Investigating the Fermi-Hubbard model by the tensor-backflow method

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 $t'$, 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. Meanwhile, changing interaction strengths in the prior unrestrictied Hartree-Fock state is helpful to bypass the local minima. When $t'$=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 $t'$=-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.