Learning spin liquids on a honeycomb lattice with artificial neural networks

Chang-Xiao Li,Jing-Bo Xu,Sheng Yang

doi:10.1038/s41598-021-95523-4

Chang-Xiao Li, Jing-Bo Xu + Show 1 more

Open Access

https://doi.org/10.1038/s41598-021-95523-4

Copy DOI

Abstract

Machine learning methods provide a new perspective on the study of many-body system in condensed matter physics and there is only limited understanding of their representational properties and limitations in quantum spin liquid systems. In this work, we investigate the ability of the machine learning method based on the restricted Boltzmann machine in capturing physical quantities including the ground-state energy, spin-structure factor, magnetization, quantum coherence, and multipartite entanglement in the two-dimensional ferromagnetic spin liquids on a honeycomb lattice. It is found that the restricted Boltzmann machine can encode the many-body wavefunction quite well by reproducing accurate ground-state energy and structure factor. Further investigation on the behavior of multipartite entanglement indicates that the residual entanglement is richer in the gapless phase than the gapped spin-liquid phase, which suggests that the residual entanglement can characterize the spin-liquid phases. Additionally, we confirm the existence of a gapped non-Abelian topological phase in the spin liquids on a honeycomb lattice with a small magnetic field and determine the corresponding phase boundary by recognizing the rapid change of the local magnetization and residual entanglement.

Highlights

The representative features and limitations of neural networks in characterizing the quantum spin liquids (QSLs) phases remain to be elucidated
We employ the machine learning approach based on RBMs to investigate the topological phase for the FM QSL honeycomb lattice, and study various physical quantities to provide a better understanding of the application of machine learning in QSLs
By investigating the accuracy of the learned energy and structure factor, we first confirmed the validity of the machine learning method in solving the QSL honeycomb lattice

Summary

Introduction

The representative features and limitations of neural networks in characterizing the QSL phases remain to be elucidated. We attempt to show the abilities and limitations of the neural network-based machine learning method to capture QSL states of the spin liquids on a honeycomb lattice with a magnetic field. We apply RBMs to learn the ground-state energy and spin-structure factor in the QSL honeycomb lattice, and compare these results with those obtained by exact diagonalization to verify the effectiveness of the neural network-based machine learning method. We investigate the performances of the quantum coherence and residual entanglement via the RBM-based machine learning method, and observe that both of these quantities can be used to determine the boundary between the gapped and gapless QSL phases. RBMs are applied to learn physical quantities including the energy, spin-structure factor, quantum coherence and multipartite entanglement for the pure QSL honeycomb lattice and FM QSL honeycomb lattice in “Results”.

Methods

Results

Conclusion