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

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.

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

Received: 17 February 2022
Revised: 23 March 2022
Accepted manuscript online: 28 March 2022

PACS:	02.70.Ss	(Quantum Monte Carlo methods)
	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)

Fund: 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.

Corresponding Authors: Rong-Qiang He, Zhong-Yi Lu
E-mail: rqhe@ruc.edu.cn;zlu@ruc.edu.cn

Cite this article:

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 2022 Chin. Phys. B 31 080203

[1] Carrasquilla J and Melko R G 2017 Nat. Phys. 13 431
[2] Nieuwenburg E, Liu Y H and Huber S D 2017 Nat. Phys. 13 435
[3] Schindler F, Regnault N and Neupert T 2017 Phys. Rev. B 95 245134
[4] Venderley J, Khemani V and Kim E A 2018 Phys. Rev. Lett. 120 257204
[5] Carleo G and Troyer M 2017 Science 355 602
[6] Hinton G E 2002 Neural Computation 14 1771
[7] Deng D L, Li X and Sarma S D 2017 Phys. Rev. X 7 021021
[8] Nomura Y, Darmawan A S, Yamaji Y and Imada M 2017 Phys. Rev. B 96 205152
[9] Chen J, Cheng S, Xie H, Wang L and Xiang T 2018 Phys. Rev. B 97 085104
[10] Glasser I, Pancotti N, August M, Rodriguez I D and Cirac J I 2018 Phys. Rev. X 8 011006
[11] Gao X and Duan L M 2017 Nat. Commun. 8 662
[12] Anderson J B 1975 J. Chem. Phys. 63 1499
[13] Becca F and Sorella S 2017 Quantum Monte Carlo Approaches for Correlated Systems (London:Cambridge University Press) p. 167
[14] Calandra Buonaura M and Sorella S 1998 Phys. Rev. B 57 11446
[15] Sandvik A W 1997 Phys. Rev. B 56 11678
[16] Neel L 1936 Annales de Physique 11 232
[17] Neel L 1932 J. Phys. Radium 3 160
[18] Wang L, Poilblanc D, Gu Z C, Wen X G and Verstraete F 2013 Phys. Rev. Lett. 111 037202
[19] Liu W Y, Dong S, Wang C, Han Y, An H, Guo G C and He L 2018 Phys. Rev. B 98 241109
[20] Hu W J, Becca F, Parola A and Sorella S 2013 Phys. Rev. B 88 060402
[21] Gong S S, Zhu W, Sheng D N, Motrunich O I and Fisher M P A 2014 Phys. Rev. Lett. 113 027201
[22] Hornik K 1991 Neural Network 4 251
[23] Roux N L and Bengio Y 2008 Neural Computation 20 1631
[24] Sorella S 1998 Phys. Rev. Lett. 80 4558
[25] Casula M and Sorella S 2003 J. Chem. Phys. 119 6500
[26] Sorella S and Capriotti L 2000 Phys. Rev. B 61 2599
[27] Reynolds P J, Ceperley D M, Alder B J and Lester W A 1982 J. Chem. Phys. 77 5593
[28] Haaf D T, Bemmel H V, Leeuwen J V, Van S and Ceperley D 1995 Phys. Rev. B 51 13039
[29] Wold S, Esbensen K and Geladi P 1987 Chemometrics and Intelligent Laboratory Systems 2 37
[30] Abdi H and Williams L J 2010 Wiley Interdisciplinary Reviews:Computational Statistics 2 433
[31] Schulz H, Ziman T and Poilblanc D 1996 Journal de Physique I 6 675

[1]	Restricted Boltzmann machine: Recent advances and mean-field theory Aurélien Decelle, Cyril Furtlehner. Chin. Phys. B, 2021, 30(4): 040202.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

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

Cite this article:

Online attention

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