and variance reduction improve the wave functions, all three yield the same ground-state picture of coexisting superconducting and stripe orders.

What carries the argument

Transformer backflow fermionic wave functions (Slater, particle-hole, Pfaffian) combined with symmetry restoration and variance reduction

If this is right

  • The ground state of the doped two-dimensional Hubbard model contains both stripe and superconducting orders.
  • Prior conflicting conclusions from different methods likely arose from variational bias encoded in the chosen ansatzes.
  • Energy minimization alone is insufficient to resolve intertwined orders; correlation functions must be tracked during systematic improvement.
  • Variational studies of competing phases require multiple distinct starting ansatzes and observable monitoring rather than energy comparison.

Where Pith is reading between the lines

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

  • The same bias-resolution procedure could be applied to other models with competing orders, such as the t-J model, to test whether similar convergence occurs.
  • If the result holds, earlier variational reports of pure stripe or pure superconducting states may have been limited by the representational capacity of their ansatzes.
  • Future calculations should prioritize ensembles of wave functions and observable convergence over single lowest-energy states when phases intertwine.

Load-bearing premise

The convergence of the three improved ansatzes to identical orders reflects the true ground state rather than a shared representational limit of the Transformer backflow family.

What would settle it

An independent high-accuracy calculation on the same doping and system size that finds either pure stripe order without superconductivity or pure superconductivity without stripes would falsify the coexistence claim.

Figures

Figures reproduced from arXiv: 2604.21978 by Anirvan Sengupta, Antoine Georges, Christopher Roth, Giuseppe Carleo, Luciano Loris Viteritti, Riccardo Rende.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the optimization land view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Real-space charge density and spin patterns for the 8 view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. In the Transformer Backflow Architecture a physi view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Real-space pairing-pairing correlation function [see view at source ↗
read the original abstract

The two-dimensional Hubbard model at finite doping hosts competing or intertwined orders, resulting in conflicting conclusions from different computational approaches regarding its ground state. We show that a key source of such discrepancies is the bias encoded in the variational ansatz. We consider three different Transformer backflow fermionic wave functions based on a Slater determinant, its particle-hole counterpart, and a Pfaffian, initialized without any mean-field pretraining. We show that, despite achieving nearly degenerate, state-of-the-art variational energies, each ansatz converges to a state with qualitatively different spin, charge, and pairing correlations. Upon improving accuracy via symmetry restoration and variance reduction, however, all three converge to the same physical picture: coexisting superconducting and stripe orders. These results demonstrate that variational energy alone is insufficient to identify the ground state in the presence of competing phases, and highlight the importance of tracking how correlation functions evolve as the wave function is systematically improved before drawing physical conclusions.

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 / 1 minor

Summary. The paper claims that three distinct Transformer backflow fermionic wave functions for the doped 2D Hubbard model (Slater-determinant-based, particle-hole-based, and Pfaffian-based), initialized without mean-field pretraining, achieve nearly degenerate state-of-the-art variational energies yet initially display qualitatively different spin, charge, and pairing correlations. After symmetry restoration and variance reduction, all three converge to the same physical picture of coexisting superconducting and stripe orders. The authors conclude that variational energy alone is insufficient to identify the ground state in the presence of competing phases and that correlation functions must be tracked during systematic improvement of the wave function.

Significance. If the central result holds, the work is significant for resolving discrepancies among computational methods for the Hubbard model by demonstrating a practical way to reduce variational bias through multiple ansatzes and accuracy improvements. The explicit comparison of how correlations evolve under symmetry restoration and variance reduction, rather than relying solely on energy, is a methodological strength that could help reconcile conflicting literature results on intertwined orders. The approach of using three differently initialized neural backflow states without pretraining also provides a useful template for future variational studies of strongly correlated systems.

major comments (3)
  1. Abstract: the claim that the three ansatzes 'converge to the same physical picture' after symmetry restoration and variance reduction is load-bearing for the central argument, yet the abstract (and implied results) provides only a qualitative statement without quantitative metrics such as differences in stripe order parameters, pairing correlations, or their statistical uncertainties before versus after improvement.
  2. Results section (implied by abstract): the conclusion that convergence indicates the true ground state rather than a shared representational limitation of the Transformer backflow family requires explicit tests showing that at least one ansatz can be optimized to represent alternative competing states (e.g., pure d-wave SC without stripes or incommensurate spirals reported by other methods); no such capability checks are described.
  3. Methods/Results: the manuscript reports state-of-the-art energies but lacks system-size scaling analysis or finite-size extrapolation for the converged correlation functions, which is necessary to support claims about the thermodynamic-limit ground state given the known strong finite-size effects in the doped Hubbard model.
minor comments (1)
  1. Abstract: the doping level and U/t value studied should be stated explicitly, as these parameters determine the regime of intertwined orders and are essential for placing the results in context with prior work.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address each of the major comments in detail below, indicating the revisions we plan to make.

read point-by-point responses

  1. Referee: Abstract: the claim that the three ansatzes 'converge to the same physical picture' after symmetry restoration and variance reduction is load-bearing for the central argument, yet the abstract (and implied results) provides only a qualitative statement without quantitative metrics such as differences in stripe order parameters, pairing correlations, or their statistical uncertainties before versus after improvement.

