Pith. sign in

REVIEW 2 major objections 6 minor 45 references

The paper argues that scaling up the Fermi Sets neural-network ansatz drives the variational energy toward the exact ground state, and that the residual gap measured by fixed-phase diffusion Monte Carlo collapses to statistical zero, certif

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

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

T0 review · deepseek-v4-flash

2026-08-01 01:15 UTC pith:N4VYI5QV

load-bearing objection Solid numerical study; the fixed-phase DMC assessment is a real step forward, but the abstract's 'convergence to ground state' overstates what ΔE can certify. the 2 major comments →

arxiv 2607.25872 v1 pith:N4VYI5QV submitted 2026-07-28 cond-mat.str-el cond-mat.mes-hall

Scaling universal Fermi network toward ground states: A diffusion-Monte-Carlo assessment

Yu-Sheng Li , Saskia Poldmaa , Tzen Ong , Ahmed Abouelkomsan , Tay-Rong Chang , Hsin Lin , Liang Fu This is my paper
classification cond-mat.str-el cond-mat.mes-hall
keywords Fermi Setsneural-network wavefunctionfixed-phase diffusion Monte Carlovariational Monte Carloquantum dottwo-dimensional electron gascomposite fermionground-state convergence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that Fermi Sets—a neural-network ansatz that provably can represent any continuous antisymmetric fermionic wavefunction—can be scaled to numerically exact interacting ground states. The authors add a fixed-phase diffusion Monte Carlo projection and use the energy drop ΔE = E_VMC − E_DMC as a quality metric. In two very different systems, a magnetic quantum dot and a strongly correlated 2D electron gas, ΔE shrinks monotonically as the network grows, reaching statistical zero in the dot and near-zero in the gas. A sympathetic reader would take this as practical evidence that the ansatz is asymptotically exact and that its own convergence can be certified without knowing the exact answer. The paper is careful that ΔE only bounds amplitude error; the phase error is not probed.

Core claim

On its own terms: Fermi Sets is a universal approximator of continuous antisymmetric wavefunctions, and this paper demonstrates empirically that the ansatz can be systematically scaled. Combining it with fixed-phase diffusion Monte Carlo yields the chain E0 ≤ E_FP ≤ E_VMC and the exact identity E_VMC − E0 = ΔE + (E_FP − E0), where both terms are nonnegative. In the magnetic quantum dot, the measured ΔE falls from about 5.6×10⁻⁴ to 0.25×10⁻⁴ effective Hartree as the network grows to 84k parameters, while the variational energy matches and sometimes beats truncated exact diagonalization. In the fully spin-polarized 2D electron gas at N=16, r_s=30, ΔE falls monotonically to about 0.57×10⁻⁵ Hart

What carries the argument

The central object is the Fermi Sets wavefunction, a complex-valued ansatz written as a sum of a small number of Slater determinants weighted by symmetric many-body functions Ω_k(R), with learnable complex orbitals; this form can approximate any continuous antisymmetric function and carries nontrivial phase structure. The companion mechanism is fixed-phase diffusion Monte Carlo: the learned phase Φθ is held fixed while only the nonnegative amplitude evolves under an effective Hamiltonian, yielding the fixed-phase energy E_FP. The identity E_VMC − E0 = ΔE + (E_FP − E0) is the machine that turns DMC into a metric: a statistically vanishing ΔE signals that the variational state has saturated al

Load-bearing premise

The load-bearing premise is that the learned phase is converging to the exact phase; the paper's ΔE metric certifies only that the amplitude has saturated, and the authors state a vanishing gap is necessary but not sufficient for the exact ground state.

What would settle it

Run the same two-stage protocol on a small system whose exact ground-state energy is known from a much larger exact-diagonalization basis than the paper used, and check whether E_DMC − E0 → 0 while ΔE → 0. A case with ΔE statistically zero but E_DMC clearly above E0 would show the gap metric is not a certificate. A complementary test: deliberately fix a wrong phase (for example, a wrong angular-momentum sector in the quantum dot) but optimize the amplitude; a small ΔE with E_FP well above E0 would demonstrate the same failure directly.

