Fig. 6: Eigenvalue spacings and eigenvector localization.

(a) The eigenvalue spacing distribution (ESD) P(z) for various values of (Moiré) twist angle θ, for 0° ≤ ϕ ≤ 2°. The short period system for ϕ = 0 and those with small twist angles 0 ≤ ϕ ≤ 1/32 are characterized by spectral measures μ with very sharp resonances leading to P(0) ≳ 0.4. However, for ϕ ≥ 1/16 the system begins to transition towards obeying Wigner-Dyson statistics with level repulsion, so that P(0) = 0. Level repulsion increases with increasing ϕ as the ESD approaches the Wigner-Dyson ESD, characterized by strong correlations and strong eigenvalue repulsion. (b) The ratio of average eigenvector \(\overline{IPR}\) with IPR_GOE = 3/N₁ is plotted versus \((r,\tan \theta )\). Yellow hues correspond to short period systems similar to the leftmost panel in Figure 2, characterized by highly extended eigenmodes (hence extended electric and displacement fields) and “mobility edges” with large localization variability. Dark green to blue hues correspond to quasiperiodic systems similar to the one shown in Fig. 2 for ϕ = 2 with material properties that resemble that of random systems with regularly distributed IPR values and tenuously connected electric and displacement field paths. This panel indicates periodic systems have a repeating pattern that turns out to be fractal in nature, as indicated in Fig. 1. Moreover, quite small changes in the Moiré parameters \((r,\tan \theta )\) result in transitions from ordered periodic systems to disordered quasiperiodic, random-like systems.