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 Letter, we introduce Gutzwiller projected hidden fermion determinant states (G-HFDS) to simulate the strongly interacting limit of the Fermi-Hubbard model, namely the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>t</a:mi><a:mtext>−</a:mtext><a:mi>J</a:mi></a:mrow></a:math> model, across the entire doping regime. We demonstrate that the G-HFDS achieve energies competitive with matrix product states on lattices as large as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mn>10</c:mn><c:mo>×</c:mo><c:mn>10</c:mn></c:mrow></c:math> 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 an inherent fermionic sign structure, opening the way for simulating large-scale fermionic systems at any particle filling.