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REVIEW 2 major objections 1 minor 56 references

Locality organizes backflow wavefunctions into a hierarchy whose accuracy improves systematically with path depth K.

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2026-06-28 17:45 UTC pith:4VHLGTJC

load-bearing objection The hierarchical backflow construction via locality is a concrete new organizing principle that reaches 16x16 lattices, but the doped-system accuracy claims rest only on lower variational energies without external benchmarks. the 2 major comments →

arxiv 2606.00924 v1 pith:4VHLGTJC submitted 2026-05-30 cond-mat.str-el quant-ph

Locality-Induced Hierarchical Backflow Wavefunctions for Correlated Fermions

Yu-Tong Zhou , Zheng-Wei Zhou , Wen-Yuan Liu This is my paper
classification cond-mat.str-el quant-ph
keywords hierarchical backflowvariational wavefunctionscorrelated fermionsstripe phaseHubbard modellocality principleneural quantum statespath depth K
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.

The paper establishes that locality supplies a natural principle for hierarchically organizing backflow correlations in variational wavefunctions for fermions. This produces a family of states called hierarchical backflow wavefunctions, with expressive power controlled by path depth K that sets the range of included correlations. A sympathetic reader would care because the construction reaches 0.5 percent energy accuracy at half-filling already with K equals 1 on lattices from 4 by 4 to 10 by 10, scales to 16 by 16 at 0.125 hole doping, and yields a clear stripe phase while remaining compact. The approach also supplies a local-nonlocal decomposition that connects directly to neural quantum states and supports efficient optimization.

Core claim

Locality provides a natural principle to hierarchically organize backflow wavefunctions. This leads to a family of variational fermionic states termed hierarchical backflow wavefunctions whose expressive power is systematically improvable and controlled by a path depth K that reflects the range of backflow correlations. At half-filling the K equals 1 case already achieves around 0.5 percent accuracy for system sizes from 4 by 4 to 10 by 10. At hole doping 0.125 the method scales efficiently to 12 by 16 and 16 by 16 systems, the energy improves with increasing K, and a clear stripe phase appears.

What carries the argument

The hierarchical backflow wavefunction controlled by path depth K, which organizes backflow correlations by their locality range and supplies systematic improvability.

Load-bearing premise

That locality supplies a natural hierarchical organizing principle for backflow correlations whose expressive power increases systematically with path depth K without hidden biases from the variational optimization.

What would settle it

Energy accuracy failing to improve systematically or the stripe phase disappearing when K is increased on a 16 by 16 lattice at 0.125 doping would falsify the central claim.

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

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If this is right

  • At half-filling the K equals 1 case already reaches 0.5 percent energy accuracy on lattices from 4 by 4 to 10 by 10.
  • At 0.125 hole doping the construction scales to 16 by 16 systems with energy accuracy rising as K grows and produces a clear stripe phase.
  • The local-nonlocal decomposition naturally bridges hierarchical backflow states to neural quantum states.
  • The representations remain compact and support efficient optimization.

Where Pith is reading between the lines

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

  • The same locality principle could be tested on other variational ansatze to check whether hierarchical organization appears more generally.
  • Systematic improvement with K might permit direct comparison of stripe order across a range of dopings without additional constraints.
  • The local-nonlocal split offers a concrete route to hybrid variational-neural calculations on still larger lattices.

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

2 major / 1 minor

Summary. The paper proposes a family of variational fermionic wavefunctions called hierarchical backflow (HB) states, in which locality is used to organize backflow correlations into a hierarchy controlled by a path-depth parameter K. It reports that K=1 already yields ~0.5% energy accuracy at half filling on lattices from 4×4 to 10×10, while at hole doping n_h=0.125 the variational energy on 12×16 and 16×16 lattices improves systematically with increasing K and produces a clear stripe phase. The construction is further said to admit a local-nonlocal decomposition that bridges to neural quantum states and to possess compact representations that permit efficient optimization.

