Abstract We explore the suitability of a feed-forward neural network (FNN) to represent the ground state of the fermionic Hubbard model in one dimension (1D). We consider the model at half-filling, represent the ground state wave function in terms of an FNN and optimize it using the variational Monte Carlo (VMC) method. The results are compared with the exact Bethe Ansatz solution. We find that for lattice sizes which give a ‘filled-shell’ condition for the non-interacting Fermi sea wave function, a simple FNN performs very well at all values of Hubbard interaction U. For lattice sizes where this condition is not obtained, the simple FNN fails and we find a modified network with a ‘sign’ component (sFNN) to work in such cases. On the flip side, though we find the FNN to be successful in providing an unbiased variational wave function for the fermionic many-body system in 1D, the computational cost for the wave function scales up rapidly with lattice size which limits its applicability.
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(N3) 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(N2) 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.
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(N3) 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 from
O(N3) to
O(N2) or less, while the overall cost depends on the number of Monte Carlo samples and the number of parameters in neural networks. 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, data preprocessing, 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.
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 use a convolutional restricted Boltzmann machine (CRBM) neural network to construct a variational wave function (WF) for the Hubbard model on a square lattice and study it using the variational Monte Carlo (VMC) method. In the wave function, the CRBM acts as a correlation factor to a mean-field BCS state. The number of variational parameters in the WF does not grow automatically with the lattice size and it is computationally much more efficient compared with other neural-network-based WFs. We find that, in the intermediate- to strong-coupling regime of the model at half filling, the wave function outperforms even the highly accurate long-range backflow-Jastrow correlated wave function. Using the WF, we study the ground state of the half filled model as a function of onsite Coulomb repulsion
U. We consider two cases for the next-nearest-neighbor hopping parameter, e.g., ${t}^{\ensuremath{'}}=0$ as well as a frustrated model case with ${t}^{\ensuremath{'}}\ensuremath{\ne}0$. By examining several quantities, e.g., double occupancy, charge gap, momentum distribution, and spin-spin correlations, we find that the weekly correlated phase in both cases is paramagnetic metallic (PM). As
U is increased, the system undergoes a first-order Mott transition to an insulating state at a critical
Uc, the value of which depends on ${t}^{\ensuremath{'}}$. The Mott state in both cases is spin gapped with long-range antiferromagnetic (AFM) order. Remarkably, the AFM order emerges spontaneously from the wave function which does not have any explicitly broken symmetry in it. Apart from some quantitative differences in the results for the two values of ${t}^{\ensuremath{'}}$, we find some interesting qualitative differences in the way the Mott transition takes place in the two cases.
We use a convolutional restricted Boltzmann machine (CRBM) neural network to construct a variational wave function (WF) for the Hubbard model on a square lattice and study it using the variational Monte Carlo (VMC) method. In the wave function, the CRBM acts as a correlation factor to a mean-field BCS state. The number of variational parameters in the WF does not grow automatically with the lattice size and it is computationally much more efficient compared to other neural network based WFs. We find that in the intermediate to strong coupling regime of the model at half-filling, the wave function outperforms even the highly accurate long range backflow-Jastrow correlated wave function. Using the WF, we study the ground state of the half-filled model as a function of onsite Coulomb repulsion
U. We consider two cases for the next-nearest-neighbor hopping parameter, e.g.,
t′=0 as well as a frustrated model case with
t′=0. By examining several quantities, e.g., double occupancy, charge gap, momentum distribution, and spin-spin correlations, we find that the weekly correlated phase in both cases is paramagnetic metallic (PM). As
U is increased, the system undergoes a first-order Mott transition to an insulating state at a critical
Uc, the value of which depends upon
t′. The Mott state in both cases is spin gapped with long range antiferromagnetic (AF) order. Remarkably, the AF order emerges spontaneously from the wave function which does not have any explicitly broken symmetry in it. Apart from some quantitative differences in the results for the two values of
t′, we find some interesting qualitative differences in the way the Mott transition takes place in the two cases.
Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this Letter, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman and Cohen [Phys. Rev. 102, 1189 (1956)10.1103/PhysRev.102.1189], adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to learn the optimal transformation via variational Monte Carlo calculations. NNB directly dresses a mean-field state, can be systematically improved, and directly alters the sign structure of the wave function. It generalizes the standard backflow [L. F. Tocchio et al., Phys. Rev. B 78, 041101(R) (2008)10.1103/PhysRevB.78.041101], which we show how to explicitly represent as a NNB. We benchmark the NNB on Hubbard models at intermediate doping, finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.