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We study one-magnon excitations in a random ferromagnetic Heisenberg chain with long-range correlations in the coupling constant distribution. By employing an exact diagonalization procedure, we compute the localization length of all one-magnon states within the band of allowed energies E. The random distribution of coupling constants was assumed to have a power spectrum decaying as S(k) ∝ 1/k α . We found that for α < 1, one-magnon excitations remain exponentially localized with the localization length ξ diverging as 1/E. For α = 1 a faster divergence of ξ is obtained. For any α > 1, a phase of delocalized magnons emerges at the bottom of the band. We characterize the scaling behavior of the localization length on all regimes and relate it with the scaling properties of the long-range correlated exchange coupling distribution.

We investigate the nonreciprocal diodelike behavior of a dimer with an asymmetric on-site potential and a saturable nonlinearity. The dimer is coupled to linear side chains. The spectra of transmission and the rectifying factor are analytically obtained using a backward iteration of the set of discrete nonlinear Schrödinger equations used to model the wave propagation through the nonlinear dimer. We show that the windows of bistable behavior leading to a pronounced nonreciprocal diodelike transmission become wider and displaced to higher input field intensities as the saturation coefficient increases. Further, saturation of the nonlinear response has opposite impacts on the rectifying action over short- and long-wavelength input signals within the second bistability window. In the first window, the rectifying action is not compromised by the saturation, thus showing that a weak contribution of high-order susceptibilities to the nonlinear response can improve the efficiency of the nonreciprocal transmission. The rectifying action of a dimer with an asymmetric nonlinearity is also discussed.

We investigate the influence of metamaterials on the scaling laws of the transmission on multilayered structures composed of random sequences of ordinary dielectric and metamaterial layers. The spectrally averaged transmission in a frequency range around the fully transparent resonant mode is shown to decay with the total number of layers as 1/N. Such thickness dependence is faster than the 1/N(1/2) decay recently reported to take place in random sequences of ordinary dielectric slabs. The interplay of strong localization and the emergence of resonant modes within the gap leads to a non-monotonous disorder dependence of the transmission that reaches a minimum at an intermediate disorder strength.

We investigate several scaling aspects of the transmission spectrum of disordered one-dimensional dielectric structures. We consider a binary stratified medium composed of a random sequence of N slabs with refraction indices satisfying the Bragg condition. The mode for which the optical thickness corresponds to half wavelength is insensitive to disorder and fully transparent. The average transmission in a frequency range around this resonance decays as 1 / N 1/2 , and the localization length diverges quadratically as this resonance mode is approached. In the vicinity of the quarter-wavelength mode, the localization length diverges logarithmically and the frequency averaged transmission exhibits an stretched exponential dependence on the total thickness. At the quarter-wavelength resonance, the Lyapunov exponent for different realizations of disorder has a Gaussian distribution leading to distinct scaling laws for the geometric and arithmetic averages of the transmission. The scaling laws for the half-and quarter-wavelength modes are analogous to those found in electronic one-dimensional Anderson models with random dimers and pure off-diagonal disorder, respectively, which are known to display similar violations of the usual exponential Anderson localization.