Neural operator-based self-learning hybrid Monte Carlo studies of electron-phonon coupled Hamiltonians

Finite-temperature quantum Monte Carlo is a workhorse for correlated lattice Hamiltonians, especially those including electron-phonon interactions, but studies of competing orders are hampered by O(N³) linear-algebra costs and long autocorrelation times. While self-learning Monte Carlo has mitigated these issues in select settings, force-based learning for quantum many-body models remains underexplored. We introduce a neural-operator surrogate that predicts fermionic-action gradients within Hybrid Monte Carlo (HMC), learning a mapping from field configurations X_il to corresponding fermionic-action derivatives ∂S_F/∂X_il. The surrogate amortizes repeated matrix solves in the HMC trajectory evaluation, substantially reducing per-trajectory wall time.We demonstrate the capabilities and limitations of neural operator self-learning using sign-problem-free electron-phonon models (e.g., the Holstein model), enabling a deeper exploration of the superconducting and charge-density-wave correlations they exhibit.