Solving the Hubbard model with Neural Quantum States

2025-07-03

AI Summary

The doped two-dimensional Hubbard model—the minimal paradigm for high-temperature superconductivity—remains challenging for neural quantum states (NQS) to capture long-range entanglement and multi-scale electronic correlations accurately. Method: We propose a Transformer-based NQS within the variational Monte Carlo framework, where distinct attention heads explicitly encode short-range local interactions and long-range spin/charge correlations; combined with fermionic sign handling and efficient optimization. Contribution/Results: Our approach precisely predicts the half-filled stripe-ordered ground state in the extended Hubbard model with next-nearest-neighbor hopping. The resulting stripe structure exhibits striking agreement with experimental observations in cuprate superconductors. This represents the first NQS method capable of faithfully representing long-range order in strongly correlated systems, overcoming a key limitation of existing approaches and achieving state-of-the-art accuracy.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data

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

Problem

• Solving the doped 2D Hubbard model using neural quantum states
• Capturing long-range correlations in strongly correlated quantum systems
• Establishing NQS as a tool for many-fermion systems

Innovation

• Transformer-based NQS for Hubbard model
• Attention heads encode multi-scale correlations
• Efficient optimization achieves state-of-the-art results