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Peer-reviewedCC BY-NC-NDNature Communications

Solving the Hubbard model with neural quantum states

  • Yuntian Gu1,2
  • Wenrui Li1,2
  • Heng Lin1,3
  • Bo Zhan1,4
  • Ruichen Li1,2
  • Yifei Huang1
  • Di He2
  • Y N Wu5
  • Tao Xiang6
  • Mingpu Qin7
  • Liwei Wang8
  • Dingshun Lv9,10
10 affiliations
  1. 1ByteDance Seed, Beijing, China
  2. 2State Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University, Beijing, China
  3. 3State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, China
  4. 4Institute of Physics, Chinese Academy of Sciences, Beijing, China
  5. 5Institute of Physics, Chinese Academy of Sciences, Beijing, China. [email protected]
  6. 6Institute of Physics, Chinese Academy of Sciences, Beijing, China. [email protected]
  7. 7Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China. [email protected]
  8. 8State Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University, Beijing, China. [email protected]
  9. 9ByteDance Seed, Beijing, China. [email protected]
  10. 10FieldQuantum Research Institute, Beijing, China. [email protected]
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Abstract

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 hopping, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.

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Cite

BibTeXRIS
BibTeX
@article{gu2026solving,
  title = {{Solving the Hubbard model with neural quantum states}},
  author = {Yuntian Gu and Wenrui Li and Heng Lin and Bo Zhan and Ruichen Li and Yifei Huang and Di He and Y N Wu and Tao Xiang and Mingpu Qin and Liwei Wang and Dingshun Lv},
  journal = {Nature Communications},
  year = {2026},
  month = jun,
  doi = {10.1038/s41467-026-74028-6},
  url = {https://doi.org/10.1038/s41467-026-74028-6},
  note = {CC BY-NC-ND},
}
RIS
TY  - JOUR
TI  - Solving the Hubbard model with neural quantum states
AU  - Yuntian Gu
AU  - Wenrui Li
AU  - Heng Lin
AU  - Bo Zhan
AU  - Ruichen Li
AU  - Yifei Huang
AU  - Di He
AU  - Y N Wu
AU  - Tao Xiang
AU  - Mingpu Qin
AU  - Liwei Wang
AU  - Dingshun Lv
PY  - 2026
DA  - 2026/06/22
JO  - Nature Communications
DO  - 10.1038/s41467-026-74028-6
UR  - https://doi.org/10.1038/s41467-026-74028-6
AB  - 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 hopping, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.
SN  - 2041-1723
ER  -