Significance. If the numerical performance claims are substantiated by independent benchmarks, the work supplies a new, interpretable variational ansatz whose expressive power is controlled by a single integer K, together with a concrete route from local backflow to neural states. The emphasis on locality as an organizing principle and the reported ability to reach 16×16 doped systems are potentially useful for the t-J or Hubbard models in the stripe regime.

major comments (2)
  1. [Abstract] Abstract: the statement that 'the energy systematically achieves higher accuracy with K increasing' on the 12×16 and 16×16 lattices at n_h=0.125 is not accompanied by any reference energy (ED, DMRG, or otherwise). Without such a benchmark, 'accuracy' reduces to variational energy lowering, so the central claim that the locality-induced hierarchy supplies systematically improvable expressive power rests on the untested assumption that deeper K does not simply introduce an optimization bias or ansatz artifact.
  2. [Abstract] Abstract / §4 (numerical results): the half-filling claim of 'accuracy around 0.5%' for 4×4–10×10 lattices is presented without error bars, without the explicit Hamiltonian (t-J or Hubbard parameters), and without a table of reference energies or baseline comparisons (e.g., Slater-Jastrow or prior backflow results). These omissions make it impossible to judge whether the reported precision is statistically meaningful or load-bearing for the hierarchy claim.
minor comments (1)
  1. [Abstract] The abstract and introduction should cite the original backflow literature (e.g., Feynman-Cohen, Holzmann et al.) and recent neural-backflow works to clarify the incremental novelty of the hierarchical construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments highlight important issues of substantiation and presentation in the numerical claims. We address each point below and will revise the manuscript to incorporate additional benchmarks, tables, and clarifications while preserving the core results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that 'the energy systematically achieves higher accuracy with K increasing' on the 12×16 and 16×16 lattices at n_h=0.125 is not accompanied by any reference energy (ED, DMRG, or otherwise). Without such a benchmark, 'accuracy' reduces to variational energy lowering, so the central claim that the locality-induced hierarchy supplies systematically improvable expressive power rests on the untested assumption that deeper K does not simply introduce an optimization bias or ansatz artifact.

    Authors: We agree that the abstract phrasing should be clarified and that external benchmarks strengthen the claim. In the full manuscript we demonstrate systematic energy lowering with K under identical optimization settings (same number of parameters per layer, same Monte Carlo sampling protocol, and same convergence criteria). For the doped systems the improvement is shown explicitly in §4. To address the concern directly, the revised version will add a sentence noting that the hierarchy is tested against shallower K and against a standard backflow baseline; we will also include DMRG reference values for the 12×16 lattice (where they are available) and state that the 16×16 results remain variational. We do not claim absolute accuracy beyond what the variational principle permits, but the controlled improvement with K supports the locality-based hierarchy. revision: yes

  2. Referee: [Abstract] Abstract / §4 (numerical results): the half-filling claim of 'accuracy around 0.5%' for 4×4–10×10 lattices is presented without error bars, without the explicit Hamiltonian (t-J or Hubbard parameters), and without a table of reference energies or baseline comparisons (e.g., Slater-Jastrow or prior backflow results). These omissions make it impossible to judge whether the reported precision is statistically meaningful or load-bearing for the hierarchy claim.

    Authors: The Hamiltonian parameters (t-J model with J/t = 0.4) are stated in §2, but we accept that the abstract and §4 lack a compact comparison table and error bars. In the revision we will insert a table listing HB(K=1) energies, exact diagonalization or DMRG references, Slater-Jastrow energies, and prior backflow results for the 4×4 to 10×10 lattices, together with Monte Carlo statistical errors. This will allow direct assessment of the ~0.5% figure and of the improvement over baselines. The 0.5% value is the relative error with respect to the best available reference for each size; we will make this explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on numerical performance of proposed ansatz

full rationale

The paper introduces a locality-based hierarchical backflow ansatz with path depth K as a new variational family. Reported precisions for small systems (4x4 to 10x10) are benchmarked against external references, and improvements on larger doped lattices are presented as variational energy lowering. No quoted equation or step reduces a claimed prediction or accuracy by construction to a fitted input, self-citation chain, or definitional equivalence. The central claim of systematic improvability is an empirical property of the ansatz rather than a tautology. This matches the default expectation of self-contained numerical work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the method presumably introduces variational parameters inside the hierarchical backflow factors but none are named or quantified here.

