Superconductivity in the two-dimensional Hubbard model revealed by neural quantum states

Simulating the 2-dimensional Hubbard model is incredibly challenging due to the small energy scales that seperate competing phases of matter. Here, I will introduce a class of variational wavefunctions that may help make progress on this problem. These wavefunctions augment Pfaffians with deep neural networks, and can be used to model phases with magnetic, superconducting or coexisting orders. Using these wavefunctions, we show we that d-wave superconductivity can emerge in various ways in the square lattice Hubbard model. For strong coupling with t'<0 we find superconductivity coexisting with magnetic order, whereas for t'=0 and intermediate coupling we find a superconductor with highly delocalized pairs.