Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.00618 · Physical review. B./Physical review. B

Solving the fractional quantum Hall problem with self-attention neural network

Yi Teng, David D. Dai, Liang Fu

Published May 13, 2025• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.111.205117

Related papers

First-Principles AI Finds Crystallization of Fractional Quantum Hall Liquids

Ahmed Abouelkomsan, Liang Fu

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 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 artificial intelligence (AI) for solving strongly interacting many-body problems and finding competing phases without external training data or physics preknowledge.

arXiv:2602.03927•PRX Intelligence•Jul 28, 2026•0 citations

Crystallization in the Fractional Quantum Hall Regime with Disorder-Aware Neural Quantum States

Jihang Zhu, Yi Huang, Xiaodong Hu +2 more

We present the first microscopic demonstration of a disorder-pinned hole Wigner crystal (WC), providing a natural explanation for the reentrant integer quantum Hall effect observed near $ν=2/3$, as well as its analogs in fractional Chern insulators. We further identify a novel crossover regime above filling $ν=2/3$ that connects this hole WC to an electron WC, characterized by a network-like electron density structure. To uncover these phenomena, we use neural-network variational Monte Carlo (NNVMC) with a disorder-aware self-attention neural quantum state that describes both fractional quantum Hall (FQH) liquids and Wigner crystals within a single unbiased variational framework. More broadly, our method establishes a unified phase diagram that exposes a fundamental asymmetry in crystallization across half-filling: near $ν=1/3$, increasing LL mixing and disorder both stabilize an electron WC, whereas near $ν=2/3$, the hole WC dominates at weak LL mixing and ultimately gives way to the electron WC at strong LL mixing.

arXiv:2604.06316•Apr 7, 2026•0 citations

Neural Network Fractional Quantum Hall States with Learnable Orbital Pair Matrix Factors: Balancing Physical Priors and Flexibility

Yubing Qian, Weiluo Ren, Ji Chen

Neural network fractional quantum Hall (FQH) states offer a promising alternative for capturing Landau level (LL) mixing and for understanding FQH physics, yet the incorporation of physical priors has so far been limited to the minimal exact constraints built into the ansatz. Working in spherical geometry, we show that the network's flexibility suffers both when only minimal additional physical priors are supplied and when priors such as flux attachment are imposed rigidly. In the former case, the monopole harmonic product orbital (MHPO) ansatz is difficult to optimize, whereas in the latter, the expressivity of the neural network is restricted. We therefore propose the orbital pair matrix factor MHPO (MHPO-M) ansatz, which balances physical priors against flexibility by not only replacing the fixed Laughlin structure with learnable pair factors but also making them dependent on both the electron configurations and the orbitals. Across $ν=1/5$, $1/3$, and $2/5$, MHPO-M achieves the lowest variational energy among the tested variants, and a factorized orbital readout based on it enables scaling to 35 electrons, with energies competitive with trends inferred from three-LL exact-diagonalization results. These results demonstrate that physical priors can be appropriately incorporated without sacrificing the flexibility of neural network wavefunctions, opening a promising path toward accurate, scalable modeling of FQH systems with LL mixing.

arXiv:2610.28095•Oct 20, 2026•0 citations

Describing Landau Level Mixing in Fractional Quantum Hall States with Deep Learning

Yubing Qian, Tongzhou Zhao, Jianxiao Zhang +3 more

Strong correlation brings a rich array of emergent phenomena, as well as a daunting challenge to theoretical physics study. In condensed matter physics, the fractional quantum Hall effect is a prominent example of strong correlation, with Landau level mixing being one of the most challenging aspects to address using traditional computational methods. Deep learning real-space neural network wave function methods have emerged as promising architectures to describe electron correlations in molecules and materials, but their power has not been fully tested for exotic quantum states. In this work, we employ real-space neural network wave function techniques to investigate fractional quantum Hall systems. On both 1/3 and 2/5 filling systems, we achieve energies consistently lower than exact diagonalization results which only consider the lowest Landau level. We also demonstrate that the real-space neural network wave function can naturally capture the extent of Landau level mixing up to a very high level, overcoming the limitations of traditional methods. Our work underscores the potential of neural networks for future studies of strongly correlated systems and opens new avenues for exploring the rich physics of the fractional quantum Hall effect.

arXiv:2412.14795•Physical Review Letters•Apr 30, 2025•12 citations

Evidence of a fractional quantum Hall nematic phase in a microscopic model

Nicolas Regnault, Joseph Maciejko, Steven Allan Kivelson +1 more

The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.

arXiv:1607.02178•Physical review. B./Physical review. B•Jul 27, 2017•41 citations

Exploring Fractional Quantum Hall States with Deep Learning: From Abelian to Non-Abelian Phases

