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

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