    Authors: We agree that the abstract would benefit from quantitative support for the convergence claim. In the revised manuscript, we will modify the abstract to include brief quantitative statements on the changes in key observables, such as the reduction in differences of stripe order parameters and pairing correlations. Additionally, we will include a new table in the results section that reports the values of these correlation functions before and after the improvements, along with their statistical uncertainties. revision: yes

  2. Referee: Results section (implied by abstract): the conclusion that convergence indicates the true ground state rather than a shared representational limitation of the Transformer backflow family requires explicit tests showing that at least one ansatz can be optimized to represent alternative competing states (e.g., pure d-wave SC without stripes or incommensurate spirals reported by other methods); no such capability checks are described.

    Authors: This comment raises a valid concern about the interpretation of our results. While the three ansatzes exhibit different initial correlation functions, demonstrating the capacity to represent varied orders, we did not explicitly optimize any ansatz to target pure d-wave superconductivity or incommensurate spirals. We will add a paragraph in the discussion section acknowledging this limitation and explaining that our evidence for the ground state comes from the convergence under accuracy improvements from multiple unbiased starting points. We believe this supports our conclusions but recognize that direct tests for alternative states would provide further validation. revision: partial

  3. Referee: Methods/Results: the manuscript reports state-of-the-art energies but lacks system-size scaling analysis or finite-size extrapolation for the converged correlation functions, which is necessary to support claims about the thermodynamic-limit ground state given the known strong finite-size effects in the doped Hubbard model.

    Authors: We concur that system-size scaling and finite-size extrapolation are important for extrapolating to the thermodynamic limit, especially given the strong finite-size effects in the doped Hubbard model. Our calculations are performed on finite lattices where we achieve state-of-the-art energies, and the convergence is observed consistently. In the revised manuscript, we will expand the methods and results sections to include more details on the system sizes studied and any observed trends with size. A full finite-size extrapolation of the correlation functions is not included in the current work due to computational constraints but will be noted as an important avenue for future research. revision: partial

Circularity Check

0 steps flagged

No significant circularity; numerical variational convergence is empirical

full rationale

The paper reports a numerical variational Monte Carlo study of the Hubbard model using three distinct Transformer backflow ansatzes (Slater, particle-hole, Pfaffian). Different initializations yield distinct correlations at comparable variational energies, but symmetry restoration and variance reduction cause convergence in spin/charge/pairing observables. This is an empirical observation from optimization and measurement of expectation values, not a mathematical derivation or self-referential definition. No equations reduce reported orders to fitted parameters by construction, no uniqueness theorems are imported via self-citation, and no ansatz is smuggled in. The central claim rests on computed observables that remain independent of the optimization inputs once the wave functions are improved.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard variational Monte Carlo framework and the representational power of Transformer backflow wave functions; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond those already standard in the field.

axioms (1)
  • standard math The variational principle: the ground-state energy is the minimum of the expectation value of the Hamiltonian over all allowed wave functions.
    Invoked implicitly throughout as the justification for minimizing the variational energy.

pith-pipeline@v0.9.0 · 5486 in / 1302 out tokens · 59344 ms · 2026-05-08T14:07:39.884391+00:00 · methodology

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

60 extracted references · 17 canonical work pages

  1. [1]

    , sN), where si ∈ {0,1,2,3}corresponds to the possible four states of the local Hilbert space of sitei: empty, singly occu- pied spin-↑or spin-↓, and doubly occupied site

    Transformer Backflow Architecture The Transformer Backflow Architecture takes as in- put a physical configurations= (s 1, . . . , sN), where si ∈ {0,1,2,3}corresponds to the possible four states of the local Hilbert space of sitei: empty, singly occu- pied spin-↑or spin-↓, and doubly occupied site. Following the standard tokenization scheme [21, 32], each...

  2. [2]

    Throughout, we consider the mean-field limit, in which the orbitals are taken to be configuration independent

    Mean Field Biases of Fermionic Ans¨ atze In this Section, we discuss the mean-field biases as- sociated with the three fermionic ans¨ atze used in this work: the determinant, the determinant in the particle- hole representation, and the Pfaffian. Throughout, we consider the mean-field limit, in which the orbitals are taken to be configuration independent....

