Neural-network quantum states for periodic systems in continuous space

We introduce a family of neural quantum states for the simulation of strongly interacting systems in the presence of spatial periodicity. Our variational state is parametrized in terms of a permutationally invariant part described by the Deep Sets neural-network architecture. The input coordinates to the Deep Sets are periodically transformed such that they are suitable to directly describe periodic bosonic systems. We show example applications to both one- and two-dimensional interacting quantum gases with Gaussian interactions, as well as to ${}^4\mathrm{He}$ confined in a one-dimensional geometry. For the one-dimensional systems we find very precise estimations of the ground-state energies and the radial distribution functions of the particles. In two dimensions we obtain good estimations of the ground-state energies, comparable to results obtained from more conventional methods.

Figure 1

Illustration of the Deep Sets neural-network architecture used in this paper. Starting from the left, we have a simulation box in $d$ spatial dimensions of extent $L$. The single-particle coordinates ${\mathbf{x}}_i\in {\mathbb{R}}^d$ are transformed to the truncated Fourier basis in Eq. (2), i.e., each single-particle coordinate is mapped to a ($2K·d$)-dimensional vector. These periodic encodings are then fed to the DSs architecture, defined by the neural networks $\varphi$ and $\rho$ and a $\mathrm{pool}$ function. The output of the DSs architecture is the logarithm of the variational wave function. The top row depicts the number of feature vectors and their dimensionality at each stage of the ansatz.

Figure 2

DMC (solid lines) and DSs (solid circles) two-body density distributions of $N=40$ particles in one dimension (1D) interacting through the Aziz potential.

Figure 3

Convergence pattern of the DSs architecture for $N=40$ helium particles in 1D. Blue shows the energy for a variational ansatz initialized with the optimized weights for the same system at $N=10$ particles. Orange shows the energy for randomly initialized weights.

Figure 4

Left: DMC (solid lines) and DSs (solid circles) two-body density distributions of $N=50$ particles in 1D at different densities interacting through the Gaussian potential. Right: the two-body density distribution of $N=64$ particles in 2D at different densities interacting through the Gaussian potential.

Figure 5

Final energy of training the DSs with single-particle input for the two-dimensional Gaussian core model with $N=32$ particles at a density $D=1/9$ as a function of the width of the neural networks used. The horizontal lines correspond to the Jastrow and DMC results, respectively.