Large Electron Model: A Universal Ground State Predictor

Motivation

• Because key material properties arise from interacting-electron dynamics, the introduction argues that scalable prediction beyond measured data requires solving many-electron systems and learning from physical principles.It identifies a foundation model that shares parameters across physical settings as the missing approach, since large systems lack ground-truth wavefunctions or energies for supervised training.Source: INTRODUCTION · p. 1
• Although density functional theory is versatile and scalable, it is nonvariational, does not produce many-body ground-state wavefunctions, and fails for strongly correlated materials.The introduction gives fractional Chern insulators and strong-coupling superconductors as examples.Source: INTRODUCTION · p. 1
• Neural-network variational Monte Carlo can recover accurate ground-state energies and wavefunctions, but typically requires optimizing a new network from scratch for each Hamiltonian and parameter choice.This prevents reusing computation across different systems and sizes.Source: INTRODUCTION · p. 1

LEM

• LEM is a single parameter-sharing network conditioned on Hamiltonian parameters and particle number, trained without reference solutions or labeled data by variational energy minimization.It directly predicts ground-state wavefunctions for Hamiltonians not seen during training.Source: INTRODUCTION · pp. 1–2

Architecture

• Its universal Fermi wavefunction ansatz combines a compact antisymmetric Slater-determinant component with an attention-based symmetric component that models electron correlations.The architecture is designed to capture global dependencies and ground states from weak to strong interaction regimes.Source: INTRODUCTION · p. 2

Results

• LEM predicts quantum-dot ground states across interaction strengths and particle-number sectors, with energy expectations beating standard-method benchmarks and real-space charge-density predictions scaling to N = 50.The model is optimized across interaction regimes and Hilbert-space sectors in a single run.Source: INTRODUCTION · p. 2

THEORETICAL FRAMEWORK

• Ground-state wavefunctions are first-quantized fermionic functions of electron positions and spin labels, with antisymmetry under relabeling indistinguishable electrons.The Hamiltonian family is indexed by Λ, including parameters such as interaction strength and particle number.Source: THEORETICAL FRAMEWORK · p. 2
• The architecture combines K Slater-determinant signature encoders with a permutation-symmetric transformer to represent electron correlations, conditioning both components on Λ.The determinant orbitals are represented by an MLP, and the transformer has a symmetric head.Source: Figure 1 · p. 2
• A shared-parameter model conditions on Hamiltonian parameters Λ to map the Hamiltonian family to ground-state wavefunctions, enabling interpolation and generalization beyond the optimization ensemble.The target is a manifold of variational ground states in continuous space and across particle-number sectors.Source: THEORETICAL FRAMEWORK · p. 2
• The model is optimized by minimizing normalized energy expectations using variational Monte Carlo, estimating them from Born-density samples and local energies for a finite ensemble of Hamiltonian settings.The workflow accumulates the energy expectations over the fixed parameter set and updates shared network weights from gradients of the total energy.Source: THEORETICAL FRAMEWORK · p. 2, Figure 2 · p. 3

APPROACH

• One conditional neural wavefunction maps Hamiltonian parameters to ground-state wavefunctions for continuum systems and is optimized once for reuse across physical settings.The paper presents this as a foundation-model formulation targeting real physical systems in the continuum.Source: APPROACH · p. 3
• Fermi Sets [1] represents continuous fermionic wavefunctions as symmetric prefactors multiplying antisymmetric cores, with Slater determinants of learned orbitals carrying antisymmetry.The approach generalizes this factorization to include parameter dependence.Source: APPROACH · p. 3
• Coordinate features relative to fixed reference centers are combined with spin, embedded, and processed by a parameter-conditioned transformer; pooling over electrons yields permutation-invariant outputs.A small MLP produces the symmetric coefficients, and permutation-equivariant weights can add expressivity.Source: APPROACH · p. 3

• The system comprises N spin-polarized interacting electrons in two dimensions, confined by an isotropic harmonic potential and subject to Coulomb repulsion.Source: PHYSICAL SYSTEM · p. 3
• The parameter vector is Λ = (N, ω, λ), where ω sets confinement curvature and λ controls interaction strength; calculations set ω = 1.Source: PHYSICAL SYSTEM · p. 4
• Results for ω ≠ 1 follow from ω = 1 solutions using dimensionless coupling eλ = λ/ω1/4 and overall energy scale √ω.Source: PHYSICAL SYSTEM · p. 4
• A single universal Fermi network is optimized to predict ground-state wavefunctions across the target Hamiltonian family.Source: PHYSICAL SYSTEM · p. 4

