Synergy Between Deep Neural Networks And The Variational Monte Carlo Method For Small $^4He

Republished By Plato

Followers: 0

Instituto de Física Gleb Wataghin, University of Campinas – UNICAMP 13083-859 Campinas – SP, Brazil

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We introduce a neural network-based approach for modeling wave functions that satisfy Bose-Einstein statistics. Applying this model to small $^4He_N$ clusters (with N ranging from 2 to 14 atoms), we accurately predict ground state energies, pair density functions, and two-body contact parameters $C^{(N)}_2$ related to weak unitarity. The results obtained via the variational Monte Carlo method exhibit remarkable agreement with previous studies using the diffusion Monte Carlo method, which is considered exact within its statistical uncertainties. This indicates the effectiveness of our neural network approach for investigating many-body systems governed by Bose-Einstein statistics.

Synergy between deep neural networks and the variational Monte Carlo method for small $^4He_N$ clusters PlatoBlockchain Data Intelligence. Vertical Search. Ai.

Featured image: A cartoon illustrating the interatomic potential among atoms, the artificial neural network representing the quantum states of helium clusters and the resultant dimer wave function.

Popular summary

Artificial neural networks, inspired by the structure of the brain, are intricate systems of interconnected artificial neurons. These computational models store information through learning algorithms. Our research delves into the application of artificial neural networks for modeling quantum systems governed by Bose-Einstein statistics. Specifically, we focus on small clusters composed of up to 14 helium atoms. The learning process, akin to how our proposed neural network adapts itself to achieve the lowest variational energy, falls under the domain of machine learning.

Remarkably, our results in obtaining a variational wave function align with previous studies that utilized established methods yielding exact results within statistical uncertainties. Once this stage is achieved, the model can comprehensively explore various quantum phenomena and properties. This capability, for instance, facilitates the investigation of quantum correlations among atoms within the cluster, providing insights into how these correlations evolve with cluster size and their implications for the quantum nature and size-dependent stability of the system. The success in describing these systems through neural networks underscores the effectiveness of this approach in exploring bosonic systems, an area that has been less explored by these networks until now.

► BibTeX data

@article{Freitas2023synergybetweendeep, doi = {10.22331/q-2023-12-18-1209}, url = {https://doi.org/10.22331/q-2023-12-18-1209}, title = {Synergy between deep neural networks and the variational {M}onte {C}arlo method for small {$^4He_N$} clusters}, author = {Freitas, William and Vitiello, S. A.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1209}, month = dec, year = {2023}<br /> }

► References

[1] Li Yang, Zhaoqi Leng, Guangyuan Yu, Ankit Patel, Wen-Jun Hu, and Han Pu. Deep learning-enhanced variational Monte Carlo method for quantum many-body physics. Physical Review Research, 2 (1): 012039, 2020-02. 10.1103/physrevresearch.2.012039.
https://doi.org/10.1103/physrevresearch.2.012039

[2] David Pfau, James S. Spencer, Alexander G. D. G. Matthews, and W. M. C. Foulkes. Ab initio solution of the many-electron schrödinger equation with deep neural networks. Physical Review Research, 2 (3): 033429, 2020-09. 10.1103/physrevresearch.2.033429.
https://doi.org/10.1103/physrevresearch.2.033429

[3] Jan Hermann, Zeno Schätzle, and Frank Noé. Deep-neural-network solution of the electronic schrödinger equation. Nature Chemistry, 12 (10): 891–897, 2020-09. 10.1038/s41557-020-0544-y.
https://doi.org/10.1038/s41557-020-0544-y

[4] Jan Kessler, Francesco Calcavecchia, and Thomas D. Kühne. Artificial neural networks as trial wave functions for quantum Monte Carlo. Advanced Theory and Simulations, 4 (4): 2000269, 2021-01. 10.1002/adts.202000269.
https://doi.org/10.1002/adts.202000269

[5] Gabriel Pescia, Jiequn Han, Alessandro Lovato, Jianfeng Lu, and Giuseppe Carleo. Neural-network quantum states for periodic systems in continuous space. Physical Review Research, 4 (2): 023138, 2022-05. 10.1103/physrevresearch.4.023138.
https://doi.org/10.1103/physrevresearch.4.023138

