Ring-Localized States, Radial Aperiodicity And Quantum Butterflies On A Cayley Tree

We present an analytical method, based on a real space decimation scheme, to extract the exact eigenvalues of a macroscopically large set of pinned localized excitations in a Cayley tree fractal network. Within a tight binding scheme we exploit the above method to scrutinize the effect of a deterministic deformation of the network, first through a hierarchical distribution in the values of the nearest neighbor hopping integrals, and then through a radial Aubry-Andre-Harper quasiperiodic modulation. With increasing generation index, the inflating loop-less tree structure hosts pinned eigenstates on the peripheral sites that spread from the outermost rings into the bulk of the sample, resembling the spread of a forest fire, 'lighting up' a predictable set of sites and leaving the rest 'un-ignited'. The penetration depth of the envelope of amplitudes can be precisely engineered. The quasiperiodic modulation yields hitherto unreported quantum butterflies, which have further been investigated by calculating the inverse participation ratio for the eigenstates, and a multifractal analysis. The applicability of the scheme to photonic fractal waveguide networks is discussed at the end.