Synergy between deep neural networks and the variational Monte Carlo method for small $^4He_N$ clusters

William Freitas and S. A. Vitiello

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

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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.

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.

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