Publications

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. On 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.

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Computational discovery of magnetic materials remains challenging because magnetism arises from the competition between kinetic energy and Coulomb interaction that is often beyond the reach of standard electronic-structure methods. Here we tackle this challenge by directly solving the many-electron Schrödinger equation with neural-network variational Monte Carlo, which provides a highly expressive variational wavefunction for strongly correlated systems. Applying this technique to transition metal dichalcogenide moiré semicondutors, we predict itinerant ferromagnetism in WS_e2/WS₂ and an antiferromagnetic insulator in twisted Γ-valley homobilayer, using the same neural network without any physics input beyond the microscopic Hamiltonian. Crucially, both types of magnetic states are obtained from a single calculation within the S_z=0 sector, removing the need to compute and compare multiple S_z sectors. This significantly reduces computational cost and paves the way for faster and more reliable magnetic material design.

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When does a fractional quantum Hall (FQH) liquid crystallize? Addressing this question requires a framework that treats fractionalization and crystallization on equal footing, especially in strong Landau-level mixing regime. Here, we introduce MagNet, a self-attention neural-network variational wavefunction designed for quantum systems in magnetic fields on the torus geometry. We show that MagNet provides a unifying and expressive ansatz capable of describing both FQH states and electron crystals within the same architecture. Trained solely by energy minimization of the microscopic Hamiltonian, MagNet discovers topological liquid and electron crystal ground states across a broad range of Landau-level mixing. Our results highlight the power of first-principles AI for solving strongly interacting many-body problems and finding competing phases without external training data or physics pre-knowledge.

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We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a compact domain can be approximated to arbitrary accuracy by a linear combination of K antisymmetric basis functions--such as pairwise products or Slater determinants--multiplied by symmetric functions. A key result is that the number of required bases is provably small: K=1 suffices in one-dimensional continua (and on lattices in any dimension), K=2 suffices in two dimensions, and in higher dimensions K grows at most linearly with particle number. The antisymmetric bases can be learned by small neural networks, while the symmetric factors are implemented by permutation-invariant networks whose width scales only linearly with particle number. Thus, Fermi Sets achieve universal approximation of fermionic wavefunctions with minimal overhead while retaining clear physical interpretability.

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We present an attention-based foundation model architecture for learning and predicting quantum states across Hamiltonian parameters, system sizes, and physical systems. Using only basis configurations and physical parameters as inputs, our trained neural network is able to produce highly accurate ground state wavefunctions. For example, we build the phase diagram for the 2D square-lattice t−V model with N particles, from only 18 parameters (V/t,N). Thus, our architecture provides a basis for building a universal foundation model for quantum matter.

The homogeneous electron gas is a cornerstone of quantum condensed matter physics, providing the foundation for developing density functional theory and understanding electronic phases in semiconductors. However, theoretical understanding of strongly-correlated electrons in realistic semiconductor systems remains limited. In this work, we develop a neural network based variational approach to study quantum wells in three dimensional geometry for a variety of electron densities and well thicknesses. Starting from first principles, our unbiased AI-powered method reveals metallic and crystalline phases with both monolayer and bilayer charge distributions. In the emergent bilayer, we discover a new quantum phase of matter: the electronic quasicrystal.

Topologically ordered states are among the most interesting quantum phases of matter that host emergent quasi-particles having fractional charge and obeying fractional quantum statistics. Theoretical study of such states is however challenging owing to their strong-coupling nature that prevents conventional mean-field treatment. Here, we demonstrate that an attention-based deep neural network provides an expressive variational wavefunction that discovers fractional Chern insulator ground states purely through energy minimization without prior knowledge and achieves remarkable accuracy. We introduce an efficient method to extract ground state topological degeneracy -- a hallmark of topological order -- from a single optimized real-space wavefunction in translation-invariant systems by decomposing it into different many-body momentum sectors. Our results establish neural network variational Monte Carlo as a versatile tool for discovering strongly correlated topological phases.

