Extracting Anyon Statistics from Neural Network Fractional Quantum Hall States

TL;DR

This paper demonstrates that neural-network wavefunctions can effectively extract and verify the topological properties and anyon statistics of fractional quantum Hall states, providing a new computational approach for studying topological order.

Contribution

The study introduces a neural-network variational Monte Carlo method to compute the modular S matrix and anyon properties in fractional quantum Hall states, a novel application of neural networks in this context.

Findings

Successfully extracted the S matrix for fractional quantum Hall states

Matched theoretical and experimental anyon properties

Established neural networks as a tool for topological order investigation

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 wavefunctions 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 nu=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 wavefunctions as a powerful new tool for investigating anyonic properties.