Artificial atoms, Wigner molecules, and emergent Kagome lattice in semiconductor moir\'e superlattices

Semiconductor moir\'e superlattices comprise an array of artificial atoms and provide a highly tunable platform for exploring novel electronic phases. We introduce a theoretical framework for studying moir\'e quantum matter that treats intra-moir\'e-atom interactions exactly and is controlled in the limit of large moir\'e period. We reveal an abundance of new physics arising from strong electron interactions when there are multiple electrons within a moir\'e unit cell. In particular, at filling factor $n=3$, the Coulomb interaction within each three-electron moir\'e atom leads to a three-lobed ``Wigner molecule''. When their size is comparable to the moir\'e period, the Wigner molecules form an emergent Kagome lattice. Our work identifies two universal length scales characterizing the kinetic and interaction energies in moir\'e materials and demonstrates a rich phase diagram due to their interplay.