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

If this is right

  • The residual gap ΔE between VMC and fixed-phase DMC energies provides a quantitative, system-agnostic measure of a neural wavefunction's residual amplitude error.
  • Scaling Fermi Sets yields near-exact energies in a time-reversal-broken system, including a composite-fermion state at filling 3/7 that a Slater-Jastrow ansatz cannot describe even qualitatively.
  • In the strongly correlated 2D electron gas, the Fermi Sets DMC energy improves on the traditional Slater-Jastrow benchmark and provides a variational upper bound on the ground-state energy at that system size.
  • The framework is stated to apply without modification to fractional quantum Hall liquids and fractional Chern insulators in moiré materials.
  • The monotone descent of E_DMC with network capacity is presented as direct evidence that the learned phase improves systematically as the network grows.

Where Pith is reading between the lines

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

  • If the scaling behavior holds beyond the two testbeds, the ΔE metric could serve as a practical stopping criterion for neural-network variational Monte Carlo, flagging when additional capacity no longer improves the amplitude.
  • A natural extension the paper leaves implicit: combining the ΔE diagnostic with twist averaging and finite-size extrapolation would carry certified accuracy to the thermodynamic limit, e.g., sharper location of the Wigner-crystallization boundary in the 2D electron gas.
  • Since the certificate only covers amplitude error, a user who adopts this method as a black-box convergence test would need an independent phase check—such as comparing E_DMC against a more complete exact-diagonalization basis or measuring a known symmetry—before claiming true ground-state convergence.
  • The paper's logic suggests a sharper test: in systems where the exact phase is known by symmetry, the same two-stage protocol should show both ΔE→0 and E_FP→E0; failure of the second would separate phase convergence from amplitude convergence.

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 / 6 minor

Summary. The paper proposes a two-stage computational framework for continuum fermionic systems: a variational Monte Carlo optimization of the complex-valued Fermi Sets neural-network wavefunction, followed by fixed-phase diffusion Monte Carlo (FPDMC) on the optimized state. The central technical object is the gap ΔE = E_VMC − E_DMC, which by Eq. (8) equals the residual amplitude error for the fixed learned phase. The authors apply the framework to a four-electron parabolic quantum dot at two magnetic fields (B = 4, 8 T) and to a 16-electron 2D uniform electron gas at r_s = 30. In both cases, as the network parameter count is increased, E_VMC and E_DMC descend monotonically, and ΔE shrinks to the order of the statistical uncertainty (0.24–0.25×10⁻⁴ H* for the dot, 0.567(7)×10⁻⁵ Ha/N for the UEG). The paper interprets this collapse as indicating convergence to the exact ground state and uses it to support the claim that Fermi Sets is an asymptotically exact, universal fermion solver.

Significance. If the convergence claim could be fully supported, the framework would be a significant advance: it would provide a systematically improvable continuum solver that, unlike fixed-node DMC, remains valid for time-reversal-broken systems. The paper's strengths are real: the decomposition in Eq. (8) is clean and correct; the DMC projection is a genuinely independent check in the sense that E_DMC is a measured projection result, not a fit to the ED or Tanatar–Ceperley benchmarks; the statistical analysis is careful, with reblocked error bars (Appendix F) and a time-step extrapolation for the UEG (Appendix H). The paper is also honest in Sec. IIB in stating that the phase error E_FP − E_0 is not probed by ΔE. However, the abstract and conclusions go beyond that caveat by asserting that the vanishing gap indicates convergence to the true ground state, which is not logically implied by the diagnostic. The missing quantum-dot time-step check is a secondary but concrete gap. With appropriate revision, the paper would be a useful contribution to the neural-quantum-state and DMC literatures.