[6] Mario Krenn, Robert Pollice, Si Yue Guo, Matteo Aldeghi, Alba Cervera-Lierta, Pascal Friederich, Gabriel dos Passos Gomes, Florian Häse, Adrian Jinich, AkshatKumar Nigam, Zhenpeng Yao, and Alán Aspuru-Guzik. On scientific understanding with artificial intelligence. Nature Reviews Physics, 4 (12): 761–769, 2022-10. 10.1038/s42254-022-00518-3.
https://doi.org/10.1038/s42254-022-00518-3

[7] Giuseppe Carleo and Matthias Troyer. Solving the quantum many-body problem with artificial neural networks. Science, 355 (6325): 602–606, feb 2017. 10.1126/science.aag2302.
https://doi.org/10.1126/science.aag2302

[8] Michele Ruggeri, Saverio Moroni, and Markus Holzmann. Nonlinear network description for many-body quantum systems in continuous space. Physical Review Letters, 120 (120): 205302, May 2018. 10.1103/physrevlett.120.205302.
https://doi.org/10.1103/physrevlett.120.205302

[9] Hiroki Saito and Masaya Kato. Machine learning technique to find quantum many-body ground states of bosons on a lattice. Journal of the Physical Society of Japan, 87 (1): 014001, 2018-01. 10.7566/jpsj.87.014001.
https://doi.org/10.7566/jpsj.87.014001

[10] A. J. Yates and D. Blume. Structural properties of $^4$He$_{N}$ (${N}$=2-10) clusters for different potential models at the physical point and at unitarity. Physical Review A, 105 (2): 022824, 2022-02. 10.1103/physreva.105.022824.
https://doi.org/10.1103/physreva.105.022824

[11] J. Peter Toennies. Helium nanodroplets: Formation, physical properties and superfluidity. In Topics in Applied Physics, pages 1–40. Springer International Publishing, 2022. 10.1007/978-3-030-94896-2_1.
https://doi.org/10.1007/978-3-030-94896-2_1

[12] P. Recchia, A. Kievsky, L. Girlanda, and M. Gattobigio. Subleading contributions to $n$-boson systems inside the universal window. Physical Review A, 106 (2): 022812, 2022-08. 10.1103/physreva.106.022812.
https://doi.org/10.1103/physreva.106.022812

[13] Elena Spreafico, Giorgio Benedek, Oleg Kornilov, and Jan Peter Toennies. Magic numbers in boson $^4$He clusters: The auger evaporation mechanism. Molecules, 26 (20): 6244, 2021-10. 10.3390/molecules26206244.
https://doi.org/10.3390/molecules26206244

[14] Daniel Odell, Arnoldas Deltuva, and Lucas Platter. van der waals interaction as the starting point for an effective field theory. Physical Review A, 104 (2): 023306, 2021-08. 10.1103/physreva.104.023306.
https://doi.org/10.1103/physreva.104.023306

[15] B. Bazak, M. Valiente, and N. Barnea. Universal short-range correlations in bosonic helium clusters. Physical Review A, 101 (1): 010501, 2020-01. 10.1103/physreva.101.010501.
https://doi.org/10.1103/physreva.101.010501

[16] A. Kievsky, A. Polls, B. Juliá-Díaz, N. K. Timofeyuk, and M. Gattobigio. Few bosons to many bosons inside the unitary window: A transition between universal and nonuniversal behavior. Physical Review A, 102 (6): 063320, 2020-12. 10.1103/physreva.102.063320.
https://doi.org/10.1103/physreva.102.063320

[17] B. Bazak, J. Kirscher, S. König, M. Pavón Valderrama, N. Barnea, and U. van Kolck. Four-body scale in universal few-boson systems. Physical Review Letters, 122 (14), apr 2019. 10.1103/physrevlett.122.143001.
https://doi.org/10.1103/physrevlett.122.143001

[18] A. Kievsky, M. Viviani, R. Álvarez-Rodríguez, M. Gattobigio, and A. Deltuva. Universal behavior of few-boson systems using potential models. Few-Body Systems, 58 (2), 2017-01. 10.1007/s00601-017-1228-z.
https://doi.org/10.1007/s00601-017-1228-z