  3. [3]

    Zero-Variance Extrapolation An estimate of the exact ground-state energy for the 8×8 Hubbard model atU= 8.0,t ′ =−0.2, and doping δ= 1/8 can be obtained by means of zero-variance ex- trapolation [37, 51, 52]. We perform this analysis using the Pfaffianansatz, since, among the variational states considered, it provides the most accurate description, as als...

  4. [4]

    8, we show the real-space pairing-pairing corre- lation function defined in Eq

    Pairing Correlation Function In Fig. 8, we show the real-space pairing-pairing corre- lation function defined in Eq. (2), obtained with the de- terminantansatzbefore and after symmetry restoration. Panel (a) shows the unsymmetrized state, which displays a pronounced stripe pattern [see Fig. 4] and only weak su- perconducting correlations [see Fig. 3]. Pan...

  5. [5]

    Spin structure factor In Fig. 9 we show the spin structure factor, defined as the Fourier transform of the real-space spin-spin correla- tionsS(k) = P r eik·r⟨ ˆSz 0 ˆSz r⟩, which we use to character- ize magnetic ordering. A dominant peak atk= (π, π) signals antiferromagnetic order. In the present case, however, the structure factor also exhibits four ad...

  6. [6]

    For thed-wave su- perconducting order parameter ∆ SC (see the definition in the main text), the determinantansatzsystematically underestimates pairing correlations

    Bounds on the Order Parameters In Table III, we report thed-wave superconducting and stripe order parameters obtained from the three fully symmetrized variational states, with both translational and rotational symmetries restored. For thed-wave su- perconducting order parameter ∆ SC (see the definition in the main text), the determinantansatzsystematicall...

  7. [7]

    Swiss AI initiative

    Optimization Setup The variational states are optimized within the Varia- tional Monte Carlo framework [53] using Stochastic Re- configuration (SR) [54] with the linear algebra trick [49, 50]. The optimization is carried out for 4×10 4 steps without the symmetries, 2×10 3 steps when restoring the translational symmetry, and an additional 10 2 steps when r...

  8. [8]

    Electron correlations in narrow energy bands,

    J. Hubbard, Electron correlations in narrow energy bands, Proceedings of the Royal Society of Lon- don. A. Mathematical and Physical Sciences276, 238 (1963), https://royalsocietypublishing.org/rspa/article- pdf/276/1365/238/54456/rspa.1963.0204.pdf

    work page arXiv 1963

  9. [9]

    S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gubernatis, and R. T. Scalettar, Numerical study of the two-dimensional hubbard model, Phys. Rev. B40, 506 (1989)

    1989

  10. [10]

    J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jim´ enez-Hoyos, E. Kozik, X.-W. Liu, A. J. Millis, N. V. Prokof’ev, M. Qin, G. E. Scuseria, H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupitsyn, S. R. White, S. Zhang, B.-X. Zheng, Z. Zhu, and E. Gull (Simons Collaboration on...

    2015

  11. [11]

    Xu, C.-M

    H. Xu, C.-M. Chung, M. Qin, U. Schollw¨ ock, S. R. White, and S. Zhang, Coexistence of superconductivity with par- tially filled stripes in the hubbard model, Science384, 10.1126/science.adh7691 (2024)

    work page doi:10.1126/science.adh7691 2024

  12. [12]

    ˇSimkovic, R

    F. ˇSimkovic, R. Rossi, A. Georges, and M. Ferrero, Origin and fate of the pseudogap in the doped hubbard model, Science385, 10.1126/science.ade9194 (2024)

    work page doi:10.1126/science.ade9194 2024

  13. [13]

    Qin, C.-M

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

    2020

  14. [14]

    Sorella, Systematically improvable mean-field varia- tional ansatz for strongly correlated systems: Applica- tion to the hubbard model, Phys

    S. Sorella, Systematically improvable mean-field varia- tional ansatz for strongly correlated systems: Applica- tion to the hubbard model, Phys. Rev. B107, 115133 (2023)

    2023

  15. [15]

    Capello, F

    M. Capello, F. Becca, M. Fabrizio, S. Sorella, and E. Tosatti, Variational description of mott insulators, Phys. Rev. Lett.94, 026406 (2005)

    2005

  16. [16]

    L. F. Tocchio, F. Becca, A. Parola, and S. Sorella, Role of backflow correlations for the nonmagnetic phase of the t–t ′ hubbard model, Phys. Rev. B78, 041101 (2008)

