Extracting Anyon Statistics from Neural Network Fractional Quantum Hall States

The fractional quantum Hall effect hosts emergent anyons with exotic exchange statistics, but direct numerical access to their topological properties in the continuum has remained limited. Most computational approaches are restricted to a single Landau level, which precludes treating realistic regimes with strong Landau-level mixing. Neural-network wavefunctions provide a flexible alternative, as they can represent highly correlated states without requiring a tailored basis. In this study, we extend these methods to the fractional quantum Hall effect on the torus. 

Using neural-network variational Monte Carlo, we obtain the 3 degenerate ground states at filling factor $\nu=1/3$. From these states, we extract the modular $S$ matrix via entanglement interferometry, a technique previously applied only to lattice models, and now, for the first time, applied in the continuum. The resulting $S$ matrix encodes the quantum dimensions, fusion rules, and exchange statistics of the emergent anyons, providing a direct numerical demonstration of their topological order. The calculated anyon properties match the well-known theoretical and experimental results. Our work establishes neural-network wavefunctions as a powerful new tool for investigating anyonic properties.