[19] J. Carlson, S. Gandolfi, U. van Kolck, and S. A. Vitiello. Ground-state properties of unitary Bosons: From clusters to matter. Phys. Rev. Lett., 119: 223002, Nov 2017. 10.1103/PhysRevLett.119.223002. URL https://link.aps.org/doi/10.1103/PhysRevLett.119.223002.
https://doi.org/10.1103/PhysRevLett.119.223002

[20] Ronald A. Aziz, Frederick R.W. McCourt, and Clement C.K. Wong. A new determination of the ground state interatomic potential for He$_2$. Molecular Physics, 61 (6): 1487–1511, 1987-08. 10.1080/00268978700101941.
https://doi.org/10.1080/00268978700101941

[21] Rafael Guardiola, Oleg Kornilov, Jesús Navarro, and J. Peter Toennies. Magic numbers, excitation levels, and other properties of small neutral he4 clusters (n$leqslant$50). The Journal of Chemical Physics, 124 (8): 084307, 2006-02. 10.1063/1.2140723.
https://doi.org/10.1063/1.2140723

[22] W. L. McMillan. Ground state of liquid $^4$He. Phys. Rev., 138 (2A): A442–A451, Apr 1965. 10.1103/PhysRev.138.A442.
https://doi.org/10.1103/PhysRev.138.A442

[23] R. P. Feynman and Michael Cohen. Energy spectrum of the excitations in liquid helium. Phys. Rev., 102: 1189–1204, Jun 1956. 10.1103/PhysRev.102.1189. URL http://link.aps.org/doi/10.1103/PhysRev.102.1189.
https://doi.org/10.1103/PhysRev.102.1189

[24] K. E. Schmidt, Michael A. Lee, M. H. Kalos, and G. V. Chester. Structure of the ground state of a fermion fluid. Phys. Rev. Lett., 47: 807–810, Sep 1981. 10.1103/PhysRevLett.47.807. URL http://link.aps.org/doi/10.1103/PhysRevLett.47.807.
https://doi.org/10.1103/PhysRevLett.47.807

[25] David Pfau James S. Spencer and FermiNet Contributors. FermiNet, 2020. URL http://github.com/deepmind/ferminet.
http://github.com/deepmind/ferminet

[26] Max Wilson, Saverio Moroni, Markus Holzmann, Nicholas Gao, Filip Wudarski, Tejs Vegge, and Arghya Bhowmik. Neural network ansatz for periodic wave functions and the homogeneous electron gas. Phys. Rev. B, 107: 235139, Jun 2023. 10.1103/PhysRevB.107.235139. URL https://link.aps.org/doi/10.1103/PhysRevB.107.235139.
https://doi.org/10.1103/PhysRevB.107.235139

[27] D. M. Ceperley and M. H. Kalos. Quantum many-body problems. In K. Binder, editor, Monte Carlo Methods in Statistics Physics, volume 7 of Topics in Current Physics, chapter Quantum Many-Body Problems, pages 145–194. Springer-Verlag, Berlin, second edition, 1986. 10.1007/978-3-642-82803-4_4.
https://doi.org/10.1007/978-3-642-82803-4_4

[28] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderón, Nikita Astrakhantsev, and Giuseppe Carleo. NetKet 3: Machine learning toolbox for many-body quantum systems. SciPost Physics Codebases, 2022-08. 10.21468/scipostphyscodeb.7.
https://doi.org/10.21468/scipostphyscodeb.7

[29] James Martens and Roger B. Grosse. Optimizing neural networks with kronecker-factored approximate curvature. In ICML’15: Proceedings of the 32nd International Conference on International Conference on Machine Learning – Volume 37, 2015. 10.48550/arXiv.1503.05671. URL https://dl.acm.org/doi/10.5555/3045118.3045374.
https://doi.org/10.48550/arXiv.1503.05671
https://dl.acm.org/doi/10.5555/3045118.3045374

[30] William Freitas. BoseNet Helium Clusters, 2023. URL https://github.com/freitas-esw/bosenet-helium-clusters.
https://github.com/freitas-esw/bosenet-helium-clusters

