Spin stiffness of spiral magnetism at low doping

We consider the low doping limit of the spin stiffness for the incommensurate spiral magnetic order. Its value determines the bare parameters for the low energy effective theory of spin fluctuations on top of the given magnetic order. In our calculation, we start from the two-dimensional Hubbard model and choose the coupling value such that the incommensurate spiral mean-field order is stable at any small finite doping, while becoming Néel exactly at half-filling. We show that at T=0 in the small doping regime the spin stiffness can be computed analytically and tends to zero in the zero doping limit, while jumping at a finite value precisely at half-filling. We analyse the validity of the non-linear sigma model for spin fluctuations in the long-wavelength limit that would lead, in principle, to quantum disordered at small doping.