major comments (2)
  1. [Abstract; Sec. IIB, Eq. (8); Sec. IIIA] The central claim that the collapse of ΔE = E_VMC − E_DMC 'indicates convergence to the ground state' is stronger than what Eq. (8) supports. A vanishing amplitude gap is necessary but not sufficient, since E_FP − E_0 is not bounded by ΔE. The paper explicitly acknowledges this in Sec. IIB, yet the abstract and Sec. IIIA phrase the result as evidence that 'the network finds the true ground state.' The indirect evidence for phase convergence—monotone descent of E_DMC and agreement with truncated Landau-level ED—is suggestive but not conclusive, especially because the ED reference is itself an upper bound on the continuum ground state and no independent reference exists for the UEG. Please rephrase the convergence claim as convergence within the learned phase manifold, and either present the phase-convergence evidence separately with an explicit statement of its indirectness or add a phase
  2. [Sec. IIB, Appendix H, Table I] The sentence in Sec. IIB stating that residual time-step and population-control biases are 'quantified in Appendix H and verified to lie below our statistical resolution' is not supported for the quantum dot. Appendix H presents a time-step scan only for the 2D UEG; no analogous scan is reported for the dot. Since the dot's largest-network ΔE values (0.24–0.25×10⁻⁴ H*, Table I) are comparable to the quoted DMC error bars, an unquantified time-step bias could systematically offset the reported ΔE collapse. Please either provide a quantum-dot time-step extrapolation or explicitly restrict the bias claim to the UEG and state what assumption is being made for the dot.
minor comments (6)
  1. [Fig. 2 caption] Typo: 'illustratses' should be 'illustrates'.
  2. [Table I caption] The unit 'H*' (effective Hartree) is used without definition; please define it at first occurrence.
  3. [Sec. IIB, Eq. (6)] The phrase 'in Hartree atomic units reads' is awkward; suggest 'reads, in Hartree atomic units,' or equivalent.
  4. [Sec. IIIA, Appendix D] The text refers to 'exact diagonalization,' but the calculation is performed in a truncated multi-Landau-level basis (7-LL or 8-LL). This is stated in Table II but could be misread in the main text; please use 'truncated ED' or 'ED in a Landau-level basis' consistently.
  5. [Appendix E, Eq. (E2)] The citation 'Ref. [14,17]' should be formatted as 'Refs. [14,17]' for consistency.
  6. [Sec. IIIB] The sentence 'the steady descent of E_DMC is direct evidence that the learned phase improves systematically with capacity' could be read as stronger than warranted; consider 'consistent with' rather than 'direct evidence,' given the caveat in Sec. IIB.

Circularity Check

0 steps flagged

No significant circularity: the DMC gap is a measured projection result, not a fitted input; the same-group citation for Fermi Sets universality is a framing dependency, not a reduction.

full rationale

The central diagnostic, ΔE = E_VMC − E_DMC, is a genuine measured quantity: E_DMC is obtained by an independent imaginary-time fixed-phase projection at the learned phase, not by fitting to the ED or Tanatar–Ceperley benchmarks. Eq. (8) is an exact identity (variational error = amplitude gap + phase error), so the decomposition itself is not a hidden definition. The paper explicitly states in Sec. II B that the phase error is 'not probed by ΔE' and that a vanishing gap is 'necessary for the exact ground state but not sufficient,' so the abstract's stronger wording 'indicating convergence to the ground state' is an interpretive overreach rather than a circular construction. External benchmarks (multi-Landau-level ED for the dot; Tanatar–Ceperley for the UEG) provide independent checks, and the DMC time-step bias is examined in Appendix H, though only for the UEG and not for the quantum dot. The one same-group load-bearing citation is Ref. [12] for the 'provably universal' property of Fermi Sets; this is a mathematical expressibility theorem independent of the fitted energy values, so under the stated rules it does not by itself constitute circularity. No step reduces to its own input; score 2 reflects only the minor self-citation in the framing.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The paper introduces no new physical entities. The central claim rests on one imported theorem (Fermi Sets universality, Ref. [12]) and one imported projection property (fixed-phase variational structure, Ref. [14]), plus a set of fitted neural parameters whose values are the fit output. The empirical content is the scaling behavior of ΔE, which is genuinely measured. The most fragile imported items are the same-group universality theorem and the use of truncated ED as the 'exact' reference.