RESULTS

• Once optimized, the model predicts wavefunctions and enables charge densities, energies, and other equal-time observables to be computed without further optimization, transfer learning, or fine-tuning.The model is optimized through unsupervised energy minimization, without problem-specific supervision.Source: RESULTS · p. 4
• For N = 7, predicted densities show sixfold structure connecting to a 1 + 6 Wigner molecule, while N = 9 correctly shows twofold structure despite never training on that angular sector.For closed-shell N = 10, density remains rotationally symmetric as increasing λ reshapes the radial profile and sharpens concentric correlation-driven rings.Source: Figure 3 · p. 5, RESULTS · pp. 5–6
• Training on λ ∈{0.0, 1.0, 2.0, 5.0, 8.0, 10.0} and N ∈{6, 8, 10}, LEM predicts accurate states across 0 ≤λ ≤10.0 and 5 ≤N ≤11, beating DMC energies while individually optimized Fermi Sets go lower.The energy grid shows out-of-sample predictions across interaction strengths and particle numbers; LEM produces explicit many-electron wavefunctions, unlike DMC’s stochastic walker distribution.Source: RESULTS · p. 4, Figure 4 · p. 6
• At N = 50, training on λ ∈{0.0, 1.0, 2.0, 3.0, 5.0, 6.0, 8.0} yields accurate unseen-coupling predictions, outperforming fresh single-system runs for most comparisons with 6.0 < λ < 8.0.One optimized model predicts across coupling strengths without reoptimizing each point; multi-task optimization may stabilize optimization and improve generalization.Source: Figure 5 · p. 6, RESULTS · p. 6
• Fermi Sets with parameter conditioning are described as applicable to other electron systems, including solids, offering a route to transferable ground-state wavefunctions across materials and configurations.The architecture requires specification of the Hamiltonian family and conditioning parameters.Source: RESULTS · pp. 6–7

CONCLUSION

• A single parameter-sharing network combines a universal fermionic representation with parameter conditioning to predict accurate wavefunctions, state-of-the-art variational energies, and expected charge densities.The model maps system and Hamiltonian parameters directly to many-body ground states.Source: CONCLUSION · p. 7
• Up to N = 50 electrons, one model generalizes across the interaction range and unseen particle-number sectors, with out-of-sample energies outperforming fresh single-system runs.The model continues to output accurate Fermi wavefunctions even as particle number changes the input dimension and configuration space.Source: CONCLUSION · p. 7
• Fully unsupervised and requiring only the Hamiltonian, the method offers a route to reusable continuum models for quantum dots, trapped ions, atoms, molecules, and materials.The authors propose that one unified model could replace single-system workflows for first-principles, large-scale studies of real materials.Source: CONCLUSION · p. 7

Table of benchmarked energies

• At N = 10, one Large Electron Model run outperforms NN–SJ backflow and Hartree–Fock energy benchmarks for seen and unseen systems with matched network size.The comparison uses NN–SJ backflow and Hartree–Fock calculations trained from scratch on a single system; the table indicates whether each parameter value was seen during foundation-model training.Source: Table of benchmarked energies · p. 10, Table II · p. 10

Neural Network Slater–Jastrow and Hartree–Fock comparison

• At λ = 7.0, Hartree–Fock produces a strongly non-uniform, irregular density through spontaneous orbital localization and mixing of total-angular-momentum sectors.Because the circular-dot Hamiltonian is rotationally invariant, a density for a single angular-momentum sector is rotationally symmetric; the broken-symmetry Hartree–Fock density is not a controlled approximation to that ground-state density.Source: Neural Network Slater–Jastrow and… · pp. 10–11
• The real Jastrow factor reshapes amplitudes and captures correlations but leaves the determinant's antisymmetric sign structure and nodal surface unchanged.The neural Jastrow uses nonlinear layers on the full configuration, giving it higher-order many-electron correlations beyond a simple pairwise form, while retaining a single-determinant backbone.Source: Neural Network Slater–Jastrow and… · p. 11
• Optional backflow can improve nodal structure and recover correlation energy, but the single-determinant ansatz remains restricted; multi-determinant expansions are outside this compact baseline comparison.Backflow uses smooth, permutation-equivariant, configuration-dependent displacements; multi-determinant expansions can improve nodal flexibility but add a multiconfigurational expansion.Source: Neural Network Slater–Jastrow and… · pp. 11–12
• Compared with Hartree–Fock, NN Slater–Jastrow densities stay closer to circular at weak and intermediate coupling and show more regular angular features at strong coupling.The baseline still exhibits residual angular-momentum mixing and cannot represent the full multiconfigurational mixing needed in the strongly correlated regime.Source: Neural Network Slater–Jastrow and… · pp. 10, 12
• Fermi Sets produces the most faithful densities, while Hartree–Fock breaks down most severely and NN Slater–Jastrow remains limited by its determinant backbone.The paper also reports that Fermi Sets has superior energy accuracy; energies are compared in Table II.Source: Figure S1 · p. 11, Neural Network Slater–Jastrow and… · p. 12

