Gradient expansion for a spiral phase in a spin-fermion model

Starting from a doped spin-fermion model for high-temperature superconductors, we derive an effective continuum theory for the spin degrees of freedom by means of a gradient expansion around a spiral spin configuration. By integrating out the fermions in a path-integral representation, we obtain an effective spin-action. An incommensurate, planar spiral configuration for the spin-background is assumed. The long-wavelength limit is obtained by expanding the effective action in powers of a short distance cutoff. The occurring infinite series can be summed to all orders of the coupling constant by exploiting the constraint that the order parameter lives on a circleS₁. It is shown that the low-energy limit of the effective action can be mapped onto a O(2) nonlinear model and an additional term due to parity breaking.