Fig. 3: Quantitative features of scarring.
a Distribution of the rescaled overlaps x between mid-spectrum eigenstates and various families of states. The overlaps with Haar random states (red) have distribution e−x (dashed), similarly to the overlaps with generic points {si} of the phase space (green). By striking contrast, the overlaps with the IS states (blue) yield a much fatter tail, that is, scarring makes some eigenstates anomalously large on the UPOs. b The distribution of the scarness parameter S for mid-spectrum eigenstates (blue) has a long tail compared to Haar random states (red). Indeed, the number Mscarred of scarred eigenstates, defined as the number of eigenstates with S larger than the 90-th percentile of the Haar states (red dashed line), minus 10% (see Supplemental Information), is large and appears to grow exponentially with system size (inset). In (a, b) the eigenstates are uniformly sampled from the middle of the spectrum, namely among the 10% eigenstates with lowest ∣E∣. In (a), the phase-space states are obtained sampling the spins {si} uniformly and independently from the sphere, and the IS states are obtained sampling s1 uniformly from the sphere and alternating the other spins at every other site as in Fig. 1(a). Here, N = 20 except where otherwise and explicitly specified. All other parameters are as in Fig. 2.