free parameters (5)
  • Fermi Sets network parameters (weights and biases of backbone, orbitals, symmetric factors) = N_p from 1.2e4 to 8.4e4 (dot) / 2.5e4 to 5.0e4 (UEG)
    All network weights are fitted by K-FAC energy minimization (Appendix A); the capacity sweep is the paper's controlled variable and the variational energies are the fit outcomes.
  • Jastrow range parameter β = learned, not quoted
    Eq. (2): learnable range parameter used in the quantum dot ansatz; affects both VMC and DMC energies.
  • Gaussian envelope exponent σ = learned, not quoted
    Eq. (3): learnable decay scale for the quantum dot; fitted to minimize the variational energy.
  • FPDMC imaginary-time step Δτ = 0.001 (verified only for UEG)
    Hand-chosen; Appendix H checks time-step bias for the UEG production network only; the quantum dot projection at Δτ=0.001 is asserted but not independently verified.
  • Zen et al. local-energy cutoff parameter α = not specified
    Imported from Ref. [22]; the cutoff value used in production runs is not quoted, so part of the DMC bias claim is unverifiable from the text.
axioms (6)
  • domain assumption Fermi Sets is a universal approximator of continuous antisymmetric functions in 2D with N_d=2 determinants, so exactness is limited only by network size.
    Sec. IIA: 'Fermi Sets can provably approximate any continuous antisymmetric function in two dimensions using only N_d = 2 determinants' — imported from same-group Ref. [12], not re-proven or machine-checked here.
  • domain assumption The fixed-phase energy E_FP is the minimum Rayleigh quotient over states with the trial phase, and the mixed estimator converges to E_FP without bias.
    Sec. IIB relies on Ortiz–Ceperley–Martin [14] and Bolton [17]; the fixed-phase variational structure is background projection-QMC theory, not re-derived here.
  • domain assumption For the dot, the true ground state lies in sectors L=14 (B=8T) and L=6 (B=4T), and the truncated multi-Landau-level ED used as reference is converged.
    Appendix D and Table II: the 'exact' reference is a 7–8 LL truncation, a variational upper bound rather than the exact continuum value; the network independently lands on the same L sectors (Table IV).
  • domain assumption The 2D UEG Hamiltonian with Ewald resummation and hexagonal cell (Eq. E2) is the correct target model, and the N=16, r_s=30 cell without twist averaging is representative.
    Appendix E; finite-size effects are not removed; the authors themselves list twist averaging and finite-size extrapolation as future work.
  • domain assumption K-FAC optimization over 500k iterations locates the variational minimum (no local-minimum trapping that would imitate capacity convergence).
    Appendix A describes a single training schedule per system; no multi-seed or optimization-convergence analysis is reported, so ΔE→0 could in principle reflect a shared amplitude/phase optimum rather than capacity saturation.
  • standard math Cusp coefficient α=1/3 for spin-polarized 2D Coulomb electrons.
    Eq. (2), per Ceperley [20]; standard 2D Coulomb cusp condition, not fitted.

pith-pipeline@v1.3.0-alltime-deepseek · 13258 in / 22669 out tokens · 201891 ms · 2026-08-01T01:15:26.535885+00:00 · methodology

0 comments
read the original abstract

In this work, we show that Fermi Sets---a provably universal neural network architecture for fermionic wavefunctions---can be systematically scaled up to find interacting ground states through energy minimization in a variational Monte Carlo framework. By further performing fixed-phase diffusion Monte Carlo (DMC) on the optimized neural network wavefunction, we demonstrate that as the network size increases, the variational energy systematically decreases while the energy improvement from DMC collapses monotonically to zero, indicating convergence to the ground state. We illustrate the scaling of Fermi Sets accompanied by the DMC assessment for interacting electrons in jellium and in a quantum dot under high magnetic fields.

