Simulating the two-dimensional $t-J$ model at finite doping with neural quantum states

Simulating large, strongly interacting fermionic systems remains a major challenge for existing numerical methods. In this work, we introduce Gutzwiller projected hidden fermion determinant states (G-HFDS) to simulate the strongly interacting limit of the Fermi-Hubbard model, namely the $t$-$J$ model, across the entire doping regime. We demonstrate that the G-HFDS achieve energies competitive with matrix product states (MPS) on lattices as large as $10 \times 10$ sites while using several orders of magnitude fewer parameters, suggesting the potential for efficient application to even larger system sizes. This remarkable efficiency enables us to probe low-energy physics across the full doping range, providing new insights into the competition between kinetic and magnetic interactions and the nature of emergent quasiparticles. Starting from the low-doping regime, where magnetic polarons dominate the low energy physics, we track their evolution with increasing doping and different next-nearest neighbor hopping amplitudes through analyses of spin and polaron correlation functions as well as the Fermi surface. Our findings demonstrate the potential of determinant-based neural quantum states with inherent fermionic sign structure, opening the way for simulating large-scale fermionic systems at any particle filling.