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arXiv:1804.06521 · Journal of the Physical Society of Japan

Method to Solve Quantum Few-Body Problems with Artificial Neural Networks

Hiroki Saito

Published Jun 19, 2018• 43 citations
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Abstract

A machine learning technique to obtain the ground states of quantum few-body systems using artificial neural networks is developed. Bosons in continuous space are considered and a neural network is optimized in such a way that when particle positions are input into the network, the ground-state wave function is output from the network. The method is applied to the Calogero-Sutherland model in one-dimensional space and Efimov bound states in three-dimensional space.

DOI
10.7566/jpsj.87.074002

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Neural-network quantum states have been developed as an efficient method for solving quantum many-body problems, not only in lattice systems but also in systems of particles in continuous space. Here, we apply this approach to strongly interacting few-body problems in continuous space at unitarity: the Efimov states and associated few-body bound states. We extend the previous work [], in which the ground states of few-boson systems were obtained, to the first excited states with a projection method, and also to a mass-imbalanced fermionic system consisting of two identical fermions and a third particle. The obtained energies of the ground and first excited states of these systems agree well with previously reported results. Furthermore, the proposed approach also reproduces the discrete scale invariance between the ground and first excited states and the critical-mass behavior in mass-imbalanced fermionic systems.
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Carleo and Troyer [3] have recently pointed out the possibility of solving quantum many-body problems by using Artificial Neural Networks (ANN). Their work is based on minimizing a variational wave function to obtain the ground states for various spin-dependent systems. This work is primarily focused on developing efficient method using ANN to solve the ground state wave function for atomic systems. We have developed a theoretical groundwork to represent the wave function of a many-electron atom by using artificial neural network while still preserving its antisymmetric property. By using the Metropolis algorithm, Variational Monte Carlo (VMC), and Stochastic Reconfiguration (SR) methods for minimization, we were able to obtain a highly accurate ground state wave function for the He atom. To verify our optimization algorithm, we reproduced the results for the ground state of a three dimensional Simple Harmonic Oscillator (SHO) given by Teng [18].
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In this paper, we demonstrate the expressibility of artificial neural networks (ANNs) in quantum many-body physics by showing that a feed-forward neural network with a small number of hidden layers can be trained to approximate with high precision the ground states of some notable quantum many-body systems. We consider the one-dimensional free bosons and fermions, spinless fermions on a square lattice away from half-filling, as well as frustrated quantum magnetism with a rapidly oscillating ground-state characteristic function. In the latter case, an ANN with a standard architecture fails, while that with a slightly modified one successfully learns the frustration-induced complex sign rule in the ground state and approximates the ground states with high precisions. As an example of practical use of our method, we also perform the variational method to explore the ground state of an antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model.
We develop a variational method to obtain many-body ground states of the Bose-Hubbard model using feedforward artificial neural networks. A fully-connected network with a single hidden layer works better than a fully-connected network with multiple hidden layers, and a multi-layer convolutional network is more efficient than a fully-connected network. AdaGrad and Adam are optimization methods that work well. Moreover, we show that many-body ground states with different numbers of atoms can be generated by a single network.