Figures

Figures reproduced from arXiv: 2607.25872 by Ahmed Abouelkomsan, Hsin Lin, Liang Fu, Saskia Poldmaa, Tay-Rong Chang, Tzen Ong, Yu-Sheng Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Two-stage framework for optimizing and assessing Fermi Sets wavefunctions. (a) Variational optimization of the Fermi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Architecture of the Fermi Sets wavefunction Ψ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left: convergence of the VMC and DMC energies [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Convergence of the Fermi Sets states for the fully [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Time-step dependence of the FPDMC energy for the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

45 extracted references · 1 canonical work pages

  1. [1]

    D. M. Ceperley and B. J. Alder, Ground state of the electron gas by a stochastic method, Phys. Rev. Lett. 45, 566 (1980)

    1980

  2. [2]

    W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Ra- jagopal, Quantum monte carlo simulations of solids, Rev. Mod. Phys.73, 33 (2001)

    2001

  3. [3]

    J. B. Anderson, A random-walk simula- tion of the schr¨ odinger equation: H+3, 10 The Journal of Chemical Physics63, 1499 (1975), https://pubs.aip.org/aip/jcp/article- pdf/63/4/1499/18897623/1499 1 online.pdf

    1975

  4. [4]

    P. J. Reynolds, D. M. Ceperley, B. J. Alder, and J. Lester, William A., Fixed-node quantum monte carlo for moleculesa) b), The Journal of Chemical Physics 77, 5593 (1982), https://pubs.aip.org/aip/jcp/article- pdf/77/11/5593/18939983/5593 1 online.pdf

    1982

  5. [5]

    von Glehn, J

    I. von Glehn, J. S. Spencer, and D. Pfau, A self- attention ansatz for ab-initio quantum chemistry (2023), arXiv:2211.13672 [physics.chem-ph]

    Pith/arXiv arXiv 2023

  6. [6]

    Geier, K

    M. Geier, K. Nazaryan, T. Zaklama, and L. Fu, Self- attention neural network for solving correlated electron problems in solids, Phys. Rev. B112, 045119 (2025)

    2025

  7. [7]

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

    2020

  8. [8]

    Cassella, H

    G. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer, and W. M. C. Foulkes, Discov- ering quantum phase transitions with fermionic neural networks, Phys. Rev. Lett.130, 036401 (2023)

    2023

  9. [9]

    Y. Teng, D. D. Dai, and L. Fu, Solving the fractional quantum hall problem with self-attention neural network, Phys. Rev. B111, 205117 (2025)

    2025

  10. [10]

    Nazaryan, F

    K. Nazaryan, F. Gaggioli, Y. Teng, and L. Fu, Artificial intelligence for quantum matter: Finding a needle in a haystack (2026), arXiv:2507.13322 [cond-mat.str-el]

    arXiv 2026

  11. [11]

    Chen and J

    Z. Chen and J. Lu, Exact and efficient repre- sentation of totally anti-symmetric functions (2025), arXiv:2311.05064 [math.CA]

    Pith/arXiv arXiv 2025

  12. [12]

    Fu, Fermi sets: Universal and interpretable neural ar- chitectures for fermions (2026), arXiv:2601.02508 [cond- mat.str-el]

    L. Fu, Fermi sets: Universal and interpretable neural ar- chitectures for fermions (2026), arXiv:2601.02508 [cond- mat.str-el]

    Pith/arXiv arXiv 2026

  13. [13]

    Zaklama, M

    T. Zaklama, M. Geier, and L. Fu, Large electron model: A universal ground state predictor (2026), arXiv:2603.02346 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  14. [14]