    2008

  17. [17]

    K. Ido, T. Ohgoe, and M. Imada, Competition among various charge-inhomogeneous states andd-wave super- conducting state in hubbard models on square lattices, Phys. Rev. B97, 045138 (2018)

    2018

  18. [18]

    Hager, G

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

    2005

  19. [19]

    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 re- gion of the two-dimensional hubbard model, Science358, 1155–1160 (2017)

    2017

  20. [20]

    Ehlers, S

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

    2017

  21. [21]

    Jiang and T

    H.-C. Jiang and T. P. Devereaux, Superconductivity in the doped hubbard model and its interplay with next- nearest hopping t’, Science365, 1424–1428 (2019)

    2019

  22. [22]

    Scheb and R

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

    2023

  23. [23]

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

    2025

  24. [24]

    Jiang, T

    H.-C. Jiang, T. P. Devereaux, and S. A. Kivelson, Com- petition between charge-density-wave and superconduct- ing orders on eight-leg square hubbard cylinders (2025), arXiv:2511.18644 [cond-mat.str-el]

    work page arXiv 2025

  25. [25]

    Nomura, A

    Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada, Restricted boltzmann machine learning for solving strongly correlated quantum systems, Phys. Rev. B96, 205152 (2017)

    2017

  26. [27]

    Robledo Moreno, G

    J. Robledo Moreno, G. Carleo, A. Georges, and J. Stokes, Fermionic wave functions from neural-network con- strained hidden states, Proceedings of the National Academy of Sciences119, 10.1073/pnas.2122059119 (2022)

    work page doi:10.1073/pnas.2122059119 2022

  27. [28]

    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 hubbard model with neural quantum states (2025), arXiv:2507.02644 [cond-mat.str-el]

    work page arXiv 2025

  28. [29]

    Chen , author 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]

    work page arXiv 2025

  29. [30]

    C. Roth, A. Chen, A. Sengupta, and A. Georges, Superconductivity in the two-dimensional hubbard model revealed by neural quantum states (2025), arXiv:2511.07566 [cond-mat.supr-con]

    work page arXiv 2025

  30. [31]

    Sharma, A

    L. Sharma, A. Shokry, R. Nutakki, O. Simard, M. Fer- rero, and F. Vicentini, Comparing symmetrized deter- minant neural quantum states for the hubbard model (2025), arXiv:2510.11710 [cond-mat.str-el]

    work page arXiv 2025

  31. [32]

    Ibarra-Garc´ ıa-Padilla, H

    E. Ibarra-Garc´ ıa-Padilla, H. Lange, R. G. Melko, R. T. Scalettar, J. Carrasquilla, A. Bohrdt, and E. Khatami, Autoregressive neural quantum states of fermi hubbard models, Phys. Rev. Res.7, 013122 (2025)

    2025

  32. [33]

    Corboz, R

    P. Corboz, R. Or´ us, B. Bauer, and G. Vidal, Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states, Physical Review B81, 10.1103/physrevb.81.165104 (2010)

    work page doi:10.1103/physrevb.81.165104 2010

  33. [34]

    Luo and B

    D. Luo and B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, Phys. Rev. Lett.122, 226401 (2019)

    2019

  34. [35]

    Ruggeri, S

    M. Ruggeri, S. Moroni, and M. Holzmann, Nonlinear net- work description for many-body quantum systems in con- tinuous space, Phys. Rev. Lett.120, 205302 (2018). 12

    2018

  35. [36]

    H. Ma, B. Kan, H. Shang, and J. Yang, Transformer- based neural networks backflow for strongly correlated electronic structure (2025), arXiv:2509.25720 [quant-ph]

    work page arXiv 2025

  36. [37]

    L. L. Viteritti, R. Rende, and F. Becca, Transformer vari- ational wave functions for frustrated quantum spin sys- tems, Phys. Rev. Lett.130, 236401 (2023)

    2023

  37. [38]

    L. L. Viteritti, R. Rende, A. Parola, S. Goldt, and F. Becca, Transformer wave function for two dimensional frustrated magnets: Emergence of a spin-liquid phase in the shastry-sutherland model, Phys. Rev. B111, 134411 (2025)

    2025

  38. [39]

    Rende, A

    R. Rende, A. Nikolaenko, L. L. Viteritti, S. Sachdev, and Y.-H. Zhang, Transformer neural-network quantum states for lattice models of spins and fermions: Applica- tion to the ancilla layer model (2026), arXiv:2603.02316 [cond-mat.str-el]

    work page arXiv 2026

  39. [40]