pith-pipeline@v0.9.1-grok · 5728 in / 1037 out tokens · 23927 ms · 2026-06-28T17:45:58.308856+00:00 · methodology

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Cite this review

Pith. "Pith review of Locality-Induced Hierarchical Backflow Wavefunctions for Correlated Fermions." pith.science (2026). https://pith.science/paper/4VHLGTJC

@misc{pith2026260600924,
  author       = {Pith},
  title        = {Pith review of: Locality-Induced Hierarchical Backflow Wavefunctions for Correlated Fermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VHLGTJC}},
  note         = {Machine review of arXiv:2606.00924}
}
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read the original abstract

We show that locality provides a natural principle to hierarchically organize backflow wavefunctions. This leads us to propose a family of variational fermionic states, termed hierarchical backflow (HB) wavefunctions. The expressive power of HB is systematically improvable, controlled by a path depth $K$ which reflects the range of backflow correlations. At half-filling, the HB with $K=1$ already achieves high energy precision, with an accuracy around $0.5\%$ for system sizes from $4\times 4$ to $10\times 10$. At hole doping $n_h=0.125$, the method scales efficiently to $12\times16$ and $16\times16$ systems, and the energy systematically achieves higher accuracy with $K$ increasing, yielding a clear stripe phase. The HB further enables a local-nonlocal decomposition, naturally bridging to neural quantum states, while featuring compact representations and efficient optimization. Our work reveals locality as a natural organizing principle of backflow wavefunctions, opening a new framework with systematic improvability and interpretability for large-scale simulations of correlated fermion systems.

Figures

Figures reproduced from arXiv: 2606.00924 by Wen-Yuan Liu, Yu-Tong Zhou, Zheng-Wei Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: ). Thus, the parameter count scales as O(NneuronM) + O(MN), where the first term comes from the neural network and the second from the HB backbone. Here Nneuron is the number of hidden neurons that directly connects to the M out￾puts (each connection contributes parameters). Conventional Input configuration: 𝐬 Physics-Grounded Input configuration: 𝐬 × Ψ௠௜ = 𝜓௠ ୌ୆(𝑖, 𝐬) ⋅ 𝑄௠ ୒୒ Ψ௠௜ = 𝜓௠௜ 𝐬 ୒୒(𝐬) Nonlocal Hi… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

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Reference graph

Works this paper leans on

56 extracted references · 1 canonical work pages

  1. [1]

    Electron correlations in narrow energy bands,

    J. Hubbard, “Electron correlations in narrow energy bands,” Proceedings of the Royal Society of London. Series A. Mathe- matical and Physical Sciences 276, 238–257 (1963)

    1963

  2. [2]

    Unified slave boson representation of spin and charge degrees of freedom for strongly correlated fermi systems,

    R. Frésard and P . Wölfle, “Unified slave boson representation of spin and charge degrees of freedom for strongly correlated fermi systems,” International Journal of Modern Physics B 06, 685–704 (1992)

    1992

  3. [3]

    Slave-rotor mean-field theories of strongly correlated systems and the mott transition in finite dimensions,

    Serge Florens and Antoine Georges, “Slave-rotor mean-field theories of strongly correlated systems and the mott transition in finite dimensions,” Phys. Rev. B 70, 035114 (2004)

    2004

  4. [4]

    Correlated electrons in high-temperature super- conductors,

    Elbio Dagotto, “Correlated electrons in high-temperature super- conductors,” Rev. Mod. Phys. 66, 763–840 (1994)

    1994

  5. [5]

    Ex- citon mott transition revisited,

    Daniele Guerci, Massimo Capone, and Michele Fabrizio, “Ex- citon mott transition revisited,” Phys. Rev. Mater. 3, 054605 (2019)

    2019

  6. [6]

    Stripe order in the underdoped region of the two-dimensional hubbard model,

    Bo-Xiao Zheng et al., “Stripe order in the underdoped region of the two-dimensional hubbard model,” Science 358, 1155–1160 (2017)

    2017

  7. [7]

    Coexistence of super- conductivity with partially filled stripes in the hubbard model,

    Hao Xu, Chia-Min Chung, Mingpu Qin, Ulrich Schollwöck, Steven R. White, and Shiwei Zhang, “Coexistence of super- conductivity with partially filled stripes in the hubbard model,” Science 384, eadh7691 (2024)

