Superconductivity in doping the quantum paramagnet on the square lattice

We report a large-scale density-matrix renormalization group study of lightly doped t-J-type model on the square lattice, which hosts a quantum paramagnetic state at half-filling. We observe an interesting interplay between d-wave superconductivity and spin and charge density wave (CDW) order. Despite the tendency of CDW order on thinner cylinders is strong, our results on wider systems suggest that the CDW order may disappear in the thermodynamic limit. The spin correlations are found to decay exponentially, on the contrary, the superconducting pair-field correlations decay power-law with an exponent significantly smaller than the spatial dimension. Our results suggest that a uniform d-wave superconductivity emerges in doping the quantum paramagnet on the square lattice. The connection of our results to two dimensions is also discussed.