Dynamic Moir\'e Potentials and Robust Wigner Crystallization in Large-Scale Twisted Transition Metal Dichalcogenides

Understanding the dynamical evolution of large-scale moir\'e systems is crucial for connecting theoretical predictions with experimental observations. Here we develop a machine-learning-based workflow, integrating DeePMD and DeepH frameworks with first-principles calculations, to efficiently investigate time-dependent structural and electronic responses in twisted bilayer transition metal dichalcogenides (TMDs) with experimentally relevant moir\'e supercells containing over 3000 atoms. Using $\mathrm{WS_2}$ as a representative system, we show that low-temperature lattice vibrations and relaxation deepen the moir\'e potential wells, narrow the lowest conduction band, and facilitate the formation of strongly localized electronic states. Based on DFT-derived moir\'e potentials that incorporate these dynamical effects, density-matrix-renormalization-group (DMRG) simulations reveal robust Wigner crystallization and a kagom\'e-patterned three-electron state, consistent with recent experimental observations. Our workflow provides a practical route for exploring large moir\'e supercells beyond static configurations and offers new insight into the interplay between lattice dynamics, electronic localization, and emergent correlated states in twisted two-dimensional materials.