Large Electron Model: A Universal Ground State Predictor

Timothy Zaklama; Max Geier; Liang Fu

cond-mat.str-elMar 2, 2026

Large Electron Model: A Universal Ground State Predictor

Authors: Timothy Zaklama, Max Geier, Liang Fu

Organizations: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Abstract

We introduce Large Electron Model, a single neural network model that produces variational wavefunctions of interacting electrons over the entire Hamiltonian parameter manifold. Our model employs the Fermi Sets architecture, a universal representation of many-body fermionic wavefunctions, which is further conditioned on Hamiltonian parameter and particle number. For interacting electrons in a two-dimensional harmonic potential, a single trained model accurately predicts the ground state wavefunction while generalizing across unseen coupling strengths and particle-number sectors, producing both accurate real-space charge densities and ground state energies, even up to $50$ particles. Our results establish a foundation model method for material discovery that is grounded in the variational principle, while accurately treating strong electron correlation beyond the capacity of density functional theory.

Figures & tables

Figure 1: Foundation model architecture. A Multilayer Perceptron (MLP) with determinant head (left) constructs a set of $K$ signature encoders formulated as Slater determinants with orbitals represented by the MLP. Electron correlations are captured by a permutation-symmetric function represented by a transformer with symmetric head (right). Conditioning both parts with the parameters $\bm{\Lambda}_{i}$ yields the continuous parameterization of the many-body wavefunction $\Psi_{\bm{\Lambda}}(\{\bm{r}_{i}\})$ .

Figure 2: Workflow. The network represents the many-body wavefunction with continuous parameter dependence $\Psi_{\mathbf{\Lambda}}(\{\mathbf{r}_{i}\})$ . The loss is computed as the total energy $\sum_{i}\langle\Psi_{\mathbf{\Lambda}_{i}}|H_{\mathbf{\Lambda}_{i}}|\Psi_{\mathbf{\Lambda}_{i}}\rangle$ accumulated over a fixed set of system parameters $\{\mathbf{\Lambda}_{i}\}$ , where the individual energy expectation values are computed using Monte-Carlo sampling. From the gradients of the total energy, an optimizer routine determines updates of network weights $\theta$ .

$\lambda$	$N$	Fermi Sets (Single Task)	Fermi Sets (LEM)	DMC [ 15 ]	FermiNet [ 22 ]
8.0	6	$\mathbf{60.374(3)}$	$\mathbf{60.3898(8)}$	$60.3924(2)$	$60.402(1)$
10.0	6	$\mathbf{68.93981(6)}$	$\mathbf{68.942(1)}$	$68.9458(4)$	$68.967(1)$
8.0	7	$\mathbf{80.50917(8)}$	$\mathbf{80.512(1)}$	$80.5146(2)$	$80.528(3)$
8.0	8	$\mathbf{103.0421(1)}$	$\mathbf{103.043(3)}$	$103.0464(3)$	$103.0467(4)$

Table 1: Benchmarked energies using Fermi Sets ansatz for both single-run and foundation model against DMC and FermiNet references. The “single task” energies are produced after optimizing for a single system and sampling the converged trial wavefunction, whereas the foundation model (LEM) is optimized on many systems at once and then predicts the trial wavefunction, from which the energy is extracted after Monte Carlo sampling. The parentheses around the final digit denote the standard error on the previous digit.

Figure 3: Real space particle density from inference for various parameter values. (a-c) are for $N=7,9,10$ , respectively, with columns corresponding to $\lambda=3.0$ (predicted), $5.0$ (optimized for $N=10$ , predicted for $N=7,9$ ), $7.0$ (predicted). Although the network is never optimized for any system with twofold angular structure, it is able to zero-shot predict the correct angular structure for the $N=9$ case.

Figure 4: Optimize and predict results across

(\lambda,N)

particle grid. The model achieves pinpoint accuracy on out-of-sample results and can generate the wavefunction and any equal-time observable for any parameter across Hilbert-space sectors at inference time. Energies are competitive with state-of-the-art benchmarks [ 15 ] , for which a detailed table is provided in the supplementary material [ 1 ] . The statistical errors on the energies are far smaller than the radius of the dots, and therefore omitted in the figure.

