Fig. 5: Averaged spectral form factor (SFF) with Gaussian filter functions.

a Averaged SFF (black line) over 500 Gaussian windows with each σ = 0.01 and the SFF of a single sample (red line) at μ = 14 with the same σ. The linear ramp predicted by random matrix theory (blue dashed line) is given in Eq. (8). b Averaged SFF (black line) over 100 Gaussian windows with each σ = 0.35 and fitted to it the “hydro-corrected" ramp (red line) of Eq. (9) in comparison with the standard linear ramp (blue dashed line). The Thouless time (green dash-dotted line) extracted from the fit is t_Th/a = 30.14 ± 0.44. The system in (a) and (b) is the same five-plaquette chain with \({j}_{\max }=\frac{3}{2}\), ag² = 1.05 and open boundary conditions j = {1, 1, 1, 0}. The sampling of the Gaussian means is performed between E = [10, 22], which for this region of the spectrum results in a Heisenberg time t_H/a ≈ 3.818 × 10⁴ (gray long-dashed line). In both cases, the averaged SFF's were further smoothed using a temporal running average, which is a common additional procedure in SFF calculations alongside spectral averaging^100,101.