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• 2D split-ring resonator array with resonant frequency disorder. • Localized modes and magnetic transport for uncorrelated and correlated disorder in 2D. • Correlated disorder can create sub-diffusive propagation. We study the effect of a resonant frequency disorder on the eigenstates and the transport of magnetic energy in a two-dimensional (square) array of split-ring resonators (SRRs). In the absence of disorder, we find the dispersion relation of magneto-inductive waves and the mean square displacement (MSD) in closed form, showing that at long times the MSD is ballistic. When disorder is present, we consider two types: the usual Anderson distribution (uncorrelated monomers) and 2 × 2 units assigned at random to lattice sites (correlated tetramers). This is a direct extension to two dimensions of the one-dimensional random dimer model (RDM). For the uncorrelated case, we see saturation of the MSD for all disorder widths, while for the correlated case we find a disorder window, inside which the MSD does not saturate at long times, with an asymptotic sub-diffusive behavior M S D ∼ t 0.26. Outside this disorder window, the MSD shows the same kind of saturation as in the monomer case. We conjecture that the sub-diffusive behavior is a remanent of a weak resonant transmission of a 2D plane wave across a tetramer unit. [ABSTRACT FROM AUTHOR]

• Some two dimensional Schrödinger equations have solutions for which the amplitudes and phases can be interchanged. • This only occurs when the Bohm potential vanishes, and for negative potentials. • These solutions represent particles with equal energy but different momenta, evolving in the same potential. • Exact solutions for physical systems, as repulsive harmonic oscillator or hydrogen atom, are found. • This system has a correspondence with optics, which implies that dual solutions can be studied in analogue optical systems. It is shown that the Schrödinger equation for a large family of pairs of two–dimensional quantum potentials possess wavefunctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Böhm potential. These solutions can be extended to three–dimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the two–dimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the eikonal limit. In this case, the potential is related to the refractive index, allowing the study of this two–dimensional dual wavefunction solutions with an optical (analogue) system. [ABSTRACT FROM AUTHOR]