Figure 5: Optimize and predict results for

N=50

over the

\lambda

parameter manifold. The model achieves pinpoint accuracy on out-of-sample results and can generate ground state energies that are even lower than single-parameter optimization and evaluation runs (highlighted in the inset). The statistical errors on the energies are far smaller than the radius of the dots, and therefore omitted in the figure.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

$\lambda$	training	Fermi Sets (LEM)	NN–SJ backflow	NN–SJ	HF
1.0	yes	$\mathbf{52.0397}$	$52.0559$	$52.0796$	$52.2161$
3.0	no	$\mathbf{87.5655}$	$87.6543$	$87.7070$	$88.7643$
5.0	yes	$\mathbf{116.9503}$	$117.6311$	$117.8350$	$120.4366$
7.0	no	$\mathbf{142.9216}$	$143.3966$	$143.8019$	$149.3424$

Appendix

Table 2: Ground-state energies for different benchmarked approaches at

N=10

. For the Neural Network Slater–Jastrow (NN–SJ) backflow, NN–SJ, and Hartree–Fock (HF) cases, energies are produced from calculations trained from scratch on a single system, using the same network size as the foundation model where applicable. The training column denotes whether the Fermi Sets network saw that parameter value during training.

Figure S1: Real-space particle density for NN Slater–Jastrow (a) and Hartree–Fock (b) calculations at $N=10,\lambda=1.0,3.0,7.0$ .

Figure S2: One-body density plots for the $N=7$ , $\lambda=1.0$ system. Left: Density predicted by the same model as in the main text (training set $\lambda\in\{0.0,1.0,2.0,5.0,8.0,10.0\}$ and $N\in\{6,8,10\}$ ). The density exhibits a two-lobed angular structure, indicating that the inferred state is not the expected weak-coupling $|L|=3$ representative. Middle: Density predicted by a model with the same network size and training points as in the main text, but with $\lambda=10$ replaced by $\lambda=0.5$ to focus more on weak coupling (training set $\lambda\in\{0.0,0.5,1.0,2.0,5.0,8.0\}$ and $N\in\{6,8,10\}$ ). Right: Density of wavefunction optimized for the single $N=7$ , $\lambda=1.0$ system (Single Fermi Sets optimization, no foundation model). The corresponding energy is 28.7929(1). Both the model with an additional weak-coupling training point and the single-system optimization predict the correct sixfold pattern evincing the $|L|=3$ representative.

Figure S3: Wavefunction angular momentum sector weights for all wavefunctions in Fig. 3 in the main text (panels (a)–(c) correspond to

N=7,9,10

, respectively, and the left, middle, and right columns correspond to

\lambda=3.0,5.0,7.0

, respectively). We project the inferred wavefunction into angular-momentum subspaces. The projection weight is dominated by the

L=3

sector for

N=7

(a),

L=1

sector for

N=9

(b), and

L=0

sector for

N=10

(c), as expected from the modified Hund rules. This provides a direct wavefunction-level diagnostic of the inferred state, beyond visual inspection of the density.

Figure S4: Total loss and variance for the small-particle-number run in the main text for (a) $N=6,8,10,\lambda=0.0,1.0,2.0,5.0,8.0,10.0$ and (b) $N=50,\lambda=0.0,1.0,2.0,3.0,5.0,6.0,8.0,10.0$ .

Figure S5: Group loss for each parameter group in the main-text runs, corresponding to the runs in Fig. S4 .

Figure S6: Conditional probability (left), phase (middle), and pair correlation plot (right) for (a) $N=10$ and (b) $N=7$ for $\lambda=8.0$ .

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QERNEL: a Scalable Large Electron Model

Khachatur Nazaryan, Liang Fu

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA-02139,USA

Neural Quantum States Materials Science Quantum Many-Body Physics Quantum Neural Networks

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We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moiré heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schrödinger equation conditioned on moiré potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moiré quantum materials and a scalable architecture toward a Large Electron Model for solids.

Abstract

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