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Representing fermionic wavefunctions efficiently is a central problem in quantum physics, chemistry and materials science. In this work, we introduce a universal and exact representation of continuous antisymmetric functions by lifting them to continuous symmetric functions defined on an enlarged space. Building on this lifting, we obtain a parity-graded representation of fermionic wavefunctions, expressed in terms of symmetric feature variables that encode particle configuration and antisymmetric feature variables that encode exchange statistics. This representation is both exact and minimal: the number of required features scales as D∼N^d (d is spatial dimension) or D∼N depending on the symmetric feature maps employed. Our results provide a rigorous mathematical foundation for efficient representations of fermionic wavefunctions and enable scalable and systematically improvable neural network solvers for many-electron systems.

Recent advances on neural quantum states have shown that correlations between quantum particles can be efficiently captured by attention-- a foundation of modern neural architectures that enables neural networks to learn the relation between objects. In this work, we show that a general-purpose self-attention Fermi neural network is able to find chiral px±ipy superconductivity in an attractive Fermi gas by energy minimization, {\it without prior knowledge or bias towards pairing}. The superconducting state is identified from the optimized wavefunction by measuring various physical observables: the pair binding energy, the total angular momentum of the ground state, and off-diagonal long-range order in the two-body reduced density matrix. Our work paves the way for AI-driven discovery of unconventional and topological superconductivity in strongly correlated quantum materials.

Neural networks (NNs) have great potential in solving the ground state of various many-bodyproblems. However, several key challenges remain to be overcome before NNs can tackle problemsand system sizes inaccessible with more established tools. Here, we present a general and efficientmethod for learning the NN representation of an arbitrary many-body complex wave function fromits N-particle probability density and probability current density. Having reached overlaps as largeas 99.9%, we employ our neural wave function for pre-training to effortlessly solve the fractionalquantum Hall problem with Coulomb interactions and realistic Landau-level mixing for as manyas 25 particles. Our work demonstrates efficient, accurate simulation of highly-entangled quantummatter using general-purpose deep NNs enhanced with physics-informed initialization.

The emergence of moiré materials, such as twisted transition-metal dichalcogenides (TMDs), has created a fertile ground for discovering novel quantum phases of matter. However, solving many-electron problems in moiré systems presents significant challenges due to strong electron correlation and strong moiré band mixing. Recent advancements in neural quantum states hold the promise for accurate and unbiased variational solutions. Here, we introduce a powerful neural wavefunction to solve ground states of twisted MoTe2 across various fractional fillings, reaching unprecedented accuracy and system size. From the full structure factor and quantum weight, we conclude that our neural wavefunction accurately captures both the electron crystal at ν=1/3 and various fractional quantum liquids in a unified manner.

The attention mechanism has transformed artificial intelligence research by its ability to learn relations between objects. In this work, we explore how a many-body wavefunction ansatz constructed from a large-parameter self-attention neural network can be used to solve the interacting electron problem in solids. By a systematic neural-network variational Monte Carlo study on a moiré quantum material, we demonstrate that the self-attention ansatz provides an accurate and efficient solution without human bias. Moreover, our numerical study finds that the required number of variational parameters scales roughly as N2 with the number of electrons, which opens a path towards efficient large-scale simulations.

We introduce an attention-based fermionic neural network (FNN) to variationally solve the problem of two-dimensional Coulomb electron gas in magnetic fields, a canonical platform for fractional quantum Hall (FQH) liquids, Wigner crystals, and other unconventional electron states. Working directly with the full Hilbert space of 𝑁 electrons confined to a disk, our FNN consistently attains energies lower than LL-projected exact diagonalization (ED) and learns the ground state wave function to high accuracy. In low LL mixing regime, our FNN reveals microscopic features in the short-distance behavior of FQH wave function beyond the Laughlin ansatz. For moderate and strong LL mixing parameters, the FNN outperforms ED significantly. Moreover, a phase transition from FQH liquid to a crystal state is found at strong LL mixing. Our study demonstrates unprecedented power and universality of FNN based variational method for solving strong-coupling many-body problems with topological order and electron fractionalization.