Fig. 4: Phase diagram and retrieved NPR(j, t) for single-site excitation.

a The phase diagram of η over the dimerization parameter Δ and on-site potential ϵ plane for a 100-site dimer-disordered SSH lattice. The blue region signifies the concurrent non-zero values of both 〈IPR〉 and 〈NPR〉, according to the definition \(\eta ={\log }_{10}[\langle {{{\rm{IPR}}}}\rangle \times \langle {{{\rm{NPR}}}}\rangle ]\). Four small yellow circles in the diagram represent the experimental sampling points we selected where the dimerization parameter Δ = 0.25, corresponds to the four regions presented in Fig. 2b. The reentrant localization is validated by the coexistence of localized and extended states. b Phase diagram of η in the on-site potential ϵ and on-site probability p plane. The near-vertical border of the reentrant localization region accentuates that the critical values remain stable as the on-site probability varies widely (0.15 < p < 0.8), confirming that our choice of on-site probability p = 0.5 does not add any specificity. c The comparison between spectral-averaged NPR based on eigenstates and the single NPR(j, t) based on the single-site excitation. In this context, we consider the parameters wherein j is assigned a value of 41, signifying the occurrence of the 41st single-site excitation, and t is set to 1000, indicating a total propagation length of 1000 μ m. The trend of the spectral-average NPR results from the ensemble of statistical behaviors of all NPRs from single-site excitations. Specifically, the NPRs from single-site excitations which are linked with extended eigenstates contribute to the peak of the spectral-averaged NPR in the reentrant localization and trigger a non-monotonic metal-insulator transition in the one-dimensional random-dimer disordered SSH system.