Xiangyu Ma, Kehuang Chen, Yuntian Gu +5 more

Fractional quantum Hall systems host strongly correlated topological phases with potential applications in topological quantum computation, yet identifying their microscopic ground states remains challenging because competing phases are often nearly degenerate and Landau-level mixing is difficult to treat. Here, we introduce a symmetry-adapted neural-network wavefunction that describes Abelian and non-Abelian states while incorporating all Landau levels. The method accurately captures composite-fermion liquids and the emergence of Wigner crystallization at low filling. When the first Landau level is partially filled, finite-size overlaps and entanglement spectra provide evidence consistent with $\bar{2}^{k}1^{k+1}$-type parton phases at $ν=4/3$ and $7/5$, while the $ν=3/2$ state exhibits anti-Pfaffian-like entanglement structure. These results demonstrate that neural-network wavefunctions offer a flexible framework for studying competing topological orders in fractional quantum Hall systems.

arXiv:2610.38533•Oct 28, 2026•0 citations
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 NNN 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.
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 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 artificial intelligence (AI) for solving strongly interacting many-body problems and finding competing phases without external training data or physics preknowledge.
We present the first microscopic demonstration of a disorder-pinned hole Wigner crystal (WC), providing a natural explanation for the reentrant integer quantum Hall effect observed near ν=2/3ν=2/3ν=2/3, as well as its analogs in fractional Chern insulators. We further identify a novel crossover regime above filling ν=2/3ν=2/3ν=2/3 that connects this hole WC to an electron WC, characterized by a network-like electron density structure. To uncover these phenomena, we use neural-network variational Monte Carlo (NNVMC) with a disorder-aware self-attention neural quantum state that describes both fractional quantum Hall (FQH) liquids and Wigner crystals within a single unbiased variational framework. More broadly, our method establishes a unified phase diagram that exposes a fundamental asymmetry in crystallization across half-filling: near ν=1/3ν=1/3ν=1/3, increasing LL mixing and disorder both stabilize an electron WC, whereas near ν=2/3ν=2/3ν=2/3, the hole WC dominates at weak LL mixing and ultimately gives way to the electron WC at strong LL mixing.
Neural network fractional quantum Hall (FQH) states offer a promising alternative for capturing Landau level (LL) mixing and for understanding FQH physics, yet the incorporation of physical priors has so far been limited to the minimal exact constraints built into the ansatz. Working in spherical geometry, we show that the network's flexibility suffers both when only minimal additional physical priors are supplied and when priors such as flux attachment are imposed rigidly. In the former case, the monopole harmonic product orbital (MHPO) ansatz is difficult to optimize, whereas in the latter, the expressivity of the neural network is restricted. We therefore propose the orbital pair matrix factor MHPO (MHPO-M) ansatz, which balances physical priors against flexibility by not only replacing the fixed Laughlin structure with learnable pair factors but also making them dependent on both the electron configurations and the orbitals. Across ν=1/5ν=1/5ν=1/5, 1/31/31/3, and 2/52/52/5, MHPO-M achieves the lowest variational energy among the tested variants, and a factorized orbital readout based on it enables scaling to 35 electrons, with energies competitive with trends inferred from three-LL exact-diagonalization results. These results demonstrate that physical priors can be appropriately incorporated without sacrificing the flexibility of neural network wavefunctions, opening a promising path toward accurate, scalable modeling of FQH systems with LL mixing.
Strong correlation brings a rich array of emergent phenomena, as well as a daunting challenge to theoretical physics study. In condensed matter physics, the fractional quantum Hall effect is a prominent example of strong correlation, with Landau level mixing being one of the most challenging aspects to address using traditional computational methods. Deep learning real-space neural network wave function methods have emerged as promising architectures to describe electron correlations in molecules and materials, but their power has not been fully tested for exotic quantum states. In this work, we employ real-space neural network wave function techniques to investigate fractional quantum Hall systems. On both 1/3 and 2/5 filling systems, we achieve energies consistently lower than exact diagonalization results which only consider the lowest Landau level. We also demonstrate that the real-space neural network wave function can naturally capture the extent of Landau level mixing up to a very high level, overcoming the limitations of traditional methods. Our work underscores the potential of neural networks for future studies of strongly correlated systems and opens new avenues for exploring the rich physics of the fractional quantum Hall effect.
The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.
Fractional quantum Hall systems host strongly correlated topological phases with potential applications in topological quantum computation, yet identifying their microscopic ground states remains challenging because competing phases are often nearly degenerate and Landau-level mixing is difficult to treat. Here, we introduce a symmetry-adapted neural-network wavefunction that describes Abelian and non-Abelian states while incorporating all Landau levels. The method accurately captures composite-fermion liquids and the emergence of Wigner crystallization at low filling. When the first Landau level is partially filled, finite-size overlaps and entanglement spectra provide evidence consistent with 2ˉk1k+1\bar{2}^{k}1^{k+1}2ˉk1k+1-type parton phases at ν=4/3ν=4/3ν=4/3 and 7/57/57/5, while the ν=3/2ν=3/2ν=3/2 state exhibits anti-Pfaffian-like entanglement structure. These results demonstrate that neural-network wavefunctions offer a flexible framework for studying competing topological orders in fractional quantum Hall systems.