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Extracting Anyon Statistics from Neural Network Fractional Quantum Hall States

Andres Perez Fadon1,*, David Pfau2, James S. Spencer2, Wan Tong Lou1, Titus Neupert3, and W. M. C. Foulkes1

  • *Contact author: andres.perez-fadon19@imperial.ac.uk
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PRX Intelligence 1, 013018 – Published 8 September, 2026

DOI: https://doi.org/10.1103/sb8g-qvn6

Abstract

Fractional quantum Hall states host emergent anyons with exotic exchange statistics, but obtaining direct access to their topological properties in real systems remains a challenge. Neural network wave functions provide a flexible computational approach, as they can represent highly correlated states without requiring a tailored basis. Here, we use the neural network variational Monte Carlo method to study the fractional quantum Hall effect on the torus and find the three degenerate ground states at filling factor ν=1/3. From these, we extract the modular S-matrix via entanglement interferometry, a technique previously only applied to lattice models. The resulting S-matrix encodes the quantum dimensions, fusion rules, and exchange statistics of the emergent anyons, providing a direct numerical demonstration of the topological order. The calculated anyon properties match the well-known theoretical and experimental results. Our work establishes neural network wave functions as a powerful tool for investigating anyonic properties.

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PRX Intelligence 1, 013018– Published 8 September, 2026
  • Received 23 December 2025
  • Accepted 28 July 2026
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