    Ortiz, D

    G. Ortiz, D. M. Ceperley, and R. M. Martin, New stochastic method for systems with broken time-reversal symmetry: 2d fermions in a magnetic field, Phys. Rev. Lett.71, 2777 (1993)

    1993

  15. [15]

    Wilson, N

    M. Wilson, N. Gao, F. Wudarski, E. Rieffel, and N. M. Tubman, Simulations of state-of-the-art fermionic neu- ral network wave functions with diffusion monte carlo (2021), arXiv:2103.12570 [physics.chem-ph]

    Pith/arXiv arXiv 2021

  16. [16]

    W. Ren, W. Fu, X. Wu, and J. Chen, Towards the ground state of molecules via diffusion monte carlo on neural networks, Nature Communications14, 1860 (2023)

    2023

  17. [17]

    Bolton, Fixed-phase quantum monte carlo method ap- plied to interacting electrons in a quantum dot, Phys

    F. Bolton, Fixed-phase quantum monte carlo method ap- plied to interacting electrons in a quantum dot, Phys. Rev. B54, 4780 (1996)

    1996

  18. [18]

    Zaheer, S

    M. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. R. Salakhutdinov, and A. Smola, Deep sets, inAdvances in Neural Information Processing Systems, Vol. 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (Curran As- sociates, Inc., 2017)

    2017

  19. [19]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. u. Kaiser, and I. Polosukhin, Attention is all you need, inAdvances in Neural Infor- mation Processing Systems, Vol. 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vish- wanathan, and R. Garnett (Curran Associates, Inc., 2017)

    2017

  20. [20]

    Ceperley, Ground state of the fermion one-component plasma: A monte carlo study in two and three dimen- sions, Phys

    D. Ceperley, Ground state of the fermion one-component plasma: A monte carlo study in two and three dimen- sions, Phys. Rev. B18, 3126 (1978)

    1978

  21. [21]

    C. J. Umrigar, M. P. Nightingale, and K. J. Runge, A diffusion monte carlo algorithm with very small time-step errors, The Journal of Chemical Physics 99, 2865 (1993), https://pubs.aip.org/aip/jcp/article- pdf/99/4/2865/19121424/2865 1 online.pdf

    1993

  22. [22]

    A. Zen, S. Sorella, M. J. Gillan, A. Michaelides, and D. Alf` e, Boosting the accuracy and speed of quantum monte carlo: Size consistency and time step, Physical Review B93, 10.1103/physrevb.93.241118 (2016)

    work page doi:10.1103/physrevb.93.241118 2016

  23. [23]

    S. M. Reimann and M. Manninen, Electronic structure of quantum dots, Rev. Mod. Phys.74, 1283 (2002)

    2002

  24. [24]

    A. D. G¨ u¸ cl¨ u and C. J. Umrigar, Maximum-density droplet to lower-density droplet transition in quantum dots, Phys. Rev. B72, 045309 (2005)

    2005

  25. [25]

    Giuliani and G

    G. Giuliani and G. Vignale,Quantum Theory of the Elec- tron Liquid(Cambridge University Press, 2005)

    2005

  26. [26]

    P. P. Ewald, Die berechnung optis- cher und elektrostatischer gitterpoten- tiale, Annalen der Physik369, 253 (1921), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19213690304

    doi:10.1002/andp.19213690304 1921

  27. [27]

    Tanatar and D

    B. Tanatar and D. M. Ceperley, Ground state of the two- dimensional electron gas, Phys. Rev. B39, 5005 (1989)

    1989

  28. [28]

    L. M. Fraser, W. M. C. Foulkes, G. Rajagopal, R. J. Needs, S. D. Kenny, and A. J. Williamson, Finite- size effects and coulomb interactions in quantum monte carlo calculations for homogeneous systems with periodic boundary conditions, Phys. Rev. B53, 1814 (1996)

    1996

  29. [29]

    N. D. Drummond and R. J. Needs, Phase diagram of the low-density two-dimensional homogeneous electron gas, Phys. Rev. Lett.102, 126402 (2009)

