Abstract
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of $O(N^3)$ for $N$ particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to $O(N^2)$ or less. We show that the accuracy of the approximation can be improved by optimizing the "variance" of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
Highlights
Recent developments in machine learning have a profound impact on the field of physics [1]
Various methods have been proposed in solid state physics far, such as the restricted Boltzmann machine [4,5,6,7], the convolutional neural network (CNN) [6,8,9,10,11,12], the graph convolutional network (GCN) [8], the Gaussian process state [13], and the fully connected neural network (FCNN) [14]
While such studies using neural networks (NNs) are successful for bosonic systems [18,19], they have encountered some difficulties for fermionic systems and frustrated spin systems due to the complicated sign structures in the wave functions [20,21]
Summary
Recent developments in machine learning have a profound impact on the field of physics [1]. Similar methods have been applied to molecules in quantum chemistry calculations [15,16,17] While such studies using NNs are successful for bosonic systems [18,19], they have encountered some difficulties for fermionic systems and frustrated spin systems due to the complicated sign structures in the wave functions [20,21]. Numerical cost for the calculation of the Slater determinant is O(N3) for N-particle systems. We propose a general framework to approximate fermionic many-body wave functions by NNs without using the Slater determinant.
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