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 as large as 256 sites, under various interaction strengths U, electron fillings n, next-nearest-neighbor hopping strengths t' and boundary conditions. Instead of considering backflow terms from all sites, competitive results are achieved by considering nearest-neighbor or next-nearest-neighbor backflow terms. Meanwhile the variational wave-function is not enforced on geometric symmetries. When t'=0, by considering nearest-neighbor backflow terms, linear stripe order is sucessfully obtained for the case of n=0. 875 and U=8 on the 16 16 lattice under periodoc boundary condition. For a similar case under open boundary condition, obtained energy is only 4. 5 10^-4 higher than the state-of-the-art method fPEPS with the bond dimension D=20. Comparing to state-of-the-art neural network results, energies are competitive and relative errors are below 5 10^-3. For cases of n=0. 8 and 0. 9375, results consistent with the phase diagram from AFQMC are obtained by direct optimizations. When t'=-0. 2, by considering next-nearest-neighbor backflow terms, obtained energies are competitive or even lower than state-of-the-art neural network results. For example, obtained energy for n=0. 875, U=8 on the 12 12 lattice under PBC is 8. 1 10^-4 lower comparing to that from the neural network state. Therefore, the Tensor-Backflow method has strong representation abilities for the Fermi-Hubbard model.