Extracting Anyon Statistics from Neural Network Fractional Quantum Hall States

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.

Figures (3)

• Figure 1: Psiformer architecture with a custom complex-valued multi-layer-perceptron-based Jastrow factor. The one-electron features depend on periodic functions of the position of each electron cassella2023discovering. The Jastrow factor is evaluated independently for each pair of electrons. The values from each pair are then multiplied together. The envelope is a sum of quasiperiodic functions with learnable coefficients.
• Figure 2: Partitions across which the second Renyi entropy is calculated. Subplots $a$, $b$, and $c$ correspond to the partitions used to obtain $S_{\leftrightarrow}$, $S_{\updownarrow}$, and $S_{\mathrel{\hbox{o}rigin=c]{45}{$↔$}}}$, respectively.
• Figure 3: Learning curve for the loss function against the number of iterations of a 6-electron calculation. The dashed line represents the energy of the Laughlin wavefunction in the torus.