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arXiv:2609.26544

Neural Network Backflow with Low-Rank Multi-Determinant Updates

Tianshu Huang, Xiaowei Ou, Vidvuds Ozolins

cond-mat.str-elcond-mat.supr-conquant-phSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
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Abstract

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 $1/8$ doping, the method achieves energies within $0.45\%$ of auxiliary-field quantum Monte Carlo at half-filling and captures intertwined charge- and spin-density stripe patterns at $1/8$ doping. These results demonstrate the potential of this framework as a scalable and interpretable approach to variational simulations of strongly correlated fermionic systems.

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16 pages, 13 figures

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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.
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Superconductivity in the ttt-t′t't′ Hubbard Model from Symmetry-Preserving Neural-Network Quantum States
Despite its fundamental importance in the theory of strongly correlated electrons, the nature of the ground state of the two-dimensional doped Hubbard model remains intensely debated. Variational approaches provide a powerful route to this problem, but their conclusions can depend sensitively on the chosen wave-function parameterization, the mean-field initialization, or the pinning fields used to guide the optimization, as well as on boundary conditions. This can favor one type of symmetry breaking over another, making it difficult to distinguish the genuine interplay of intertwined or competing orders from biases induced by the variational parameterization. Here, we introduce the Symmetry-Preserving Backflow Pairing (SBP) ansatz, a neural-network wave function that respects translational symmetry by construction and thereby avoids these broken-symmetry minima. The SBP ansatz reaches state-of-the-art variational energies for the ttt-t′t't′ Hubbard model on lattices up to 24×2424\times2424×24 with 504504504 electrons, below those of competing pure stripe solutions. By extrapolating to the thermodynamic limit, we find robust evidence for ddd-wave superconducting order, resolving a long-standing question about the 1/81/81/8-doped model at t′/t=−0.2t'/t=-0.2t′/t=−0.2 and U/t=8.0U/t=8.0U/t=8.0. Built on general principles of symmetry and locality, the SBP wave function provides a broadly applicable variational representation for challenging interacting fermionic systems.
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