Weak coupling density and angular momentum for the open-shell case

• For spin-polarized N = 7, the seventh electron enters a shell permitting |L| = 1 and 3; Coulomb-flattened confinement lowers the (0, ±3) pair, favoring L = ±3 at weak coupling.The noninteracting ground state includes both |L| = 1 and |L| = 3 sectors, and the interaction lifts their degeneracy.Source: Weak coupling density and angular… · p. 12
• At N = 7, λ = 1.0, replacing λ = 10.0 with λ = 0.5 in the same-sized training set changes the predicted density from twofold to the correct sixfold |L| = 3 pattern, matching single-system optimization.The original twofold result was an inference error from insufficiently resolved weak-coupling training, not a physical transition or structural limitation of the ansatz.Source: Weak coupling density and angular… · pp. 12–13, Figure S2 · p. 13
• Angular-momentum projections show dominant weights in L = ±3 (N = 7), L = ±1 (N = 9), and L = 0 (N = 10), diagnosing expected sectors beyond density inspection.The projections provide a direct wavefunction-level diagnostic for the main-text density results.Source: Figure S3 · p. 13, Weak coupling density and angular… · p. 13
• Weak-coupling open-shell states are demanding because small energetic splittings determine the sector; large or anomalous local energy variance can identify regions needing additional variational data.Adding one weak-coupling training point corrects the angular-momentum sector for the unseen N = 7, λ = 1.0 system.Source: Weak coupling density and angular… · p. 14
• Because the model outputs an explicit many-electron wavefunction, its sector weights and other equal-time observables—including pair correlations, structure factors, and conditional densities—can be computed from the same state.The sector weights are obtained by projection into subspaces with total angular momentum L.Source: Weak coupling density and angular… · p. 14

Optimization stability across system size

• Small-N co-training converges smoothly and plateaus early, while the N = 50 run converges more slowly with occasional energy and variance spikes.The large-electron regime constrains batch size and model width on a single 96 GB GPU because pairwise features and determinant construction scale roughly as N^2.Source: Optimization stability across system… · p. 14
• In N = 50 co-training, lower-λ groups fluctuate more than higher-λ groups because shared updates across disparate energy scales can give low-λ tasks an overly aggressive effective step size.Larger-energy groups contribute larger-magnitude gradients to the averaged objective; the fluctuations are an optimization effect, not increased physical difficulty at weak coupling.Source: Optimization stability across system… · pp. 14–15, Figure S5 · p. 15
• At N = 50, co-training reaches stable, accurate final energies with typical standard errors below 10^-3 on average and faithful wavefunctions across the full coupling range.The large-N optimization is noisier and slower, but remains well-behaved at long times despite memory and compute constraints.Source: Optimization stability across system… · pp. 14–15
• Balancing task scales through loss reweighting or variance regularization, or increasing model capacity, may reduce low-λ noise and gradient interference.The reported noise does not prevent accurate convergence or faithful wavefunctions across the coupling range.Source: Optimization stability across system… · p. 15

Extended data

• The model’s full many-body wavefunction is visualized by fixing N −1 coordinates at a high-probability configuration and scanning the remaining electron across the plane.The figure presents conditional probability, phase, and pair correlation for N = 10 and N = 7 at λ = 8.0.Source: Extended data · pp. 15–16, Figure S6 · p. 16
• Phase-domain boundaries coincide with suppressed amplitude, revealing nodal contours and the ground state’s sign/phase structure, including fermionic antisymmetry.The phase structure is information inaccessible from one-body densities alone and is not fixed by matching energies.Source: Extended data · p. 16, Figure S6 · p. 16
• At λ = 8, N = 10 shows an approximately isotropic correlation ring, while N = 7 has stronger angular modulation and enhanced localization in conditional pair correlations.Both cases show a short-distance exchange-correlation hole; averaging over particle labels and configurations restores rotational symmetry in the one-body densities.Source: Extended data · p. 16, Figure S6 · p. 16