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Solving the Hubbard model with Neural Quantum States

Yuntian Gu, Wenrui Li, Heng Lin, Bo Zhan, Ruichen Li, Yifei Huang, Di He, Yantao Wu, Tao Xiang, Mingpu Qin, Liwei Wang, Dingshun Lv

arXiv:2507.02644v2cond-mat.str-elcs.AIquant-ph

TL;DR

Exact energy calculations for the 2D Hubbard model are infeasible at large sizes, motivating accurate approaches to its complex energy landscapes. The paper develops an NQS framework that achieves state-of-the-art results and identifies stripe ground states, including half-filled stripes with next-nearest-neighbor hopping.

  • Problem

    For the 2D Hubbard model, exact energy calculations become infeasible at large sizes, while its energy landscapes are complex and high-dimensional.

    Source: Methods and benchmarks · p. 7, §1 · p. 25

  • Method

    The paper develops an NQS framework by leveraging cutting-edge machine-learning architectures and highly accurate optimization strategies.

    Source: Conclusions and outlook · p. 14

  • Results

    The method achieves state-of-the-art results at unprecedented sizes, confirming filled stripes for the doped pure Hubbard model and finding half-filled stripes with t' = −0.2.

    Source: Conclusions and outlook · p. 14

  • Takeaways & Limitations

    Large-system calculations under periodic boundary conditions support reliable study of stripe phases in the 2D Hubbard model toward the thermodynamic limit.

    Source: Conclusions and outlook · p. 14

    Dynamic and finite-temperature properties still require more sophisticated NQS ansätze.

    Source: Conclusions and outlook · p. 15

Abstract

from arXiv · show

The rapid development of neural quantum states (NQS) has established it as a promising framework for studying quantum many-body systems. In this work, by leveraging the cutting-edge transformer-based architectures and developing highly efficient optimization algorithms, we achieve the state-of-the-art results for the doped two-dimensional (2D) Hubbard model, arguably the minimum model for high-Tc superconductivity. Interestingly, we find different attention heads in the NQS ansatz can directly encode correlations at different scales, making it capable of capturing long-range correlations and entanglements in strongly correlated systems. With these advances, we establish the half-filled stripe in the ground state of 2D Hubbard model with the next nearest neighboring hoppings, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.

The Hubbard model

This section examines transformer-based NQS on the Hubbard model, covering its capture of correlations across scales and the reported stripe phases, orientations, and attention patterns. It also outlines the measurement procedure and directions for future development.

The Hubbard model

  • The study uses square lattices with OBC and PBC, sets t = 1 and U = 8, and considers t′ = 0 or −0.2 at hole concentration δ = 1/8.The paper identifies U = 8 as a value relevant to cuprates.Source: The Hubbard model · p. 5

Methods and benchmarks

  • MARCH adds gradient second-moment estimates to SPRING, adapting each parameter’s learning rate and producing a more direct path to the energy minimum.The second-moment information estimates gradient variance and is described as helping MARCH avoid the overshooting associated with momentum-based SPRING.Source: Methods and benchmarks · pp. 7–8
  • At half-filling, NQS energies for the pure Hubbard model under PBC agree with AFQMC within twice the statistical errors.The benchmark uses systems for which numerical exact ground-state energies can be obtained by AFQMC.Source: Methods and benchmarks · p. 8
  • NQS energies are lower than PEPS across tested widths, surpass DMRG in wider systems, and are consistent with AFQMC within twice the statistical errors at half-filling.DMRG has the lowest energy for narrow systems with Ly ≤ 6; the NQS comparison uses PEPS with bond dimension D ≥ 20.Source: Figure 2 · p. 6, Methods and benchmarks · p. 8
  • The transformer’s attention heads specialize in different physical correlations, including short-range interactions, nearest-neighbor hopping, antiferromagnetic order, and delocalized phase information.The first head prioritizes local interactions, while the other heads attend to adjacent sites, a checkerboard pattern, and more complex delocalized patterns, respectively.Source: Methods and benchmarks · pp. 8–10

