Enhancing Neural Network Backflow

TL;DR

This work examines neural network backflow (NNBF) as a variational tool for strongly correlated fermions and shows that practical gains saturate with larger networks. It demonstrates that efficient multi-determinant expansions generated by a single Lanczos step or symmetry projection substantially improve energies with minimal extra parameters, outperforming prior approaches on the doped square-lattice Hubbard model. By directly optimizing symmetrized multi-determinant states, the authors achieve state-of-the-art variational energies on $4\times16$ and $4\times8$ lattices and reveal how spin, charge, and pairing correlations evolve with improved wavefunctions. The findings offer a scalable strategy for high-accuracy ground-state calculations in challenging correlated systems and underscore the value of symmetry and Krylov-projected refinements alongside NNBF expressivity.

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.

Figures (8)

• Figure 1: Energy vs. variance of different variational wave-functions with a linear extrapolation to zero variance taken through the base NNBF and all symmetrized versions (e.g. red, blue, orange and purple points) whose energy was less then -0.7615. Symbols denote $n_h$.
• Figure 2: Variational energy of NNBF approaches as a function of GPU training hours. Symbols denotes $n_h$. Comparison with other works shown as dotted lines. The lowest variational energy achieved in this work is shown as a solid line. The Lanczos step optimization and DMC for $n_h = 512$ failed and are excluded.
• Figure 3: Variational energy versus GPU training hours for different symmetric operations at $n_h=2048$. We consider both optimization with a fixed number of symmetries (orange), and projection from these points to the full symmetry sector (purple). The numbers next to each point correspond to the number of symmetries. The projection from the non-symmetrized (red point) to full symmetries (blue point) is also seen in Fig. \ref{['fig:NeuronEnhancements']}. Comparison with other works shown as dotted lines and a pink cross are all higher in energy then the projected optimized symmetries. The optimization curve is the optimization for the optimized fully symmetric state.
• Figure 4: Observables on the $4\times 16$ lattice. (a-c) The correlation functions $C_s$, $C_c$, and $C_p$ in real space for the best variational model, $n_h=2048$ with optimized symmetries. To show weaker correlation, the values are truncated such that $\lvert C_s\rvert \leq 0.1$ and $\lvert C_c\rvert, \lvert C_p\rvert \leq 0.01$. The central site is marked with a star. (d,e) The spin and charge structure factors of $n_h=2048$ models with no symmetries, projected symmetries, and optimized symmetries. Peaks appear at $(7\pi/8, \pi)$ and $(\pi/4, 0)$ respectively. (f) The pair-pair correlation $C_p(x,2)$ on the same models in $(d,e)$. Similar benchmarks for HFPS are shown in Ref. HFPS.
• Figure 5: Energies on the $4\times 8$ lattice. Symbols denote $n_h$. (a) Energy vs variance of the optimized symmetries at various $n_h$ with variance extrapolated best-fit line which is used as the bottom for the y-axis in (b) and (c). (b) Energy vs GPU time for different methods which optimize only the single mean-field as well as variational energies from other works (dotted lines) and our lowest variational energy (solid orange). (c) Energy vs GPU time for optimized symmetries (orange) and projected optimized symmetries (purple) at $n_h=1024$ with numbers indicating the number of symmetries. The orange optimization curve is for the projected optimized symmetry with 128 symmetries. Inset contains projected optimized symmetries with 128 symmetries at various $n_h$.