Dynamics in Systems with Modulated Symmetries

We extend the notions of multipole and subsystem symmetries to more general spatially modulated symmetries. We uncover two instances with exponential and (quasi)periodic modulations and provide simple microscopic models in one, two, and three dimensions. Seeking to understand their effect on the long-time dynamics, we numerically study a stochastic cellular automaton evolution that obeys such symmetries. We prove that, in one dimension, the periodically modulated symmetries lead to a diffusive scaling of correlations modulated by a finite microscopic momentum. In higher dimensions, these symmetries take the form of lines and surfaces of conserved momenta. These give rise to exotic forms of subdiffusive behavior with a rich spatial structure influenced by lattice-scale features. Exponential modulation, on the other hand, can lead to correlations that are infinitely long-lived at the boundary while decaying exponentially in the bulk.