From continuum neural quantum states to effective lattice models: Fluctuating Wigner molecules in moir'e superlattices

Effective descriptions of quantum materials rely on a separation of energy scales that identifies their low-energy degrees of freedom. When kinetic energy, lattice potential, and interactions compete, this hierarchy breaks down, and both the microscopic state and its effective description are unknown. Continuum neural quantum states (NQS) can find the ground state directly without projecting onto prescribed bands, orbitals, or spins, but their microscopic wavefunctions are difficult to interpret in terms of effective variables. Here, we introduce a wavefunction readout that maps the continuum NQS onto site-resolved charge and spin statistics, allowing effective degrees of freedom to emerge from the microscopic solution. Focusing on moiré transition metal dichalcogenides at filling ν= 3, the NQS reveals a low-spin ground state below the ferromagnetic crystal of spin-polarized Wigner molecules. Although the charge density retains triangular molecular motifs, our readout reveals significant charge fluctuations through bound nearest-neighbor doublon--hole pairs. Each molecular charge state has an inert spin-singlet core and rim electrons that carry the spin. Charge and spin fluctuations are therefore intertwined, with antiferromagnetic spin-correlations between sites, motivating an effective description by a spinful two-orbital Hubbard model. More generally, our results demonstrate a route from continuum wavefunctions to effective lattice models when competing scales preclude a controlled reduction.