Solving and visualizing fractional quantum Hall wavefunctions with neural network

TL;DR

This paper presents an attention-based neural network approach to accurately solve and visualize fractional quantum Hall wavefunctions, outperforming traditional methods and revealing new microscopic features.

Contribution

Introduces a novel fermionic neural network that effectively solves complex quantum many-body problems in FQH systems, surpassing existing techniques in accuracy and insight.

Findings

FNN achieves lower energies than exact diagonalization.

Reveals microscopic features beyond Laughlin wavefunction.

Identifies phase transition from FQH liquid to crystal at strong LL mixing.

Abstract

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 $N$ electrons confined to a disk, our FNN consistently attains energies lower than LL-projected exact diagonalization (ED) and learns the ground state wavefunction to high accuracy. In low LL mixing regime, our FNN reveals microscopic features in the short-distance behavior of FQH wavefunction 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.