Recently, a variational wave function based on the tensor representation of backflow corrections along with a Lanczos step has achieved state-of-the-art energy precision for Fermi-Hubbard-type models. However, solving the Fermi-Hubbard model remains challenging, and the validity of any method must be assessed through the ground state's physical properties. For simplicity, we refer this method as tensor-backflow. In this work, we apply the tensor-backflow method to investigate the Fermi-Hubbard model on two-dimensional lattices up to 256 sites, exploring various interaction strengths $U$, electron fillings $n$, next-nearest-neighbor hopping $t^{\prime }$, and boundary conditions. By considering backflow terms from nearest- or next-nearest-neighbor sites, we achieve competitive results without enforcing geometric symmetries on the variational wave function. The optimizations were stable from a prior unrestrictied Hartree-Fock state, followed by adding backflow corrections. However, changing interaction strengths in the prior unrestrictied Hartree-Fock state is helpful to bypass the local minima. When $t^{\prime }=0$, by considering nearest-neighbor backflow terms, linear stripe order emerges successfully for the case of $n=0.875$ and $U=8$ on a $16\times 16$ lattice with periodic boundary conditions. In a similar case with open boundary conditions, the energy obtained is only $4.5\times {10}^{-4}$ higher than the state-of-the-art method fPEPS with bond dimension $D=20$. Compared to state-of-the-art neural network methods, the energies obtained using the tensor-backflow approach are competitive, with relative errors below $5\times {10}^{-3}$. For $n=0.8$ and $n=0.9375$, direct optimizations yield results consistent with the phase diagram from AFQMC. When $t^{\prime }=-0.2$, considering next-nearest-neighbor backflow terms leads to energies that are either competitive with or even lower than those from state-of-the-art neural network approaches. For instance, for $n=0.875$ and $U=8$ on a $12\times 12$ lattice with periodic boundary conditions, the energy obtained is $8.1\times {10}^{-4}$ lower than that from the neural network result. Thus, the tensor-backflow method demonstrates strong representational capabilities for solving the Fermi-Hubbard model.

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• Charge density waves
• Spin density waves
• Superconductivity
• 2-dimensional systems
• Approximation methods for many-body systems
• Hubbard model
• Monte Carlo methods
• Trial wave functions
• Variational wave functional methods