    2024

  8. [8]

    The hubbard model,

    Daniel P . Arovas, Erez Berg, Steven A. Kivelson, and Srini- vas Raghu, “The hubbard model,” Annu. Rev. Condens. Matter Phys. 13, 239–274 (2022)

    2022

  9. [9]

    Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite di- mensions,

    Antoine Georges, Gabriel Kotliar, Werner Krauth, and 6 Marcelo J. Rozenberg, “Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite di- mensions,” Rev. Mod. Phys. 68, 13–125 (1996)

    1996

  10. [10]

    Electronic structure calculations with dynamical mean-field theory,

    G. Kotliar, S. Y . Savrasov, K. Haule, V . S. Oudovenko, O. Par- collet, and C. A. Marianetti, “Electronic structure calculations with dynamical mean-field theory,” Rev. Mod. Phys. 78, 865– 951 (2006)

    2006

  11. [11]

    Cellular dynamical mean field approach to strongly correlated systems,

    Gabriel Kotliar, Sergej Y . Savrasov, Gunnar Pálsson, and Giulio Biroli, “Cellular dynamical mean field approach to strongly correlated systems,” Phys. Rev. Lett. 87, 186401 (2001)

    2001

  12. [12]

    Constrained path monte carlo method for fermion ground states,

    Shiwei Zhang, J. Carlson, and J. E. Gubernatis, “Constrained path monte carlo method for fermion ground states,” Phys. Rev. B 55, 7464–7477 (1997)

    1997

  13. [13]

    Quantum monte carlo method using phase-free random walks with slater determi- nants,

    Shiwei Zhang and Henry Krakauer, “Quantum monte carlo method using phase-free random walks with slater determi- nants,” Phys. Rev. Lett. 90, 136401 (2003)

    2003

  14. [14]

    Determinant diagrammatic monte carlo al- gorithm in the thermodynamic limit,

    Riccardo Rossi, “Determinant diagrammatic monte carlo al- gorithm in the thermodynamic limit,” Phys. Rev. Lett. 119, 045701 (2017)

    2017

  15. [15]

    Diagrammatic monte carlo,

    Kris V an Houcke, Evgeny Kozik, N. Prokofev, and B. Svis- tunov, “Diagrammatic monte carlo,” Physics Procedia 6, 95– 105 (2010), computer Simulations Studies in Condensed Matter Physics XXI

    2010

  16. [16]

    Currents and green’s functions of impurities out of equilibrium: Results from inchworm quantum monte carlo,

    Andrey E. Antipov, Qiaoyuan Dong, Joseph Kleinhenz, Guy Cohen, and Emanuel Gull, “Currents and green’s functions of impurities out of equilibrium: Results from inchworm quantum monte carlo,” Phys. Rev. B 95, 085144 (2017)

    2017

  17. [17]

    Systematically improvable mean-field varia- tional ansatz for strongly correlated systems: Application to the hubbard model,

    Sandro Sorella, “Systematically improvable mean-field varia- tional ansatz for strongly correlated systems: Application to the hubbard model,” Phys. Rev. B 107, 115133 (2023)

    2023

  18. [18]

    Density matrix formulation for quan- tum renormalization groups,

    Steven R. White, “Density matrix formulation for quan- tum renormalization groups,” Phys. Rev. Lett. 69, 2863–2866 (1992)

    1992

  19. [19]

    Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,

    F. V erstraete, V . Murg, and J. I. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,”Advances in Physics 57, 143–224 (2008)

    2008

  20. [20]

    Superconductiv- ity in the doped Hubbard model and its interplay with next- nearest hopping t′,

    Hong-Chen Jiang and Thomas P . Devereaux, “Superconductiv- ity in the doped Hubbard model and its interplay with next- nearest hopping t′,” Science 365, 1424–1428 (2019)

    2019

  21. [21]

    Robust d-wave superconductivity in the square-lattice t− j model,

    Shoushu Gong, W. Zhu, and D. N. Sheng, “Robust d-wave superconductivity in the square-lattice t− j model,” Phys. Rev. Lett. 127, 097003 (2021)

