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arXiv:2304.04725 · Physical Review A

Machine learning one-dimensional spinless trapped fermionic systems with neural-network quantum states

J. W. T. Keeble, M. Drissi, Abel Rojo-Francàs, Bruno Juliá-Díaz, Arnau Rios

Published Dec 29, 2023• 5 citations
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Abstract

We compute the ground-state properties of fully polarized, trapped, one-dimensional fermionic systems interacting through a Gaussian potential. We use an antisymmetric artificial neural network, or neural quantum state, as an Ansatz for the wave function and use machine learning techniques to variationally minimize the energy of systems from two to six particles. We provide extensive benchmarks for this toy model with other many-body methods, including exact diagonalization and the Hartree-Fock approximation. The neural quantum state provides the best energies across a wide range of interaction strengths. We find very different ground states depending on the sign of the interaction. In the nonperturbative repulsive regime, the system asymptotically reaches crystalline order. In contrast, the strongly attractive regime shows signs of bosonization. The neural quantum state continuously learns these different phases with an almost constant number of parameters and a very modest increase in computational time with the number of particles.

DOI
10.1103/physreva.108.063320

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We compute the ground-state properties of fully polarized, trapped, one-dimensional fermionic systems interacting through a Gaussian potential. We use an antisymmetric artificial neural network, or neural quantum state, as an Ansatz for the wave function and use machine learning techniques to variationally minimize the energy of systems from two to six particles. We provide extensive benchmarks for this toy model with other many-body methods, including exact diagonalization and the Hartree-Fock approximation. The neural quantum state provides the best energies across a wide range of interaction strengths. We find very different ground states depending on the sign of the interaction. In the nonperturbative repulsive regime, the system asymptotically reaches crystalline order. In contrast, the strongly attractive regime shows signs of bosonization. The neural quantum state continuously learns these different phases with an almost constant number of parameters and a very modest increase in computational time with the number of particles.
The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.
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.
First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow-inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared with exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge-ordered phases as a function of the interaction strength and the particle density.
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To analyze quantum many-body Hamiltonians, recently, machine learning techniques have been shown to be quite useful and powerful. However, the applicability of such machine learning solvers is still limited. Here, we propose schemes that make it possible to apply machine learning techniques to analyze fermion-boson coupled Hamiltonians and to calculate excited states. As for the extension to fermion-boson coupled systems, we study the Holstein model as a representative of the fermion-boson coupled Hamiltonians. We show that the machine-learning solver achieves highly accurate ground-state energy, improving the accuracy substantially compared to that obtained by the variational Monte Carlo method. As for the calculations of excited states, we propose a different approach than that proposed in K. Choo et al., Phys. Rev. Lett. 121 (2018) 167204. We discuss the difference in detail and compare the accuracy of two methods using the one-dimensional S=1/2S=1/2S=1/2 Heisenberg chain. We also show the benchmark for the frustrated two-dimensional S=1/2S=1/2S=1/2 J1J_1J1​-J2J_2J2​ Heisenberg model and show an excellent agreement with the results obtained by the exact diagonalization. The extensions shown here open a way to analyze general quantum many-body problems using machine learning techniques.