The connection between Hilbert-space return probability and real-space autocorrelations in quantum spin chains

The dynamics of interacting quantum many-body systems has two seemingly disparate but fundamental facets. The first is the dynamics of real-space local observables, and if and how they thermalise. The second is to interpret the dynamics of the many-body state as that of a fictitious particle on the underlying Hilbert-space graph. In this work, we derive an explicit connection between these two aspects of the dynamics. We show that the temporal decay of the autocorrelation in a disordered quantum spin chain is explicitly encoded in how the return probability on Hilbert space approaches its late-time saturation. As such, the latter has the same functional form in time as the decay of autocorrelations but with renormalised parameters. Our analytical treatment is rooted in an understanding of the morphology of the time-evolving state on the Hilbert-space graph, and corroborated by exact numerical results.