Fig. 2: Average distribution of anomalously accumulating eigenstates and their spectra.

a Average distribution of all the anomalously accumulating eigenstates \(\bar{\rho }(x)=\mathop{\sum }\nolimits_{n = 1}^{L-1}{\rho }_{n}(x)/(L-1)\) with ρ_n(x) = ∣ψ_x,n∣², ψ_x,n the wave-function value of the nth eigenstate at x, and α = 4, L = 20. The blue, orange, and green cases correspond to reversed scale-free accumulation (SFA), quasi-periodic boundary conditions (PBCs) and SFA. b Spectra for these anomalously accumulating eigenstates, excluding the two isolated eigenstates induced by the strong impurity. The black dashed curve is the true PBC spectrum (μ = 1), which overlaps with the quasi-PBC orange curve at μ = e^2α = e⁸. In both panels, the circles, squares, and triangles are numerical data points, and the colored solid curves are approximations from Eqs. (1) and (2).