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arXiv:2606.05293

Thermalization Dynamics in the Two-Dimensional Hubbard Model with Neural-Network Quantum States

Alessandro Sinibaldi, Luciano Loris Viteritti, Riccardo Rende, Fakher F. Assaad, Giuseppe Carleo

cond-mat.str-elcond-mat.dis-nnquant-phSubmitted Jun 3, 2026• Announced Jun 5, 2026• 0 citations• Revised Jul 31, 2026 (v2)
v1 · Jun 3, 2026v2 · Jul 31, 2026
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Abstract

Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction $U$, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two dynamical behaviors: for weak to intermediate interactions, the double occupancy rapidly approaches the thermal prediction, consistent with ergodic evolution; beyond a critical interaction $U_{C}$, the dynamics remains distinct from the thermal expectation on the timescales investigated, revealing signatures of a prethermal plateau delaying fast relaxation. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.

Comments
10 pages, 7 figures

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Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction UUU, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two dynamical behaviors: for weak to intermediate interactions, the double occupancy rapidly approaches the thermal prediction, consistent with ergodic evolution; beyond a critical interaction UCU_{C}UC​, the dynamics remains distinct from the thermal expectation on the timescales investigated, revealing signatures of a prethermal plateau delaying fast relaxation. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.
We introduce fermionic neural Gibbs states (fNGS), a variational framework for modeling finite-temperature properties of strongly interacting fermions. fNGS starts from a reference mean-field thermofield-double state and uses neural-network transformations together with imaginary-time evolution to systematically build strong correlations. Applied to the doped Fermi-Hubbard model, a minimal lattice model capturing essential features of strong electronic correlations, fNGS accurately reproduces thermal energies over a broad range of temperatures, interaction strengths, even at large dopings, for system sizes beyond the reach of exact methods. These results demonstrate a scalable route to studying finite-temperature properties of strongly correlated fermionic systems beyond one dimension with neural-network representations of quantum states.
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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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