Rigorous bounds on dynamical response functions and time-translation symmetry breaking

Dynamical response functions are standard tools for probing local physics near the equi-librium. They provide information about relaxation properties after the equilibrium state is weakly perturbed. In this paper we focus on systems which break the assumption of thermalization by exhibiting persistent temporal oscillations. We provide rigorous bounds on the Fourier components of dynamical response functions in terms of exten-sive or local dynamical symmetries, i.e., extensive or local operators with periodic time dependence. Additionally, we discuss the effects of spatially inhomogeneous dynamical symmetries. The bounds are explicitly implemented on the example of an interacting Flo-quet system, specifically in the integrable Trotterization of the Heisenberg XXZ model.