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arXiv:2507.01856 · Physical review. B./Physical review. B

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

Xiao Liang

Published Mar 25, 2026• 0 citations
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Abstract

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.

DOI
10.1103/dqc7-zsvc

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We apply the Tensor-Backflow method to investigate the Fermi-Hubbard model on two-dimensional lattices up to 256 sites, exploring various interaction strengths UUU, electron fillings nnn, next-nearest-neighbor hopping t′t'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′t't′=0, by considering nearest-neighbor backflow terms, linear stripe order emerges successfully for the case of nnn=0.875 and UUU=8 on a 16×1616 \times 1616×16 lattice with periodic boundary conditions. In a similar case with open boundary conditions, the energy obtained is only 4.5×10−44.5 \times 10^{-4}4.5×10−4 higher than the state-of-the-art method fPEPS with bond dimension DDD=20. Compared to state-of-the-art neural network methods, the energies obtained using the Tensor-Backflow approach are competitive, with relative errors below 5×10−35 \times 10^{-3}5×10−3. For nnn=0.8 and nnn=0.9375, direct optimizations yield results consistent with the phase diagram from AFQMC. When t′t'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 nnn=0.875 and UUU=8 on a 12×1212 \times 1212×12 lattice with periodic boundary conditions, the energy obtained is 8.1×10−48.1 \times 10^{-4}8.1×10−4 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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The quantum many-body problem is an important topic in condensed matter physics. To efficiently solve the problem, several methods have been developed to improve the representation ability of wave functions. For the Fermi-Hubbard model under periodic boundary conditions, current state-of-the-art methods are neural network backflows and the hidden fermion Slater determinant. The backflow correction is an efficient way to improve the Slater determinant of free particles. In this work we propose a tensor representation of the backflow-corrected wave function; we show that for the spinless $t\text{\ensuremath{-}}V$ model, the energy precision is competitive or even lower than current state-of-the-art fermionic tensor network methods. For models with spin, we further improve the representation ability by considering backflows on fictitious particles with different spins, thus naturally introducing nonzero backflow corrections when the orbital and the particle have opposite spins. We benchmark our method on molecules in the STO-3G basis and the Fermi-Hubbard model with periodic and cylindrical boundary conditions. We show that the tensor representation of backflow corrections achieves competitive or even lower-energy results than current state-of-the-art neural network methods.
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.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
The two-dimensional Hubbard model was originally proposed as the minimal model of uniform ddd-wave superconductivity in the cuprate high-temperature superconductors, but recent calculations with relevant parameters in a realistic doping regime have often found other ground states. Further, uncertainty remains over such results due to the intrinsic bias of different computational methods. Here we address this uncertainty through state-of-the-art neural tensor network and neural backflow state simulations, enabled by large-scale GPU computation, that together reach an unprecedented variational accuracy on lattices with up to 1024 sites. At 1/81/81/8 doping in the pure Hubbard model, as well as with next-nearest-neighbor hopping relevant to the hole-doped cuprates, we reveal a near degeneracy among a low-energy uniform ddd-wave state, superconducting stripes, and non-superconducting stripe states on the milli-ttt energy scale. Our work rebalances the narrative that the Hubbard model supports a single dominant order in relevant parameter regimes towards a picture of extreme competition between stripes and superconductivity.
Abstract This study extended the fixed-node QMC method to a lattice model for hopping fermions using Green’s function techniques. In general, the modifications of the procedures for continuous systems to adapt them to a lattice system were minor, and there appeared to be no surprises. Applications were made to the two-dimensional Hubbard model with different on-site interactions, lattice dimensions, boundary conditions, node specifications, and trial wavefunctions for importance sampling. Results were compared with those of Hartree-Fock and variational QMC calculations. For the HF calculations with zero interaction, the exact solutions are known, and the Green’s function QMC results agreed with those within statistical error. Trial wavefunctions specifying the nodes were obtained directly as those of HF determinantal solutions for zero interaction, or as variationally optimized determinantal functions with correlation terms. The procedure for fixed nodes was to eliminate walkers crossing the boundaries of a nodal region, a procedure not quite correct, but probably a reasonable approximation. The results in all cases were consistent with the HF and variational QMC calculations, giving lower energies than either for nonzero interaction energies. The existence of a domain-wall phase suggested by the HF and variational QMC results was confirmed by the fixed-node Green’s function results. It was noted that the fixed-node method would be applicable to systems other than the Hubbard model.