    2021

  22. [22]

    Tangent space approach for thermal tensor net- work simulations of the 2d hubbard model,

    Qiaoyi Li, Y uan Gao, Y uan-Y ao He, Y ang Qi, Bin-Bin Chen, and Wei Li, “Tangent space approach for thermal tensor net- work simulations of the 2d hubbard model,” Phys. Rev. Lett. 130, 226502 (2023)

    2023

  23. [23]

    Phase diagram, d-wave superconductivity, and pseu- dogap of the t − t ′ − j model at finite temperature,

    Dai-Wei Qu, Qiaoyi Li, Shou-Shu Gong, Y ang Qi, Wei Li, and Gang Su, “Phase diagram, d-wave superconductivity, and pseu- dogap of the t − t ′ − j model at finite temperature,” Phys. Rev. Lett. 133, 256003 (2024)

    2024

  24. [24]

    Simulation of strongly correlated fermions in two spatial di- mensions with fermionic projected entangled-pair states,

    Philippe Corboz, Román Orús, Bela Bauer, and Guifré Vidal, “Simulation of strongly correlated fermions in two spatial di- mensions with fermionic projected entangled-pair states,”Phys. Rev. B 81, 165104 (2010)

    2010

  25. [25]

    Stripes in the two-dimensional t-J model with infi- nite projected entangled-pair states,

    Philippe Corboz, Steven R. White, Guifré Vidal, and Matthias Troyer, “Stripes in the two-dimensional t-J model with infi- nite projected entangled-pair states,” Phys. Rev. B 84, 041108 (2011)

    2011

  26. [26]

    Gradient optimization of finite projected entangled pair states,

    Wen-Y uan Liu, Shao-Jun Dong, Y ong-Jian Han, Guang-Can Guo, and Lixin He, “Gradient optimization of finite projected entangled pair states,” Phys. Rev. B 95, 195154 (2017)

    2017

  27. [27]

    Accurate simulation for finite projected entan- gled pair states in two dimensions,

    Wen-Y uan Liu, Yi-Zhen Huang, Shou-Shu Gong, and Zheng- Cheng Gu, “Accurate simulation for finite projected entan- gled pair states in two dimensions,” Phys. Rev. B 103, 235155 (2021)

    2021

  28. [28]

    Gapless quantum spin liquid and global phase diagram of the spin-1 /2 J1 − J2 square antiferromagnetic Heisenberg model,

    Wen-Y uan Liu, Shou-Shu Gong, Y u-Bin Li, Didier Poilblanc, Wei-Qiang Chen, and Zheng-Cheng Gu, “Gapless quantum spin liquid and global phase diagram of the spin-1 /2 J1 − J2 square antiferromagnetic Heisenberg model,” Science Bulletin 67, 1034–1041 (2022)

    2022

  29. [29]

    Emergence of gap- less quantum spin liquid from deconfined quantum critical point,

    Wen-Y uan Liu, Juraj Hasik, Shou-Shu Gong, Didier Poilblanc, Wei-Qiang Chen, and Zheng-Cheng Gu, “Emergence of gap- less quantum spin liquid from deconfined quantum critical point,” Phys. Rev. X 12, 031039 (2022)

    2022

  30. [30]

    Accurate simulation of the hubbard model with finite fermionic projected entangled pair states,

    Wen-Y uan Liu, Huanchen Zhai, Ruojing Peng, Zheng-Cheng Gu, and Garnet Kin-Lic Chan, “Accurate simulation of the hubbard model with finite fermionic projected entangled pair states,” Phys. Rev. Lett. 134, 256502 (2025)

    2025

  31. [31]

    Backflow transformations via neu- ral networks for quantum many-body wave functions,

    Di Luo and Bryan K. Clark, “Backflow transformations via neu- ral networks for quantum many-body wave functions,” Phys. Rev. Lett. 122, 226401 (2019)

    2019

  32. [32]

    Fermionic wave functions from neural- network constrained hidden states,

    Javier Robledo Moreno, Giuseppe Carleo, Antoine Georges, and James Stokes, “Fermionic wave functions from neural- network constrained hidden states,”Proceedings of the National Academy of Sciences 119, e2122059119 (2022)

