Enhancing Neural Network Backflow

TL;DR

This paper improves neural network-based variational wave functions for strongly correlated systems by combining multi-determinant expansions with techniques like Lanczos steps and symmetry projection, achieving state-of-the-art energies.

Contribution

It introduces efficient multi-determinant ansatz enhancements for Neural Network Backflow, surpassing network size limitations and achieving lower energies in Hubbard model simulations.

Findings

Multi-determinant expansions improve variational energies.

Lanczos and symmetry projection techniques yield similar performance.

Achieved lowest energies for 4x16 and 4x8 Hubbard lattices.

Abstract

Accurately describing the ground state of strongly correlated systems is essential for understanding their emergent properties. Neural Network Backflow (NNBF) is a powerful variational ansatz that enhances mean-field wave functions by introducing configuration-dependent modifications to single-particle orbitals. Although NNBF is theoretically universal in the limit of large networks, we find that practical gains saturate with increasing network size. Instead, significant improvements can be achieved by using a multi-determinant ansatz. We explore efficient ways to generate these multi-determinant expansions without increasing the number of variational parameters. In particular, we study single-step Lanczos and symmetry projection techniques, benchmarking their performance against diffusion Monte Carlo and NNBF applied to alternative mean fields. Benchmarking on a doped periodic square Hubbard model near optimal doping, we find that a Lanczos step, diffusion Monte Carlo, and projection onto a symmetry sector all give similar improvements achieving state-of-the-art energies at minimal cost. By further optimizing the projected symmetrized states directly, we gain significantly in energy. Using this technique we report the lowest variational energies for this Hamiltonian on $4\times 16$ and $4 \times 8$ lattices as well as accurate variance extrapolated energies. We also show the evolution of spin, charge, and pair correlation functions as the quality of the variational ansatz improves.