Local Ward identities for collective excitations in fermionic systems with spontaneously broken symmetries

We derive Ward identities for fermionic systems exhibiting a gauge symmetry that gets globally broken. In particular, we focus on the relation that connects the gauge field response functions to the transverse susceptibilities of the order parameter. We find that the long-wavelength and zero-energy limit of the former are related to the coefficients of a low-energy expansion of the latter. We examine three physical cases: the superconductor, the N\'eel antiferromagnet, and the spiral magnet. In the case of a metallic spiral magnet that completely breaks the SU(2) spin symmetry, we explicitly show that the Ward identities are fulfilled within the random phase approximation. We subsequently derive microscopic expressions for the spin stiffnesses and spin-wave velocities, which can be plugged into low-energy models to study the effect of long-wavelength bosonic fluctuations on top of mean-field solutions.