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Investigating the Fermi-Hubbard model by the Tensor-Backflow method

Xiao Liang

TL;DR

The paper tackles the challenge of accurately solving the two-dimensional Fermi-Hubbard model on sizeable lattices by introducing the Tensor-Backflow variational approach, which augments a Slater determinant with tensor-encoded backflow corrections and avoids enforcing geometric symmetries. Ground states are obtained via variational Monte Carlo with a Lanczos refinement, achieving energies competitive with state-of-the-art methods such as fPEPS and neural-network-based approaches across various $U$, $n$, and $t'$, under both periodic and open boundaries. Notably, at $t'=0$ and $n=0.875$, the method reveals a linear stripe order with SDW and CDW periods, with structure factors consistent with the expected ordering, and it extends competitive performance to $t'=-0.2$ where NNN backflow yields improvements over NN and benchmarks. Overall, Tensor-Backflow demonstrates strong representation capabilities for lattice fermion systems, offering a scalable, symmetry-agnostic framework that can compete with leading techniques while maintaining relatively modest parameter counts.

Abstract

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 $16\times 16$ lattice under periodoc boundary condition. For a similar case under open boundary condition, obtained energy is only $4.5\times 10^{-4}$ 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 $5\times 10^{-3}$. 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 $12\times 12$ lattice under PBC is $8.1\times 10^{-4}$ lower comparing to that from the neural network state. Therefore, the Tensor-Backflow method has strong representation abilities for the Fermi-Hubbard model.

Investigating the Fermi-Hubbard model by the Tensor-Backflow method

TL;DR

The paper tackles the challenge of accurately solving the two-dimensional Fermi-Hubbard model on sizeable lattices by introducing the Tensor-Backflow variational approach, which augments a Slater determinant with tensor-encoded backflow corrections and avoids enforcing geometric symmetries. Ground states are obtained via variational Monte Carlo with a Lanczos refinement, achieving energies competitive with state-of-the-art methods such as fPEPS and neural-network-based approaches across various , , and , under both periodic and open boundaries. Notably, at and , the method reveals a linear stripe order with SDW and CDW periods, with structure factors consistent with the expected ordering, and it extends competitive performance to where NNN backflow yields improvements over NN and benchmarks. Overall, Tensor-Backflow demonstrates strong representation capabilities for lattice fermion systems, offering a scalable, symmetry-agnostic framework that can compete with leading techniques while maintaining relatively modest parameter counts.

Abstract

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 , electron fillings , next-nearest-neighbor hopping strengths 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 =0, by considering nearest-neighbor backflow terms, linear stripe order is sucessfully obtained for the case of =0.875 and =8 on the lattice under periodoc boundary condition. For a similar case under open boundary condition, obtained energy is only higher than the state-of-the-art method fPEPS with the bond dimension =20. Comparing to state-of-the-art neural network results, energies are competitive and relative errors are below . For cases of =0.8 and 0.9375, results consistent with the phase diagram from AFQMC are obtained by direct optimizations. When =-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 =0.875, =8 on the lattice under PBC is lower comparing to that from the neural network state. Therefore, the Tensor-Backflow method has strong representation abilities for the Fermi-Hubbard model.

Paper Structure

This paper contains 16 sections, 7 equations, 14 figures, 7 tables.

Table of Contents

  1. Introduction
  2. the Tensor-Backflow method
  3. Wave-function definitions
  4. Optimization method
  5. Computational complexity
  6. Numerical results
  7. Optimization details
  8. Comparisons with previous results
  9. n=0.875
  10. n=0.8 and n=0.9375
  11. Discussions and conclusions
  12. Acknowledgement
  13. Comparing ground state patterns of two optimization strategies
  14. Comparisons between initial UHFs and subsequent Tensor-Backflow wave-functions
  15. Energy convergence of transferring from $U$=8
  16. ...and 1 more sections

Figures (14)

  • Figure 1: Comparisons of two optimization strategies demonstrated by $n$=0.875, $U$=8 and $t'$=0 on the $4\times 16$ lattice under OBC, with NN backflow in wave-functions. Both strategies start from the same initial state. For strategy 1, sample number is fixed as $\mathcal{O}(4.5\times 10^4)$, the step size is reduced from $2\times 10^{-3}$ to $2\times 10^{-4}$ at the 6000-th step. For strategy 2, sample number is increased from $\mathcal{O}(4.5\times 10^4)$ to $\mathcal{O}(4.5\times 10^5)$ at around 3000-th step, meanwhile maintaining the step size to $2\times 10^{-3}$. After about another 2500 steps, the step size is reduced to $2\times 10^{-4}$. Strategy 1 and 2 converge to -0.6773 and -0.6774, respectively.
  • Figure 2: Energy extrapolations and comparisons respect to fPEPS Hubbard_fPEPS under OBC for the lattice of $8\times 8$(a) and $4\times 16$(b). All cases are for $n$=0.875, $U$=8 and $t'$=0. Energy extrapolation for the lattice of $4\times 16$ under PBC(c) yields to the neural Pfaffian result Hubbard_NN4. $\sigma^2$ is the energy variance. Red and blue solid dots depict Tensor-Backflow with NN and all-site backflow terms. For solid dots of the same color, low and high energies depict results with and without one Lanczos optimization.
  • Figure 3: Patterns of ground states for $n$=0.875, $U$=8 and $t'$=0 on the $16\times 16$ lattice under PBC(a)(b) and OBC(c)(d). All results are from $p$=1 wave-functions with NN backflow. Left-sided figures denote the average spin per site: $s_i^z=(n_i^\uparrow-n_i^\downarrow)/2$, and right-sided figures denote the average density per site: $n_i=n_i^\uparrow+n_i^\downarrow$.
  • Figure 4: Charge and spin structure factors for $n$=0.875 and $t'$=0 under various $U$ and boundary conditions. All results are from $p$=1 wave-functions, with NN backflow in wave-functions. (a)(b)On the $16\times 16$ lattice, results of $U$=8 under PBC are obtained by a direct optimization, meanwhile other results are obtained by the initialization with the $U$=8 PBC wave-function. (c)(d)Results of $U$=8 under PBC by a direct optimization on the $8\times 32$ lattice.
  • Figure 5: Patterns of ground states for $n$=0.875, $U$=8 and $t'$=-0.2 under PBC on the $8\times 16$ lattice, from the $p$=1 wave-function with NNN backflow. Left-sided (right-sided) figure denotes the average spin (density) per site, within definitions from Fig.(\ref{['fig:Sz_ni']}).
  • ...and 9 more figures