Effective Field Theory of Spontaneously Broken Modulated Symmetries

Modulated symmetries are spatially dependent transformations whose generators carry nontrivial spatial weights, such as exponential or multipole forms. We develop a unified effective field theory framework for systems that spontaneously break such symmetries, and classify their general types. We show that breaking certain modulated symmetries gives rise to massive Goldstone modes, revealing new forms of collective excitations.