Fig. 3: Duality between strong and weak impurity strengths.
a and b Average inverse participation ratios (IPRs) defined as \(\bar{I}(\mu )={\sum }_{n}{I}_{n}/N\) and \({\bar{I}}_{d}(\mu )={\sum }_{n}{I}_{n,d}/N\), with In = \({\sum }_{x}\)∣ψx,n∣4 and Id,n = \({\sum }_{x}\)(xc − x)∣ψx,n∣4/(L/2), ψx,n the wave-function value of the nth eigenstate at x, N and L the total number of states and lattice sites, and xc = L/2 being the center of the system. The impurity strength μ is parametrized by a variable A, i.e. \(\mu ={\mu }_{\alpha }^{\pm }={e}^{\alpha }{A}^{\pm 1}\) and \(\mu ={\mu }_{0}^{\pm }={A}^{\pm 1}\) for (a) and (b), respectively. In (a), \({\mu }_{\alpha }^{+}\) and \({\mu }_{\alpha }^{-}\) form a pair of duality, whereas in (b) with much larger A values, \({\mu }_{0}^{+}\) and \({\mu }_{0}^{-}\) form a pair of duality. The summation of n runs over all continuous eigenstates, and N = L−1 (L + 1) is their total number in the presence (absence) of the pair of isolated states Eiso ≈ ± μ. Colors of the curves indicate IPRs for different choices of μ. Blue and orange curves are almost identical in (a). c Spectra with dual parameters, as indicated by the gray arrows. A phase transition to OBC-like line-spectrum at μ = e±(L+1)α = e±21, with the parameters L = 20 and α = 1. The duality indicated in (a) is further confirmed by the identical spectra (blue circles and red stars) of \(\mu ={\mu }_{\alpha }^{\pm }\) (with α = 1), at A = 0.5 and A = 2 for the left two panels, respectively. Similarly, the two panels on the right depict the duality between \(\mu ={\mu }_{0}^{\pm }\) indicated in (b), for the two points of A = 20.5 and A = 21.
