SO(3) nonlinear $\sigma$ model for a doped quantum helimagnet

A field theory describing the low-energy, long-wavelength sector of an incommensurate, spiral magnetic phase is derived from a spin-fermion model that is commonly used as a microscopic model for high-temperature superconductors. After integrating out the fermions in a path-integral representation, a gradient expansion of the fermionic determinant is performed. This leads to an O(3)$\otimes$O(2)-symmetric quantum nonlinear $\sigma$ model, where the doping dependence is explicitly given by generalized fermionic susceptibilities which enter into the coupling constants of the $\sigma$ model and contain the fermionic band-structure that results from the spiral background. A stability condition of the field theory self-consistently determines the spiral wavevector as a function of the doping concentration. Furthermore, terms of topological nature like the $\theta$-vacuum term in (1+1)-dimensional nonlinear $\sigma$ models are obtained for the plane of the spiral.