Peak structure of the infinite-temperature spectral function in scarred spin models

The spectral function (transverse component of the dynamical structure factor) in spin models that exhibit long-lived scars exhibit peaks at the energy spacing of the scarred ladder at all temperatures. In quench experiments, this translates to underdamped oscillations in the corresponding local observable starting from typical initial states at the same frequency as that observed in quenches from the special scarred states. This is puzzling as the (exact) scars constitute a vanishing part of the many-body spectrum at large system sizes. In this talk, I will discuss how the peak structure arises from the proximity of the scarred models to models with an integer spectrum. Specifically, we attempt to quantitatively predict the quality factor of the oscillation and the widths of the peaks using Schrieffer-Wolff transformation near a paramagnet.