    2009

  30. [30]

    Abouelkomsan and L

    A. Abouelkomsan and L. Fu, First-principles ai finds crystallization of fractional quantum hall liquids (2026), arXiv:2602.03927 [cond-mat.mes-hall]

    arXiv 2026

  31. [31]

    Gattu, Dressing composite fermions with artificial in- telligence (2025), arXiv:2512.00527 [cond-mat.str-el]

    M. Gattu, Dressing composite fermions with artificial in- telligence (2025), arXiv:2512.00527 [cond-mat.str-el]

    arXiv 2025

  32. [32]

    A. P. Fadon, D. Pfau, J. S. Spencer, W. T. Lou, T. Ne- upert, and W. M. C. Foulkes, Extracting anyon statis- tics from neural network fractional quantum hall states (2025), arXiv:2512.15872 [cond-mat.str-el]

    arXiv 2025

  33. [33]

    X. Li, Y. Chen, B. Li, H. Chen, F. Wu, J. Chen, and W. Ren, Deep learning sheds light on integer and fractional topological insulators (2025), arXiv:2503.11756 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  34. [34]

    D. Luo, T. Zaklama, and L. Fu, Solving fractional elec- tron states in twisted mote 2 with deep neural network (2025), arXiv:2503.13585 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  35. [35]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosen- bluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing ma- chines, The Journal of Chemical Physics21, 1087 (1953), https://pubs.aip.org/aip/jcp/article- pdf/21/6/1087/18802390/1087 1 online.pdf

    1953

  36. [36]

    W. K. Hastings, Monte carlo sampling methods using markov chains and their applications, Biometrika57, 97 (1970), https://academic.oup.com/biomet/article- pdf/57/1/97/23940249/57-1-97.pdf

    1970

  37. [37]

    Becca and S

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

    2017

  38. [38]

    Sorella, Green function monte carlo with stochastic reconfiguration, Phys

    S. Sorella, Green function monte carlo with stochastic reconfiguration, Phys. Rev. Lett.80, 4558 (1998). 11

    1998

  39. [39]

    Martens and R

    J. Martens and R. Grosse, Optimizing neural networks with kronecker-factored approximate curvature, inPro- ceedings of the 32nd International Conference on Ma- chine Learning, Proceedings of Machine Learning Re- search, Vol. 37, edited by F. Bach and D. Blei (PMLR, Lille, France, 2015) pp. 2408–2417

    2015

  40. [40]

    Attaccalite, S

    C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Correlation energy and spin polarization in the 2d electron gas, Phys. Rev. Lett.88, 256601 (2002)

    2002

  41. [41]

    Y. Kwon, D. M. Ceperley, and R. M. Martin, Effects of three-body and backflow correlations in the two- dimensional electron gas, Phys. Rev. B48, 12037 (1993)

    1993

  42. [42]

    Wilson, S

    M. Wilson, S. Moroni, M. Holzmann, N. Gao, F. Wu- darski, T. Vegge, and A. Bhowmik, Neural network ansatz for periodic wave functions and the homogeneous electron gas, Phys. Rev. B107, 235139 (2023)

    2023

  43. [43]

    Smith, Y

    C. Smith, Y. Chen, R. Levy, Y. Yang, M. A. Morales, and S. Zhang, Unified variational approach description of ground-state phases of the two-dimensional electron gas, Phys. Rev. Lett.133, 266504 (2024)

    2024

  44. [44]

    M. P. Allen and D. J. Tildesley,Computer Simulation of Liquids(Oxford University Press, 2017)

    2017

  45. [45]

    Flyvbjerg and H

    H. Flyvbjerg and H. G. Petersen, Error estimates on aver- ages of correlated data, The Journal of Chemical Physics 91, 461 (1989), https://pubs.aip.org/aip/jcp/article- pdf/91/1/461/18981401/461 1 online.pdf

    1989