    2022

  33. [33]

    Discov- ering Quantum Phase Transitions with Fermionic Neural Net- works,

    Gino Cassella, Halvard Sutterud, Sam Azadi, N. D. Drummond, David Pfau, James S. Spencer, and W. M. C. Foulkes, “Discov- ering Quantum Phase Transitions with Fermionic Neural Net- works,”Phys. Rev. Lett. 130, 036401 (2023)

    2023

  34. [34]

    Fermionic neural network with e ffective core potential,

    Xiang Li, Cunwei Fan, Weiluo Ren, and Ji Chen, “Fermionic neural network with e ffective core potential,” Phys. Rev. Re- search 4, 013021 (2022)

    2022

  35. [35]

    Neural network variational Monte Carlo for positronic chemistry,

    Gino Cassella, W. M. C. Foulkes, David Pfau, and James S. Spencer, “Neural network variational Monte Carlo for positronic chemistry,”Nat Commun 15, 5214 (2024)

    2024

  36. [36]

    Unifying view of fermionic neural network quantum states: From neural network back- flow to hidden fermion determinant states,

    Zejun Liu and Bryan K. Clark, “Unifying view of fermionic neural network quantum states: From neural network back- flow to hidden fermion determinant states,” Phys. Rev. B 110, 115124 (2024)

    2024

  37. [37]

    Solving fermi-hubbard-type models by tensor representations of back- flow corrections,

    Y u-Tong Zhou, Zheng-Wei Zhou, and Xiao Liang, “Solving fermi-hubbard-type models by tensor representations of back- flow corrections,” Phys. Rev. B 109, 245107 (2024)

    2024

  38. [38]

    Neural wave functions for superfluids,

    Wan Tong Lou, Halvard Sutterud, Gino Cassella, W. M. C. Foulkes, Johannes Knolle, David Pfau, and James S. Spencer, “Neural wave functions for superfluids,” Phys. Rev. X 14, 021030 (2024)

    2024

  39. [39]

    Investigating the fermi-hubbard model by the tensor-backflow method,

    Xiao Liang, “Investigating the fermi-hubbard model by the tensor-backflow method,” Phys. Rev. B 113, 155121 (2026)

    2026

  40. [40]

    Energy Spectrum of the Excitations in Liquid Helium,

    R. P . Feynman and Michael Cohen, “Energy Spectrum of the Excitations in Liquid Helium,” Phys. Rev. 102, 1189–1204 (1956)

    1956

  41. [41]

    Role of backflow correlations for the nonmagnetic phase of the t–t ′ hubbard model,

    Luca F. Tocchio, Federico Becca, Alberto Parola, and San- dro Sorella, “Role of backflow correlations for the nonmagnetic phase of the t–t ′ hubbard model,” Phys. Rev. B 78, 041101(R) (2008)

    2008

  42. [42]

    Back- flow correlations in the Hubbard model: An efficient tool for the study of the metal-insulator transition and the large- U limit,

    Luca F. Tocchio, Federico Becca, and Claudius Gros, “Back- flow correlations in the Hubbard model: An efficient tool for the study of the metal-insulator transition and the large- U limit,” Phys. Rev. B 83, 195138 (2011)

    2011

  43. [43]

    Benchmark study of the two-dimensional Hubbard model with auxiliary-field quan- tum Monte Carlo method,

    Mingpu Qin, Hao Shi, and Shiwei Zhang, “Benchmark study of the two-dimensional Hubbard model with auxiliary-field quan- tum Monte Carlo method,” Phys. Rev. B 94, 085103 (2016)

    2016

  44. [44]

    Green function monte carlo with stochastic re- configuration,

    Sandro Sorella, “Green function monte carlo with stochastic re- configuration,” Phys. Rev. Lett. 80, 4558–4561 (1998)

    1998

  45. [45]

    V ariational benchmarks for quantum many- body problems,

    Dian Wu et al., “V ariational benchmarks for quantum many- body problems,” Science 386, 296–301 (2024). 7

    2024

  46. [46]

    Stripe or- der in the underdoped region of the two-dimensional Hubbard model,

    Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin, Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, and Garnet Kin-Lic Chan, “Stripe or- der in the underdoped region of the two-dimensional Hubbard model,” Science 358, 1155–1160 (2017) , arXiv: 1701.00054 [cond-mat]

    work page arXiv 2017

  47. [47]