Filled stripe order in pure Hubbard model

  • Under PBC, the pure Hubbard model exhibits a robust filled stripe phase with wavelength λ = 8, while the OBC stripe-direction crossover disappears.Under OBC, the crossover occurs as width increases from Ly = 8 to Ly = 12, indicating a finite-size and boundary effect.Source: Filled stripe order in pure Hubbard… · p. 10
  • At t'=-0.2 and δ=1/8, horizontal stripes have lower energy on studied Lx×8 lattices (Lx=8–32; 8×8 states coincide); the Hubbard-model stripe is half-filled (λ=4), consistent with cuprate observations.Source: Filled stripe order in pure Hubbard… · p. 12, Figure 5 · p. 13, Conclusions and outlook · p. 14
  • For t'=-0.2, horizontal stripes are favored over vertical ones, with evidence suggesting a tendency toward fewer long stripes rather than fragmented ones.The reported energy advantage comes mainly from kinetic energy; the authors suggest that a long river of holes benefits kinetic energy, while noting that a comprehensive explanation remains open.Source: Figure 5 · p. 13, The direction of the stripe state · p. 14
  • When width constraints frustrate λ=4 horizontal stripes, the 16×10 ground state instead has λ=5 horizontal stripes, while the 16×12 ground state has vertical λ=4 stripes.A horizontal λ=4 stripe is impossible when the rectangular system’s width is not divisible by 8.Source: The direction of the stripe state · p. 14
  • The doped pure model shows filled stripes, including λ=8 on a 16×16 lattice, but determining whether its true ground state is horizontal requires larger systems than the study could reach.The smallest periodic system accommodating wavelength-8 stripes in both directions is 32×16, beyond the authors’ capacity.Source: Filled stripe order in pure Hubbard… · p. 11, Conclusions and outlook · p. 14, The direction of the stripe state · p. 14

Conclusions and outlook

  • After optimization, different NQS attention heads specialize in encoding correlations at different scales, enabling the ansatz to capture long-range correlations and entanglements.The authors identify more efficient optimization and ansätze for dynamic and finite-temperature properties as future needs.Source: Conclusions and outlook · p. 15

Methods summary

  • In the VMC procedure, MCMC samples electron configurations during optimization, followed by a separate inference run with the optimized NQS to compute observables to the desired statistical accuracy.Network architecture, hyperparameters, and algorithmic specifics are deferred to the supplementary materials.Source: Methods summary · p. 15

Supplementary materials

The supplementary materials identify the paper’s authors and note that four authors contributed equally.

Supplementary materials

1 Variational Monte Carlo

The section describes variational Monte Carlo for optimizing a neural wavefunction, the transformer-based ansatz, and MARCH, an adaptive stochastic-reconfiguration optimizer.

1 Variational Monte Carlo

  • VMC minimizes the neural wavefunction’s energy, estimating its expectation through Monte Carlo sampling because exact summation is infeasible in the exponentially large Hilbert space.The local-energy summation scales linearly with system size, and configurations are sampled according to the wavefunction’s squared amplitude.Source: §1 · p. 25

1.1 Neural Quantum State Wavefunction Ansatz

  • The NQS encodes each site’s four-state occupation as a token, adds learnable positional encodings, and processes the sequence through Transformer blocks.The output produces backflow orbitals; selecting occupied rows forms Slater matrices whose determinants are combined linearly into the wavefunction.Source: §1.1 · pp. 25–26

1.2 Optimization

  • MARCH combines Adam-like moment estimates with SR and ideas from SPRING and Min-SR.The first moment acts as momentum, while the second moment tracks changes in the update and can suppress volatile directions.Source: §1.2 · pp. 26, 28
  • Stochastic reconfiguration uses the Fubini–Study distance to choose parameter updates that approximate imaginary-time evolution within the variational family.The method relates the variational state to the evolving state using first-order expansions and a batch of Monte Carlo samples.Source: §1.2 · pp. 26–27
  • MARCH adaptively translates and scales parameter updates using first- and second-moment estimates.Its second moment tracks gradient volatility: erratic changes suppress updates, while stable gradients allow more aggressive updates.Source: §1.2 · p. 28

2 Pretrain

The pretraining procedure teaches the transformer orbital matrices generated by a variationally optimized neural-network backflow ansatz; the pretrained transformer is reported to be more stable and to converge faster than random initialization.

