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arXiv:2610.28095

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

Yubing Qian, Weiluo Ren, Ji Chen

Published Oct 20, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2610.28095

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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.
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