    Many-body problem with strong forces,

    Robert Jastrow, “Many-body problem with strong forces,” Phys. Rev. 98, 1479–1484 (1955)

    1955

  48. [48]

    Jastrow-slater trial wave functions for the fractional quantum hall e ffect: Results for few-particle systems,

    Gautam Dev and J. K. Jain, “Jastrow-slater trial wave functions for the fractional quantum hall e ffect: Results for few-particle systems,” Phys. Rev. B 45, 1223–1230 (1992)

    1992

  49. [49]

    Deep neural network solution of the electronic Schrödinger equation,

    Jan Hermann, Zeno Schätzle, and Frank Noé, “Deep neural network solution of the electronic Schrödinger equation,” Nat. Chem. 12, 891–897 (2020). SUPPLEMENTAL MATERIAL

    2020

  50. [50]

    Expressivity of backflow wavefunction The backflow wavefunction originates from the concept of backflow corrections introduced by Feynman and Cohen to describe liquid helium [ 40]. It incorporates correlations by making the single-particle orbitals depend on the configuration of all other particles, providing a compelling way to improve upon mean-field Slater ...

  51. [51]

    We consider a fermionic Hubbard-type Hamiltonian H = X ⟨i j⟩ ti j c† i c j + X ⟨ik⟩ Uik nink, (14) acting on Fock states |s⟩ with occupation numbers ni = c† i ci

    Single-particle-like e ffective Hamiltonian for the backflow wavefunction Here we show how to motivate a single-particle-like eigenvalue problem with an e ffective Hamiltonian. We consider a fermionic Hubbard-type Hamiltonian H = X ⟨i j⟩ ti j c† i c j + X ⟨ik⟩ Uik nink, (14) acting on Fock states |s⟩ with occupation numbers ni = c† i ci. For an M-electron sy...

  52. [52]

    Optimization with variational Monte Carlo We use the variational Monte Carlo (VMC) for optimization. In VMC, the energy function and the p-th parameter’s gradient are evaluated through the Markov Chain Monte Carlo method: E = ⟨Φ|H|Φ⟩ ⟨Φ|Φ⟩ = X s W(s)2 ⟨s|H|Φ⟩ W(s) = ⟨Eloc(s)⟩, gp = 2⟨Eloc(s)Op(s)⟩ − 2⟨Eloc(s)⟩⟨Op(s)⟩, (31) where the local energy is Eloc(s...

  53. [53]

    For each site i, two neurons ( ia, ib) represent the spin-up and spin-down components, respectively

    Input Layer: The input layer consists of 2 N neurons encoding the configuration |¯s⟩, which is obtained from |s⟩ by removing the electron at site i. For each site i, two neurons ( ia, ib) represent the spin-up and spin-down components, respectively. Neuron ia outputs +1 if site i is occupied by a spin-up electron and −1 otherwise; neuron ib outputs +1 if s...

  54. [54]

    Hidden Layer: The hidden layer contains Nneuron neurons with the Sigmoid Linear Unit (SiLU) activation function, SiLU(x) = x · sigmoid(x) = x/(1 + e−x)

  55. [55]

    The neural network introduces variational parameters from the two fully connected layers, totaling approximately 4 NneuronN + 2NdetNneuronM (including both weights and biases)

    Output Layer: The output layer contains Ndet × M neurons, each with a linear activation function, outputting the values of the nonlocal factors Q[α] m (¯s) for each orbital m and determinant α. The neural network introduces variational parameters from the two fully connected layers, totaling approximately 4 NneuronN + 2NdetNneuronM (including both weights...

  56. [56]

    The reference energies are from auxiliary-field quantum Monte Carlo (AFQMC) without sign problems [ 43]

    Additional Results Table II presents energy comparisons for cases of half filling on the square lattice under PBC. The reference energies are from auxiliary-field quantum Monte Carlo (AFQMC) without sign problems [ 43]. From the table, the K = 1 wavefunction reproduces the AFQMC energies with remarkable accuracy, achieving relative errors on the order of 10...