Absence of Dynamical Steps in the Exact Correlation Potential in Linear Response

Recent work showed that the exact exchange-correlation potential of time-dependent density functional theory generically displays dynamical step structures. These have a spatially non-local and time-non-local dependence on the density in real time dynamics. The steps are missing in the usual approximations which consequently yield inaccurate dynamics. Yet these same approximations typically yield good linear response spectra. Here we investigate whether the steps appear in the linear response regime, when the response is calculated from a real-time dynamics simulation, by examining the exact correlation potential of model two-electron systems at various times. We find there are no step structures in regions where the system response is linear. Step structures appear in the correlation potential only in regions of space where the density response is non-linear; these regions, having exponentially small density, do not contribute to the observables measured in linear response.