    Deep convolutional and fully-connected DNA neural networks,

    R. Rende, L. L. Viteritti, F. Becca, A. Scardicchio, A. Laio, and G. Carleo, Foundation neural-networks quantum states as a unified ansatz for multiple hamilto- nians, Nature Communications16, 10.1038/s41467-025- 62098-x (2025)

    work page doi:10.1038/s41467-025- 2025

  40. [41]

    Loehr and B

    K. Loehr and B. K. Clark, Enhancing neural network backflow (2025), arXiv:2510.26906 [cond-mat.str-el]

    work page arXiv 2025

  41. [42]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev.108, 1175 (1957)

    1957

  42. [43]

    Rende, F

    R. Rende, F. Gerace, A. Laio, and S. Goldt, Mapping of attention mechanisms to a generalized potts model, Phys. Rev. Res.6, 023057 (2024)

    2024

  43. [44]

    L. L. Viteritti, R. Rende, S. Sachdev, and G. Carleo, Ap- proaching the thermodynamic limit with neural-network quantum states (2026), arXiv:2602.02665 [cond-mat.str- el]

    work page arXiv 2026

  44. [45]

    Blankenbecler, D

    R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Monte carlo calculations of coupled boson-fermion sys- tems. i, Phys. Rev. D24, 2278 (1981)

    1981

  45. [46]

    J. E. Hirsch, Two-dimensional hubbard model: Numeri- cal simulation study, Phys. Rev. B31, 4403 (1985)

    1985

  46. [47]

    D. J. Scalapino, S. R. White, and S. Zhang, Insulator, metal, or superconductor: The criteria, Phys. Rev. B47, 7995 (1993)

    1993

  47. [48]

    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

  48. [49]

    R. Levy, M. A. Morales, and S. Zhang, Automatic or- der detection and restoration through systematically im- provable variational wave functions, Phys. Rev. Res.6, 013237 (2024)

    2024

  49. [50]

    Gao and S

    N. Gao and S. G¨ unnemann, Neural pfaffians: Solv- ing many many-electron schr¨ odinger equations (2024), arXiv:2405.14762 [cs.LG]

    work page arXiv 2024

  50. [51]

    Mizusaki and M

    T. Mizusaki and M. Imada, Quantum-number projection in the path-integral renormalization group method, Phys. Rev. B69, 125110 (2004)

    2004

  51. [52]

    Tahara and M

    D. Tahara and M. Imada, Variational monte carlo method combined with quantum-number projection and multi-variable optimization, Journal of the Physical So- ciety of Japan77, 114701 (2008)

    2008

  52. [53]

    Nomura, Helping restricted boltzmann machines with quantum-state representation by restoring symmetry, Journal of Physics: Condensed Matter33, 174003 (2021)

    Y. Nomura, Helping restricted boltzmann machines with quantum-state representation by restoring symmetry, Journal of Physics: Condensed Matter33, 174003 (2021)

    2021

  53. [54]

    Viteritti, F

    L. Viteritti, F. Ferrari, and F. Becca, Accuracy of re- stricted Boltzmann machines for the one-dimensional J1 −J 2 Heisenberg model, SciPost Phys.12, 166 (2022)

    2022

  54. [55]

    M. Reh, M. Schmitt, and M. G¨ arttner, Optimizing design choices for neural quantum states, Phys. Rev. B107, 195115 (2023)

    2023

  55. [56]

    Rende, L

    R. Rende, L. L. Viteritti, L. Bardone, F. Becca, and S. Goldt, A simple linear algebra identity to optimize large-scale neural network quantum states, Communica- tions Physics7, 10.1038/s42005-024-01732-4 (2024)

    work page doi:10.1038/s42005-024-01732-4 2024

  56. [57]

    Chen and M

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

    2024

  57. [58]

    W.-J. Hu, F. Becca, A. Parola, and S. Sorella, Direct evidence for a gaplessZ 2 spin liquid by frustrating n´ eel antiferromagnetism, Phys. Rev. B88, 060402 (2013)

    2013

  58. [59]

    Becca, W.-J

    F. Becca, W.-J. Hu, Y. Iqbal, A. Parola, D. Poilblanc, and S. Sorella, Lanczos steps to improve variational wave functions, Journal of Physics: Conference Series640, 012039 (2015)

    2015

  59. [60]

    Becca and S

    F. Becca and S. Sorella,Quantum Monte Carlo Ap- proaches for Correlated Systems(Cambridge University Press, 2017)

    2017

  60. [61]

    Sorella, Green function monte carlo with stochastic reconfiguration, Phys

    S. Sorella, Green function monte carlo with stochastic reconfiguration, Phys. Rev. Lett.80, 4558 (1998)

    1998