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Communications in Mathematical Sciences

Geometry of backflow transformation ansatze for quantum many-body fermionic wavefunctions

Hang Huang, Joseph M. Landsberg, Jianfeng Lu

Published Jan 1, 2023• 1 citation
Publisher
DOI
10.4310/cms.2023.v21.n5.a12

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Wave function ansatz based on the backflow transformation are widely used to parametrize anti-symmetric multivariable functions for many-body quantum problems. We study the geometric aspects of such ansatz, in particular we show that in general totally antisymmetric polynomials cannot be efficiently represented by backflow transformation ansatz at least in the category of polynomials. In fact, one needs a linear combination of at least O(N3N−3)O(N^{3N-3})O(N3N−3) determinants to represent a generic totally antisymmetric polynomial. Our proof is based on bounding the dimension of the source of the ansatz from above and bounding the dimension of the target from below.
Neural networks have shown to be a powerful tool to represent the ground state of quantum many-body systems, including fermionic systems. However, efficiently integrating lattice symmetries into neural representations remains a significant challenge. In this work, we introduce a framework for embedding lattice symmetries in fermionic wavefunctions and demonstrate its ability to target both ground states and low-lying excitations. Using group-equivariant neural backflow transformations, we study the t-V model on a square lattice away from half-filling. Our symmetry-aware backflow significantly improves ground-state energies and yields accurate low-energy excitations for lattices up to 10 × 10. We also compute accurate two-point density-correlation functions and the structure factor to identify phase transitions and critical points. These findings introduce a symmetry-aware framework important for studying quantum materials and phase transitions. In this manuscript, the authors introduce a method for numerically modelling fermionic systems using a neural network wavefunction ansatz. They demonstrate that this method can efficiently and accurately find ground and low-lying excited states in two-dimensional models, outperforming existing approaches.
Simulating strongly correlated fermions remains a long-standing challenge due to the exponential complexity of the Hilbert space and the intricate sign structure of many-body wavefunctions. We introduce a variational framework centered on a neural network backflow transformation that combines deep learning with variational Monte Carlo. The proposed ansatz employs a multi-determinant expansion with low-rank shifts to capture non-local correlations and complex sign structures. Applied to the two-dimensional Hubbard model at both half-filling and 1/81/81/8 doping, the method achieves energies within 0.45%0.45\%0.45% of auxiliary-field quantum Monte Carlo at half-filling and captures intertwined charge- and spin-density stripe patterns at 1/81/81/8 doping. These results demonstrate the potential of this framework as a scalable and interpretable approach to variational simulations of strongly correlated fermionic systems.