Spin stiffnesses and stability of magnetic order in the lightly doped two-dimensional Hubbard model

Abstract

We analyze the density dependence of the spin stiffnesses and the stability of magnetic order with respect to quantum fluctuations in the two-dimensional Hubbard model close to half-filling. The stiffnesses are computed from the spin susceptibility obtained from a random phase approximation in a magnetically ordered state. For a sizable next-to-nearest neighbor hopping amplitude and a moderate Hubbard interaction, the mean-field ground state is a N\'eel antiferromagnet in the electron doped regime at and above half-filling, and a planar circular spiral state in the hole doped regime below half-filling. Upon electron doping, the N\'eel stiffness decreases smoothly and not very steeply. By contrast, the in-plane and out-of-plane stiffnesses in the spiral state drop abruptly at half-filling. The out-of-plane stiffness even drops to zero, and then increases again very slowly upon increasing hole doping. At finite temperatures, the N\'eel-to-spiral transition is shifted into the hole doped regime, the stiffnesses are continuous functions of the density, and they vanish at the transition. For small hole doping, the spin stiffnesses describe the quantum fluctuations only in a small momentum range, which shrinks to zero upon approaching half-filling. Using the above results, we show that the quantum ground state of the lightly electron doped Hubbard model remains N\'eel ordered, while quantum fluctuations probably destroy the spiral long-range order in the hole doped regime, giving rise to a quantum disordered state with a spin gap.