Green's function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J1-J2 Heisenberg model

doi:10.1088/1674-1056/ac615f

中国物理B ›› 2022, Vol. 31 ›› Issue (8): 80203-080203.doi: 10.1088/1674-1056/ac615f

Green's function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J₁-J₂ Heisenberg model

He-Yu Lin(林赫羽), Rong-Qiang He(贺荣强)^†, and Zhong-Yi Lu(卢仲毅)^‡

Department of Physics, Renmin University of China, Beijing 100872, China

收稿日期:2022-02-17 修回日期:2022-03-23 接受日期:2022-03-28 出版日期:2022-07-18 发布日期:2022-07-29
通讯作者: Rong-Qiang He, Zhong-Yi Lu E-mail:rqhe@ruc.edu.cn;zlu@ruc.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11934020 and 11874421) and the Natural Science Foundation of Beijing (Grant No. Z180013). Computational resources were provided by National Supercomputer Center in Guangzhou with Tianhe-2 Supercomputer and Physical Laboratory of High Performance Computing in RUC.

Green's function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J₁-J₂ Heisenberg model

He-Yu Lin(林赫羽), Rong-Qiang He(贺荣强)^†, and Zhong-Yi Lu(卢仲毅)^‡

Department of Physics, Renmin University of China, Beijing 100872, China

Received:2022-02-17 Revised:2022-03-23 Accepted:2022-03-28 Online:2022-07-18 Published:2022-07-29
Contact: Rong-Qiang He, Zhong-Yi Lu E-mail:rqhe@ruc.edu.cn;zlu@ruc.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11934020 and 11874421) and the Natural Science Foundation of Beijing (Grant No. Z180013). Computational resources were provided by National Supercomputer Center in Guangzhou with Tianhe-2 Supercomputer and Physical Laboratory of High Performance Computing in RUC.

摘要/Abstract

摘要： Restricted Boltzmann machine (RBM) has been proposed as a powerful variational ansatz to represent the ground state of a given quantum many-body system. On the other hand, as a shallow neural network, it is found that the RBM is still hardly able to capture the characteristics of systems with large sizes or complicated interactions. In order to find a way out of the dilemma, here, we propose to adopt the Green's function Monte Carlo (GFMC) method for which the RBM is used as a guiding wave function. To demonstrate the implementation and effectiveness of the proposal, we have applied the proposal to study the frustrated J₁-J₂ Heisenberg model on a square lattice, which is considered as a typical model with sign problem for quantum Monte Carlo simulations. The calculation results demonstrate that the GFMC method can significantly further reduce the relative error of the ground-state energy on the basis of the RBM variational results. This encourages to combine the GFMC method with other neural networks like convolutional neural networks for dealing with more models with sign problem in the future.

关键词: restricted Boltzmann machine, Green's function Monte Carlo, frustrated J₁-J₂ Heisenberg model

Abstract: Restricted Boltzmann machine (RBM) has been proposed as a powerful variational ansatz to represent the ground state of a given quantum many-body system. On the other hand, as a shallow neural network, it is found that the RBM is still hardly able to capture the characteristics of systems with large sizes or complicated interactions. In order to find a way out of the dilemma, here, we propose to adopt the Green's function Monte Carlo (GFMC) method for which the RBM is used as a guiding wave function. To demonstrate the implementation and effectiveness of the proposal, we have applied the proposal to study the frustrated J₁-J₂ Heisenberg model on a square lattice, which is considered as a typical model with sign problem for quantum Monte Carlo simulations. The calculation results demonstrate that the GFMC method can significantly further reduce the relative error of the ground-state energy on the basis of the RBM variational results. This encourages to combine the GFMC method with other neural networks like convolutional neural networks for dealing with more models with sign problem in the future.

Key words: restricted Boltzmann machine, Green's function Monte Carlo, frustrated J₁-J₂ Heisenberg model

中图分类号: (Quantum Monte Carlo methods)

02.70.Ss

07.05.Mh (Neural networks, fuzzy logic, artificial intelligence) 75.10.Jm (Quantized spin models, including quantum spin frustration) 73.43.Nq (Quantum phase transitions)

He-Yu Lin(林赫羽), Rong-Qiang He(贺荣强), and Zhong-Yi Lu(卢仲毅). Green's function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J₁-J₂ Heisenberg model[J]. 中国物理B, 2022, 31(8): 80203-080203.

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Green's function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J₁-J₂ Heisenberg model

Green's function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J₁-J₂ Heisenberg model

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