[31] Nicholas Gao and Stephan Günnemann. Sampling-free inference for ab-initio potential energy surface networks. arXiv:2205.14962, 2022. 10.48550/arXiv.2205.14962.
https://doi.org/10.48550/arXiv.2205.14962
arXiv:2205.14962

[32] Ingrid von Glehn, James S. Spencer, and David Pfau. A self-attention ansatz for ab-initio quantum chemistry. axXiv:2211.13672, 2023. 10.48550/arXiv.2211.13672.
https://doi.org/10.48550/arXiv.2211.13672

[33] M. Przybytek, W. Cencek, J. Komasa, G. Łach, B. Jeziorski, and K. Szalewicz. Relativistic and quantum electrodynamics effects in the helium pair potential. Physical Review Letters, 104 (18): 183003, 2010-05. 10.1103/physrevlett.104.183003.
https://doi.org/10.1103/physrevlett.104.183003

[34] Stefan Zeller and et al. Imaging the He$_2$ quantum halo state using a free electron laser. Proceedings of the National Academy of Sciences, 113 (51): 14651–14655, 2016-12. 10.1073/pnas.1610688113.
https://doi.org/10.1073/pnas.1610688113

[35] Shina Tan. Energetics of a strongly correlated Fermi gas. Ann. Phys., 323 (12): 2952 – 2970, 2008a. ISSN 0003-4916. http://dx.doi.org/10.1016/j.aop.2008.03.004. URL http://www.sciencedirect.com/science/article/pii/S0003491608000456.
https://doi.org/10.1016/j.aop.2008.03.004
http://www.sciencedirect.com/science/article/pii/S0003491608000456

[36] Shina Tan. Large momentum part of a strongly correlated Fermi gas. Ann. Phys., 323 (12): 2971 – 2986, 2008b. ISSN 0003-4916. http://dx.doi.org/10.1016/j.aop.2008.03.005. URL http://www.sciencedirect.com/science/article/pii/S0003491608000432.
https://doi.org/10.1016/j.aop.2008.03.005
http://www.sciencedirect.com/science/article/pii/S0003491608000432

[37] Shina Tan. Generalized virial theorem and pressure relation for a strongly correlated Fermi gas. Ann. Phys., 323 (12): 2987 – 2990, 2008c. ISSN 0003-4916. http://dx.doi.org/10.1016/j.aop.2008.03.003. URL http://www.sciencedirect.com/science/article/pii/S0003491608000420.
https://doi.org/10.1016/j.aop.2008.03.003
http://www.sciencedirect.com/science/article/pii/S0003491608000420

[38] Gerald A. Miller. Non-universal and universal aspects of the large scattering length limit. Physics Letters B, 777: 442–446, 2018-02. 10.1016/j.physletb.2017.12.063.
https://doi.org/10.1016/j.physletb.2017.12.063

[39] Félix Werner and Yvan Castin. General relations for quantum gases in two and three dimensions. II. bosons and mixtures. Physical Review A, 86 (5): 053633, 2012-11. 10.1103/physreva.86.053633.
https://doi.org/10.1103/physreva.86.053633

[40] Félix Werner and Yvan Castin. General relations for quantum gases in two and three dimensions: Two-component fermions. Physical Review A, 86 (1): 013626, 2012-07. 10.1103/physreva.86.013626.
https://doi.org/10.1103/physreva.86.013626

[41] Yaroslav Lutsyshyn. Weakly parametrized jastrow ansatz for a strongly correlated bose system. J. Chem. Phys., 146 (12): 124102, Mar 2017. 10.1063/1.4978707.
https://doi.org/10.1063/1.4978707

[42] S. A. Vitiello and K. E. Schmidt. Optimization of $^4$He wave functions for the liquid and solid phases. Phys. Rev. B, 46: 5442–5447, Sep 1992. 10.1103/PhysRevB.46.5442. URL http://link.aps.org/doi/10.1103/PhysRevB.46.5442.
https://doi.org/10.1103/PhysRevB.46.5442

Cited by

Could not fetch Crossref cited-by data during last attempt 2023-12-19 03:48:44: Could not fetch cited-by data for 10.22331/q-2023-12-18-1209 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2023-12-19 03:48:44).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.