Pretraining

  • The procedure first variationally optimizes a two-hidden-layer MLP backflow ansatz that maps occupation numbers to block-diagonal orbital matrices.Occupied rows form a block-diagonal square orbital matrix, whose determinant defines the backflow ansatz amplitude.Source: §2 · p. 29
  • The optimized backflow ansatz generates configuration–orbital-matrix examples, and the transformer is pretrained to predict those matrices with mean squared error and Adam.The dataset can be generated online.Source: §2 · p. 29
  • A pretrained transformer is more stable and converges faster than the randomly initialized ansatz.Source: §2 · p. 29

3 Pinning Field

A temporary antiferromagnetic pinning field guides the ansatz toward a chosen stripe pattern, which is transferred to a second transformer before optimization without the field.

Pinning Field

  • The method applies an antiferromagnetic pinning field with h_m = 0.2 and h_i ∈ {−1, 0, 1} to guide the first transformer toward a selected magnetic pattern.Source: §3 · p. 30
  • A second transformer is pretrained to maximize fidelity with the pinned state, then optimized after removing the field so it can relax toward the ground state.Source: §3 · p. 30
  • On a 32 × 8 lattice with t′ = −0.2 and δ = 1/8, the horizontal stripe is lower in energy than the vertical stripe.Source: §3 · p. 30

4 Benchmark Results

The section benchmarks the NQS approach against leading numerical methods across Hubbard-model configurations and tests MARCH against other optimizers. It also describes accuracy checks on systems with exact solutions.

4 Benchmark Results

  • The section reports extensive comparisons across Hubbard-model configurations and system parameters, alongside an optimizer ablation.The benchmark suite compares the NQS methodology with several leading numerical techniques.Source: §4 · p. 30

4.1 Comparison with Hidden Fermion Determinantal State; 4.2 Comparison with DMRG

  • With PBC, NQS reports a ground-state energy of −0.76298, below HFDS; separately, its variational energy −0.72791 is below DMRG’s −0.72747, obtained at bond dimension m = 40000.The PBC result represents the largest calculation reported in their paper.Source: §4.1 · pp. 30–31, §4.2 · p. 31

4.3 Comparison with PEPS

  • NQS has expressive power similar to fermion PEPS with D = 10 and is more computationally efficient than PEPS under the same hardware conditions.The computational-efficiency comparison is reported in Figure S33 and Table S3.Source: §4.3 · p. 31

4.4 Optimizer Ablation Study

  • For the tested systems, MARCH converges significantly faster and more stably than SPRING and Min-SR.The convergence behavior is presented in Figure S34.Source: §4.4 · p. 31

4.5 Accuracy Estimation

  • Accuracy comparisons use relatively simple systems with known exact solutions, including a 4 × 4 lattice and the half-filled pure Hubbard model.These systems can be solved using Exact Diagonalization or AFQMC; the attention maps are organized by layer and head.Source: §4.5 · pp. 31–32, Figure S1 · p. 32

5 Hyperparameter

This section gives NQS training and pretraining settings, describes parts of the wavefunction architecture, and reports comparisons with PEPS and AFQMC. It also includes stripe visualizations and an optimizer comparison.

5 Hyperparameter

  • Training, pretraining, and MARCH settings are tabulated, and the authors note that hyperparameters may differ between systems.Source: §5 · p. 32
  • NQS results are reported as comparable to numerically exact AFQMC values.Source: §5 · p. 33
  • A temporary pinning field is used to obtain a vertical stripe; Figure S32 compares vertical and horizontal stripes for a 32 × 8 system with t′ = −0.2 and δ = 1/8.Source: §5 · p. 53, Figure S32 · p. 56
  • The embedding produces one token per lattice site; the transformer outputs K sets of occupation-selected backflow orbitals, and the wavefunction amplitude sums determinants.Source: §5 · p. 55
  • NQS’s expressive power is similar to PEPS; reported results say it surpasses PEPS, and it is significantly more efficient on an 8 × 8 lattice.Source: §5 · pp. 34, 35, 57
  • MARCH is reported to converge faster than SPRING and Min-SR.Source: §5 · p. 58
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