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 thermalize. 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 relation 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 renormalized 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.

Figure 1

The quantity $L^{-1}ln[\overline{\mathcal{G}(r,t)}/N_r\overline{\mathcal{G}(0,t)}]$ as a function of $r/L$ for different $L$ and $t$, numerically computed for the disordered Floquet spin chain described in Eq. (23) and Eq. (24). Different intensities/markers correspond to different $L$ and the colors to different $t$ as noted in the legends. The two panels correspond to different parameter values, both in the ergodic phase. The collapse of the data for different $L$ implies the large-deviation form in Eq. (15).

Figure 2

Logarithms of the functions $f(x_{*}\left(t\right),t)$ and ${\partial }_x{f(x,t)|}_{x_{*}\left(t\right)}$, corresponding to the plots in Fig. 1, shown as scatter plots where $x_{*}\left(t\right)$ denotes the saddle points given by Eq. (17). Different colors correspond to different $L$ and different intensities denote data points for different times in the range $t\in [1,1000]$ with darker colors denoting later times. The linear behavior supports the conjecture in Eq. (21).